# ImageMagick v6 Examples -- Image Transformations

Index
ImageMagick Examples Preface and Index
Art-like Transformations
Computer Vision Transformations
Shade 3D Highlighting
Using FX, the DIY Image Operator
Evaluate and Function, Fast FX Operators
Mathematics on Gradients

These operations produce major changes to the overall appearance of the image either for visual, or art-like effects. However while the overall look of the image has changed, often dramatically, the original image itself is still generally visible in the result.

## Art-Like Transformations

### Raise or Sunk Borders

The "`-raise`" operator is such a simple image transformation, that it almost isn't. All it does is as a rectangular bevel highlight to an existing image.
 `````` convert rose: -raise 5 rose_raise.gif ``````

A inverted sunken effect can be generated using the 'plus' form of the operator...
 `````` convert rose: +raise 5 rose_sunken.gif ``````

This operator is a bit like Framing an image, but instead of adding extra pixels as a border, the "`-raise`" operator re-colors the edge pixels of the image. This makes it an image transform.

 In actual fact Image Framing is achieved by adding a Border, then raising it!

 The operator only works on rectangular images, and will fail for images with a transparent background, as the color modifications will also be transparent. Basically is it a rather dumb operator!

### Adding an Inside Border

Rather than adding a border around the outside of an image a user wanted to add one to overlay the edges of an image. The solution was to draw a rectangle around the image. As the built in rose in is 70x46 pixels, this is the result.
 `````` convert rose: -fill none -stroke navy -strokewidth 11 \ -draw 'rectangle 0,0 69,45' inside_border.jpg ``````

The width of the border added is controlled by the "`-strokewidth`" of the rectangle. That is
`{stroke width}  =  {border width} * 2 - 1`

As such the above 6 pixel border needed a "`-strokewidth`" of 11.

If you don't know the size of the image, then you can Shave the image then add the Border as normal. This is probably easier, though prehaps not as versatile.
 `````` convert rose: -bordercolor green -shave 6x6 -border 6x6 inside_border2.jpg ``````

### Random Pixel Spread

The "`-spread`" would replace each pixel the color of a random nearby color from the source image. This random selection was made as per the use of Pixel Interpolation and Virtual-Pixel Setting.

For example...
 `````` convert -size 80x40 xc:red xc:blue -append -spread 5 spread_interpolated.png ``````

If you were to examine the pixels image you will see that some pixels may have a mix of red and blue colors. That is they are interpolated, not simply spread or swapped. This is more pronounce smaller distance values.

Also note that "`-spread`" also makes use of the Virtual-Pixel Setting.
 `````` convert -size 80x80 xc: -virtual-pixel black -spread 10 spread_virtual.png ``````

As you can see you get a randomized border, mostly of pure black virtual pixels Though there are a few grey pixels interpolated from the border between the real pixels of the image and virtual pixels.

To get a more traditional spread pixels effect, you can prevent this color mixing by forcing the color lookup of specific pixels by using "`-interpolate Nearest`". To avoid the problems with virtual pixels and posible 'edge color bias', I recommend you use "`-virtual-pixel Mirror`".

As such this is a more traditonal random 'spread' of pixels...
 `````` convert rose: -interpolate nearest -virtual-pixel mirror \ -spread 5 spread_rose.png ``````

Under Construction

The main problem with the above is that you can lose some pixel data from the image. That is the pixels are not 'swapped' but randomally copied, which means a specific pixel in the image may become duplicated or lost.

As of IM v6.9.2-2 you can use "`+spread`" to actually swap pixels within the image, meaning that no pixel in the image will be duplicated or lost. Every pixel in the original image is still present, just displaced to to new location.

However due to the way pixels are processed, pixels may be 'double-swapped'. That is a specific pixel may be swapped, but then selected to be swapped again with a later pixel. That means a specific pixel could drift further than was requested by the spread argument.

This double swapping also mean that pixels were likely to spread further toward the lower right corner. That movement is of course balanced but a smaller drift of a large number of pixels toward the upper-left.

For example, here I spread pixels with the original prepended as a referance.
 `````` convert -size 40x40 xc:red xc:blue -append \ \( +clone +spread 5 \) +append spread_bias.png ``````

Note how some red pixels are spread downward more, though you also get a few blue pixels spreading upward more than expected too ( though toward the left side of the image). This problem is more pronounce when you use a smaller distance argument to spread.

A solution to this double-swap problem is not easy, and we are looking for a 'limited area shuffle' algorithm to solve it. But in the mean time you can at least mitigate the directional bias by doing the spread twice, with a the Transverse (top-left to bottom right diagonal mirror) distortion.
 `````` convert -size 40x40 xc:red xc:blue -append \ \( +clone +spread 5 -transverse -spread 5 -transverse \) \ +append spread_no_bias.png ``````

Of course this does makes the spread more pronounced, and less linear, but at least it is without a directional bias, or pixel duplication/loss.

The above addition was developed from a Forum Discussion: t=28043

IM Forum Discussion rearrange vertical pixel row .

### Vignette Photo Transform

A special operator to make a image circular with a soft blurry outline.
 `````` convert rose: -background black -vignette 0x5 rose_vignette.gif ``````

By using a zero (or very small) sigma you can remove the blur, and generate ellipse or oval frames. However note that it does not actually use the largest ellipse posible, so you may want to DIY it.
 `````` convert rose: -background black -vignette 0x0 rose_vignette_0.gif ``````

You can use it with transparency (and PNG format)...
 `````` convert rose: -alpha Set -background none -vignette 0x3 rose_vignette.png ``````

An alternative argument method is to use a very large number for the second sigma component, and then use the first radius to define the spread of the blur. This produces a 'linear' distribution rather than a more common Gaussian distribution to the vignette blurring.
 `````` convert rose: -background black -vignette 5x65000 rose_vignette_linear.gif ``````

Another technique for a more rectangular vignette, producing soft edges to the image is demonstrated in Thumbnails with Soft Edges.

### Complex Polaroid Transformation

Thanks to the work done by Timothy Hunter, (of RMagick fame), a "`-polaroid`" transformation operator, was added to IM v6.3.2.
Polaroid® is a registered trademark of the Polaroid Corporation.

For example, here I give a polaroid look to a photo thumbnail. The image is looking up the spiral staircase (downward), inside the Arc de Triumph, Paris. It is a very long staircase!.
 `````` convert spiral_stairs_sm.jpg -thumbnail 120x120 \ -bordercolor white -background black +polaroid poloroid.png ``````
 Note the resulting image has a semi-transparent shadow, so you either have to use a PNG format image, or "`-flatten`" the result onto a fixed background color for GIF or JPG formats.

This operator is very complex, as it adds border (as per the "`-bordercolor`" setting), 'curls' the paper, and adds a inverse curl to the shadow. The shadow color can be controlled by the "`-background`" color setting.

As you saw above the plus form of the operator will rotate the result by a random amount. This operator makes a Index of Photos much more interesting and less static than you would otherwise get.

The minus form of the operator lets you control the angle of rotation of the image.
 `````` convert spiral_stairs_sm.jpg -thumbnail 120x120 \ -bordercolor AliceBlue -background SteelBlue4 -polaroid 5 \ poloroid_5.png ``````

If the image has "`-caption`" meta-data, that text will also be added into the lower border of the polaroid frame, via the "`caption:`" image creation operator. That is it will be word wrapped to the width of the photo.
 `````` convert -caption '%c %f\n%wx%h' spiral_stairs_sm.jpg -thumbnail 120x120 \ -bordercolor Lavender -background gray40 +polaroid \ poloroid_captioned.png ``````

The other standard text settings (as per "`caption:`"), allows you to control the look of the added caption.
 `````` convert spiral_stairs_sm.jpg -thumbnail 120x120 -font Candice -pointsize 18 \ -bordercolor Snow -background black -fill dodgerblue -stroke navy \ -gravity center -set caption "Spiral Stairs\!" -polaroid 10 \ poloroid_controls.png ``````

 The image meta-data attribute "`-caption`" was used due to the internal use of "`caption:`" text to image generator. On the other hand the IM command "`montage`" uses "`-label`" as it uses the non-word wrapping "`label:`" text to image generator.

The transforms use of Rotate and Wave shearing distortions to add a little 'curl' to the photo image has a tendency to produce horizontal lines of fuzziness in text of the image generated. This is a well known Image Distortion problem (see Rotating a Thin Line), and one that can be solved by using a super sampling technique.

Basically we generate the polaroid twice as large as what we really want, then we just resize the image to its final normal size. The reduction in the image size effectively sharpens the resulting image, and more importantly the caption text.

However to make this work we not only need a image at least twice the final size, but also we may need to a larger border to the image, and draw the text at twice its normal "`-density`". Do not increase the fonts "`-pointsize`" as that does not enlarge the text in quite the same way.
 `````` convert -caption 'Spiral Staircase, Arc de Triumph, Paris, April 2006' \ spiral_stairs_sm.jpg -thumbnail 240x240 \ -bordercolor Lavender -border 5x5 -density 144 \ -gravity center -pointsize 8 -background black \ -polaroid -15 -resize 50% poloroid_modified.png ``````

As you can see, even though we used a much smaller font pointsize, the caption text is very sharp, clear and readable. The same for any other fine detail that may have been present in the original image. The only disadvantage of this is that the shadow of the resulting image will be smaller, and less fuzzy.

For total control of the polaroid transformation, you can do all the steps involved yourself. The original technique documented on Tim Hunter's page, RMagick Polaroid Effect. The steps are: create and append caption, add borders, curl photo with wave, add a reversed curled shadow, and finally rotate image.

For more examples, and other DIY methods, see Polaroid Thumbnail Examples, and A Montage of Polaroid Photos. You may also be interested in some of the polaroid examples in RubbleWeb IM Examples, Other.

### Oil Painting, blobs of color

The "`-paint`" operator is designed to convert pictures into paintings made by applying thick 'blobs' of paint to a canvas. The result is a merging of neighbourhood colors into larger single color areas.

 `````` convert rose: -paint 1 rose_paint_1.gif convert rose: -paint 3 rose_paint_3.gif convert rose: -paint 5 rose_paint_5.gif convert rose: -paint 10 rose_paint_10.gif convert rose: -blur 0x3 -paint 10 rose_blur_paint_10.gif ``````

Notice that at a high radius for the paint blobs, the blobs start to get a squarish look to them. This effect can be smoothed somewhat by blurring the image slightly before hand, as shown in the last image above.

It is an interesting effect and could be used to make some weird and wonderful background images. For example see its use in Background Examples.

On final warning. While "`-paint`" is supposed to produce areas of a single solid color, at large radius values, it has a tendency to produce a vertical gradient in some areas. This is most annoying, and may be a bug. Does anyone know?

There are alternative to using "`-paint`". One is to use "`-statistic Mode`" instead, which assigns each pixel with the 'predominate color' within the given rectangular neighbourhood, and can produce a nicer result.
 `````` convert rose: -statistic Mode 10 rose_paint_mode.gif ``````

Another is to use some of the Morphology Methods, and more specifically the Intensity Varient for Color Images. Here for example is an 'OpenIntensity' Morphology on the rose.
 `````` convert rose: -morphology OpenI Disk rose_paint_open.gif ``````

And here I use 'CloseIntensity' with a slightly smaller 'Disk'.
 `````` convert rose: -morphology CloseI Disk:2.5 rose_paint_close.gif ``````

You don't have to use 'disks', but can design your own 'brush' shaped kernel for the blobs that it creates. For example what about using a diagonal line brush.

### Charcoal, artists sketch of a scene

The charcoal effect is meant to simulate artist's charcoal sketch of the given image.

The "`-charcoal`" operator is in some respects similar to edge detection transforms used by Computer Vision. Basically it tries to convert the major borders and edges of object in the image into pencil and charcoal shades.

The one argument is supposed to represent the thickness of the edge lines.
 `````` convert rose: -charcoal 1 rose_charcoal_1.gif convert rose: -charcoal 3 rose_charcoal_3.gif convert rose: -charcoal 5 rose_charcoal_5.gif ``````

For a better example of using a charcoal transformation on a real image see Charcoal Sketch of a Photo.

 Technically the "`-charcoal`" operator is a "`-edge`" operator with some thresholding applied to a grey-scale conversion of the original image.

### Pencil Sketch Transform

The "`-sketch`" operator basically applies a pattern of line strokes to an image to generate what looks like an artistic pencil sketch. Arguments control the length and angle of the strokes.

However it is best applied to a larger image with distinct and shadings.

See Pencil Sketch for a full example of this operator and how it works internally.

### Emboss, creating a metallic impression

The "`-emboss`" operator tries to generate the effect of a acid impression of a grey-scale image on a sheet of metal. It is in many respects very similar to the "`-shade`" operator we will look at below, but without the 3D looking edges.

Its argument is a radius/sigma, with only the sigma being important. I not found the argument very useful, and may in fact be buggy. The argument has also changed in a recent version of IM. I just don't know what is going on. Help me understand if you can.
 `````` convert rose: -emboss 0x.5 rose_emboss_0x05.gif convert rose: -emboss 0x.9 rose_emboss_0x09.gif convert rose: -emboss 0x1 rose_emboss_0x10.gif convert rose: -emboss 0x1.1 rose_emboss_0x11.gif convert rose: -emboss 0x1.2 rose_emboss_0x12.gif convert rose: -emboss 0x2 rose_emboss_0x20.gif ``````

The operator is a grey-scale operator, meaning it will be applied to the three color channels, separately. As such should only be applied to grey-scale images. As you saw above, color images can produce some weird effects.
 `````` convert rose: -colorspace Gray -emboss 0x.5 rose_g_emboss_0x05.gif convert rose: -colorspace Gray -emboss 0x.9 rose_g_emboss_0x09.gif convert rose: -colorspace Gray -emboss 0x1 rose_g_emboss_0x10.gif convert rose: -colorspace Gray -emboss 0x1.1 rose_g_emboss_0x11.gif convert rose: -colorspace Gray -emboss 0x1.2 rose_g_emboss_0x12.gif convert rose: -colorspace Gray -emboss 0x2 rose_g_emboss_0x20.gif ``````

``` If anyone knows exactly what the emboss algorithm is supposed to do,
please let me know.

```

### Stegano, hiding a secret image within an image

The "`-stegano`" operator is really more of a 'fun' operator. For example it could be use by a spy to hide info in the 'chaos' of a random image.

First a warning...
Do not use JPEG, GIF, or any other 'lossy' image encoding with Stegano

For example, lets generate a cryptic message (image) that you want to send to your fellow spy...
 `````` convert -gravity center -size 50x40 label:"Watch\nthe\nPidgeon" message.gif `````` identify message.gif ``````

Note that we will also need the size of the message image (36x43 pixels), thus the identify in the above.

Next the put it into some image with some offset. The offset (and message size) used is the cryptographic 'key' for the hidden message.
 `````` composite message.gif rose: -stegano +15+2 rose_message.png ``````

Now you can send that image to your compatriot, who presumably already knows the messages size and offset.

We can the recover the message hidden in the image...
 `````` convert -size 50x40+15+2 stegano:rose_message.png message_recovered.gif ``````

The larger the containing image the better the recovered image will be, so hiding small images in larger ones is better than the example shown above.

Just to show you how the hidden message was distributed throughout the container image, lets do a comparison of the combined image against the original.
 `````` compare -metric PAE rose: rose_message.png rose_difference.png ``````

Which shows how the message image was encrypted and distributed all over the container image to hide it.

Also the '`PAE`' metric returned by the above shows that the largest difference was only a single color value out of the 8 bit color values used for this image.

That is tiny. So tiny that a small change or modification to the image will destroy the message hidden within. It is such a small difference, you can't even use JPEG with its lossy compression as the image format, or any other lossy image format (including GIF) for the container image.

Also if you had the wrong 'offset key' you will not get the message...
 `````` convert -size 50x40+14+2 stegano:rose_message.png message_bad.gif ``````

You can also limit what area of the image the message is to be hidden by using a Region setting. The same setting will also be needed when attempting to recover the message. This would probably make finding and decoding the hidden message in a large image, especially if restricted to a 'busy' area, a order of magnatitude harder to determine.

However be warned that this is not a very cryptographically secure technique. Especially if the original source image is also available. Frequency analysis of the image will generally let an attacker know there is a hidden message.

As a method of image copyright protection, the Stegano Operator is also useless, as the smallest change to the image will destroy the message, and thus its effectiveness.

As a spy tool it is also not very good, with such a small number of 'combinations' for a reasonable sized image. Anyone who knows roughly what you are doing could probably crack it quickly. Better to stick to well known and time tested cryptographic methods.

It's only real practical use is as a fun tool, or as a way to add very small amounts of noise to an existing image.

### Encrypting Image Data

The operators "`-encipher`" and "`-decipher`" will basically encrypt image data into a garbled mess.

That is the image content itself is no longer recognisable at all until the image is later decrypted. This can be used for example to protect sensitive images on public services, so that only others with the secret pass-phrase can later view it.

But first a warning...
Do not use JPEG, GIF, or any other 'lossy' image encoding with Encryption

For example lets encrypt that secret message image we created above, using a pass-phrase I have saved in a, not quite so 'secret', file "pass_phrase.txt".
 `````` convert message.gif -encipher pass_phrase.txt \ -depth 8 png24:message_hidden.png ``````

 The encrypted image assumes it is saved using a 8 bit image file format. As such it is recommended to enforce that limitation by setting "`-depth 8`" before the final save to the output file. The "`png24`" was also needed in the above to ensure that the output is not a palette or colormapped "png8:" image, which also does not work properly.

As you can see the resulting image looks like complete garbage, with no indication of the images real content.

Now you can publish that image on the web, and only someone who knows the exact original pass-phrase can restore the image data...
 `````` convert message_hidden.png -decipher pass_phrase.txt message_restored.gif ``````

However be warned that if the image data is corrupted in some way, you will not be able to restore it. That includes if the PNG saved using a gray-scale image format type. As such, only a non-lossy image format can be used, such as PNG, MIFF, TIFF, or even Pixel Enumeration Text. However using a lossy image format, such as JPEG, PNG8, and GIF, will corrupt the image data, thus destroy the resulting encryption.

Note that any meta-data that may be describing the image, will still be in the clear. That means, you could encrypt images using the images own 'comment' string as the pass-phrase or use that comment encrypted using some smaller password. Its a simple idea that could make the pass-phrase more variable.

Encrypting an image can be just one step. Taking the result just that little further can produce a image that will not simply decrypt, without some extra processing. For example here I use some Simple Non-Destructive Distorts to confuse anyone trying to decrypt the image in the normal way.
 `````` echo "password" | convert message.gif -encipher - \ -transpose -depth 8 png24:message_obfuscate.png echo "password" | convert message_obfuscate.png -transpose \ -decipher - message_restored_2.png ``````

If you did not include the "`-transpose`" in the decryption command above, the image will not have deciphered correctly. Also note that due to the streaming cipher used (see the expert note below) using just a "`-roll`" of some sort will not prevent the image from being decrypting, at least partially.

Note that in the above I did not use a file to hold the 'pass-phrase' but fed the phrase into the "`convert`" command using standard input, which allows you use some other program or command to get the phrase from the user, generate it, or some other method, instead of using text file with the pass-phrase in the clear.

The pass-phrase could also be generated from other freely downloadable files and images. For example you could decrypt your image using the signature, or comment string of a well known, freely downloadable reference image. For instance here I use the signature of the "`rose.gif`" image to encrypt and later decrypt the "`message.gif`" image.
 `````` identify -format %# rose.gif |\ convert message.gif -encipher - -depth 8 png24:message_signed.png identify -format %# rose.gif |\ convert message_signed.png -decipher - message_restored_3.png ``````

As of IM v6.4.8-0 the file used by "`-encipher`" and "`-decipher`" can be a binary file. As such you could even directly use an image itself as the the passphrase.
 `````` convert message.gif -encipher rose.gif -depth 8 png24:message_binary.png convert message_binary.png -decipher rose.gif message_restored_4.png ``````

 Before IM v6.4.8-0 a binary file would stop at the first 'NULL' character it finds. Something that would happen rather early if a PNG image was used.

This technique is exact (unless some of the data was destroyed in transmission). And as such you can use it to encrypt an image containing other hidden information such as a Stegano Image. This means even if authorities do decrypt the image, or force you to reveal the password, they will see actual image, but that image may not be the final hidden one.

 The "`-encipher`" and "`-decipher`" operators was added to IM v6.3.8-6, but required you to include a "`--enable-cipher`" option in the build configuration. However by IM v6.4.6 (when did it change?) this configuration item was no longer needed and it became a standard configuration setting. As such you can probably use it immedaitally.

 The cipher was implemented using a self-synchronizing stream cipher implemented from a block cipher. This means that you can still decipher even a partial download of the image, which was destroyed by transmission error, even though some part of the image may have been destroyed. You also do not need to downloaded the whole image to decrypt and examine the parts that was successfully downloaded. But you do need the pass-phase to have any chance at all of successfully decrypting the image, as it is a very very strong encryption.

### Pixelate an Image

Pixelating an image is basicaly used to convert a image into a set of large colored 'pixels' that only shows a vague outline of the original image.

Both techniques involve shrinking the image (to generate fewer pixels), then enlarging them in such a way so as to create 'pixel block' using either a Scaling Operator or Sampling Operator to generate the block of color. It is just how the image is reduced that determines exactly what color wil be used. A single pixel sample, or a merged average color.
 `````` convert rose: -sample 25% -scale 70x46\! rose_pixelate_sampled.gif convert rose: -scale 25% -scale 70x46\! rose_pixelate_scaled.gif convert rose: -resize 25% -scale 70x46\! rose_pixelate_resized.gif ``````

As you can see, the 'sampled' areas will have much more distinct (aliased) 'pixels', while the other two uses a merged or averaged color, which tends to produce more muted, but more accurite color representation for each 'pixel'.

See also Protect Someones Anonymity for a example of using this on just a smaller masked area of the image, such as a persons face.

### Grids of Pixels

Gridding an image is very similar to pixelating an image. In this case we want only want to enlarge the image, to generate distinct pixel-level view of a image's details. Typically a very small image.

The simplest way is like the previous example, simply Scale a small image, to enlarge the pixels.
 `````` convert rose: -crop 10x10+12+20 +resize grid_input.png convert grid_input.png -scale 1000% grid_scale.png ``````

The problem with a simple scaling, is that in areas where pixels are similar in color, you can have trouble seeing the individual 'pixel blocks'.

What we need to add a border around the pixels, to separate them.

For this we need to overlay a generated tile mask. See Tiling with an Image already In Memory for various methods of using a generated tiling image, in a single command.

Here we generate a white on black 'grid' which is overlayed using Screen Composition (overlay white, while leaving black areas as-is).
 `````` convert -size 10x10 xc: -draw 'rectangle 1,1 9,9' -write mpr:block +delete \ grid_input.png -scale 1000% -size 101x101 tile:mpr:block \ +swap -compose screen -composite grid_blocks.png ``````

Note that the size used to generate the tile is `scale*image_size+gap_size` (in this case 10*10+1 => 101).

I also Swapped the two images so that the final image size comes from the tile image, rather than from the scaled image which is one pixel smaller in size. However this may lose any image meta-data that was in the original image as I used the tiled image for the destination.

Here I generate circular 'spots' of color, but this time used a Multiply Composition (overlay black, while leaving white areas as-is).
 `````` convert -size 10x10 xc: -draw 'circle 5,5 1,3' -negate \ -write mpr:spot +delete \ grid_input.png -scale 1000% -size 101x101 tile:mpr:spot \ +swap -compose multiply -composite grid_spots.png ``````

You can make the grid border transparent by also negating the tiled overlay (black areas become transparent) and use a CopyOpacity Composition instead of Multiply.

Other colors can also be added, but for this to work you have to use a tile image that actually contains real transparency. For this you need to convert the black and white tile image into a Shaped Mask.

For example, here I use Basic Morphology Operator to generate a diamond shaped 'holes' in a colored overlay. For this a single 'seed' pixel is drawn and expanded using a Diamond Morphology Kernel.
 `````` convert -size 10x10 xc: -draw 'point 5,5' -morphology Erode:4 Diamond \ -background Navy -alpha shape -write mpr:diamond +delete \ grid_input.png -scale 1000% -splice 1x1+0+0 \ -size 101x101 -background none tile:mpr:diamond \ -alpha set -compose Over -composite grid_diamonds.png ``````

 Note that the "`tile:`" coder will replace any transparency in the image with the current background color. If you want to preserve the transparency of the tiling image, either set "`-background none`" or "`-compose Src`". The former is easier.

Note techniqually, the alpha shaping can be done either before saving the tile image, or after tiling the tile image, before overlaying it. The choice is yours.

One final technique is to use a bad resampling filter to produce a Resampling Failure to generate aliased circles of each pixel. This is not great technique (mis-using image processing failure), but it does form a Grid of Pixels.

### Spacing Out Tiles

A similar problem is spacing out a grid of tiles in an image. this is not simply scaling up individual pixels into 'pixel blocks' but inserting space between rectangluar areas of an image. That is Splicing extra pixels into an image at regular intervals.

Currently the best solution is to break up the image into Rows and Columns and Splicing in the extra spacing onto each tile before Appending the tiles back together.

For example...
 `````` convert rose: -background SkyBlue \ -crop 10x0 +repage -splice 3x0 +append \ -crop 0x10 +repage -splice 0x3 -append \ grid_tile.png ``````

Here is another method which also separates the original image into tiles, but then uses some DIY FX Expressions to calculate the new postion of a tile, from its old position.
 `````` convert rose: -crop 10x10 \ -set page '+%[fx:page.x+3*page.x/10]+%[fx:page.y+3*page.y/10]' \ -background skyblue -layers merge +repage grid_tile_fx.png ``````

The number '`3`' in the above is the gap width to add, and '`10`' is the tile size. All that you need is to add a border or other edge to the result.

With a little more work you can even add some random 'jitter' to the placement of each tile in the above grid, for a less regular effect.

The problem with both these methods is that generating lots of small images, only to join them back together does generate a lot of work. Especially for very small tiles sizes.

A better method that has been proposed is a special extension to the Splice Operator, in the IM Forum Discussion Splice (adding tile gridding gaps).

## Computer Vision Transformations

### Edge Detection

The "`-edge`" operator highlights areas of color gradients within an image. It is a grey-scale operator, so is applied to each of the three color channels separately.
 `````` convert mask.gif -edge 1 mask_edge_1.gif convert mask.gif -edge 2 mask_edge_2.gif convert mask.gif -edge 3 mask_edge_3.gif convert mask.gif -edge 10 mask_edge_10.gif ``````

As you can see, the edge is added only to areas with a color gradient that is more than 50% white! I don't know if this is a bug or intentional, but it means that the edge in the above is located almost completely in the white parts of the original mask image. This fact can be extremely important when making use of the results of the "`-edge`" operator.

For example if you are edge detecting an image containing an black outline, the "`-edge`" operator will 'twin' the black lines, producing a weird result.
 `````` convert piglet.gif -colorspace Gray -edge 1 -negate piglet_edge.gif ``````

However by negating the image before doing the edge detecting, the twined lines go inward and join together, removing the 'twin line' effect.
 `````` convert piglet.gif -colorspace Gray \ -negate -edge 1 -negate piglet_edge_neg.gif ``````

I have found that the edges tend to be too sharp, generating a non-smooth edge to the resulting images. As such I find a very very slight blur to the result improves the look quite a bit.
 `````` convert piglet_edge_neg.gif -blur 0x.5 piglet_edge_blur.gif ``````

Here I have applied edge detection to a color image, and a grey-scale version to show you its effects on photo-like images.
 `````` convert rose: -edge 1 rose_edge.gif convert rose: -colorspace Gray -edge 1 rose_edge_grey.gif ``````

As you can see without converting the image to grey-scale the edges for the different color channels are generated completely independent of each other.

### Canny Edge Detector

As on IM v6.8.9-0, IM now supports the canny edge detector. (See Announcment Examples on the IM Forum). This is a very advanced edge detection algorithm, that produces a very strong (binary) single pixel wide lines at all sharp edges, with very little noise interferance.

For example, here we apply it to the test images we used above..
 `````` convert mask.gif -canny 0x1+10%+30% mask_canny.gif convert piglet.gif -canny 0x1+10%+30% piglet_canny.gif convert piglet.gif -negate -canny 0x1+10%+30% piglet_canny_neg.gif convert rose: -canny 0x1+10%+30% rose_canny.gif ``````

As you can see it produces a much sharper result than the Edge Operator above. The fuzzy anti-aliased edge has little to no effect, on the result producing thin bitmap lines. Also as the piglet image shows it is not placed on one specific side as the previous edge operator did. As a result, negating the input image has no effect. But like all edge detectors it can have problems with real world images with 'busy' backgrounds, such as the built-in rose image.

This clean result is very important later in Hough Line Detection.

### Edge Outlines from Anti-Aliased Shapes

The biggest problem with normal edge detection methods is that the result is highly aliased. That is it generates a very staircase like pixel effects, regardless of if the shape is smooth (anti-aliased) or aliased.

For example here is a smooth anti-aliased voice balloon ("WebDings" font character '`(`' ).
 `````` convert -size 80x80 -gravity center -font WebDings label:')' voice.gif ``````

And here is its edge detected image...
 `````` convert voice.gif -edge 1 -negate voice_edge.gif ``````

As you can see it looks horrible, with some minor anti-aliasing on the outside of the edge, and a total aliased (staircase) look on the inside of the line.

The negating the image generated a similar outline around the outside of the image, but also has strong aliasing outside of the line.
 `````` convert voice.gif -negate -edge 1 -negate voice_edge_negate.gif ``````

An alternative when you already have an image with an anti-aliased edge, is to generate the difference image of a 'jittered' clone of the original shape. For example here we find the difference image between the original, image and one that has been offset (or jittered) to the right by 1 pixel.
 `````` convert voice.gif \( +clone -roll +1+0 \) -compose difference -composite \ -negate voice_jitter_horiz.gif ``````

Note that the this does not produce a good edge for horizontal sloped edges. However by combining both a horizontal and a vertical jittered difference image, we can get a very good anti-aliased outline of the shape.
 `````` convert voice.gif \ \( -clone 0 -roll +1+0 -clone 0 -compose difference -composite \) \ \( -clone 0 -roll +0+1 -clone 0 -compose difference -composite \) \ -delete 0 -compose screen -composite -negate voice_jitter_edge.gif ``````

This technique also has the advantage of working regardless of if the mask is negated or not.

Note however that the result has a 1/2 pixel offset relative to the original image, so it may require some further 'distortion' processing to re-align either the original shape, or the outline if the two needs to be combined to get the result you want.

### Edge Outlines from Bitmap Shapes

Bitmap images are much harder, as they don't have any anti-aliased pixels that can be used to produce a smooth outline.

For example here is a fancy 'Heart' shape that was extracted from the "WebDings" font (character '`Y`'). However I purposefully generated it as a aliased bitmap, to simulate a horrible bitmap image downloaded from the network. Such as the outline of a GIF image containing transparency.
 `````` convert +antialias -size 80x80 -gravity center \ -font WebDings label:Y heart.gif ``````

So we have this horrible image, but we want to find the images outline rather than its shape. Direct use of edge detection will only generate a pure bitmap edge around the outside of the bitmap shape.
 `````` convert heart.gif -edge 1 -negate heart_edge.gif ``````

A negated edge generates a edge image but for the inside of the black area.
 `````` convert heart.gif -negate -edge 1 -negate heart_edge_negate.gif ``````

By adding both of the above you get a 2 pixel edge centered on the bitmap shapes edge.
 `````` convert heart.gif \( +clone -negate \) -edge 1 \ -compose add -composite -negate heart_edge_double.gif ``````

As you can see the resulting image is highly aliased with 'staircase' like effects in the outline, even though the original image is itself not too bad in this regard. This is not a good solution.

A slightly better edge can be created by using an '`EdgeIn`' morphology method, or others like it.
 `````` convert heart.gif -negate -morphology EdgeIn Diamond -negate heart_edgein.gif ``````

And a similar effect can be achieved by just using resizes to blur the edge in the right way, before using a Solarize to extract the mid-gray pixels that form the edge. A thicker edge can be generated by adding a "`-filter Cubic`" setting, or some other Resampling Filters.
 `````` convert heart.gif -resize 400% -resize 25% \ -solarize 50% -evaluate multiply 2 -negate heart_resize.gif ``````

Or for more controlled fuzzier effect you can just blur the shape and extract the edge in a similar way. I find a blur of '`0.7`' about the best, with a 3 pixel limit to speed things up.
 `````` convert heart.gif -blur 3x.7 -solarize 50% -level 50%,0 heart_blur.gif ``````

Note the use of Level Operator, which is the equivalent of the "`-evaluate multiply 2 -negate`" used in the previous example.

With an anti-aliased border you can now re-add the original shape if you just want to smooth the original shape rather than get its outline. Just remember that the outline is positioned exactly along the edge of the original image, so will be half a pixel larger in size that the previous examples.

Do you know of any other ways of generating a anti-aliased outline from a shape (anti-aliased, or bitmap). If so please mail it to me, or the IM forum. You will be credited.

### Edging using a Raster to Vector Converter

One of the most ideal solutions is to use a non-IM 'raster to vector' conversion program to convert this bitmap shape into a vector outline. Programs that can do this include: "`ScanFont`", "`CorelTrace`", "`Streamline`" by Abobe, and "`Vector Magic`". Most of these however cost you at least some money. "VectorMagick" and another tracing program "AutoTracer" have free to use online image converters available. Other free solutions are "`AutoTrace`", or "`PoTrace`". More suggestions are welcome.

These trace programs are simple to use, but typically requires some form of pre and post image setup. They have a limited number of input formats, and outputs a vector image which will create a 'smoothed' form of the input image. I prefer the "`AutoTrace`" as it does not scale the resulting SVG data, and thus producing a standard line thickness, however you can not use it in a 'pipeline'.

For best results it is a good idea to ensure we only feed it a basic bitmap image, which we can ensure by thresholding the input image, while we convert it to a image format autotrace understands. I can then convert that image into a SVG vector image.
 `````` convert heart.gif -colorspace gray -threshold 50% heart_tmp.pbm autotrace -output-format svg -output-file heart.svg heart_tmp.pbm convert heart.svg heart_svg.gif rm -f heart_tmp.pbm ``````

As of IM v6.4.2-6 you can do the above sequence directly using the "`autotrace:`" image input delegate. This only requires the "`autotrace`" command to be installed. For example
 `````` convert autotrace:heart.gif heart_traced.gif ``````

If your IM was built with the "`AutoTrace`" delegate library, you can also have IM directly generate the SVG image from an image in memory. For details of this see SVG Output Handling. For example....
 `````` convert heart.gif heart_2.svg ``````

Now the SVG output will of course represent a smoothed version of the original image, which is not what we actually want in this example. But as we now have the shape of the bitmap in vector form, we can simply adjust the SVG '`style`' attributes so as to '`stroke`' the outline, rather than '`fill`' the shape. The modified SVG can then be fed back into ImageMagick again to recreate the clean outline raster image.

For example...
 `````` cat heart.svg | sed 's/"fill:#000000[^"]*"/"fill:none; stroke:black;"/' | convert svg:- heart_outline.gif ``````

Yes it is a little awkward, but the smooth anti-aliased result is well worth the effort. It would be nice if outline or some other modifications could be specified as options to the "`autotrace`" command itself, but that is currently not a feature.

You can also further modify the SVG output to thickening the edge, or specify some other stroke or background color, change the fill color of the vector edge shape. For example here we generate a thicker outline of the shape with a red fill, all nicly anti-aliased.
 `````` cat heart.svg | sed 's/"fill:#000000;[^"]*"/"fill:red; stroke:black; stroke-width:5;"/' | convert svg:- heart_outline_thick.gif ``````

Such a perfect looking heart could not have been generated from a bitmap shape in any other way.

We could also just extract the '`d="..."`' path element to use directly as a SVG Path String in the Draw Command. This would then allow you to use any of the other IM draw settings with that vector outline, giving you complete control of the final result.

For another example of using the "`AutoTrace`" program, see Skeleton using Autotrace.

### Hough Line Detector

The Hough Line Detector ("`-hough-lines`" added IM v6.8.9-1), is a very complex transform with a lot of stages (for details see Wikipedia, Hough Transform). Basically it is designed to examine an image, looking for white lines on a black background, and try to return the exact location of any line segments (linear sequences pixels) present in the image. This can be very important for things like removing image rotations, or determining the perspective transformation in a image, so it can be repeated, or removed.

Here is the full set of options to the operator
 `````` -background {background} -stroke {line_color} -hough-lines {W}x{H}+{threshold} ``````

The colors (background and line_color) is used to set the colors of lines in the resulting image (if you actually draw them). The argument to the Hough Operator (W}x{H}+{threshold) is used to define the size and treshold of the filter used to find 'peaks' in the intermedite 'search image'. that is is controls how well it actuall 'finds' the lines we are trying to detect (see below). You would adjust these help find the line detection.

For example lets try to find the lines in a rectangular shaped image. First we need to reduce the image to lines, and for a clean result the Canny Edge Detector is recommened.
 `````` convert shape_rectangle.gif -canny 0x1+10%+30% rectangle.gif ``````

Now lets apply the Hough Line Detector to this image.
 `````` convert rectangle.gif -background black -stroke red \ -hough-lines 5x5+20 rectangle_lines.gif ``````

As you can see 5 lines were found, 2 of which are very close together. The reason for the extra line is that the rectangle in the image is not perfect.

Now while I am displaying an raster (GIF) image result, the Hough Operator actually generates a vector image in Magick Vector Graphics Format. This means you can list the line information for further processing.
 `````` convert rectangle.gif -background black -stroke red \ -hough-lines 5x5+20 rectangle_lines.mvg ``````

Note that the lines are drawn from one edge (with floating point values) to another edge of the image. And from this you can see that the second and third lines are the two that are close.

The comment in the MVG output gives you the accumulated number of pixels that the line 'hits' in the image, and thus is a good indication of how strong the line is in the image. This value will always be larger than the treshold value you give to the "`-hough-line`" operator. From this you can see the first and last lines (which are close matches) are both roughly equal in strength, so it would be hard to pick one of them over another.

If this happens to you, I suggest you try to improve your edge detection step.

Note that MVG image does not have any colors defined. The color settings in this example were not actually used. The colors are only used if you actually 'draw' the vectors when converting the above result into a 'raster' image, as we did before.

You can also see the intermedate 'search image', or 'accumulator' that is looking for white pixels in every orientation, by using a special define.
 `````` convert rectangle.gif \ -define hough-lines:accumulator=true -hough-lines 5x5+20 \ -delete 0 -contrast-stretch 0.1% rectangle_accumulator.gif ``````

The define will just append the 'search image' to the image result, which in this case we delete. We also applied some contast to the 'accumulated values' to make them more visible. This is the image that the arguments to the Hough Detector is searching.

The image is always 180 pixels wide (1 pixel per degree of line angle), while the height is twice the diagonal length of the image. As such the location of the peak will directly define the angle of the line, and the perpendicular distance of the line relative to the center point of the input image. That is the X coordinate is the angle in degrees, and the Y coordinate the distance from center from -diagonal distance, to +diagonal distance.

If you look closely at the lower-right peak you can see why we ended up with two lines instead of one. The peak here is 'twined' with a slight gap between them.

The algorithm is based on the script "`houghlines`" by Fred Wienhaus, though with a different 'perpendicular distance' accumulation handling.

### Local Adaptive Thresholding

Under Construction

The "`-lat`" operator, tries to adaptively threshold each pixel based on the value of pixels in a surrounding window. This is commonly used to threshold images with an uneven background (i.e., uneven illumination). It is based on the assumption that pixels in a small window will have roughly the same background color and roughly the same foreground color.

```For example.
convert input.png -lat 17 output.png
```
In the above a 17 pixel square 'window' is used to determine the average color of the image at each point, if the pixel is darker than this average it is made black, if lighter than this average it is made white.

A small window size will make the threshold more sensitive to small changes in illuminations, is faster to compute but is adversely affected by noise in the image.
`Example`
A larger window size will make the threshold less sensitive to small changes in illumination, is slower to compute and less affected by noise in the image. This has the effect of making the threshold value selection more or less sensitive to small changes in pixel values.
`Example`
```The window does not need to be square.  for example...
convert input.png -lat 15x25 output.png
```
You can also provide a offset which will be added to the calculated average color, making the local threshold value for each pixel either lighter or darker. This can be used for example to reduce the effect of noise or the effect of small changes in pixel values.

```  convert input.png -lat 15x25+2%
```
These small changes normally occur when a scanner or digital camera is used to acquire the image. Use a positive offset value to make the adaptive thresholding less sensitive to small variations in pixel values. Use a negative threshold to make the adaptive threshold more sensitive to small variations in pixel values.

Alternatively, one could reduce the noise in the image before processing it with "-lat".

```In summary, each pixel is thresholded using the following logic:
AVG = average value of each pixel in the window
IF (input pixel is > AVG + OFFSET)
Output pixel is BLACK
else
Output pixel is WHITE

---

An alternative is to subtract a blurred copy of the original image
using (Modulus) Subtraction, then thresholding.

convert rose: -colorspace gray -lat 10x10+0% x:

is roughly equivalent to...

convert rose: -colorspace gray \( +clone -blur 10x65535 \) \
-compose subtract -composite -threshold 50%  x:

The special "-blur 10x65535"  is a linear averaging blur limiting itself to a
10x10 window.

The 'Subtract' composition being a mathematical modulus type of operation will
wrap the values that goes negative back round to a value greater than 50%.

If you want to include an offset you can do so by also subtracting a solid
color background image by using a -flatten...  for example

convert rose: -colorspace gray -lat 10x10+10% x:

is roughly equivalent to...

convert rose: -colorspace gray \( +clone -blur 10x65535 \) \
-compose subtract -background gray10 -flatten -threshold 50%  x:

```
The above was modified from initial notes provided by D Hobson <dhobson@yahoo.com>

```   -adaptive-sharpen
Sharpen images only around the edges of the images

-segment cluster-threshold x smoothing-threshold
Segmentation of the color space (not image objects)
This can produce very verbose output.
This applies the "fuzzy c-means algorithm" if you want to know more.

Also related is -despeckle. to remove single off color pixels.

Generate a 3d stereogram of two images (one for each eye)
This is also known as a anaglyph
composite left.jpg right.jpg -stereo anaglyph.jpg

```

## Shade 3D Highlighting

### Shade Usage

The "`-shade`" operator I have always thought one of the most interesting operators provided by ImageMagick. The documentation of this operator only gave a rough hint as to its capabilities. It too me a lot of personal research to make sense of the operator, and even figure out how best to use the power that it can provide IM users.

Basically what this operator does is assume that the given image is something called a 'height field'. That is a grey-scale image representing the surface of some object, or terrain. The color '`white`' represents the highest point in an image, while '`black`' the lowest point.

This representation come out of the 1980 computer vision research, where a photo with a strong 'camera light' was used, making near points bright, and points far away dark.

 As a grey-scale image is needed by "`-shade`", the operator will automatically remove any color from the input image. Similarly any transparency that may be present in the image is completely useless and ignored by the operator.

Now "`-shade`" takes this grey-scale height field and shines a light down onto it. The result is a representation of light shades that would thus be produced. Remember you must think of the input image as a 'surface' for the output to make any sense.

For our demonstrations we will need a 'height field' image so lets draw one.
 `````` convert -font Candice -pointsize 64 -background black -fill white \ label:A -trim +repage -bordercolor black -border 10x5 \ shade_a_mask.gif ``````

This image is also equivalent of a 'mask' of an shape, is often not only used as input to "`-shade`", but also for Masking Images to cut out the same shape from the shaded results. See Masking a Shade Image below.

To the "`-shade`" operator this image will look like a flat black plain, with a flat white plateau rising vertically upward. Only the edges of this image will thus produce interesting effects.

To this effect the two arguments defines the direction from which the light is shining.

The first argument is the direction from which the light comes. As such a '`0`' degree angle will be from the east (of left), '`90`' is anti-clockwise from the north (or top), and so on. For example...
 `````` convert shade_a_mask.gif -shade 0x45 shade_direction_0.gif convert shade_a_mask.gif -shade 45x45 shade_direction_45.gif convert shade_a_mask.gif -shade 90x45 shade_direction_90.gif convert shade_a_mask.gif -shade 135x45 shade_direction_135.gif convert shade_a_mask.gif -shade 180x45 shade_direction_180.gif ``````

You get the idea. The light can come from any direction.

The other argument is the elevation, and represents angle the light source makes with the ground. You can think of it as how high the sun is during the day, so that '0' is dawn, and '90' is directly overhead.

 `````` convert shade_a_mask.gif -shade 90x0 shade_elevation_0.gif convert shade_a_mask.gif -shade 90x15 shade_elevation_15.gif convert shade_a_mask.gif -shade 90x30 shade_elevation_30.gif convert shade_a_mask.gif -shade 90x45 shade_elevation_45.gif convert shade_a_mask.gif -shade 90x60 shade_elevation_60.gif convert shade_a_mask.gif -shade 90x75 shade_elevation_75.gif convert shade_a_mask.gif -shade 90x90 shade_elevation_90.gif ``````

As you can see with an elevation of '`0`' the shape is only highlighted on the side from which the light is coming. Everything else is black, as no light shines on any other surface. I call this a 'Dawn Highlight' and has its own special uses.

This brings us to the first item of note. A image that is "`-shade`" will often have more dark, or shadowed areas, than highlighted areas. The shading is not equal.

As the light gets higher over the 'height field' image. The overall brightness of the image will become whiter, until at 'high-noon' or an elevation of '`90`' any flat areas are brilliantly white, and only slopes and edges are shaded to a grey color, with a mid grey as a maximum, or 'cliff-like' slope change.

This 'noon' image is another special case that is a bit like a edge detection system, though it is between 2 and 4 pixels wide for sharp edges. I have used this image in the past for generating a mask for the the beveled edge of "`-shade`" images, so as to make flat areas transparent.

If the elevation angle goes beyond '`90`' degrees, you will get the same result as if the light was from the other direction. As such the argument '`0x135`' will produce exactly the same result as '`180x45`'. A negative elevation angle will also produce the same results, as if the light is coming up from below, onto a 'translucent' like surface. As such '`0x-45`' will be the same as '`0x45`'. In other words for a particular shade there are usually 4 other arguments that will also produce the same result.

From the above I would consider an argument of '`120x45`' to be about the best for direct use of the shade output. For example here it creates some beveled text...
 `````` convert -size 320x100 xc:black \ -font Candice -pointsize 72 -fill white \ -draw "text 25,65 'Anthony'" \ -shade 120x45 shade_anthony.jpg ``````

One of the major problems with "`-shade`" is the thickness of the bevel that is actually produced. A sharp edge such as I used above will always produce a bevel of about 4 pixels wide, both into and out of the masked area. There is no way to adjust this thickness, short of resizing images before and after using the "`-shade`" operator..

If you would like to find out just how bright 'flat areas' will be from a specific elevation lighting angle, then you can use the following command, to shade a flat solid color surface.
 `````` convert -size 50x50 xc:white -draw 'circle 25,25 20,10' \ -blur 0x2 -shade 0x45 -gravity center -crop 1x1+0+0 txt:- ``````

As you can see a elevation of '`45`' degrees produces a quite bright flat color of about 70% grey, which is a reasonable grey level for general viewing. However if you plan to use shade for generating 3-D highlights of various shapes, then the actual grey level becomes very important. This will be looking at later in Creating Overlay Highlights.

That is basically it, for the "`-shade`" operator. However using it effectively presents a whole range of techniques and possibilities, which we will look at next.

### Masking Shaded Shapes

As mentioned above, a simple 'mask' shape is often used with "`-shade`" to generate complex 3-D effects from a simple shape. For example lets do this to a directly shaded mask image.

 `````` convert shade_direction_135.gif shade_a_mask.gif \ -alpha Off -compose CopyOpacity -composite shade_beveled.png ``````

Notice that about half the bevel generated by the "`-shade`" operator, actually falls outside the masked area. In other words, a straight bevel is halved when masked.

On the other hand the vertical or 'midday' shade image (using '`90`' degree elevation angle) can be used to just extract the beveled edge, leaving the center of the image hollow.
 `````` convert shade_direction_135.gif \ \( shade_elevation_90.gif -normalize -negate \) \ -alpha Off -compose CopyOpacity -composite shade_beveled_edge.png ``````

Note however that the 'midday' shade image, while providing a way to mask the location (and intensity) of the effects of the "`-shade`" operator does not actually cover those effects completely.

By combining the 'midday' shade image with the original mask you can increase the size of that mask slightly to produce a better masked beveled image.
 `````` convert shade_direction_135.gif \ \( shade_elevation_90.gif -normalize -negate \ shade_a_mask.gif -compose screen -composite \) \ -alpha Off -compose CopyOpacity -composite shade_beveled_plus.png ``````

Remember with IM v6 you can generate the 'shade' image I generated previously all in the same command. As such the above could have been completely generated from scratch. For example.
 `````` convert -font Candice -pointsize 72 -background black -fill white \ label:X -trim +repage -bordercolor black -border 10x5 \ \( -clone 0 -shade 135x45 \) \ \( -clone 0 -shade 0x90 -normalize -negate \ -clone 0 -compose screen -composite \) \ -delete 0 -alpha Off -compose CopyOpacity -composite \ shade_beveled_X.png ``````

### Shaded Shape Images

The Alpha Extract Operator will not only extract the alpha channel from a shaped images as a gray scale mask, but also has the side effect of preserving the shape in the 'turned-off' alpha channel. As it is 'turned off' it will not be touched by many image processing operators, including "`-shade`", preserving its detail.

What this means is that for shaped images, you can extract the shape, do the work, then simply recover the transparency AFTER you have finished all the image processing, simply by turning the Alpha On again!

For example here I draw a 'Heart' on a transparent background, do some blurring and shading of the image, then restore the original shaped outline of the image.
 `````` convert -size 100x100 -gravity center -background None \ -font WebDings label:Y \ -alpha Extract -blur 0x6 -shade 120x21 -alpha On \ -normalize +level 15% -fill Red -tint 100% shade_heart.png ``````

All I can say is WOW, what a time saver! A lot simplier than the previous command with all its extra processing and image cloning.

### Rounding Shade Edges

As you saw in the last example, by blurring the image shape mask, the 'slope' of edge 'cliffs' will be smoothed out, as if worn down by time. This produces a nice rounded effect to the shade image.
 `````` convert -size 50x50 xc:black -fill white -draw 'circle 25,25 20,10' \ shade_circle_mask.gif convert shade_circle_mask.gif -shade 120x45 shade_blur_0.gif convert shade_circle_mask.gif -blur 0x1 -shade 120x45 shade_blur_1.gif convert shade_circle_mask.gif -blur 0x2 -shade 120x45 shade_blur_2.gif convert shade_circle_mask.gif -blur 0x3 -shade 120x45 shade_blur_3.gif convert shade_circle_mask.gif -blur 0x4 -shade 120x45 shade_blur_4.gif convert shade_circle_mask.gif -blur 0x5 -shade 120x45 shade_blur_5.gif ``````

As you can see blurring not only rounds-off the edges, but makes the lighting effects dimmer. You can maximize the contrast of the result by normalizing the it, so as to bring the brightest and darkest points back to pure white and black colors respectively.
 `````` convert shade_blur_3.gif -normalize shade_blur_3n.gif ``````

The only draw back with this is that this also generally darkens the shaded image. This is something which we'll need to take into account in Creating Overlay Highlights.

Lets finish off this shade image by directly masking it as well..
 `````` convert shade_blur_3n.gif shade_circle_mask.gif \ -alpha Off -compose CopyOpacity -composite shade_blur_3n_mask.png ``````

As you can see blurring the mask image will round off the edges of the resulting shape very nicely.

### Creating Overlay Highlighting

The output from the "`-shade`" operator is very nice, but it is rare that you actually want a plain grey scale image of your shape. What is needs is some color.

This however is not so easy as the two major ways of adding color, Color Tinting Mid-Tones to just recolor a grey-scale, or 'Overlay' alpha composition, to replace the grey areas with a image, both rely on a special form of grey-scale image. That is a perfect mid-tone grey ('`grey50`') is replaced by the color or image, while whiter or darker greys, whiten and darken the color or image as appropriate.

These special grey-scale 'overlay highlight' images with perfect mid-tone greys for un-modified areas is not so straight forward to create using "`-shade`". However the following are some of the more simpler ways I have discovered.

Using a 30 degree elevation lighting angle with "`-shade`", is one way of producing a perfect mid-tone grey for flat areas of the shape being shaded.

For example here I shade an image, then extract the top-left pixel to check the resulting color of a 'flat' part of the image.
 `````` convert -size 50x50 xc:black -fill white -draw 'circle 25,25 20,10' \ -blur 0x2 -shade 120x30 shade_30.png convert shade_30.png -gravity center -crop 1x1+0+0 txt:- ``````

Unfortunately changing the rounding effect of the "`-blur`" in the above command tends to also vary the result highlight intensity of the shade image. That is using a large blur not only produces a well rounded looking edge, but also made the highlight so dim as to be near invisible.

This means that you need to add lots more contrast to the output of the "`-shade`" image produced, to make the highlight effective as an overlay image. To fix this we need a way remove this contrast effect from the rounding adjustment. The typical way to do this is to just "`-normalize`" the image, but doing this to 30 degree shade image, results in the 'flat' areas will no longer being a perfect grey. For example...
 `````` convert -size 50x50 xc:black -fill white -draw 'circle 25,25 20,10' \ -blur 0x2 -shade 120x30 -normalize shade_30_norm.png convert shade_30_norm.png -gravity center -crop 1x1+0+0 txt:- ``````

After some further experimentation however I found that using a 21.78 degree shade elevation angle, will after being normalized, produce the desired perfect mid-tone grey level as well as a good strong highlighting effect.
 `````` convert -size 50x50 xc:black -fill white -draw 'circle 25,25 20,10' \ -blur 0x2 -shade 120x21.78 -normalize shade_21_norm.png convert shade_21_norm.png -gravity center -crop 1x1+0+0 txt:- ``````

As the shade image is now run though the "`-normalize`" operator, the "`-blur`" value used for 'rounding edges' will no longer effect final intensity of the result. A much better method.

In summery, normalizing a shade image will shift the mid-tones away from a perfect-grey color.

Now we can adjust the output intensity of the highlights produces output completely independent to the other adjustments. Typically as the normalized result is extreme, we will need a controlled de-normalization, or anti-contrast control, to reduce the highlight to the desired level.

The simplest method for adjusting the resulting highlight, is to color tint the image with a perfect grey. This will shift all the color levels in the image toward the central pure mid-tone grey color.

For example...
 `````` convert -size 50x50 xc:black -fill white -draw 'circle 25,25 20,10' \ \( +clone -blur 0x2 -shade 120x21.78 -normalize \) \ +swap -alpha Off -compose CopyOpacity -composite shade_tint_0.png convert shade_tint_0.png -fill grey50 -colorize 10% shade_tint_10.png convert shade_tint_0.png -fill grey50 -colorize 30% shade_tint_30.png convert shade_tint_0.png -fill grey50 -colorize 50% shade_tint_50.png convert shade_tint_0.png -fill grey50 -colorize 80% shade_tint_80.png ``````

An alternative to just linearly tinting the highlight, is to reduce its general effect while preserving the extreme bright/dark spots of the highlight by using Sigmoidal Non-liner Contrast instead. This should give a more 'natural' look to the highlight effect, and can make the highlight brighter, as if the surface was more reflective.

However to make this technique more effective, we need make sure we do not have pure white and black colors in the shade result. This can be achieved by first using a "`-contrast-stretch`" of '`0%`' rather than "`-normalize`", and also de-normalizing that result by a small amount, as we did above.

This may seem to be just adding complexity to the generation of the highlight overlay image, but emphasizing the bright spots in the highlight makes the extra processing worth the effort.

For example...
 `````` convert -size 50x50 xc:black -fill white -draw 'circle 25,25 20,10' \ \( +clone -blur 0x2 -shade 120x21.78 -contrast-stretch 0% \) \ +swap -alpha Off -compose CopyOpacity -composite shade_sig_0.png convert shade_sig_0.png -sigmoidal-contrast 10x50% shade_sig-10.png convert shade_sig_0.png -sigmoidal-contrast 5x50% shade_sig-5.png convert shade_sig_0.png -sigmoidal-contrast 2x50% shade_sig-2.png convert shade_sig_0.png +sigmoidal-contrast 2x50% shade_sig+2.png convert shade_sig_0.png +sigmoidal-contrast 5x50% shade_sig+5.png convert shade_sig_0.png +sigmoidal-contrast 10x50% shade_sig+10.png ``````

As you can see that the overall highlighting is reduced in intensity, but the bright spot from reflected light remains as bright as ever, just reduced in size. The result is a much more natural 'shiny' look to the shape.

The only drawback with this technique is that a shadow 'spot' is also generated though this is often not as noticeable.

Finally we can combine the a 'highlight spot' with a general highlight reduction to produce a highly configurable set of highlight overlay generator controls...
 `````` convert -size 50x50 xc:black -fill white -draw 'circle 25,25 20,10' \ \( +clone -blur 0x4 -shade 120x21.78 -contrast-stretch 0% \ +sigmoidal-contrast 7x50% -fill grey50 -colorize 10% \) \ +swap -alpha Off -compose CopyOpacity -composite shade_overlay.png ``````

In summary, the above example has four separate controls...
"`blur`" : Rounding the shape edges (0.001=beveled 2=smoothed 10=rounded)
"`shade`" : The direction the light is coming from (120=top-left 60=top-right)
"`sigmoidal`" : surface reflective control highlight spots (1=flat 5=good 10=reflective )
"`colorize`" : Overall contrast of the highlight ( 0%=bright 10%=good 50%=dim )

Note while the above examples have been shaped to the original 'circle' shape, the transparency should only be restored AFTER 'Overlay' compositing has been applied, not before.

Also if you plan to use a highlight repeatedly on the same shape (after any rotation is performed), you can pre-generate the highlight overlay once for each shape you plan to use, saving the result for multiple re-use. An example of this re-use of shading overlay is with the generation of 3D DVD covers from flat source images in the IM Discussion Forums.

I also highly recommend you experiment with the above techniques, as they are key to making your flat shaped images, much more realistic looking. If you come up with other ideas for highlighting, please let me know.

```FUTURE:
Color Tinting the Overlay image
Overlay Alpha Composition with an Image
```

### Using a Dawn Shade Highlight

In Masking Shade Images above we showed how useful a 'mid-day' or 'high-noon' shade image (using an elevation of '`90`'), can be useful for masking and location and extent of the effects produced by "`-shade`. However the horizontal or 'dawn' shade images (using an elevation of '`0`')of a shape can also be quite useful as well.

It can for example be used as a mask for either white or black images to generate separate highlight and shading effects on shapes. This also can be used ensure a shape gets roughly equal amounts of light and dark areas (or even unequal amounts), as I produce them in seperatally but in a completely controled way.

`FUTURE: more detail here`
See the first Advanced 3D Logo for an example of using this technique.

## Using FX, The DIY Image Operator

The new IM version 6 image list operator "`-fx`" is a general DIY operator that does not fit into any specific category of IM operators, as it can be used to create just about any image operation. Examples of its use are thought these pages, but here we will look specifically at its capabilities and how you can use them.

The command is so generic in its abilities, that it can,
• create canvases, gradients, mathematical colormaps.
• move color values between images and channels.
• adjust image colors in just about any way imaginable
• translate, flip, mirror, rotate, scale, shear and generally distort images.
• merge or composite multiple images together.
• tile image(s) in weird and wonderful ways.
• convolve or merge neighboring pixels together.
• generate image metrics or 'fingerprints'
• compare images in unusual ways.

Of course many of these techniques are already part of IM, producing a faster and more flexible result. But if it isn't built-in the "`-fx`" allows you to generate your own version of the desired operation. In fact I and others have often used it to prototype new operations that are later built into IM's core library. As an example see DIY New Ordered Dither Replacement where I used "`-fx`" to develop a revised version of the `-ordered-dither`" operator.

The operator is essentially allows you to perform free-form mathematical operations on one or more images. For the official summary of the command see FX, The Special Effects Image Operator on the ImageMagick Web Site.

### FX Basic Usage

The command takes a image sequence of as many input images you like. Typically one or two images, and replaces ALL the input images with a copy of the first image, which has been modified by the results of the "`-fx`" function. That is any meta-data that is in the first image will be preserved in the result of the "`-fx`" operator.

For mathematical ease of use, all color values provided are normalized into a 0.0 to 1.0 range of values. Results are also expected to be in this range.

This includes the transparency or alpha channel, which goes from 0.0 (meaning fully transparent) to 1.0 (meaning fully opaque). The values represent 'alpha transparency' and is actually the negative of how IM normally stores the transparency internally (as matte values). It is however more mathematically correct and easier to use in this form.

The "`-channel`" setting defines what channel(s) in the first (also called the 'zeroth' or "`u`") image, is replaced with the result of the "`-fx`" operator. This is limited, by default, to just the color channels ('`RGB`') of the original image. Any existing transparency in that image will not be modified, unless the "`-channel`" setting is changed, to include the alpha ('`A`') channel.

The expression is executed once for each pixel, as well a once for each color channel in the pixel that is being processed. Also as the expression is re-parsed each time it is executed, a complex expression could take some time to process on a large image.

For example, here we define a black image, but then set the blue channel to be half-bright to form a 'navy blue' color instead.
 `````` convert -size 64x64 xc:black -channel blue -fx '1/2' fx_navy.gif ``````

And here we we take a black-white gradient, and then set the blue and green channels to zero, so it becomes a black-red gradient.
 `````` convert -size 64x64 gradient:black-white \ -channel blue,green -fx '0' fx_red.gif ``````

 To make the "`-channel`" setting more like the "`-fx`" operator, it will accept any combinations of the letters '`RGBA`' to specify the channels to which operators are to confine their actions. This means that to limit the output of "`-fx`" to just the blue and green channels you can now say "`-channel BG`" instead of the longer "```-channel blue,green```".

We could have generated the above examples without using "`-fx`", but being able to do this to an existing image is what makes this a powerful image operator.

The function can in fact read and use ANY pixel, or specific color from ANY of the images already in the current image sequence in memory. The first 'zero' image, is given the special name of "`u`". The second image "`v`". Other images in memory can be referenced by an index. As such "`u[3]`" is the fourth image in the current image sequence, while "`u[-1]`" is the last image in the sequence. This is the same indexing scheme used by the Image List Operators, so you should be right at home.

If no other qualifiers are given, the color value used is same color used in the image specified. That is unless you specifically say you want to use the red color, it will use the color value for the color channel the command is processing at that time. That is it will apply the expression for the blue color value when it is processing the blue channel.

Unless told otherwise it will process each of the RGB color values (as set by the default "`-channel`" setting), for each and every pixel in the image. That is 3*w*h calculations which modifies all the values in the image by the expression given.

For example here we take the IM built-in "`rose:`" image and multiply all pixel values by 50%.
 `````` convert rose: -fx 'u*1.5' fx_rose_brighten.gif ``````

In the above example, each of the individual red, green and blue values was multiplied by 1.5. If the resulting value is outside the 0 to 1 range it, will be limited to the appropriate bound (1.0 in this case), unless you are using a specially built HDRI version of ImageMagick.

Lots of other "`-fx`" formulas to recolor images are explored in Mathematical Color Adjustments and Histogram Curves.

As we can also reference any image in the current image sequence, as part of the expression for modifing the first image, we can merge two, or even more images, in just about any way we want.

Here we generate a black-red-blue color chart image, by copying the blue channel from a black-blue gradient (rotated), into the previous black-red gradient we generated above.
 `````` convert -size 64x64 gradient:black-blue -rotate -90 fx_blue.gif convert fx_red.gif fx_blue.gif \ -channel B -fx 'v' fx_combine.gif ``````

 Of course we could have just used a Channel Coping Composition Method instead which would be a lot faster. But that is not point. Though the reverse is also true. Just about every IM image operation could be replaced by a FX equivelent function.

Now the second image in the above is only used as a source image. What really happens is that "`-fx`" first creates a copy of just the first image. It then modifies that image according to the formula, using all the other images given. And finally it junks all the input images replacing them with the modified copy of the first image.

You can also calculate values based on each pixel location within the image. values '`i,j`' is the current position of the pixel being processed, while '`w,h`' gives the size of the image (the first image unless a specific image qualifier is given). For example here we generate a DIY Gradient Image.
 `````` convert rose: -channel G -fx 'sin(pi*i/w)' -separate fx_sine_gradient.gif ``````

Or something more complex using both '`i,j`' position values.
 `````` convert -size 80x80 xc: -channel G -fx 'sin((i-w/2)*(j-h/2)/w)/2+.5'\ -separate fx_2d_gradient.gif ``````

When generating gray-scale gradients, you can make the -fx operator work about 3 times faster, simply by asking it to only process one color channel, such as the '`G`' or green channel in the above example. This channel can then be Separated to generate the final gray-scale image. This can represent a very large speed boost, especially when using a very complex "`-fx`" formula.

For more FX generated gradients, see examples Roll your own Gradients.

You can use the position information to lookup specific pixels from the source image using the '`p{x,y}`' syntax.

For example you can easily make your own 'mirror image' type function (like the "`-flop`" image operator), that replaces each pixel, with the color values from the 'mirror' position of the original source.
 `````` convert rose: -fx 'p{w-i-1,j}' fx_rose_mirror.gif ``````

This type of 'image distortion' was made more powerful by creating Distortion Image Mapping, or other types of Value Lookup Tables, in the form of images. Examples of doing this has been provided in DIY Dither Patterns and Threshold Maps, where FX is used to replace specific colors with patterns from other images.

Now the size of the final image generated by an FX expression is the same as the first image given, as such to generate a larger image, you will need to set the first image to the size you want.

In this type of situation a second image (or even a third image) can be used as a color source (hence the Swap in the next example).

For example here we resize rose image (using Interpolated Scaling or Resize) to generate a larger image.
 `````` convert rose: -size 120x80 xc: +swap \ -fx 'v.p{ (i+.5)*v.w/w-.5, (j+.5)*v.h/h-.5 }' \ fx_scaled.png ``````

Note how the pixel lookup is performed, it may seem complex but it is the proper way to scale (distort) an image. Basically all the extra '`0.5`' values added to the expression is needed to correctly convert between Pixel Coodinates used for input coordinates '`i,j`' and location lookup '`v.p{...}`, while the more mathematically correct Image Coordinates is needed for the actual mathematical calculations (scaling).

The above is actually the exact methodology used by any form of Image Distortion. You can see this FX equivelent for most distortions by turning on the Verbose Distortion Summery. This reports a FX equivelent for most image distortions, as a way to double check the distortion is doing what it is expected to do.

The use of the FX DIY Operator to do image distortions, shows just how powerful (though slow) this operator really is. If it wasn't for this operator I doubt may of the new operations, such as distortions, sparse-color, or ordered dithers would have been added to the ImageMagick Core Library.

Here is something a little simplier, swapping the red and blue channels of the rose image. See if you can figure how it works.
 `````` convert rose: \( +clone -channel R -fx B \) \ +swap -channel B -fx v.R fx_rb_swap.gif ``````

 A faster and better way to do the same thing, is to use "`-separate`" and "`-combine`"). See Combining RGB Channel Images. Alternatively you can also use a "`-color-matrix`" to do the same thing faster still. Do you see a trend here?

As the default "`-channel`" setting, it limits the output of the "`-fx`" operator to just the three color channels. This means that if you want to effect the alpha or transparency channel, you must explicitly specify it, by changing the channel setting.

For example lets make a semi-transparent "`rose:`" image, by setting all the alpha channel values to half.
 `````` convert rose: -alpha set -channel A -fx '0.5' fx_rose_trans.png ``````

Note the for the above to work properly I needed to ensure that the "`rose:`" actually had a alpha channel for the "`-fx`" to work with. I did this with the Alpha Channel Control Operator.

This ability of the "`-fx`" operator to manipulate the RGBA channels of an image makes this operator perfect for manipulating Channels and Masks.

As of IM 6.2.10 you can add variable assignments to "`-fx`" expressions, which allows you to reduce the complexity of some expressions, that would basically be impossible any other way.

For example, here I create a gradient based on the distance from a particular point (assigned to the variables '`xx`' and '`yy`'). Without the use of the variables this formula could have become very hard to read.
 `````` convert -size 100x100 xc: -channel G \ -fx 'xx=i-w/2; yy=j-h/2; rr=hypot(xx,yy); (.5-rr/70)*1.2+.5' \ -separate fx_radial_gradient.png ``````

 Due the simple tokenization handling used by "`-fx`", variable names can only consist of letters, and must not contain numbers. Also as a lot of single letters are used for internal variables accessing image information, it is recommended that variable names be at least two letters long. As such I use '`xx`' and '`yy`' rather than just '`x`' or '`y`'.

 The "`-fx`" function 'rr=`hypot(xx,yy)`' was added to IM v6.3.6 to speed up the very commonly used expression '`rr=sqrt(xx*xx+yy*yy)`'. Of course if you need the distance squared, you should avoid the '`hypot()`' function, and the sqrt() function it implies.

For more examples of some really complex expressions see More Complex DIY Gradients, which would be impossible with out multiple statement assignments. The same is true for FX form of Perspective Distortion.

As of IM version 6.3.0-1, the complexity of "`-fx`" expressions started to require external files, so the standard '`@filename`' can now be used to read the expression from a file.
 `````` echo "u*2" | convert rose: -fx "@-" fx_file.png ``````

This also means you can use more complex scripts to generate the specific FX expressions for a particular job. Internally the file is simply read into a string and interpreted as usual.

Other settings that are important to "`-fx`" are "`-virtual-pixel`" and "`-interpolate`".

The Virtual Pixel Setting allows one to set what colors or image results should be returned when the lookup coordinates go outside the area covered by the input image. This allows one to set edge effects for things like blurs, as well as tile image over a larger area.

The Interpolate Setting allows one to specify how IM should mix colors of neighbouring pixels when the lookup coordinates (floating point values) fall between the integer coordinates of the pixels in the input image. For more information see Interpolated Pixel Lookup.

 Some More functions were added at various times IM v6.3.6 : hypot() IM v6.7.3-4 : while(), not(), guass(), squish()

### FX Debugging

The '`debug(expr)`' is essentially a way of printing a floating point value, each time the FX expression is calculated. This in turn provides a method of debugging your expressions.

However you can limit the output from the "`debug()`" by using a tertiary if-else expression. For example this will print the floating point color values for pixel 10,10 from the built-in "`rose:`" image. The actual image result is ignored by using the '`NULL:`' image handler.
 `````` convert rose: -fx 'i==10&&j==10?debug(u):1; u' null: ``````

Remember the output is on standard error, not the normal standard output, that way you can use this in a command pipeline, without problems.

Note how the FX expression was executed three times, once for each channel for just that one pixel. Multiply that by the number of pixels, and you can imagine the length of the output if "`debug()`" was not limited to just one pixel, even for this small image.

### FX-like Built-in Operations

Well "`-fx`" operations is slow, very slow when processing large images with very complex expressions. However it represents a way to develop new image processing functions that previously did not exist in ImageMagick. The result of such development by users has allows ImageMagick to expand, with new functions and methods, such as the Color Lookup Table ("`-clut`").

Generally however once a new method has stabilized using "`-fx`", the expression is converted into a faster built-in operation, usually added as part of a group of similar operators.

These include the follow general image operator and there methods...
 -evaluate Direct pixel, color values, channel modification functions. (See Evaluate below). -function More Complex pixel, color value, channel modification functions. (See Function below). -evaluate-sequence Merge a multi-image sequence of images mathematically (See Evaluate-Sequence below). -sparse-color General Image Re-Coloring Operator. (See Sparse Color Gradients ) -compose General Multi-Image combining and overlaying method. (See Alpha Composition). -distort General Image Distortion operator, using reversed pixel mapping. (See the Distort Operator ) -morphology General Area Effect Convolution/Morphology function. (See the Morphology Operator and the Convolve Operator )

As people developed new types of image operations, they usually prototype it using a slow "`-fx`" operator first. When they have it worked out that 'method' is then converted into a new fast built-in operator in the ImageMagick Core library. Users are welcome to contribute their own "`-fx`" expressions (or other defined functions) that they feel would be a useful addition to IM, but which are not yet covered by other image operators, if they can be handled by one of the above generalized operators, it should be reasonably easy to add it.

For example I myself needed a 'mask if color similar' type operation for comparing two images. This has been added as a new "`-compose`" method "`ChangeMask`". This in turn allowed me to then add a more complex Transparency Optimization for GIF animations.

What is really needed at this time is a FX expression compiler, that will pre-interpret the expression into a tighter and faster executable form. Someone was going to look into this but has since disappeared.

However if speed and complexity is starting to become a problem then it is probably better to move on to a API scripting language such as PerlMagick. An example of this using PerlMagick "`pixel_fx.pl`" is part of that API's distribution.

### FX Expressions as Format and Annotate Escapes

As of IM version 6.2.10 you can now use FX Expressions within Image Property Escaped strings such as used by "`-format`" and "`-annotate`" arguments.

The escape sequence '`%[fx:...]`' is replaced by a number as a floating point value, calculated once for each image in the current image sequence.

The FX Expression however is modified slightly during processing. Specifically...

• The current pixel coordinates '`i`', '`j`' is fixed to the value 0, so on its own an image variable only returns the value from pixel 0,0, unless a '`p{}`' index is used.
• Unless a color channel is selected only the red channel value is returned.
• The default image reference '`s`' is set to current image, being annotated or identified.
• The index '`t`' returns the index of the image referred to by '`s`'.
 Before IM v6.6.8-6 both FX expression values of "`t`" image index and "`n`" total number of images, were broken, and only returned a value of 0 and 1 respectively for ALL images. The same goes for the equivalent percent escapes '`%p`' and '`%n`'.

For example here I "`-annotate`" each image with the color of the top left corner of each image.
 `````` convert -size 150x25 xc:DarkRed xc:Green xc:Blue \ -fill white -gravity center \ -annotate 0 '%[fx:t] / %[fx:n] : %[fx:r],%[fx:g],%[fx:b]' \ annotate_fx_%d.gif ``````

Notice how the text that is written is different for each image, as '`r`' is actually equivalent to '`s.p{0,0}.r`'. The same goes for the '`g`' and '`b`' color channel values. Of course each one returns a normalized value in the range of 0.0 to 1.0.

To make the output of specific pixel color values easier, a '`%[pixel:...]`' escape was also added in IM v6.3.0. This operator calls the given FX expression once for each channel in each image, and formats the returned value into a color that IM can handle as a color argument.
 `````` convert -size 300x100 gradient:yellow-limegreen \ -gravity NorthWest -annotate 0 '%[pixel:s.p{0,0}]' \ -gravity Center -annotate 0 '%[pixel:s.p{0,50}]' \ -gravity SouthEast -annotate 0 '%[pixel:s.p{0,99}]' \ annotate_pixel.gif ``````

You can just output the result directly using a "`-format`" with the "`identify`" command.
 `````` identify -format '%[fx:atan(1)*4]' null: ``````

This will mathematically calculate and return the value of PI, though this value is available as the built-in variable '`pi`'.

You can generate random numbers. For example to generate a integer between -5 and 10 inclusive. Here I use the "`info:`" equivalent to the "`identify`" command.
 `````` convert xc: -format '%[fx:int(rand()*16)-5]' info: ``````

For more methods see Identify Alternatives: Text Output Options.

Also see Border with Rounded Corner which used a FX Expressions to generate a draw string based on image width and height information.

You can Calculate Positions Images using FX formulas or even position using the size and location of other images (See Incrementally Calculated Positions).

You can also use FX Escapes in Filename Percent Escapes to generate new files based on calculated values. For an example, see the final example in Tile Cropping.

All the above will essentially run the "`-format`" and thus any containing FX Expression one for each image in the current image sequence.

The "`-print`" operator will work much like "`-identify`" except that it is only run once, with access to ALL the images in the current image sequence. With this operator you can use '`u[{i}]`' to access values from any image, unlike the above.

Fx Expressions can be applied to images in other colorspaces, so I can for example find out the 'Hue' value (in the 'red' channel) for three different colors.
 `````` convert xc:red xc:green xc:blue -colorspace HSL \ -format '%[fx: s.r ]\n' info: ``````

You can also use IM for some direct color maths, such as find out the average color of '`gold`', '`yellow`', and '`khaki`'.
 `````` convert xc: -format '%[pixel:(gold+yellow+khaki)/3]' info: ``````

While this shows what the color looks like compared to the three source colors...
 `````` convert xc:gold xc:yellow xc:khaki +append \ \( xc: -fx '(gold+yellow+khaki)/3' \) \ -scale 90x30\! -append fx_hues.png ``````

You can also use "`-print`" to print information. This is applied only once against the whole image sequence. That means you can use this operator to calculate much more complex '`%[fx:...]`' expressions involving multiple images.

#### Accessing data from other images

There is one serious problem with using FX escaped expressions however. IM does not have direct access to the other images in the current image sequence when you are creating images. This is just generally not needed, in typical image creation, as new images generally do not depend on previous in-memory images.

Basically if you want to gather the color of a specific pixel in a different image to the one you are drawing on (as above), or are creating a new image, then the IM core functions have no direct link to the desired info.

For example if you try to create a label with the color of the built-in "`rose:`" image pixel 12,26 (a bluish pixel), the direct approach will fail!
 `````` convert rose: label:'%[pixel:p{12,26}]' -delete 0 label_fx_direct.gif ``````

Well the rose image does not actually contain any black pixels, so the above result was wrong.

The way to fix this is to extract the wanted information and save it into the global IM meta-data. This is passed to all sub-routines in the library core, including those for image creation.
 `````` convert rose: -set option:mylabel '%[pixel:u.p{12,26}]' -delete 0 \ label:'%[mylabel]' label_fx_indirect.gif ``````

This is not intuitive but we now get the correct result.

The special '`option:`' tag, tells the "`-set`" option that you want the given setting saved as a global Artifact, rather than as a image 'Attribute' or 'Properties' string, just as "`-define`" can. However the "`-set`" form allows you to expand Percent Escapes in setting the Artifact, where as "`-define`" does not.

When the "`label:`" operator expands its percent escapes, the given 'key' is looked for first as a per image 'attribute' or 'proprieties', but if it fails to find anything, it will then look for the 'key' in the global Artifact settings. As such the global 'artifact' we created from the previous image is used, even though that image is no longer present at the time the Artifact was created.

Basically 'Artifact' settings are global during the life time of the "`convert`" command, and thus can be used to pass information from one image to another.

For programmed API's this situation can be avoided as you can read the required data directly from the image and generate the label string yourself, without needing IM to store that information in such a convoluted way.

## Evaluate and Function, Freeform Channel Modifiers

Because the FX Operator (see above) is so slow, the "`-evaluate`" operator was added to let you make simple image modifications more quickly.

Later a more complex "`-function`" operator was added in IM v6.4.8-8, to allow greater flexibility in complex image adjustments. These two operators, along with other Image Level Adjustment Operators such as "`-negate`", "`-level`", will probably be most useful for minor tweaks to grey-scale images, before you apply those images.

Especially in gray-scale images such as used for Background Removal, Highlight and Shadow Overlays, and the generation and fine-tuning of Image Maps.

### Evaluate, Simple Math Operations

The "`-evaluate`" operator is basically a fast, but very simple version of "`-fx`" operator (actually pre-dates its addition to IM by just a couple of months). However it is limited to just one simple operation, using a single user provided constant number.

You can find out what functions have been built into evaluate using
 `````` convert -list evaluate ``````

This includes the typical mathematical functions '`add`', '`subtract`', '`multiply`', and '`divide`'. against constant values.

Unlike the `-fx` operator the values are not normalised to a 0 to 1 range, but remain the real color values of the image. As such subtracting a value of 50 in a Q8 IM (See Quality and Depth will result in a large subtraction, but for a Q16 version of IM, it will only be a small hardly noticeable change.

However if you add a '%' to the argument, that argument will represent a percentage of the maximum color value (known as '`QuantumRange`' which is equal to ('`2quality-1`'). This means you can make your "`-evaluate`" arguments IM quality level independent, by the appropriate use of percentages for the appropriate evaluate methods.

For example to just simply replace all color values in an image to a 50% gray level is very simple and very fast, using '`Set`'
 `````` convert rose: -evaluate set 50% rose_set_gray.gif ``````

The "`-evaluate`" operator also includes the typical mathematical functions '`add`', '`subtract`', '`multiply`', and '`divide`'.

For example, to half the contrast of the image, you can '`divide`' it by '`2`' then '`add`' '`25%` to re-center it around a the perfect grey.

 `````` convert rose: -evaluate divide 2 -evaluate add 25% rose_de-constrast.gif ``````

This is a couple of orders of magnitude faster than directly using the "`-fx`" operator with '`u/2+.25`'. As such you should use this operator in preference to "`-fx`" if at all possible.

The major problem with "`-evaluate`" is that all results are clipped to the 0 to '`QuantumRange`' limits (unless you are using a HDRI version of ImageMagick), as each modified value is saved back into the image data. That means that after any individual "`-evaluate`" operation, the values could be clipped by the '`QuantumRange`'.

As such if you try to apply a contrast enhancement function (equivalent to "`-fx '2*u-.25' `") directly as it stands, you will fail to get the correct results, as the doubled value will be clipped, before the subtraction is made.
 `````` convert rose: -evaluate multiply 2 -evaluate subtract 25% \ rose_contrast.gif ``````

First the '`multiply`' will clip all the large color values to the maximum value, then the '`subtract`' will clip the lower bound values. the result is an incorrect clipping of the upper bounds, producing a dark and color distorted result.

The direct solution is to '`subtract`' the appropriate constant first (doing the final but correct clip of the lower bounds), before multiplying, effectively using the equivalent formula '`(u-.125)*2`'
 `````` convert rose: -evaluate subtract 12.5% -evaluate multiply 2 \ rose_contrast2.gif ``````

However there are lots of alternatives to this 'clipping' problem. The first logical one being the newer Polynomial Function Method (see below). Other alternatives also include using Level Adjustment Operators or even a Level Adjustment by Color, to simply specify the original color values that you want to stretch out, to fill whole color range.

Basically be careful with regards to color value clipping when using multiple "`-evaluate`" methods.

The "`-evaluate`" operator, like "`-fx`" (and most other low level IM operators) is "`-channel`" effected. This allows you to control an images alpha transparency separately to the color channels.

And yes, like "`-fx`", transparency is treated as 'alpha values' and not a 'matte' value.

For example to make an image 50% transparent, as part of a Dissolve type operation.
 `````` convert rose: -alpha set -channel A -evaluate divide 2 rose_transparent.png ``````

The result is a semi-transparent image, which means when displayed, half the color you see is the web-pages background color. As such the image shown is dimmed toward the background color.

Often I have also found that it is often easier to use "`-evaluate`" on the individual color channels before separating the various channels into separate images for specific purposes, (See Separating Channels).

For example here I use it to do a fast, but unusual form of gray-scaling. Basically I multiply each channel by the appropriate amount, then separate and add the channels together to produce a image that has been gray-scaled using a specific set of color ratios.
 `````` convert rose: -channel R -evaluate multiply .2 \ -channel G -evaluate multiply .5 \ -channel B -evaluate multiply .3 \ +channel -separate \ -background black -compose plus -flatten grey_253.png ``````

### Evaluate Math Functions

Included in Evaluate, there are also a set of special purpose mathematical functions.

These functions are implemented to generally use a normalized color value (0 to 1 range) with the output again normalized so as to fit the full color range of the image. The Sigmoidal Contrast function is also an example of this math function fitting.

#### Power Of

The '`Pow`' function (added IM v6.4.1-9) for example works with normalized color values, and allows users to do image brightness modifications.

It is exactly equivalent to the pow() C function, (using normalize color values in a 0 - 1 range)
`    value = pow(value, constant) `
As such to create a 'parabolic' gradient you can use an argument of '`2`'. Or use a value of '`0.5`' to create a 'square root' gradient. For example...

 `````` convert -size 20x600 gradient: -rotate 90 gradient.png convert gradient.png -evaluate Pow 2 eval_pow_parabola.png convert gradient.png -evaluate Pow 0.5 eval_pow_sq_root.png ``````

 The three lower images show the profile of the gradient produced both a graph and the original image itself. This makes it easier to see how one gradient image was modified to become another. It was generated using the Gnuplot graph ploting program, via the script "`im_profile`" in the IM Examples, Scripts directory.

This is actually equivalent to the Gamma Adjustment operator but with the argument inverted. For example a "`-gamma 2`" operation would be equivalent to an "```-evaluate pow 0.5```" or a 'square root' operation function. Similarly "`-gamma 0.5`" is equivelent to squaring using "```-evaluate pow 2```"

By doing some special gradient manipulations, you can use this method to convert a linear gradient into a complex circular arc.
 `````` convert -size 20x300 gradient: -rotate 90 \ -evaluate Pow 2 -negate -evaluate Pow 0.5 \ -flop \( +clone -flop \) +append eval_circle_arc.png ``````

For those wanting to figure this out, the second line in the above is equivelent to the FX expression '`sqrt(1-u^2)`'. This generates a single quarter circle arc, which is then Flopped, and Appended together, to produce a half circular arc.

It is also a lot faster than using an FX expression, even though it requires many more individual (smaller) steps.

See also the more advanced Polynomial Function.

#### Logrithmic

The '`Log`' function (added IM v6.4.2-1) also works with normalized values (with a 1.0 added to avoid infinities), with the given constant being used as the logarithmic base. The actual formula (with normalized values) is thus...
`    value = log(value*constant+1.0)/log(constant+1.0) `
For example...
 `````` convert gradient.png -evaluate Log 10 eval_log.png ``````

This may seem very simular to the previous Pow Evaluate Method, but it isn't quite the same. '`Log`' will produce a appreciable slope as it approaches '`0`', where '`Pow`' will produce a vertial slope. The value controls the slope.

A logrithmic function is also closely related to a exponential function, which is currently only implemented as Sigmoidal Contrast Adjustment operator. This contains the same slope features you can see in the above logrithmic curves. This explains why "`-sigmoidal-contrast`" is a better technique for enhancing images involving low light conditions, than a Gamma Adjustment or 'power of' curve.

#### Sine and Cosine

As of IM v6.4.8-8 the '`sin`' and '`cos`' methods were added. These methods take the value given in the image and normalize it into a angle so the full range will cover a full circle of angles. The result is given a 50% bias and scaled to again fit into the normal range of values. The constant is used as a multiplier for the value (and thus the angle) so that 'N' means the function will go around the circle 'N' times over the full value range.

Specifically it defines these function (using normalized values) as...
```   value = 0.5 * sin( constant*value*2*PI ) + 0.5
value = 0.5 * cos( constant*value*2*PI ) + 0.5 ```
In essence what these functions do is re-map the image values (usually gray-scale values) into a sine/cosine curve.

For example here I take a gradient image and modify it using these evaluate methods.
 `````` convert gradient.png -evaluate sin 1 eval_sin_1.png convert gradient.png -evaluate cos 1 eval_cos_1.png ``````

Now as the constant parameter is a angle multiplier, the value given to the evaluate method will create that many peaks over the whole gradient within an image.
 `````` convert gradient.png -evaluate cos 5 -negate eval_cos_5.png ``````

This is perfect for many tasks, from generating ripple or dispersion effects to generate ripple looking displacement curves.

By using a multiplier constant of '0.5' you can simply convert a linear gradient into a sine curve gradient, which still has the same slope as the original. By negating the result you can ensure that the gradient also slopes correctly.
 `````` convert gradient.png -evaluate cos 0.5 -negate eval_cos.5.png ``````

Which is great generating smooth gradients for use in overlapping photos.

However these last two "`-evaluate`" methods are rarely used as they have been superseded by a more general Sinusoid Function (see below) that provide more control options, beyond that of a simple frequency option.

## Function, Multi-Argument Evaluate

The above wave generators proved to be extremely useful, especially with Distortion Image Mapping. But it was found that a much finer control of the functions was needed, requiring more than one parameter.

Because of this the "`-function`" operator was added in IM v6.4.8-9.

Basically "`-function`" is a multi-argument form of "`-evaluate`". However unlike the Evaluate Operator, these operators like the mathematical operators, all the functions above work only on normalised channel values (0.0 to 1.0 range) of the image, which in most cases makes them easier to use.

#### Polynomial Function

The '`polynomial`' method will take any number of values, and will modify the color values in an image according the exact expression given, much faster than the FX Operator can.

`-function   Polynomial   a,b,c,...`
Each value will be used as a coefficient from the highest order to the lowest, to produce a polynomial with the number of terms given.

For example a argument of '`4,-4,1`' will generate the polynomial expression equivalent to the "`-fx`" expression "` 4*u^2 - 4*u + 1 `".

If you know your high school maths you should know then that this polynomial function produces a parabolic curve going from 1.0 to 0.0 then back to 1.0, over the input ('`u`') color range 0.0 to 1.0. That is it will make, black and white colors 'white', and make perfect grays, 'black'.
 `````` convert gradient.png -function Polynomial 4,-4,1 func_parabola.png ``````

You can even make a much more complex gradient, for example a quartic polynomial, which was the result of generating a Curve Level Adjustment, using a set of 'level control points'. This is typically used to adjust the colors of an image to give it various shading effects.
 `````` convert gradient.png -function Polynomial '-25, 53, -36, 8.3, 0.2' \ func_quartic.png ``````

Of course simple linear modification is also possible, exactly as you get if you used a Level Operator...
 `````` convert gradient.png -function Polynomial '4, -1.5' func_linear.png ``````

Note however that you can not use '`Polynomial`' to do a full Threshold operation, due to the need for infinite coefficients to do so, though you can get pretty close.

A single value is naturally just a constant, and results in a direct assignment of that value. In other words it is just like the "`-evaluate Set `" method, in this case to a 33% gray value.
 `````` convert gradient.png -function Polynomial 0.33 func_constant.png ``````

By combining a '`Polynomial`' with other math functions you can create even more complex gradient modifications.

For example by taking the square root of a polynomial, I can create a true circular arc over a linear gradient. The equivalent of the very slow "`-fx`" expression '`sqrt( -4*u^2 + 4*u + 0 )`'...
 `````` convert gradient.png -function Polynomial -4,4,0 -evaluate Pow 0.5 \ func_circle_arc.png ``````

See also the Pow Evaluate Method for an alternative to the above.

#### Sinusoid Function

The '`Sinusoid`' function method is a much more advanced version of the "`-evaluate`" methods '`sin`' and '`cos`', and can in fact replicate those functions, but you have much better controls over how it modifies the color values in an image.

`-function   Sinusoid   frequency,phase,amplitude,bias`
And is implemented using the formula...
`  value = ampl * sin(2*PI( freq*value + phase/360 ) ) + bias`
This may seem complex but it ensures the function is easily to use.

Only the first value 'frequency', which works exactly as per above, is required with all the other parameters being optional.

By default it will generate a Sine Curve.
 `````` convert gradient.png -function Sinusoid 1 func_sine.png ``````

By adding a 'phase' argument in degrees, you can specify the starting angle for the curve. Allowing you convert the default sine curve into a cosine.

 `````` convert gradient.png -function Sinusoid 1,90 func_cosine.png ``````

By adjusting the 'frequency', and 'phase' I can directly convert a linear gradient into a smooth sinusoidal gradient going from black to white (minimum to maximum along a Sine curve). See Evaluate Cosine Method for a less direct method.
 `````` convert gradient.png -function Sinusoid 0.5,-90 func_sine_grad.png ``````

The next two optional values, 'amplitude' and 'bias' controls the scale and center-line of sinusoidal curve. For example, here I make a wave (negated cosine curve) that oscillates between white and gray (values ranging from `0.75 ±0.25`, or `0.5 to 1.0`), starting and finishing on white.
 `````` convert gradient.png -function Sinusoid 5,90,.25,.75 func_sine_bias.png ``````

Becareful with these last parameters as they could easy cause the waveform to exceed the bounds of the color value range, and thus be clipped (unless you are using a HDRI version of ImageMagick).

#### Arcsin Function

The inverse sinusoid function '`Arcsin`' was added to IM v6.5.3-0. This is a special curve that was needed to generate a Cylindrical Displacement Map.

It parameters are...
`-function   Arcsin   width,center,range,bias`
And is implemented using the formula...
`  value = range/PI * asin(2/width*( value - center ) ) + bias`
By default values (if not defined) '`1, 0.5, 1, 0.5`' ensures the the function is centered so as to cover the whole color range from `0,0` to `1,1`.
 `````` convert gradient.png -function Arcsin 1 func_arcsin.png ``````

By halving the 'width' of the resulting curve you get...
 `````` convert gradient.png -function Arcsin 0.5 func_arcsin_width.png ``````

The 'center' will let you reposition the curve according to the input grey values.
 `````` convert gradient.png -function Arcsin 0.4,0.7 func_arcsin_center.png ``````

The 'range' argument allows to reduce the output range of the color values, and the 'bias' will adjust the center of that range.
 `````` convert gradient.png -function Arcsin 0.5,0.5,0.5,0.5 func_arcsin_range.png ``````

Note how the values that are invalid as a result of the function are handled. This this allows better control when the function is used in displacements, and provides ways in which to clean them up. The actual values used are is '`bias ±range/2`', as you would expect.

Note that if either the 'width' or the 'range' are made negative the slope of the function will be flipped, as a result of that negative value.
 `````` convert gradient.png -function Arcsin -1 func_arcsin_neg.png ``````

#### Arctan Function

The '`Arctan`' method was added to IM v6.5.3-1.

Its parameters are...
`-function   Arctan   slope,center,range,bias`
And is implemented using the formula...
`  value = range/PI * atan(slope*PI*( value - center ) ) + bias`
As you can see it is almost exactly the same as the 'Arcsin' function, with only a small change to make it more useful. It even has the same set of default values (if not defined) '`1, 0.5, 1.0, 0.5`'.

This means that if you specify a slope value of '`1.0`' the slope of the histogram change will produce a 1:1 change around pure gray, (no scaling) while making white and black a more gray value. For example
 `````` convert gradient.png -function Arctan 1 func_arctan.png ``````

That is the middle part of the gradient is actually left unchanged, with only the black and white ends becoming de-contrasted.

As the 'slope' of the curve becomes larger, the gradient in the center will become stronger (more compressed in the middle), by that amount.
 `````` convert gradient.png -function Arctan 10 func_arctan_10.png ``````

This in many ways is very similar to a Sigmoidal Contrast color modification operator. However a '`Arctan`' function will NEVER actually reach the output range limits of pure black and white. It will approach those limits but never cross them.

Similarly to the previous functions, (and Sigmoidal Contrast) the second argument will adjust the position of the curve relative to the input gradient values.
 `````` convert gradient.png -function Arctan 10,.7 func_arctan_center.png ``````

And the last two arguments 'range' will let you adjust output range of values that will be generated. For example by expanding this value slightly you can ensure that it will completely cover the whole range of possible values.
 `````` convert gradient.png -function Arctan 5,0.7,1.2 func_arctan_range.png ``````

However if you are really wanting to generate curve to modify the whole contrast of an image in this way, it is more typical to use the Sigmoidal Contrast Operator, which is designed for this purpose.

The more typical use of an '`Arctan`' gradient function to create a curve that will very quickly approach a specific value but not exceed that value. It is these limiting values that the 'range' and 'bias' arguments control.

For example, this curve will modify the gradient in an image to produce very sharp threshold around the input gray level of 0.7, but with the values changing between the range limits of 0.5 and 1.0
 `````` convert gradient.png -function Arctan 15,0.7,0.5,0.75 func_arctan_typ.png ``````

This is something that Sigmoidal Contrast can not generate.

## Mathematics on Gradient Images

Now the above functions provide some very basic transformations for gradient images. But what if you want to do some mathematics with two or more gradient images. That is modify one gradient using the gradient of another image.

For this you need to use the special Mathematical Compose Methods (such as "`Plus`" and "`Divide`").

Before we start however, I would like to give you one word of warning. If your gradient images are purely grey-scale images, with no alpha channels, then you can use the Mathematical Compose Methods directly. However if you want to limit these methods to a specific channel, or apply them to the alpha (transparency) channel, then you need to ensure that you set the appropriate "`-channel`" setting, with no special '`Sync`' channel flag. See Image Mathematics using Image Composition for more details.

Normally using Mathematical Compose Methods is not really that difficult. The complications arise when you have gradients that also contain a 'bias'. That is the gradient should represent a value of 'zero' at '50% grey, and cover a range from -1 (black) to +1 (white). Such images are often used for Distortion Image Mapping.

As such performing maths on 'biased gradients' is the real problem, and what will be looked at more specifically here.

### Attenuate a Biased Gradient

For example, here I want to create a sine wave, but one that starts out small, but then gets larger in amplitude.

This known as 'attenuating' a biased gradient. Or putting it another way, multiply a biased gradient by another absolute gradient. It is also how 'Amplitude Modulation' such as in AM radio works!

So first we need a sine wave, which we can simply generate from a linear gradient...
 `````` convert -size 5x300 gradient: -rotate 90 math_linear.png convert math_linear.png -evaluate sine 12 math_sine.png ``````

Now to attenuate this we multiply the sine wave with a linear gradient, using a Multiply alpha composition...
 `````` convert math_sine.png math_linear.png \ -compose Multiply -composite math_sine_2.png ``````

But to use this in say a Water Ripples, Displacement Map the wave must remain centered around a perfect gray. To do this we need to add a bias to the original image. This happened to be the same function we used to multiply the original image, negated and divided by two....
 `````` convert math_linear.png -negate -evaluate divide 2 math_bias.png convert math_sine_2.png math_bias.png \ -compose Plus -composite math_attenuated.png ``````

And so we have a linearly attenuated Sine Wave Gradient, suitable for use in a displacement map.

Of course you can do the whole process all in the one command, and it does not have to be a simple linear attenuation either. For example here I attenuate the high frequency Sine wave, using a negated Cosine wave, instead of a linear gradient.

 `````` convert math_linear.png -evaluate cos 1 -negate math_cosine_peak.png convert math_sine.png math_cosine_peak.png \ \( -clone 0,1 -compose multiply -composite \) \ \( -clone 1 +level 50%,0 \ -clone 2 -compose plus -composite \) \ -delete 0--2 math_cosine_atten.png ``````
attenuate

As of IM v6.5.4-3 it is now possible to do the all the steps above all in one compose method, using the special Mathematics Compose Method. Basically by recognising that an attenuation operation is the formula `Sc*Dc-.5*Sc+.5` or the arguments, "`1,-.5,0,.5`".
 `````` convert math_sine.png math_cosine_peak.png \ -compose Mathematics -set option:compose:args 1,-.5,0,.5 \ -composite math_attenuate.png ``````

The same result can also be achieved by first adjusting the attenuation gradient using a Polynomial Function and then using a Exclusion compose operator, to merge the images.
 `````` convert math_sine.png \( math_cosine_peak.png -function polynomial -.5,.5 \) \ -compose Exclusion -composite math_poly_excl.png ``````

### Multiply Biased Gradients

But what if BOTH functions are biased so a perfect gray means zero, and black and white represent the range from -1 to +1?

Well this is a little more complex as you can't just multiply them and expect it to come out right, as the multiplication can consist of a negative values. This requires some care so as to ensure you don't end up clipping the values and getting the right negation of the curve in the resulting image.

The trick is to break up the multiplication into multiple steps. That is   ` A × B `   can also be written as   ` A × abs(B) × sign(B)`. By doing this you avoid multiplying by a negative value, which can't be stored in a normal gradient image. So all we need to do is take one of the bias gradients and separate it into two parts so they can be applied to the other gradient appropriately.

The '`sign()`' of a biased gradient, or getting a mask of what parts are negative, can get extracted by using a Threshold on the gradient at the bias level. You can later selectively negate the other gradient using a Composite Difference, with that threshold image.

The '`abs()`' of a biased gradient can be extracted easily using Solarize, then negating and doubling (using Level) that to get the absolute value of the gradient ranging from 0.0 to 1.0. As we will also need the bias offset as part of the multiply (as per Attenuate above), you can directly use the negated and half-scaled solarize output, before it is converted into the gradients absolute value.

So lets convert one gradient into these three components.
 `````` convert math_cosine_peak.png -threshold 50% -negate math_m_sign.png convert math_cosine_peak.png -solarize 50% math_m_bias.png convert math_m_bias.png -level 50%,0 math_m_abs.png ``````
 Sign of Gradient white = negative Bias Offset Absolute Value

Now that we have these three parts of one of the gradient images, we can merge them with the other gradient. To do this we multiply by the absolute value, re-add the bias, and then negate the parts that should be made negative.
 `````` convert math_sine.png math_m_abs.png \ -compose Multiply -composite math_m_1.png convert math_m_1.png math_m_bias.png \ -compose Plus -composite math_m_2.png convert math_m_2.png math_m_sign.png \ -compose Difference -composite math_multiply.png ``````
 sign

And that is a perfect multiplication of two bias gradient images!

Here it is again but all in the one command...
 `````` convert math_sine.png math_cosine_peak.png \ \( -clone 1 -threshold 50% -negate \) \ \( -clone 1 -solarize 50% \) \ \( -clone 3 -level 50%,0 \) \ \( -clone 0,4 -compose multiply -composite \ -clone 3 -compose plus -composite \ -clone 2 -compose difference -composite \) \ -delete 0--2 math_multiply_2.png ``````

One final note, unlike Attenuation, this Multiply of biased gradients is commutative. That is swapping the input images does not effect the final result.

As the above is equivelent to the formula `2*Sc*Dc-Sc-Dc+1`, as of IM v6.5.4-3, you can implement the above complex steps as a single '`Mathematics`' compose method using the argument "`2,-1,-1,1`".
 `````` convert math_sine.png math_cosine_peak.png \ -compose Mathematics -set option:compose:args 2,-1,-1,1 \ -composite math_bias_multiply.png ``````

That is vastly easier and faster method than the dozen or more steps needed without this argumented compose method.

It so happens that once I saw that formula, I realised that this happens to be simply the negation of the '`Exclusion`' compose method. Weird but true. As such the following will also generate the same zero baised multiply.
 `````` convert math_sine.png math_cosine_peak.png \ -compose exclusion -composite -negate math_excl_neg.png ``````

### Adding Biased Gradients

With the advent of the '`Mathematics`' compose method, adding biased gradients is also relativally easy. The equivelent FX formula is "`u+v-0.5`" or a compose argument of "`0,1,1,-.5`".

For example the following was a Fourier Transform Example that I had hand generated, requiring the addition of 3 biased sinusoids, and a constant DC value.
 `````` convert math_linear.png -function sinusoid 3.5,0,.25 wave_1.png convert math_linear.png -function sinusoid 1.5,-90,.13 wave_2.png convert math_linear.png -function sinusoid 0.6,-90,.07 wave_3.png convert wave_1.png wave_2.png wave_3.png -background gray40 \ -compose Mathematics -set option:compose:args 0,1,1,-.5 \ -flatten added_waves.png ``````

Note in the above how I used the "`-flatten`" operator with a "`-background`" setting to implement a mutliple image composition. Or in this case a 'Biased Sum' of all the given images plus the background constant.

### Frequency Modulation

By applying a function directly to the output of another function, you do NOT produce a simple result. The reason is that all these math functions are applied to the gradient 'value' of individual pixels, and not against the x value of the pixel in the gradient.

For example....
 `````` convert gradient.png -evaluate sin 0.5 -normalize \ -evaluate cos 8 math_cos_var.png ``````

This generates a very complex function that is essentually equivelent to
``` cos( 8 * sin( {value}/2 ) ) ```
In other words a variable frequency, where the frequency varies with the gradient of the first sine curve.

Basically the faster the gradient changes in the original image, the smaller the distance between the peaks. However the height (amplitude) of the peak do not vary.

This is actually how 'Frequency Modulation' works, where a seemingly simple function produces a very complex result.

Under Construction

```Miscellaneous Image Transformation Techniques.

These have not been exampled yet, but are some basic IM developed transforms
that may provide useful.  If you have an interesting effect please contribute.

pixelize an image
resize an image down 10 then scale the image 10 to produce blocks
of roughly averaged color.
For example...
convert input.jpg -resize 10% -sample 1000% output.jpg

De-skew slightly rotated images

-deskew {threshold}
straighten an image. A threshold of 40% works for most images.

Use -set option:deskew:auto-crop {width} to auto crop the image. The set
argument is the pixel width of the image background (e.g 40).

Programmically we auto crop by running a median filter across the image
to eliminate salt-n-pepper noise.  Next we get the image bounds of the
median filter image with a fuzz factor (e.g. -fuzz 5%).  Finally we
crop the original image by the bounds.  The code looks like this:

median_image=MedianFilterImage(image,0.0,exception);
geometry=GetImageBoundingBox(median_image,exception);
median_image-DestoryImage(median_image);

print("  Auto-crop geometry: %lux%lu%+ld%+ld",
geometry.width,geometry.height, geometry.x,geometry.y);
crop_image=CropImage(rotate_image,&geometry,exception);

See Trimming 'Noisy' Images

Segmentation
look at scripts
divide_vert
segment_image
for some simple scripts I wrote to segment well defined images into
smaller parts.   I hope to get simple segmentation functions like this
into the core library, to allow for things like automatic sub-division of
GIF animations, and seperating images and diagrams from scanned documents.

```

Created: 15 March 2004
Updated: 18 June 2009
Author: Anthony Thyssen, <A.Thyssen@griffith.edu.au>
Examples Generated with:
URL: `http://www.imagemagick.org/Usage/transform/`