Color Modifications -- IM v6 Examples

Here we look at techniques for modifying all the colors in an image as a whole. Whether it is to lighten or darken the image, or more drastic color modifications.

To explore these technqiues we will need a test image...
Don't worry above how I actually generated this image, it is not important for the exercise. I did design it to contain a range of colors, transparencies and other features, specifically to give IM a good workout when used.

If you are really interested in the commands used to generate this image you can look at the special script, "generate_test", I use to create it.

WARNING: The color processes below generally assumes the image is using a linear colorspace. Most images are however saved using a sRGB or Gamma corrected colorspace, as such to get things right colorspace correction should also applied first.

Converting Color to Gray-Scale

Gray scale images can be very useful for many uses, such as, furthering the processing of the original image or for use in background compositions. The best method of converting an image to gray-scale is to just ask IM to convert the image into a gray-scale Color Space representation for the image.


  convert  test.png  -colorspace Gray   gray_colorspace.png

Note how the blue is much darker than the red, due the weighting to match the intensity as they seem to appear to the human eye. That is, 'red' is quite a bright color compared to 'blue' which looks darker.

This is equivelent to the use of the 'rec709luma' conversion formula with the dedicated "-grayscale" operator (Added IM v6.8.3-10).


  convert test.png  -grayscale rec709luma  gray_grayscale.png

The 'rec709luma' value is just one of many greyscaling formula that has been defined for use by the "-intensity" setting (see below).

Here for example is the other common greyscaling formula 'rec601luma'


  convert test.png  -grayscale rec601luma  gray_grayscale_601.png

As you can see there is a slight different in intensity levels for the different red, green and blue color channels.

However there a many other methods, and meanings of 'gray-scale'...

For example, you can drain all the color out of the image by using the Modulate Operator, to set all color saturation levels to zero.


  convert test.png  -modulate 100,0  gray_modulate.png

This essentually converts the image to the HSL colorspace, and extracts the grayscale 'Lightness' value from that colorspace. However using a "-define modulate:colorspace" you can specify other colorspace models to use. See Modulate in Other Colorspaces below.

Note how the IM 'green' color I used for the center colored disk in my test image is not actually a pure green, such as used in the colored rainbow, but the half-bright green defined by the new SVG -- Scalable Vector Graphics standard. If you need a pure RGB green you can use the color 'lime' instead. See Color Name Conflicts for more detail.

Another way is to use the FX DIY operator to average the three channels together to get a pure mathematical meaning of gray-scale.


  convert test.png -fx '(r+g+b)/3' gray_fx_average.png

The average of sRGB channel values also equivelent to the intensity channel of 'OHTA' colorspace (red channel). Or the 'I' channel of HSI. colorspace.

Another technique is to simply add all three channels together (a color measure known as manhatten distance) and while the resulting image will not loose information due to 'quantum rounding' effects, you may loose information about the brightest colors. Unfortunately, you also loose the transparency channel, too.


  convert test.png -separate \
          -background black -compose plus -flatten   gray_added.png

You can use the same adding channels technique to control the weighting of the individual color channels. For example this is one DIY formula that you can use...


  convert test.png -fx '0.3*r+0.6*g+0.1*b' gray_diy.png

You can also use 'intensity' if you want the same meaning within the "-fx" operator.


  convert  test.png  -fx intensity  gray_intensity.png

However as the FX DIY operator is interpreted, it can run very very slowly. For more complex operations you can use the simpler Evaluate Operator, "-evaluate".

For example here is a 2/5/3 ratio gray-scaled image, though again I make no attempt to preserve the transparency channel of the original image.


  convert test.png -channel R -evaluate multiply .2 \
                   -channel G -evaluate multiply .5 \
                   -channel B -evaluate multiply .3 \
                   -channel RGB -separate -compose add -flatten gray_253.png

The above would suffer from 'quantization' effects for an ImageMagick compiled at a 'Q8' Quality Level. That is because the results of the "-evaluate" will be saved into a small 8 bit integer, used for image values. Only later are those values added together with the resulting loss of accuracy.
An ImageMagick compiled with 'Q16', or better still the HDRI, quality compile options will produce a much more exact result. Another new alternative is the Poly - Weighted Image Merging Operator, which will do the weighting and addition of the separated channel images in one operation, so avoiding 'quantum rounding' effects.

A similar technique can be used to generate a pure mathematical gray-scale, by directly averaging the three RGB channels equally.


  convert test.png -separate -average  gray_average.png

However as you can see, I did not attempt to preserve the alpha channel of the resulting image.

Another fast alternative is to use the "-recolor" color matrix operator, which will let you specify the weighting of the three color channels.


  convert test.png -recolor '.2 .5 .3
                             .2 .5 .3
                             .2 .5 .3'   gray_recolor.png

This doesn't affect transparency, but makes it a much better way of converting colors using a specific weighting.

Basically the first row of numbers is the channel weighting for the resulting images red channel, next 3 for green, and the final three numbers for blue.

You can also use "-type" to tell IM to treat the image as gray-scale, when either reading or writing the image.


  convert  test.png  -type GrayScaleMatte  gray_type.png

The "-type" setting is generally only used as a guide when an image is being read or written to a file. As such its action is delayed to the final write of the image. Its effect is also highly dependant on the capabilities of the image file format involved and is used to override ImageMagick's normal determination during that process. See the Type examples for more information.

Before IM v6.3.5-9 the above will have removed any transparency in the written image (equivalent of a "-type Grayscale") due to a bug. This was fixed as soon as I noted the problem and reported it. (There is a lesson here :-)

A much more interesting technique is to extract a variety of different meanings of brightness by extracting the appropriate Color Channel from various Colorspace representations of the image. Examples see Grayscale Channels from Colorspace Representations.

Image Level Adjustments

The most basic form of adjustment you can make to images are known as 'level' adjustments. This basically means taking the individual RGB color values (or even the alpha channel values) and adjusting them so as to either stretch or compress those values.

As only channel values are being adjusted, they are best demonstrated on a gray-scale image, rather than a color image. However if you adjust all the color channels of an image by the same amount you can use them with color images, for the purposes of either enhancing, or adjusting the image. Do not confuse this with the more automatic form of level adjustments, which we will look at in the next major section of examples below, Normalize Adjustments. This function will do exactly the same operation regardless of the actual content of the image. It does not matter if the image is bright, or dark, or has a blue, or yellow tint. The operations are blind to the actual image content.

In demonstrating these operations I will be using a modified "gnuplot" graph such as shown to the right, which I generate using a special script "im_graph". The graph has a red line which maps the given original 'x' value (representing the gray-scale value of the top most gradient) to the 'y' value shown. The resulting color gradient is also shown underneath the input linear gradient.

The graph shown to right is of the IM "-noop" operator which actually does nothing to an image. As such each of the image's color values are just mapped to exactly the same value without change. The lower gradient is thus the same as the upper gradient.

Image Negation

The simplest and most basic global level adjustment you can make is to negate the image, using the "-negate" image operator.

Essentially this makes white, black, and black, white, , adjusting all the colors to match. That is, it will make the color red, its complementary color of cyan, and blue, yellow, etc.

You can see this with the mapping graph shown below, as I use the "-negate" operator on both the 'test' image and the standard IM 'rose' built-in image. Note how the lower gradient in the mapping graph image is now reversed, so that black and white are swapped, and the same reversal appearing in the negated 'test' image.


  convert  test.png  -negate  test_negate.png
  convert  rose:     -negate  rose_negate.gif

Internally negate is actually rather stupid. It handles the three color channels independently, and by default ignores the alpha channel. If this was not the case, you would get a very silly result like this...


  convert  test.png -channel RGBA  -negate  negate_rgba.png

The image is negated, as you can see by the semi-transparent color gradient. But as the transparency channel has also been negated you loose all the opaque colors in the image. This is why the default setting for "-channel" is 'RGB'. See Color Channels for more information.

You can limit the negation to just one channel, say the green color channel. This may not seem very useful, but at times it is vitality important.


  convert  test.png -channel green  -negate  negate_green.png

The "-negate" operator is actually its own inverse. Doing two negations with the same "-channel" setting cancels each other out.


  convert  negate_green.png  -channel green  -negate  negate_restore.png

Negation is extremely common in image processing, particularly when dealing with gray-scale images as a step before or after other processing options. As such I recommend you play with it and keep it in mind whenever you are doing anything, as working with negated images can solve some otherwise difficult problems.

Direct Level Adjustments

The "-level" operator is the more general level adjustment operator. You basically give it two values a 'black_point' and a 'white_point', as well as an optional third value (gamma adjustment), which I will look at later.

What it does is map any color values in the image that is equal to or less than the 'black_point', and make them black (or a 0 value). Similarly, any color values that are equal to or brighter that the 'white_point' will make them white (or a Maximum value). The colors in between these two points are then 'stretched' linearly to fill the complete range of values.

The effect of this is to improve the contrast, enhancing the colors within an image. For example here is a 25% contrast enhancement of our test image, using the same values as shown by the graph.

As you commonly adjust both the black and white points by the same amount from the 0% and 100% amounts, you can just specify the 'black_point' only. The white point will be adjusted by the same amount inward.


  convert  test.png  -level 25%,75%  test_level.png
  convert  rose:     -level 25%      rose_level.gif

Note that 25% is a huge contrast enhancement for any image, but it clearly shows what it does.

You don't have to change both the 'black' and 'white' points. Instead it is quite permissible to just adjust only one end of the color range. For example we can make a very light, or a very dark rose image.


  convert  rose:  -level 0,75%     rose_level_light.gif
  convert  rose:  -level 25%,100%  rose_level_dark.gif

However I again warn you that the colors outside the given range are 'clipped' or 'burned', and as such will no longer be available for later image processing. This is the biggest problem with using a "-level" operator.

By using a negative value you can do some rough de-contrasting of an image.

What this means is that rather than providing a color value for the values to be mapped to 'black' and 'white' and thus stretching out the colors in between, you instead compress the color values so as to map the imaginary negative color to black or white. The result is a general graying of the image.


  convert  rose:  -level -25%  rose_decontrast.gif

This method of de-contrasting an image however is very inaccurate and not recommended, unless you have an IM older than version 6.4.2 where you don't have access to the new Reversed Level Operator. [IM Graph]

You can use the "-level" operator to negate an image (as previously shown above, just by swapping the 'black' and 'white' point values given, using "-level 100%,0".


  convert  rose:  -level 100%,0  rose_level_neg.gif

Or by setting them to the same value, you can effectively call all the color values in the image to be thresholded. Using "-level" to threshold an image is exactly the same as if you used a Threshold Operator with that value. The mapping graph shown right, shows the results of a "-level 50%,50%" operation, and its effect on a grayscale gradient.


  convert  rose:  -level 50%,50%  rose_level_thres.gif

Note that unlike "-threshold" the image is not automatically converted to a grayscale image when used with the default "-channel" setting.

The general nature of using level to linearly modify an image, makes the "-level" operator good for general gray-scale image modifications, and mask adjustments. Add the fact that you can modify individual channels (using the "-channel" setting) as opposed to the whole image, makes it one of the best color modification operators available to IM users.

Note you can also use the Evaluate and Function Operators for a more direct mathematical modification of the color values, to achieve the same results for -level both + and - forms).

Be warned that the "-level" operator treats the transparency channel as 'matte' values. As such 100% is fully transparent and 0% is opaque. Please take this into account when using "-level" with a blurred shape image. This is most typically done after blurring an 'shape' image, to expand and stretch the results. For examples of this see Soft Edges, and Shadow Outlines.

Reversed Level Adjustments -- Decontrasting Images

As of IM version 6.4.2 the Level Operator was expanded to provide a 'reversed' form "+level" (note the 'plus'). Alternatively you can use the original "-level" form of the operator but add a '!' to the level argument given (for older API interfaces).

The arguments for this variant is exactly the same, but instead of stretching the values so as to map the 'black_point' and 'white_point' to 'black' and 'white', it maps 'black' and 'white' to the given points. In other words "+level" is the exact reverse of "-level".

For example here we map 'black' to a 25% gray, and white to 75% gray, effectively de-contrasting the image in a very exact way, using the two methods of specifying the 'reversed' form.


  convert  test.png   +level 25%    test_level_plus.png
  convert  rose:      -level 25%\!  rose_level_plus.gif

If you compare the above "+level 25%" operation with the use of an a negative de-contrasting, "-level -25%" operator we showed previously, you will see that are not the same. The 'plus' version produces a much stronger de-contrasted image (it is greyer), but does so by mapping to the exact values you give the operator, and not the 'imaginary' values the 'minus' form used. This exact value usage is important, and one of the reasons why the 'plus' form of the operator was added.

Of course a '25%' is again a very large value, and it is not recommended for use with typical image work.

Note that the "-level" and "+level", are in actual fact the exact reverse of each other when given the same argument. That is, one maps values to the range extremes, while the other maps from the range extremes.

For example here we compress the colors of the test image using "+level", then decompress them again using "-level", so as to restore the image close to its original appearance.


  convert  test.png  +level 20%  -level 20%  test_level_undo.png

The two images appear to be very very similar, and as I am using a high quality 'Q16' version of IM, you will be hard pressed to notice any difference at all. However the values may not be exactly the same, as you have effectively compressed the color values of the image to a smaller range of integers, and then restored them again. In extreme cases this can result in Quantum Rounding Effects.

Doing these two operations in the opposite order (stretch, then compress the color values) will produce Quantum Clipping Effects.

One other useful aspect of the "+level" operator is that you can completely compress all the color values in an image to the same gray-scale level.


  convert  test.png  +level 30%,30%  test_level_const.png

By specifying levels according to the values of specific colors for each individual channel, you can effectively convert a greyscale gradient into a specific color gradient. However this is rather difficult to calculate and do. As such a "-level-colors" operator has also been provided that will let you specify the black and white points in terms of specific colors rather than 'level' values. See Level by Color below.

Level Gamma Adjustments

Both the above "-level" variants also allow you to use a third setting. The 'gamma' adjustment value. By default this is set to a value of 1.0', which does not do any sort of mid-tone adjustment of the resulting image, producing a pure linear mapping of the values from the old image to the new image.

However by making this value larger, you will curve the resulting line so as to brighten the image, while shrinking that value will darken the image.

For example here I use just the 'gamma' setting to brighten and darken just the mid-tones of the image.


  convert  rose:  -level 0%,100%,2.0   rose_level_gamma_light.gif
  convert  rose:  -level 0%,100%,0.5   rose_level_gamma_dark.gif

Values generally range from 10 for a blinding bright image, to .2 for very dark image. As mentioned a value of 1.0 will make no 'gamma' changes to the image. However the special value of '2.0' (see above) can be used to get the square root of the images normalized color.

Both versions of the "-level" operate handles 'gamma' in the same way. This means you can combine the level adjustment of the 'black' and 'white' ends with a non-linear 'gamma' adjustment. You can also only adjust a single channel of an image. For example, here we give an image a subtle tint at the black end of just the blue channel, while using gamma to preserve the mid-tone color levels of the image.


  convert  test.png  -channel B +level 25%,100%,.6 test_blue_tint.png

This specific example could be used to tint a weather satellite photo, where only the sea is pure black, while land is more grey. Other alternatives to this blue channel adjustment are given below in DIY Mathematical Non-linear Adjustments.

Gamma Operation Adjustments

The "-gamma" operator is also provided, and has exactly the same effect as the 'gamma' setting in the "-level" operator. However it will let you adjust the 'gamma' adjustment level for each individual channel as well.

Its real use is in adjusting the 'gamma' function of an image before performing linear operations on them. For more details see Human Color Perception and Gamma Correction.

We can also use this function to brighten the image differently for each individual RGB channel.


  convert  rose:  -gamma 0.8,1.3,1.0  gamma_channel.gif

As you can see this can be used to do some subtle tinting and color adjustments to an image, or correct images with contain too much of a specific color.

For reasons about why you should used this function see Gamma Correction.

This function actually equivelent to the Evaluate POW function but with the argument inverted. As such a "-evaluate POW 2.2" will actually do a "

-gamma
0.45455

" (0.45455 is equal it 1/2.2) operation, which is the reverse of a "-gamma 2.2".

One of the less obvious uses of "-gamma" is to zero out specific image channels (see Zeroing Color Channels). Or color an image completely 'black', 'white' or some other primary color (see Primary Colored Canvases).

Level Adjustment by Color

The "-level-colors" operator was added to IM v6.2.4-1. Essentially, it is exactly the same as the Level Operator we discussed above, but with the value for each channel specified as a color value.

That is, the "-level-colors" option will map the given colors to 'black' and 'white' and stretching all the other colors between them linearly. This effectively removes the range of colors given from the image.

And while this works, it is not particularly useful, as it is prone to fail for colors that have common values in some channel. For example, the colors 'DodgerBlue' and 'White' have the same color values in the blue channel. As such, "-level-colors DodgerBlue,White" may not always convert those colors to black and white.

The better technique in that case is to extract a greyscale image of the channel with the highest differences (such as red) and level or normalize that channel.

WARNING: watch out for 'transparent' colors.

The plus form of the operator "+level-colors" on the other hand is extremely useful as it will map the 'black' and 'white' color to the given values compressing all the other colors linearly to fit the two colors you give.

For example lets map 'black' and 'white' to 'green', and 'gold'...


  convert  test.png  +level-colors green,gold   levelc_grn-gold.png

As you can see the grayscale gradient is remapped into a gradient bound by the colors given, and although colors outside a gray-scale range are also modified, they will also follow the basic style of the color range specified. This makes the "+level-colors" operator an extremely useful one, especially when mapping grayscale images.

If you only supply one colorname but include a comma, the missing color will default either to 'black' or 'white' as appropriate.


  convert test.png  +level-colors ,DodgerBlue   levelc_dodger.png
  convert test.png  +level-colors ,Gold         levelc_gold.png
  convert test.png  +level-colors ,Lime         levelc_lime.png
  convert test.png  +level-colors ,Red          levelc_red.png

  convert test.png  +level-colors Navy,         levelc_navy.png
  convert test.png  +level-colors DarkGreen,    levelc_darkgreen.png
  convert test.png  +level-colors Firebrick,    levelc_firebrick.png

This makes it easy to convert grayscale images into a gradient for any color you like. For example here I remap a black and white gradient to a red and white gradient, (note the ',' in the argument)...


  convert cow.gif   +level-colors red,   cow_red.gif

This didn't just replace 'black' with 'red' but also re-mapped all the anti-aliased gray colors to an appropriate mix of 'red' and 'white', producing a very smooth result.

If I had just performed a simple Direct Color Replacement converting pure black colors to red, I would end up with the horrible image (showen right). See Fuzz Factor for the example used to generate that image.

Of course if you want one of the colors to be made transparent instead you are better off using the -alpha Shape operator instead, as this requires you to transfer the gradient into the alpha channel.

If you only specify a single color, without any 'comma' separator, that color will be used for both black and white points. That means all the colors in the image will be reset to that one color. (according to the current "-channel" setting limitations).


  convert  test.png  +level-colors dodgerblue  levelc_blue.png

This is an identical result to using "

-fill DodgerBlue -colorize
100%

" to Colorize Images (see below).

If you want to set the images transparency setting as well you will need to set "-channel" to include the transparency channel, OR set the Alpha Channel to fully-opaque, using either "-alpha opaque" or "-alpha off".


  convert  test.png -channel ALL +level-colors dodgerblue levelc_blue2.png

Also see Blanking Existing Images.

Here are a few more examples of using this to adjust or 'tint' a colorful image, rather than a gray-scale image.


  convert rose: +level-colors             navy,lemonchiffon  levelc_faded.gif
  convert rose: +level-colors        firebrick,yellow        levelc_fire.gif
  convert rose: +level-colors 'rgb(102,75,25)',lemonchiffon  levelc_tan.gif

In summary the "+level-colors" is a gradient color replacement, a linear tinting operator, and can also completely reset colors.

Sigmoidal Non-linearity Contrast

From a PDF paper on 'Fundamentals of Image Processing' (page 44) they present an alternative to using a linear contrast control (level), with one using gamma correction known as 'sigmoidal non-linearity contrast control'.

The result is a non-linear, smooth contrast change (a 'Sigmoidal Function' in mathematical terms) over the whole color range, preserving the white and black colors, much better for photo color adjustments.

The exact formula from the paper is very complex, and even has a mistake, but essentially requires with two adjustment values. A threshold level for the contrast function to center on (typically centered at '50%'), and a contrast factor ('10' being very high, and '0.5' very low).

For those interested, the corrected formula for the 'sigmoidal non-linearity contrast control' is...
( 1/(1+exp(β*(α-u))) - 1/(1+exp(β*(α)) ) / ( 1/(1+exp(β*(α-1))) - 1/(1+exp(β*α)) )
Where α is the threshold level, and β the contrast factor to be applied.
Here is an alternate version of the formula using intermedite variables.
x = exp(β * (α - u)) y = exp(β + 1 result (x / y + 1) * (1 / (x + 1) - 1 / y)
The formula is actually very simple exponential curve, with the bulk of the above formula is designed to ensure that 0 remains zero and 1 remains one. That is, the graph always goes though the points 0,0 and 1,1. And the highest gradient of change is at the given threshold.

Here for example is a "-fx" implementation of the above formula, resulting from a very high contrast value of '10' and a '50%' threshold value. These values have been rolled into the floating point constants, to speed up the function.


  convert test.png  -fx '(1/(1+exp(10*(.5-u)))-0.006693)*1.013567' \
              sigmoidal.png

Lucky for us IM v6.2.1 had this complex function built in as a new operator "-sigmoidal-contrast", allowing a much simpler application.


    convert test.png  -sigmoidal-contrast 10,50% test_sigmoidal.png

As a bonus IM also provides the inverse, a 'sigmoidal contrast reduction' function (as plus '+' form of the operator), which if applied with the same arguments restores our original image (almost exactly).


    convert test_sigmoidal.png +sigmoidal-contrast 10,50% \
                                             test_sigmoidal_inv.png

And here we apply it to the rose image...


    convert  rose:  -sigmoidal-contrast 10,50%  rose_sigmoidal.gif

I did say '10' was a very heavy contrast factor. In fact anything higher than this value can be considered to be more like a fuzzy threshold operation, rather than a contrast enhancement.

For a practical example of using this operator see the advanced "Gel" Effects Example, where it is used to sharpen the bright area being added to a shaped area color.

Miscellaneous Contrast Operators

Under Construction

   -contrast  and   +contrast
         Rather useless minor contrast adjustment operator

-threshold
   Threshold the image, any value less than or equal to the given value is
   set to 0 and anything greater is set to the maximum value.

   Note that like level, this is a channel operator, but if the default
   'channel setting' is used only the gray-scale intensity of the image is
   thresholded producing a black and white image.

   convert rose: -threshold 45%  x:

   You can force normal channel behaviour, where each channel is thresholded
   individually buy using "-channel All"

   convert rose: -channel All -threshold 45%  x:

-black-threshold
-white-threshold
   This is like -threshold except that only one side of the threshold value is
   actually modified.

   For example, here anything that is darker than 30% is set to black.

   convert rose: -black-threshold 30%  x:
   convert rose: -white-threshold 50%  x:

   These operators however do not seem to be channel effected, so may only be
   suitable for gray-scale images!

Adjustments Using Histogram Modification

This section was a joint effort by Fred Weinhaus and Anthony Thyssen.

What is a histogram?

A histogram is a special type of graph. It simply sorts the color levels of the pixels in an image into a fixed number of 'bins' each of which span some small range of values. As such each bin contains a count of the number of color levels (pixel values) in the image that fall into that range.

The result is a representation of how the color values that make up an image are distributed, from black at the left, to white at at the right.

The histogram can be generated for each channel separately or as a global histogram which looks at values from all the channels combined. The result is often displayed as an image of a bar chart. In IM, this is done using the special Histogram: output format. For example...


  convert rose: histogram:histogram.gif

But it can also be displayed as a line graph where the line connects the tops of the bars. This will be demonstrated later in the discussion below.

See Histogram: for more details of this special output format. This is recommended reading at this point as it is the best way to extract histogram information about images using IM.

A histogram chart's actual height has little actual meaning, since it is usually scaled so that the highest peak touches the top of the image. As such the height of each individual 'bar' is not relevant. What is much more important is the distribution of the histogram over the whole range, and how the relative heights relate to each other over the whole of the chart.

When looking at a histogram you would consider the following factors.

Does the histogram form one wide band of values? This means that the image makes wide use of the colorspace and thus has good contrast.
Or is it all in a tight group in the middle or at one end of the range? This means the image has a low contrast, making it look 'fogged' or 'grayed', or perhaps overly light or dark.
Does it form two or more peaks? As a result of highly different areas or regions in the image.
Where are most of the pixels? At the left, meaning the image is very dark. Or at the right, meaning it is very bright. Or spread out around the middle?
Are there regular gaps or empty spaces between individual bars? This usually means either the image has very few pixels, so it could not completely fill out the whole histogram, or the image was color reduced, or modified in some way, so as to produce those gaps.

Essentially a histogram is a simpler representation of an image, and as such it is much easier to change or adjust an image in terms of its histogram.

Almost any mathematical color transformation that one applies to an image will normally cause not only the image to be modified, but its histogram as well. These include linear operations such as the Level Operator or non-linear operations such as the Gamma Operator, (see above). The mapping graphs we saw above represent how the graylevels in an image and thus how the image's histogram is to be transformed.

For example, lets make a low contrast image to demonstrate. However, the final result is that it not only modifies the image, but does so by modifying the image's histogram (by compressing it).


  convert chinese_chess.jpg -contrast -contrast -contrast -contrast \
          chinese_contrast.png

  convert chinese_chess.jpg     histogram:chinese_chess_hist.gif
  convert chinese_contrast.png  histogram:chinese_contrast_hist.gif

In the above case, "-contrast, is a simple Level type operator that adds just a little more contrast to the image. the result of this is that the histogram itself is spread out more, causing it to cover the whole of the possible color range better.

You can also see from the histograms, before and after, that the colors will also end up with gaps and holes between the 'bins', due to the way in which the stretching was performed. Specifically it creates a 'histogram' with all the colors being places into 'bin'. These 'binned' colors are then modified as a whole, causing the image colors to be grouped together. It is not a particularly good way of handling image colors.

This operator however works blindly, without any knowledge of the image content or color distribution. It thus cannot be done without some user control, as the operator could very easily make any image it is applied to, worse, rather than better.

In this section we will look at image processing operators that examine the image's histogram as part of its decision making process. It then modifies images using the result this study, so as to enhance some quality of the image color distribution. As these operators make use of actual information coming from the image being processed, they can often be used more globally over many images with much checking by the user.

Operators of this type include automatic linear 'level' type operators such as "-normalize", "-contrast-stretch", and "-linear-stretch", but also non-linear ones such as "-equalize", and others that may eventually be included into ImageMagick such as Fred Weinhaus's script, "redist".

Histogram Stretching

The simplest techniques, like the previous example simply stretch the histogram of the image outward to improve the color range. However instead of just blindly picking the black-point and white-point for Level operation, they select points based on the images histogram.

Basically they count up the number of color values in each histogram bin, from each of the two ends, inward until they reach some threshold. These points will then be used as the black-point and white-point for the histogram (level) stretching.

Diagram needed

Basically the histogram counts provide the graylevel values that the stretch will force to black and white. This means that all pixels in the image that fall within the range of bins from pure black to the selected black-point bin's corresponding graylevel will end up pure black.

Likewise those pixels in the image that fall within the range of bins from from pure white to the white-point bin's corresponding graylevel will end up pure white.

The pixels that are outside these points however will have been stretched outside the possible color range of values, and as a result they will be simply be set to the range limits. That is these pixels are 'clipped' 'burned-in' as they are converted to the extreme of pure black or pure white color values.

As a result if the 'threshold' limits for selecting the black-point and white-point is set too high, you will get lots of black and white areas in the image, with the resulting histogram having large counts (tall bars) at the extreme end bins.

Example of severe burn-in -- Chinese Chess Image? Summary of 'stretch' operators...

-contrast-stretch, and -linear-stretch all generate a histogram (using 1024 bins) to determine the color position to stretch. as such it is not 'exact'. The other difference is how 'zero' is handled, and that -linear-stretch actually does a -level operation to do the stretch, while -contrast-stretch uses histogram bin values for color replacement stretching (which introduces a 1024 quantum rounding effect. -normalize uses -contrast-stretch internally.

A mathematically perfect normalization stretching operator is -auto-level. While a perfect 'white-point only' or 'black-point only' version is posible it has not been implemented at this time.

Auto-Level - perfect mathematical normalization

The "-auto-level" finds the largest and smalled values in the image to use to stretch the image to the full Quantum Range. At no time will values become 'clipped' or 'burned' as a result of the histogram being stretch beyond the value range.

The "-channel" setting will determine if all channels are stretch equally 'in sync' (using the maximum and minimum over all channels) or separatally (each individual channel as a separate entity).

At this time the hidden color of fully-transparent pixels, are also used in determining the levels, which can cause some problems when transparency is involved. This is regarded as a bug.

FUTURE: We actually need three modes of operation...
  synced color channels with 'alpha' (and 'read') masking.
  synced channels (as defined by channel)       (current default)
  individual separate channels   (currently if -channel is set by user)

It is a pure-mathematical histogram stretch just as the manual Level Operator is. That is the minimum will be adjusted to zero and maximum to Quantum range, and a linear equation is used to adjust all other values in the image.

It does not use 'histogram bins' or any other 'grouping of values' that other methods may use for either determining the levels to be used, or other histogram adjustments.

Normalize

The "-normalize" operator is the simplest of these three operators. It simply expands the grayscale histogram so that it occupies the full dynamic range of gray values, while clipping or burning 2% on the low (black) end and 1% on the on the high (white) end of the histogram. That is, 2% of the darkest grays in the image will become black and 1% of the lightest grays will become white.

This is not a large loss in most images, and the overall result is that the contrast (intensity range) of the image will be automatically maximized.

An idealized diagram is needed here!

Example using chinese chess?

Here we create a gray-scale gradient, and expand it to the full black and white range.


  convert -size 150x100 gradient:gray70-gray30 gray_range.jpg
  convert gray_range.jpg  -normalize  normalize_gray.jpg

For practical reasons to do with JPEG color inaccuracies (see JPEG Color Distortion for more details) and scanned image noise, "-normalize" does not expand the very brightest and darkest colors, but a little beyond those values. That is, it is equivalent to a "-contrast-stretch" with a value of '2%,99%' (see below).
This means if highest and lowest color values are very close together, "-normalize" will fail, an no action will be taken.
If you really want to expand the exact brightest and darkest color values to their extremes use "-auto-level" instead.

Up until IM version 6.2.5-5, "-normalize" worked purely as a grayscale operator. That is, each of the red, green, blue, and alpha channels were expanded independently of each other according to the "-channel" setting. As of IM version 6.2.5-5, if only the default "+channel" setting is given, then "-normalize" will tie together all the color channels, and normalizes them all by the same amount. This ensures that pixel colors within the image are not shifted. However, it also means that you may not get a pure white or black color pixel.

For example here we added some extra colors (a blue to navy gradient) to our normalization test image.


  convert -size 100x100 gradient:gray70-gray30 \
          -size  50x100 gradient:blue-navy  +append  color_range.jpg
  convert color_range.jpg -normalize  normalize.jpg

As you can see from the last example, for color images "-normalize" maximized all the channels together so one channel has a zero value, and another channel has a maximum value. That is, no black pixels were generated, as all the added blue colors already contain 'zero' values in the 'red' and 'green' channels. As such the lower bounds of the image did not expand.

If you want the old "-normalize" behaviour (before IM v6.2.5-5), you will need to specify any non-default "-channel" setting. For images that contain no alpha (or matte) channel, you can just use the 'all' channel setting.


  convert color_range.jpg -channel all  -normalize   normalize_all.jpg

Alternatively, you can normalize each channel as a separate image using the "-separate" operator (as of IM v6.2.9-2), then "-combine" them back into a single image again.


  convert color_range.jpg -separate -normalize -combine normalize_sep.jpg

In these last two examples, we see that the grayscale areas of the image turned yellow, since the 'red' and 'green' channels were lightened, while the 'blue' channel is only darkened slightly.

This brings use to an important point

Normalise and other Histogram operators are really grayscale operators,
caution is needed when using it with color images.

In actual fact, "-normalize" is just a subset of the more general "-contrast-stretch" with default values for black-point 2% and white-point=1%. So what is "-contrast-stretch"?

contrast-stretch

The "-contrast-stretch" operator (added IM v6.2.6) is similar to "-normalize", except it allows the user to specify the number of pixels that will be clipped or burned-in. That is it provides you with some control over its selection of the 'black-point' and 'white-point' it will use for the histogram stretching. Thus the user specifies a count (or percent counts) of the darkest grays in the image become black and the count of the lightest greys to become white.

For example, this will replace both the top and bottom 15% of colors with their extremes (white and black), stretching the rest of the 70% of colors appropriately. The final result is to try to improve the overall contrast of the image.


  convert gray_range.jpg  -contrast-stretch 15%  stretch_gray.jpg

You can also easily see the 'burn' and 'clip' effects at the top and bottom of the above gradient, as those gray colors get stretched well beyond the limits of the color range.

And here I purposefully 'burn' 90% of the darker grays, leaving just 10% of the brightest pixels to be stretched into a tight linear gradient at the top of the image.


  convert gray_range.jpg  -contrast-stretch 90%x0%  stretch_black.jpg

This can be quite useful in order to find the brightest 'N' pixels in an image, as they will be the only ones not 'burned' to a value of zero. (A better way is to use "-threshold-black")

One important aspect of "-contrast-stretch" is the use of zero for the black-point and white-point threshold counts. In this case, "

-contast-stretch
0

", will locate the minimum and maximum bins in the image's histogram. Since the counts actually begins at these bins, the result is simply to stretch the min and max bins to full black and full white. This will result in a contrast stretch with a minimum or possibly zero amount of clipping, with all the values in those 'bins' becoming 0 and maximum values.

Under Construction

Linear-Stretch

In many ways "-linear-stretch" is very similar to the previous "-contrast-stretch" operator. Both functions can take black-point and white-point arguments as either raw counts or as percentages of the total number of pixels involved. However there are several important differences.

One difference has to do with how the default black-point and white-point is computed. With "-contrast-stretch". If only one value, the black-point, is provided, then the white point will be the same value. Thus "-contrast-stretch 1" is equivalent to "

-contrast-stretch
1x1

" and "-contrast-stretch 1%" is equivalent to "-contrast-stretch 1x1%"". However, with "-linear-stretch", if only one value, the black-point, is provided, then the white point will be the complement value.

That is, if the black-point is specified as a raw count, then the white-point will be the total pixels in the image minus the black-point count. Likewise, if the black-point is specified as a percent count, then the white-point will be, 100% minus the black-point percentage count. Thus "-linear-stretch 1%" will be equivalent to "

-linear-stretch
1x99%

The second difference has to do with where counts begin. Consider a histogram with 256 bins (some 'bins' which may have zero counts) going from graylevel 0 to graylevel 255. In "-contrast-stretch", counts start at zero with the lowest (min) and highest (max) populated bins in the image (which may or may not be at bin 0 or bin 255 in the histogram). Thus a black-point of 10% will cumulate counts from all bins after the min bin until it reaches 10% and stretch the black side from that graylevel. Thus the amount burned-in at the black side of the histogram will end up being 10% plus what was already found in the darker 'bins' before it. Likewise with counting from the bright side of the histogram.

With "-linear-stretch", the count starts at the ends of the histogram, namely, at bin 0 and bin 255. Thus the amount burned-in at the dark side will always be the black-point value and the amount of burn-in at the bright side will always be the white-point value.

As an example, lets take a gradient of 100 pixels and look at its histogram.


  convert -size 1x100 gradient: \
          -depth 8 -format "%c" histogram:info:

As expected every bin is equally populated with a single pixel, producing a count of 1. (To see the full listing click on the output text image above).

Now lets do the same after using "-contrast-stretch 10x10%"


  convert -size 1x100 gradient:   -contrast-stretch 10x10%  \
          -depth 8 -format "%c" histogram:info:

And now "-linear-stretch 10x10%".


  convert -size 1x100 gradient:   -linear-stretch 10x10%  \
          -depth 8 -format "%c" histogram:info:

So we confirm that for "-contrast-stretch 10x10%" we get 11 pixels at each end. That is equivalent to the count in the end bins plus 10% of the image pixels, which is equal to 10 pixels. So 10+1=11 pixels burned-in. On the other hand, in "-linear-stretch", the end bins end up containing containing only 10 pixels or 10% of the image.

One consequence of the aforementioned difference is that "-contrast-stretch 0x0" may change the image, if the lowest and/or highest populated bins are not the end bins at 0 and 255. In this case, the image will be stretched between the graylevels corresponding to those bins. On the other hand, "-linear-stretch 0x0" will never change the image.

For example, lets take the gradient and compress its graylevels by 10% on each end. That is, we will move black-point up 10% to graylevel 26 and white-point down 10% to graylevel 230.


  convert -size 1x100 gradient:   +level 10x90%  \
          -depth 8 -format "%c" histogram:info:

Now, lets apply "-contrast-stretch 0x0" to the above de-contrasted gradient


  convert -size 1x100 gradient: -level 10x90%  -contrast-stretch 0x0  \
          -depth 8 -format "%c" histogram:info:

And now "-linear-stretch 0x0"


  convert -size 1x100 gradient: -level 10x90%  -linear-stretch 10x10% \
          -depth 8 -format "%c" histogram:info:

So we see that the original image had a histogram that did not span the full dynamic range of 0 to 255. It only went between graylevels 26 and 230. But after applying "-contrast-stretch 0x0", it was stretched to full dynamic range. On the other hand, "-linear-stretch 0x0" made no change in the resulting histogram.

The third difference is that "-contrast-stretch" is channel sensitive, whereas "-linear-stretch" is not.

That means that with "-contrast-stretch" any one or more channels can be changed without affecting the others. Thus if no channel is specified, the overall histogram from all the channels will be used to modify all the channels in the same manner so that no color shifts are produced.

However, if "-channel RGB" is specified, then each channel will be stretched separately and the result will depend upon the end bins in each channel. If they are different, then a color shift will be produced between the individual channels in the resulting image.

With "-linear-stretch", all the channels will be processed in a common way, thus assuring that no color shifts of the channels relative to each other will be produced.

So lets get a verbose identify and the histogram of a real image.


  convert port.png  -verbose -identify +verbose  histogram:port_hist.gif

We see that none of the channels of the above image span the full dynamic range. Also note that each of the channels spans an uniquely different range of values.

Now lets apply "-contrast-stretch 1x1%" without a "-channel" setting.


  convert port.png -contrast-stretch 1x1% \
          -write histogram:port_cs1_hist.gif   port_cs1.png

In the above result, the image is stretched consistently across all the channels. Thus, there are no color shifts between channels. Now let's do the same but with "-channel RGB".


  convert port.png  -channel RGB  -contrast-stretch 1x1% \
          -write histogram:port_cs1rgb_hist.gif    port_cs1rgb.png

In the above result, because we set "-channel RGB", rather than use the default channel setting, the image is stretched differently for each channel. This causes a color shift between channels.

Now let's apply "-linear-stretch" without a "-channel" setting.


  convert port.png   -linear-stretch 1x1% \
          -write histogram:port_ls1_hist.gif \
          port_ls1.png

In the above result, the image is stretched consistently across all the channels. So there is no color shift between channels. Now let's do the same, but with "-channel RGB".


  convert port.png  -channel RGB  -linear-stretch 1x1% \
          -write histogram:port_ls1rgb_hist.gif    port_ls1rgb.png

In the above result with "-linear-stretch", the image is stretched consistently across all the channels and "-channel RGB" is ignored. Thus there is no color shift between channels and the result is identical to that above without "-channel RGB".

Histogram Redistribution

Histogram redistribution is a non-linear technique that redistributes the bins in a histogram in order to achieve some particular shape. The two most common shapes are uniform (flat) and Gaussian (bell-shaped), although Hyperbolic and Rayleigh are other types of distributions have also been used.

Equalize - Uniform Histogram Redistribution

For the case of an uniform distribution, the histogram bins are shifted, spaced and combined so that on average the histogram has a flat or constant height across the whole range. This is called histogram equalization. The IM function, "-equalize", does this.

Unfortunately, it operates on each channel separately, rather than applying the same operation to all channels. As such, color shifts are possible, when it is applied to RGB colorspace.

Here is an example of histogram equalization using the IM function -equalize. Notice the color balance shift from the equalization on each channel independently.


  convert zelda.png  -write histogram:zelda_hist.gif \
          -equalize  -write histogram:zelda_equal_hist.gif \
          zelda_equal.png

You may note that the histogram does not look very uniform. But if we convert the resulting image to grayscale and display its histogram, its histogram looks a bit more uniform in comparison to the original image's grayscale histogram


  convert zelda.png  -colorspace gray   histogram:zelda_ghist.gif

  convert zelda_equal.png  -colorspace gray \
          histogram:zelda_equal_ghist.gif

The other way to approach redistributing the bins is by using a transformation look up table that is generated from the separate cumulative histograms of each channel and the desired integrated distribution curve. If one does not want any color shifts between channels, then one uses the combined histogram from all the channels of the image. An approximation is simply to use the histogram of the image after converting it to grayscale.

Fred Weinhaus has developed a script, called "redist" that does just that. It redistributes the histogram of an image into an uniform distribution, while appling the same change to all color channels equally.


  redist -s uniform zelda.png  zelda_uniform.png

  convert zelda_uniform.png   histogram:zelda_uniform_hist.gif

Note how the results different from the IM built-in "-equalize" operator. Specifically, all the colors are preserved, without the color shift you saw previously.

What the script does is work on the grayscale histogram, which it then applies to all the color channels, so that all the colors are kept together.

For comparison with the IM "-equalize" histograms, lets show the grayscale histogram results here, too. Note that the redistributed histogram appears to be a bit more leveled out (flat, or uniform) than that of IM equalize.


  convert zelda.png  -colorspace gray   histogram:zelda_ghist.gif

  convert zelda_uniform.png  -colorspace gray \
          histogram:zelda_uniform_ghist.gif

FUTURE: Add examples of equalizing in other colorspaces! That is, the grayscale channel in HSL, HSB and CMYK colorspaces.

Gaussian Redistribution

Equalizing a histogram is not the only way of changing the histogram distribution of an image. Actually it isn't normally very useful, except in computer vision applications.

Here is the same image, but transformed so its histogram has a Gaussian (bell-shaped) distribution. The values used here are a 60% gray mean, with a 60 sigma roll-off to either side of that mean.


  redist -s gaussian 60,60,60  zelda.png \
         zelda_gaussian.png

  convert zelda_gaussian.png -colorspace gray \
          histogram:zelda_gaussian_ghist.gif

From the resulting grayscale histogram, you can see that the image is modified so its colors follow a Gaussian bell curve type of distribution.

For photos, this produces a more 'natural' looking result. The image will not only have been contrast optimized, but also adjusted in brightness so most of the pixels in the image have about a 60% grayscale brightness.

Histogram Redistribution Methodology

So how does this type of direct histogram adjustment work?

Basically it computes the histogram of the current image and that of the desired distribution. It then works out how the graylevel value of each 'bin' needs to be changed so that the counts in the bins best follow the desired distribution. Some bins may be shifted darker, while others may be shifted lighter.

This is actually quite an involved process, so lets go though it step by step.

First, we need to get the actual histogram data from ImageMagick, rather than a graphic image of the histogram. Note that the data is from all the color values, combined into a grayscale. This was done so as to distribute all the channels together, and adjust the image overall brightness to follow to the desired curve.


  convert zelda.png -colorspace gray \
         -depth 8 -format "%c" histogram:info:- |\
    tr -cs '0-9\012' ' ' |\
      awk '# collect the histogram data.
           { bin[$2] += $1; }
           END { for ( i=0; i<256; i++ ) {
                   print bin[i]+0;
                 }
               } ' > zelda_hist_data.txt

  # get the maximum count for any one histogram 'bin'
  max_count=`sort -n zelda_hist_data.txt | tail -n 1`

  # convert histogram into a profile graph of the data
  echo "P2 256 1 $max_count" | cat - zelda_hist_data.txt |\
    im_profile -s - zelda_hist_graph.gif

To collect the data I take the 'comment' meta-data from the histogram image, which IM includes just for this purpose. The data is then cleaned to leave just the raw numbers (using a program called "tr", short for 'translate'). This raw data is then given to another utility program called "awk", which is used to collect the actual histogram counts for each bin.

So that we can look at the results, I also process the histogram counts into a gradient image (via the NetPBM, PGM text grayscale image file format, and display it as a line graph using the "im_profile" script. Essentially this is just a different way of generating a histogram image, though this time directly from a numerical data file.

Now that we have the histogram data in a text file, we also need the histogram of the function we want the redistributed data to match. In this case, it is a Gaussian distribution with a mean value of 153 (60% gray) and sigma width of 60. Both values are in terms of the 256 range of the histogram 'bins'.


  awk '# AWK to generate gaussian distribution graph
        BEGIN { mean = 153;   sigma = 60;
                fact = 1/(2*(sigma/256)^2);
                expo = exp(1);
                for ( i=0; i<256; i++ ) {
                  print int(65535*expo^(-(((i-mean)/256)^2)*fact));
                }
              }' /dev/null  > gaussian_hist_data.txt

   # convert gaussian data into a profile graph
   echo "P2 256 1 65535" | cat - gaussian_hist_data.txt |\
     im_profile -s -b - gaussian_hist_graph.gif

The histograms above are interesting and reflect the image's original histogram distribution and the histogram's desired state. But for conversion purposes, this form of histogram, while good for us to understand, is not very useful for our purposes.

Actually, what we really need are the cumulative histograms. These histograms are very similar to a normal histogram, except that each 'bin' in the histogram is a count of its 'bin' plus all the 'bins' that came before it, starting at 0. That is, each 'bin' is an 'accumulation' or count of all the darker 'bins'.

These are actually easier to generate directly from the original image. So lets repeat the process, but computing and saving the 'cumulative' counts.


  convert zelda.png -colorspace gray \
         -depth 8 -format "%c" histogram:info:- |\
    tr -cs '0-9\012' ' ' |\
      awk '# Collect the cumulative histogram for an image
               { bin[$2] += $1; }
           END { for ( i=0; i<256; i++ ) {
                   cum += bin[i];
                   print cum;
                 }
               } ' > zelda_cumhist_data.txt

  total_count=`tail -n 1 zelda_cumhist_data.txt`
  echo "P2 256 1 $total_count" | cat - zelda_cumhist_data.txt |\
    im_profile -s - zelda_cumhist_graph.gif

  awk '# AWK to generate gaussian distribution cumulative graph
        BEGIN { mean = 153;   sigma = 60;
                fact = 1/(2*(sigma/256)^2);
                expo = exp(1);
                for ( i=0; i<256; i++ ) {
                  gas[i] = expo^(-(((i-mean)/256)^2)*fact);
                  total += gas[i]
                }
                for ( i=0; i<256; i++ ) {
                  cum += gas[i];
                  print int(65535*cum/total);
                }
              }' /dev/null  > gaussian_cumhist_data.txt

  total_count=`tail -n 1 gaussian_cumhist_data.txt`
  echo "P2 256 1 $total_count" | cat - gaussian_cumhist_data.txt |\
    im_profile -s -b - gaussian_cumhist_graph.gif

Image Cumulative
Histogram

Gaussian Cumulative
Histogram

Now what we need to do is convert the image's cumulative histogram into the gaussian cumulative histogram. To do this, each gray value in the input image is used to find its 'normalized' cumulative value. This is then mapped to the same cumulative value in the gaussian distribution and then its corresponding gray value is found.

This diagram should make the mapping process clearer...

The following command does the lookup for every possible 8-bit color value, in order to generate a Color Look Up Table, or CLUT. This special image can then be used to map the color values in the original image to the new values needed to redistribute the image's histogram.


  # Generate a CLUT to Redistribute the Histogram
  paste  zelda_cumhist_data.txt   gaussian_cumhist_data.txt |\
    awk '# AWK to generate gaussian distribution graph
              { bin[NR] = $1;   gas[NR] = $2;  }
          END { k=0;  # number of pixels less than this value
                print "P2 256 1 65535";
                for ( j=0; j<256; j++ ) {
                  while ( k<255 &&
                            gas[k]/gas[255] <= bin[j]/bin[255] ) {
                    k++;
                  }
                  print 65535*k/255;
                }
              }' |\
      convert pgm:- gaussian_clut.png

  convert zelda.png   gaussian_clut.png -clut   zelda_redist.png

As you can see, converting a histogram of an image to attempt to follow a specific distribution function, such a gaussian bell curve, is quite an involved and highly numerical process.

Here it is all in one rather long and complex command...


  convert zelda.png -colorspace gray \
         -depth 8 -format "%c" histogram:info:- |\
    tr -cs '0-9\012' ' ' |\
      awk '# AWK to generate gaussian distribution graph
            { # just read in image histogram into a 'bin' table
                  bin[$2] += $1;
                }
            END { # Generate Gaussian Histogram
                  mean = 153;   sigma = 60;
                  fact = 1/(2*(sigma/256)^2);
                  expo = exp(1);
                  for ( i=0; i<256; i++ ) {
                    gas[i] = expo^(-(((i-mean)/256)^2)*fact);
                  }
                  # Convert normal histograms to cumulative histograms
                  for ( i=0; i<256; i++ ) {
                    gas[i] += gas[i-1];
                    bin[i] += bin[i-1];
                  }
                 # Generate Redistributed Histogram
                 k=0;  # number of pixels less than this value
                 print "P2 256 1 65535";
                 for ( j=0; j<256; j++ ) {
                   while ( k<255 &&
                            gas[k]/gas[255] <= bin[j]/bin[255] ) {
                     k++;
                   }
                   print 65535*k/255;
                 }
                }' |\
        convert zelda.png   pgm:-  -clut   zelda_gaussian_redist.png

Just some final words on the above technique.

Using "awk" to do the calculations to speed up Fred Weinhaus's "redist" script were suggested and contributed by Anthony Thyssen.
To apply the above redistribution technique to generate a 'uniform', or 'equalized' distribution, the function histogram is simply a constant. This in turn results in an an integrated distribution that is simply the formula y = x, or simply a diagonal straight line. Applying the same conversion technique leads to a CLUT image that turns out to be identical to the input image's cumulative histogram.
In other words, for an equalization of the histogram, you can simply convert the image's cumulative histogram into a CLUT and apply it to the image directly.
Most Image processing packages, including ImageMagick at this time, apply the transformation formulae directly to the values in the image itself, rather than generate an intermediate CLUT. However as histograms and thus cumulative histograms have a limited size (256 'bins' typically), that can lead to serious errors, since the image color values may be rounded off, during the process.
However with ImageMagick, we generate an intermediate CLUT (containing those same round off errors), and then convert the original un-rounded image values though the prepared CLUT using a linear interpolation of the values. As a result of this interpolation, the color values of the new image is more accurate, as they have not been rounded off or 'bin'ed during processing.

The above will hopefully eventually be built into ImageMagick. In the mean time Fred Weinhaus's "redist" script is available to do the task.

You may also be interested in Fred's "retinex" script, which attempts to make similar automatic enhancements to images, in localized regions of the image, rather than globally as this technique does.

DIY Level Adjustments

Mathematical Linear Histogram Adjustments

The various basic forms of Level Adjustments shown above linearly adjust the colors of the image.

These changes can be applied mathematically as well. For example by multiplying the image with a specific color, we set all pure white areas to that color. So lets just read in our image, create an image containing the color we want, then multiply the original image with this color using the IM free-form "-fx" or DIY Operator.


  convert test.png  -size 1x1 xc:Yellow \
          -fx 'u*v.p{0,0}'    fx_linear_white.png

By getting "-fx" to read the color from a second 'v' image makes it easy to change the color, without needing to convert colors to RGB values for use in the mathematics.

If you were using a fancy graphical image processing package like "Gimp" and "Photoshop" the above operation would have been applied to an image by adjusting the images color histogram graph 'curve'.

For example to the right is a "gnuplot" generated graph (See the script "im_histogram") of the mathematical formula showing what happens to just one of the three RGB channels. The original color (green line) is remapped to a darker color (red line) linearly.

Linearly tinting the black colors is also quite simple. For example to linear map 'black' to a gold like color 'rgb(204,153,51)', (while leaving 'white' as 'white'), would require a mathematical formula such as...

          result = 1-(1-color)*(1-intensity)

This formula negates the colors, multiples the image with the negated color wanted, and negates the image back again. The result is tinting of the black side of the gray scale, leaving white unchanged.


  convert test.png  -size 1x1 xc:'rgb(204,153,51)'  \
          -fx '1-(1-v.p{0,0})*(1-u)'   fx_linear_black.png

A "gnuplot" histogram graph of the remapping formula is also displayed in the above for your reference.

With a slightly more complicated formula you can linearly replace both the 'black' and 'white' end of the grayscale with specific colors.


  convert test.png  -size 1x2  gradient:gold-firebrick \
          -fx 'v.p{0,0}*u+v.p{0,1}*(1-u)'   fx_linear_color.png

The "-size 1x2 gradient:color1-color2" in the above is only used to generate a two color pixel image for the "-fx" formula to reference. The first color replaces white, while the second replaces black, while all others are interpolated between white and black. As is typical of a gray-scale operator, each RGB channel is treated as a separate gray scale channel, though the linear interpolation is different for each channel.

This by the way is exactly equivalent to the Level Adjustments by Color operator "+level-colors"

However unlike "+level-colors", the colors to use can of course come from any image source, and not just the color names provided as an argument. However even direct use of color names is possible.


  convert test.png   -fx "yellow*u+green*(1-u)"  fx_linear.png

Mathematical Non-linear Histogram Adjustments

While linear color adjustments are important, and faster methods are available, there are many situations where a linear 'level' adjustment, is not what is wanted, and this is where the "-fx" DIY Operator, becomes more useful.

Well an alternative formula for linear adjustment is "

-fx
'v.p{0,1}+(v.p{0,0}-v.p{0,1})*u'

", which has the advantage that the 'u' can be replaced by a single random function 'f(u)' to produce non-linear color change.

This lets you do more interesting things. For example what if in the last example you wanted to push all the colors toward the 'black' side, resulting in the image being a more 'firebrick' color.


  convert test.png -size 1x2  gradient:gold-firebrick \
          -fx 'v.p{0,1}+(v.p{0,0}-v.p{0,1})*u^4'  fx_non-linear.png

In a more practical example, Adelmo Gomes needed a color adjustment for a automated Weather Map Recoloring script he was developing.

In this case he wanted to tint pure black parts of the image to a .25 blue, but leave the rest of the gray-scale alone, especially the white and mid-tone grays of the image. Only the blue color needed such adjustment, which he currently was doing by hand in an image editor.

For example you could use a quadratic formula like 'u^2' to tint the black end of the histogram to a '.25' blue color. Only the blue channel needs to be modified, so the value was inserted directly into the formula.


  convert test.png  -channel B  -fx '.25+(1-.25)*u^2'  fx_quadratic.png

However while this produced a reasonable result it does darken the mid-tone grays slightly, producing a sickly off-yellow color.

To avoid this a 'exponential' function can be used instead, to give better control of the tinting process.


  convert test.png  -channel B  -fx '.3*exp(-u*4.9)+u'  fx_expotential.png

Again the graph show how blue channel was modified to give black a distinctive dark blue tint.

The second value ('4.9') is the falloff back to a linear '+u' graph. The smaller this value is the slower the fall off, and the more linear the adjustment becomes. The larger the value, the more dramatic the 'fall-off'. The value may need to be adjusted for different color values, so this is not a good general formula for general black color tinting, but perfect for tinting weather maps.

Generally if you can express the color adjustment you want mathematically, you can then use "-fx" operator to achieve the results you want.

'Curves' Adjustments

Normally in a graphical photo editor you would be presented with a histogram 'curves' chart such as I have shown to the left. The user can then edit the 'curve' by moving four (or more) control points, and the histogram adjustment function will follow those points.

The control points generally specify that the first grayscale level is after adjustment to become the second grayscale level. So a point like 0.0,0.2 basically means that a 0% gray (black) should after adjustment be a 20% gray level.

Now IM does not allow you to directly specify 'control points' to generate a 'curve' adjustment, what it wants is the mathematical formula of that 'curve' generated. Lucky for us there are programs that can generate that curve formula from the control points, including "gnuplot", "fudgit", "mathematica", and "matlab", as well as many more mathematical software packages.

The following is one method you can use to generate the formula from four control points using "gnuplot" which is a standard extra package you can install on most linux distributions, as well as windows.


  ( echo "0.0 0.2";  echo "1.0 0.9"; \
    echo "0.2 0.8";  echo "0.7 0.5"; )   > fx_control.txt

  ( echo 'f(x) = a*x**3 + b*x**2 + c*x + d'; \
    echo 'fit f(x) "fx_control.txt" via a, b, c, d'; \
    echo 'print a,"*u^3 + ",b,"*u^2 + ",c,"*u + ",d'; \
  ) | gnuplot 2>&1 | tail -1             > fx_funct.txt