**Common Names:** Histogram Modeling, Histogram Equalization

Histogram modeling techniques (*e.g.* histogram equalization) provide a
sophisticated method for modifying the dynamic range and contrast of
an image by altering that image such that its intensity
histogram has a desired shape. Unlike contrast
stretching, histogram modeling operators may employ *non-linear*
and *non-monotonic* transfer functions to map between pixel
intensity values in the input and output images. Histogram
equalization employs a monotonic, non-linear mapping which re-assigns
the intensity values of pixels in the input image such that the output
image contains a uniform distribution of intensities (*i.e.* a flat
histogram). This technique is used in image comparison processes
(because it is effective in detail enhancement) and in the correction
of non-linear effects introduced by, say, a digitizer or display
system.

Histogram modeling is usually introduced using continuous, rather
than discrete, process functions. Therefore, we suppose that the
images of interest contain continuous intensity levels (in the
interval [0,1]) and that the transformation function *f*
which maps an input image onto an output image
is continuous within this interval. Further,
it will be assumed that the transfer law (which may also be written in
terms of intensity density levels, *e.g.* )
is single-valued and monotonically increasing (as is the case in
histogram equalization) so that it is possible to define the inverse
law . An example of such a transfer
function is illustrated in Figure 1.

Figure 1A histogram transformation function.

All pixels in the input image with densities in the region to will have their pixel values re-assigned such that they assume an output pixel density value in the range from to . The surface areas and will therefore be equal, yielding:

where .

This result can be written in the language of probability theory if
the histogram *h* is regarded as a continuous
probability density function *p* describing the
distribution of the (assumed random) intensity levels:

In the case of histogram equalization, the output probability densities should all be an equal fraction of the maximum number of intensity levels in the input image (where the minimum level considered is 0). The transfer function (or point operator) necessary to achieve this result is simply:

Therefore,

where is simply the cumulative probability
distribution (*i.e.* cumulative histogram) of the original image.
*Thus, an image which is transformed using its cumulative histogram
yields an output histogram which is flat!*

A digital implementation of histogram equalization is usually performed by defining a transfer function of the form:

where *N* is the number of image pixels and
is the number of pixels at intensity level k or
less.

In the digital implementation, the output image will not necessarily
be fully equalized and there may be `holes' in the histogram (*i.e.*
unused intensity levels). These effects are likely to decrease as the
number of pixels and intensity quantization levels in the input image
are increased.

To illustrate the utility of histogram equalization, consider

which shows an 8-bit grayscale image of the surface of the moon. The histogram

confirms what we can see
by visual inspection: this image has poor dynamic range. (Note that we
can view this histogram as a description of pixel probability
densities by simply scaling the vertical axis by the total number of
image pixels and normalizing the horizontal axis using the number of
intensity density levels (*i.e.* 256). However, the shape of the
distribution will be the same in either case.)

In order to improve the contrast of this image, without affecting the
structure (*i.e.* geometry) of the information contained therein, we can
apply the histogram equalization operator. The resulting image is

and its histogram is shown

Note that the histogram is not flat (as in the examples from the continuous case) but that the dynamic range and contrast have been enhanced. Note also that when equalizing images with narrow histograms and relatively few gray levels, increasing the dynamic range has the adverse effect of increasing visual grainyness. Compare this result with that produced by the linear contrast stretching operator

In order to further explore the transformation defined by the histogram equalization operator, consider the image of the Scott Monument in Edinburgh, Scotland

Although the contrast on the building is acceptable, the sky region is represented almost entirely by light pixels. This causes most histogram pixels

to be pushed into a narrow peak in the upper graylevel region. The histogram equalization operator defines a mapping based on the cumulative histogram

which results in the image

While histogram equalization
has enhanced the contrast of the sky regions in the image, the picture
now looks artificial because there is very little variety in the middle
graylevel range.
This occurs because the transfer
function is based on the shallow slope of the cumulative histogram in
the middle graylevel regions (*i.e.* intensity density levels 100 - 230)
and causes many pixels from this region in the original image to be
mapped to similar graylevels in the output image.

We can improve on this if we define a mapping based on a sub-section of the image which contains a better distribution of intensity densities from the low and middle range graylevels. If we crop the image so as to isolate a region which contains more building than sky

we can then define a histogram equalization mapping for the whole image based on the cumulative histogram

of this smaller region. Since the cropped image contains a more even distribution of dark and light pixels, the slope of the transfer function is steeper and smoother, and the contrast of the resulting image

is more natural.
This idea of defining mappings based upon particular sub-sections of
the image is taken up by another class of operators which perform
*Local Enhancements* as discussed below.

Histogram equalization is limited in that it is capable of producing only one result: an image with a uniform intensity distribution. Sometimes it is desirable to be able to control the shape of the output histogram in order to highlight certain intensity levels in an image. This can be accomplished by the histogram specialization operator which maps a given intensity distribution into a desired distribution using a histogram equalized image as an intermediate stage.

The first step in histogram specialization, is to specify the desired
output density function and write a transformation
*g(c)*. If is single-valued
(which is true when there are no unfilled levels in the specified
histogram or errors in the process of rounding off
to the nearest intensity level), then defines a mapping from the equalized levels of the
original image, . It is possible to
combine these two transformations such that the image need not be
histogram equalized explicitly:

The histogram processing methods discussed above are global in the
sense that they apply a transformation function whose form is based
on the intensity level distribution of an entire image. Although this
method can enhance the overall contrast and dynamic range of an image
(thereby making certain details more visible), there are cases in
which enhancement of details over small areas (*i.e.* areas whose total
pixel contribution to the total number of image pixels has a
negligible influence on the global transform) is desired. The solution
in these cases is to derive a transformation based upon the intensity
distribution in the local neighborhood of every pixel in the image.

The histogram processes described above can be adapted for local
enhancement. The procedure involves defining a neighborhood around
each pixel and, using the histogram characteristics of this
neighborhood, to derive a transfer function which maps that pixel into
an output intensity level. This is performed for each pixel in the
image. (Since moving across rows or down columns only adds one new
pixel to the local histogram, updating the histogram from the previous
calculation with new data introduced at each motion is possible.)
Local enhancement may also define transforms based on pixel attributes
other than histogram, *e.g.* intensity mean (to control variance) and
variance (to control contrast) are common.

You can interactively experiment with this operator by clicking here.

- Suppose that you have a 128×128 square pixel image with
an 8 gray level intensity range, within which the lighter intensity
levels predominate as shown in the table below.
**A)**Sketch the histogram (number of pixels vs gray level) to describe this distribution.**B)**How many pixels/gray levels would there be in an equalized version of this histogram?**C)**Apply the discrete transformation described above and plot the new (equalized) histogram. (How well does the histogram approximate a uniform distribution of intensity values?)------------------------------- | Gray Level | Number of Pixels | |------------+------------------| | 0 | 34 | |------------+------------------| | 1 | 50 | |------------+------------------| | 2 | 500 | |------------+------------------| | 3 | 1500 | |------------+------------------| | 4 | 2700 | |------------+------------------| | 5 | 4500 | |------------+------------------| | 6 | 4000 | |------------+------------------| | 7 | 3100 | -------------------------------

- Suppose you have equalized an image once. Show that a second pass of histogram equalization will produce exactly the same result as the first.
- Interpreting images derived by means of a non-monotonic or
non-continuous mapping can be difficult. Describe the effects of the
following transfer functions:
(a)

*f*has a horizontal plateau,(b)

*f*contains a vertical jump,(c)

*f*has a negative slope.(Hint: it can be useful to sketch the curve, as in Figure 1, and then map a few points from histogram A to histogram B.)

- Apply local histogram equalization to the image
Compare this result with those derived by means of the global transfer function shown in the above examples.

- Apply global and local histogram equalization to the montage
image
Compare your results.

**R. Boyle and R. Thomas** *Computer Vision: A First Course*,
Blackwell Scientific Publications, 1988, pp 35 - 41.

**R. Gonzalez and R. Woods** *Digital Image Processing*, Addison-Wesley Publishing Company, 1992, Chap. 4.

**A. Jain** *Fundamentals of Digital Image Processing*,
Prentice-Hall, 1986, pp 241 - 243.

**A. Marion** *An Introduction to Image Processing*, Chapman
and Hall, 1991, Chap. 6.

Specific information about this operator may be found here.

More general advice about the local HIPR installation is available in the
*Local Information* introductory section.