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 1 A 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 Specification
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:
Local Enhancements
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.
------------------------------- | Gray Level | Number of Pixels | |------------+------------------| | 0 | 34 | |------------+------------------| | 1 | 50 | |------------+------------------| | 2 | 500 | |------------+------------------| | 3 | 1500 | |------------+------------------| | 4 | 2700 | |------------+------------------| | 5 | 4500 | |------------+------------------| | 6 | 4000 | |------------+------------------| | 7 | 3100 | -------------------------------
(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.)
Compare this result with those derived by means of the global transfer function shown in the above examples.
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.
©2003 R. Fisher, S. Perkins,
A. Walker and E. Wolfart.
