Image Processing Fundamentals - Histogram-based Operations

Histogram-based Operations

Contrast stretching
Equalization
Other histogram-based operations

An important class of point operations is based upon the manipulation of an image histogram or a region histogram. The most important examples are described below.

Contrast stretching

Frequently, an image is scanned in such a way that the resulting brightness values do not make full use of the available dynamic range. This can be easily observed in the histogram of the brightness values shown in Figure 6. By stretching the histogram over the available dynamic range we attempt to correct this situation. If the image is intended to go from brightness 0 to brightness 2^B-1 (see Section 2.1), then one generally maps the 0% value (or minimum as defined in Section 3.5.2) to the value 0 and the 100% value (or maximum) to the value 2^B-1. The appropriate transformation is given by:

This formula, however, can be somewhat sensitive to outliers and a less sensitive and more general version is given by:

In this second version one might choose the 1% and 99% values for p_low% and p_high%, respectively, instead of the 0% and 100% values represented by eq. . It is also possible to apply the contrast-stretching operation on a regional basis using the histogram from a region to determine the appropriate limits for the algorithm. Note that in eqs. and it is possible to suppress the term 2^B-1 and simply normalize the brightness range to 0 <= b[m,n] <= 1. This means representing the final pixel brightnesses as reals instead of integers but modern computer speeds and RAM capacities make this quite feasible.

Equalization

When one wishes to compare two or more images on a specific basis, such as texture, it is common to first normalize their histograms to a "standard" histogram. This can be especially useful when the images have been acquired under different circumstances. The most common histogram normalization technique is histogram equalization where one attempts to change the histogram through the use of a function b =

(a) into a histogram that is constant for all brightness values. This would correspond to a brightness distribution where all values are equally probable. Unfortunately, for an arbitrary image, one can only approximate this result.

For a "suitable" function (*) the relation between the input probability density function, the output probability density function, and the function (*) is given by:

From eq. we see that "suitable" means that (*) is differentiable and that d/da >= 0. For histogram equalization we desire that p_b(b) = constant and this means that:

where P(a) is the probability distribution function defined in Section 3.5.1 and illustrated in Figure 6a. In other words, the quantized probability distribution function normalized from 0 to 2^B-1 is the look-up table required for histogram equalization. Figures 21a-c illustrate the effect of contrast stretching and histogram equalization on a standard image. The histogram equalization procedure can also be applied on a regional basis.

Figure 21a Figure 21b Figure 21c Original Contrast Stretched istogram Equalized

Other histogram-based operations

The histogram derived from a local region can also be used to drive local filters that are to be applied to that region. Examples include minimum filtering, median filtering, and maximum filtering . The concepts minimum, median, and maximum were introduced in Figure 6. The filters based on these concepts will be presented formally in Sections 9.4.2 and 9.6.10.