Histogram matching

Implementation

Consider a grayscale input image X. It has a probability density function pr(r), where r is a grayscale value, and pr(r) is the probability of that value. This probability can easily be computed from the histogram of the image by

p_r (r_j )= {n_j \over n}

where nj is the frequency of the grayscale value rj, and n is the total number of pixels in the image.

Now consider a desired output probability density function pz(z). A transformation of pr(r) is needed to convert it to pz(z).

Each pdf (probability density function) can easily be mapped to its cumulative distribution function by

: S(r_k )= \sum_{j=0}^k p_r(r_j),\qquad k = 0,1,2,3,\ldots,L-1

: G(z_k )= \sum_{j=0}^k p_z(z_j),\qquad k = 0,1,2,3,\ldots,L-1

where L is the total number possible of gray levels (256 for a standard image).

The idea is to map each r value in X to the z value that has the same probability in the desired pdf. I.e. S(r**j) = G(z**i) or z = G−1(S(r)).

Example

The following input grayscale image is to be changed to match the reference histogram.

The input image has the following histogram:

It will be matched to this reference histogram to emphasize the lower gray levels.

Algorithm

Given two images, the reference and the target images, we compute their histograms. Following, we calculate the cumulative distribution functions of the two images' histograms: F_1(), for the reference image and F_2(), for the target image. Then for each gray level G_1\in[0,255], we find the gray level G_2, for which F_1(G_1)=F_2(G_2),, and this is the result of histogram matching function: M(G_1)=G_2,. Finally, we apply the function M() on each pixel of the reference image.

Exact histogram matching

In typical real-world applications, histogram matching can only approximate the specified histogram. All pixels of a particular value in the original image must be transformed to just one value in the output image.

Exact histogram matching is the problem of finding a transformation for a discrete image so that its histogram exactly matches the specified histogram.{{cite journal

Because of this there may be holes or open spots in the output matched histogram.

Multiple histogram matching

The histogram matching algorithm can be extended to find a monotonic mapping between two sets of histograms. Given two sets of histograms P={p_i}{i=1}^k and Q = {q_i }{i=1}^k , the optimal monotonic color mapping M is calculated to minimize the distance between the two sets simultaneously, namely \operatorname{argmin}_M \sum_k d(M(p_k),q_k) where d(\cdot,\cdot) is a distance metric between two histograms. The optimal solution is calculated using dynamic programming.

References

| book-title = Proceedings of the IEEE International Conference on Image Processing

References

1. (2008). "Digital Image Processing". Prentice Hall.
2. Gonzalez, Rafael. (2017). "Digital image processing 4th Edition". Pearson.