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What is the main purpose of histogram equalization on image? Is it used to enhance contrast of image or is its goal finding a transformation which gets a more uniformly histogram?

Although Wikipedia says:

Histogram equalization is a method in image processing of contrast adjustment using the image's histogram.

But it references this file which uses $$T(r) = (L-1) \int\limits_0^rp_r(w)\ dw$$ that returns a uniform random variable which its $PDF$ is $\frac{1}{L-1}$. This try for getting a uniform random variable and knowing that entropy of a uniform random variable in $[0,1]$ is $0$, doubt me in about the main purpose of histogram equalization. Does it depend on used algorithm for equalizing?

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Contrast enhancement is usually the goal here, and making the histogram more uniform is the means to achieve it.

Additionally, histogram equalization is useful for reducing the effects of varying lighting or auto exposure, to help computer vision algorithms like SLAM.

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Histogram equalization is used for contrast enhancement. The more classical way of doing this is trying to convert the histogram as a uniform distribution so the cumulative histogram is a line.

In several algorithms this is a preprocessing step for something more, for example for thresholding. This would "normalize" the images for when the thesholding is happening all images are more similar or in the same "domain".

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