Exact Histogram Specification for Digital Images Using a Variational Approach
Résumé
We consider the problem of exact histogram specification for digital (quantized) images. The goal is to transform the input digital image into an output (also digital) image that follows a prescribed histogram. Classical histogram modification methods are designed for real-valued images where all pixels have different values, so exact histogram specification is straightforward. Digital images typically have numerous pixels sharing the same value. If one imposes the prescribed histogram to a digital image, usually there are numerous ways of assigning the prescribed values to the quantized values of the image. Therefore, exact histogram specification for digital images is an ill-posed problem. In order to satisfy the prescribed histogram, all pixels of the input digital image should be rearranged in a strictly ordered way, while preserving the specific features of the input image. Such a task can be realized if we are able to extract additional representative information (called auxiliary attributes) from the input digital image. This is a real challenge in exact histogram specification for digital images. We propose a new method that efficiently provides a strict ordering for all pixel values. It is based on a well designed variational approach. Noticing that the input digital image contains quantization noise, we minimize a specialized objective function whose solution is a real-valued image with slightly reduced quantization noise that remains very close to the input digital image. We show that all the pixels of this real-valued image can be ordered in a strict way with a very high probability. Then transforming the latter image into another digital image satisfying a specified histogram is an easy task. Numerical results show that our method outperforms by far the existing competing methods.
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