First, an algorithm for histogram modification via image evoluwhere β½(x, y, t): R 2 Ο« [0, ) Η R is the evolving image, tion equations is presented. We show that the image histogram F : R Η R is a given functional which depends on the can be modified to achieve any given distribution as the steady specific algorithm, and the image β½ 0 is the initial condition.
state solution of this differential equation. We then prove that
The solution β½(x, y, t) of the evolution equation gives the the proposed evolution equation solves an energy minimization processed image. 3 problem. This gives a new interpretation to histogram modifi-Partial differential equations may be obtained as the cation and contrast enhancement in general. This interpretation is completely formulated in the image domain, in contrast with gradient descent flow of a variational problem. Since a classical techniques for histogram modification which are forlarge number of problems can be formulated in a variamulated in a probabilistic domain. From this, new algorithms tional framework, a variety of PDEs can be found in the for contrast enhancement, including, for example, image and literature. What is relatively new as a major research area perception models, can be derived. Based on the energy formuis the explicit use of differential equations. 4 That is, the lation and its corresponding differential form, we show that evolution equation is derived directly, and not just as the the proposed histogram modification algorithm can be comgradient descent of an energy minimization problem. The bined with image regularization schemes. This allows us to energy minimization interpretation many times is given a perform simulations contrast enhancement and denoising, avoiding common noise sharpening effects in classical schemes. posteriori, as in this paper. Theoretical results regarding the existence of solutions to the Most of the explicit PDEs for image processing were proposed equations are presented.