Variational numerical methods in image processing and analysis

Zuzana Kriva

Department of Mathematics, Slovak University of Technology, Radlinskeho 11, 813 68 Bratislava, Slovakia

Basic tasks of image processing and computer vision - such as selective image smoothing, enhancement, restoration, segmentation, edge detection, shape analysis, optic flow computations have been recetnly modeled by the application of evolutionary PDEs to the initial data. The well known example is the nonlinear diffusion equation of Perona-Malik type, which denoises the data in such way, that edges stay preserved. One of well established variational techniques is the finite volume method (FVM), in which the approximation of the solution is assumed to be piecewise constant on control volumes (i.e. elements of the computational grid, which directly correspond to the pixel structure of the image). Recently, the finite volume method (FVM)has been widely used in computational sciences and engineering, since it is based on physical principles as conservation laws, is local and easy to implement.

For the Perona-Malik equation, we propose the coarsening strategy for the finite volume computational method given by K.Mikula and N. Ramarosy (Numer. Math.89,2001,561-590) applied to the regular computational grid. The solution tends to be more flat with the increasing scale in large regions of the image and we can improve the efficiency of the method considerably by using adaptivity, i.e. choosing nonuniform grids with less number of finite volumes. In other words, we can say, that we are introducing a simultaneous compression of the images, while they undergo the smoothing evolution. This access reduces computational effort, because the coarsening of th ecomputational grid reduces the number of unknowns in the linear systems to be solved at discrete scale steps of the method.

Some computational results:

The noisy exlibris from the lycean library is denoised by adaptive FVM algorithm for the Perona-Malik equation, where the diffusion is restricted to certain intensities.
Picture1 Picture2
Example of denoising by an adaptive algorithm in 3D.
a) shows the original data, b) shows the data perturbed by noise, c) and data shows results of smoothing with slices of adaptive grid in various scale steps.