Variational numerical methods in image processing and analysis

Karol Mikula

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

In many applications computers analyse images or image sequences which are often contaminated by noise, and their quality can be poor (e.g. in medical imaging). We discuss how nonlinear partial differential equations (PDEs) can be used to automatically produce an image of much higher quality, enhance its sharpness, filter out the noise, segment and extract shapes, etc. We also give a general, robust and efficient approach for numerical solutions of partial differential equations (PDEs) arising in image processing and computer vision. The models are based on the well-known Perona-Malik image selective smoothing equation and on geometrical equations of mean curvature flow type. Since the images are given on discrete grids, PDEs are discretized by variational techniques, namely by the semi-implicit finite element, finite volume and complementary volume methods in order to get fast and stable solutions. Since such methods are based on principles like minimization of energy (finite element method) or conservation laws (finite and complemetary volume methods), they allow clear and physically meaningful derivation of difference equations which are local and easy to implement. The variational methods are combined with semi-implicit discretization in scale, which gives favourable stability and efficiency properties of computations. Convergence of the schemes to variational solutions of these strongly nonlinear problems and the extension of the methods to adaptive scheme strategies improving computational efficiency are also discussed. Computational results with artificial and real 2D, 3D images and image sequences are presented.

Some computational results:

Processing of image sequences - dynamic filtering and visualization of the left ventricular volume:


Segmentation of 2D slice of the left ventricle:


Extraction of the Kanizsa triangle (subjective contour):


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