Order Elliptic Equations In Image Processing

This paper mainly introduces subelliptic equations’ application in image processing. Subelliptic equation is a kind of second order degenerate elliptic equation. This kind of equation and its related sub-Riemannian geometry in recent decades have caused a lot of attention and it is one of the hot research topics in partial differential equation. The image processing technology based on the subelliptic equation and sub-Riemannian geometry in recent years begins to be a research topic. Current research shows that this technique in image inpainting with occlusion has unique advantages. The thesis is summarized as follows:1. In this paper, sub-Riemannian diffusion equation and concentration algorithm are proposed. Sub-Riemannian diffusion equation restores the image through the known information in the image region, spreading to each point on the damaged region, which is related to sub-Riemannian geometry’s connectivity property. So sub-Riemannian geometry’s connectivity property is very important. In this paper, connectivity property in sub-Riemannian geometry is discussed and concentration algorithm is proposed. Concentration algorithm can make the restored image smooth and multiple concentration algorithm based on concentration algorithm in repairing co-existence of occluded image has unique advantages. In the lifted image with occlusion, minimal surface is calculated.2. By comparison with the classic PDE method, some advantages of the subelliptic equations method are revealed.3. Subelliptic equations’ numerical algorithms and the experimental results in image inpainting are studied. With the help of some principles in image processing using subelliptic equations, we get numerical algorithms of repairing model of subelliptic equations. Finally we obtain some significant experimental results.