THE EVOLUTION OF MEAN CURVATURE IN IMAGE FILTERING

Adel I. El-Fallah - Gary E. Ford

Abstract:

A new formulation for inhomogeneous image diffusion is presented in which the image is regarded as a surface in 3-space. The evolution of this surface under diffusion is analyzed by classical methods of differential geometry. A nonlinear filtering theory is introduced in which only the divergence of the direction of the surface gradient is averaged. This averaging preserves edges and lines, as their direction is non-divergent, while noise is averaged since it does not have non-divergent consistency. Our approach achieves this objective by evolving the surface at a speed proportional to mean curvature leading to the minimization of the surface area and the imposition of regularity everywhere. Furthermore, we introduce a new filter that renders corners, as well as edges, invariant to the diffusion process. Experiments demonstrating the adequacy of this new theory are presented.

• INTRODUCTION
• THE DIFFUSION MODEL
• SURFACE REPRESENTATION
• NORMAL VECTOR FIELD
• MAIN THEOREM
• PROOF
• THE EDGE
• GAUSSAIN VERSUS MCD
• MCD EDGE DYNAMICS.1.
• MCD EDGE DYNAMICS.2.
• SURFACE SMOOTHING
• RECOVERY & ENHANCEMENT
• DEVELOPED ENHANCEMENT
• MEAN CURV. DYNAMICS I
• MEAN CURV. DYNAMICS II
• MEAN CURV. DYNAMICS III
• MEAN CURV. DYNAMICS IV
• MEAN CURV. DYNAMICS V
• THE CORNER
• EXPERIMENT
• CORNER'S FEATURE
• CORNER PRESERVING FILTER
• EXPERIMENT
• IMAGE FILTERING
• CONCLUSIONS