SOME MATHEMATICAL PROBLEMS IN COMPUTER VISION

Several interesting mathematical problems arising in computer vision are discussed. Computer vision deals with image understanding at various levels. At the low level, it addresses issues like segmentation, edge detection, planar shape recognition and analysis. Classical results on differential invariants associated to planar curves are relevant to planar object recognition under partial occlusion, and recent results concerning the evolution of closed planar shapes under curvature controlled diffusion have found applications in shape decomposition and analysis. At higher levels, computer vision problems deal with attempts to invert imaging projections and shading processes toward depth recovery, spatial shape recognition and motion analysis. In this context. the recovery of depth from shaded images of objects with smooth, diffuse surfaces require the solution of nonlinear partial differential equations. Here results on differential equations, as well as interesting results from low-dimensional topology and differential geometry are the necessary tools of the trade. We are still far from being able to equip our computers with brains capable to analyze and understand the images that can easily be acquired with camera-eyes; however the research effort in this area often calls for both classical and recent mathematical results.