A study of fast marching method for shape from shading under orthographic and perspective projection

Abstract

A great feat of vision is the extraction of a description of the world from pictures or sequences of pictures. People would like to organize their photographs and make 3D models of the world around them. Techniques of computer vision are able to solve these problems. In this thesis, applying the Fast Marching method to solve the problem of shape from shading (SFS) under orthographic and perspective projection is studied. SFS is one of the best-known techniques for the shape reconstruction in computer vision. It uses the pattern of lights and shades but not shadows in an image to infer the shape of the surfaces in view. To simplify most vision problems, orthographic projection is usually assumed. It presumes that the viewer and the light sources are far enough away from the objects being viewed. However, perspective projection is undoubtedly a more appropriate viewing system that people use in reality. The optimization approach is commonly used to solve the shading problem. This includes the Newton's method which takes care of function derivatives and the Nelder and Mead's method which solely minimizes the function cost. In the case of frontal illumination, experimental results of these two methods under the perspective projection model are reported. It is found that a weakness of these methods is the non-convergence problem. Recently, a new propagation approach: the Fast Marching method is successful when applied to the directly illuminated shading model under orthographic projection. Building on this, a novel shading model under perspective projection is proposed. Both synthetic and real testing images are used to justify the performance. Experimental results show that the proposed method can successfully converge to the correct solution. It overcomes the weakness of the Newton's method and the Nelder and Mead's method. A simple method involving a ray casting algorithm is introduced for generating accurate perspective images for the experiments. Finally, the shading model under the orthographic model is extended to oblique illumination and experimental results are reported. It is found that the fast marching method is workable for both frontal as well as non-frontal illumination except at the shadow region. It is predicted that the shading model under perspective projection may also be successful.