Global Shape from Shading

direction of the camera, the local shape recovered assuming the extremum is a minimum is the reflection of the A global shape from shading algorithm which develops a technique to merge local shape from shading results obtained shape recovered assuming the extremum is a maximum, around singular points into the complete shape using the mounand therefore the same saddle point will be found. This taineers theorem was recently presented in Kimmel and is not the case when the illumination comes from other Bruckstein (1995). In this comment, we enhance this result by directions and the closest saddle point may be different. proving the completeness and uniqueness obtained by the global In both cases the algorithm is run on all possible assignshape from shading algorithm.  1996 Academic Press, Inc. ments of extrema as minima or maxima. The global algorithm merges the results of the local algorithm by merging two local surfaces which have the In [2], the following global shape from shading algorithm same closest saddle point. After two surfaces have been was presented. A local shape from shading algorithm is merged, the local algorithm extends the merged surface used to recover the shape around each singular point. The until another saddle point is met. This process continues algorithm inspects the behavior of iso-height contours until all local surfaces have been merged together and the around each singular point. The contours are monitored global shape has been recovered. from the singular point ‘‘outward’’ until another singular We assume that on the image boundary the downward point is encountered. An underlying assumption is that direction is everywhere outward (or inward). Under this the shape to be recovered is a Morse function. For such condition we will prove that only when the singular points functions, according to the mountaineers theorem [1, 3], are correctly classified will the algorithm be able to comthe number of extrema located within a closed equal height plete the recovery of the surface, and that when the singular contour of a smooth surface exceeds by one the number points are correctly classified there exists a unique solution of saddle points within that contour. Therefore, when (modulo the fact that minima and maxima cannot be differtracking iso-height contours that start as a small circle entiated). Thus the algorithm is run on different classificaaround an extremum, the first singular point that the extions of the singular points and when it is able to merge tending contours meet must be a saddle point. This saddle all the subsurfaces, the correct surface has been found. point is the singular point whose height is closest to the The first step of the proof will show that when the singuheight of the extremum. When the illumination is in the lar points are correctly classified, the algorithm recovers a unique shape. In the second step of the proof we will show that when the singular points are misclassified the 1 E-mail: [email protected]. algorithm cannot find a way to merge all the local surfaces. 2 E-mail: [email protected]. 3 E-mail: [email protected]. The results of the global algorithm can be described as