The USC Andrew and Erna Viterbi School of Engineering USC Signal and Image Processing Institute USC Ming Hsieh Department of Electrical Engineering University of Southern California

Technical Report USC-SIPI-105

“A Method for Enforcing Integrability in Shape from Shading Algorithms”

by Robert Frankot and Rama Chellappa

June 1987

Several recently developed techniques for reconstructing surface shape from shading information estimate surface slopes without ensuring that they are integrable. This paper presents an approach for enforcing integrability, a particular implementation of the approach, an example of its application is extending an existing shape from shading algorithm and experimental results showing the improvement that results from enforcing integrability.

A possibly nonintegrable estimate of surface slopes is represented by a finite set of basis functions, and integrability is enforced by calculating the orthogonal projection onto a vector subspace spanning the set of integrable slopes. This projection maps closed convex sets onto closed convex sets and, hence, is attractive as a constraint. The special case of Fourier basis functions is formulated. This provides an intuitive frequency domain interpretation of shape from shading, a computationally efficient implementation using the FFT, and a convenient method for introducing low resolution information into the shape from shading solution. Reconstruction of surface height by integrating surface slope estimates is obtained as a byproduct of the integrability constraint. Other possible applications of this method to computer vision problems such as shape from texture and surface reconstruction from synthetic aperture radar imagery are discussed.

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