A graph-spectral approach to shape-from-shading

In this paper, we explore how graph-spectral methods can be used to develop a new shape-from-shading algorithm. We characterize the field of surface normals using a weight matrix whose elements are computed from the sectional curvature between different image locations and penalize large changes in surface normal direction. Modeling the blocks of the weight matrix as distinct surface patches, we use a graph seriation method to find a surface integration path that maximizes the sum of curvature-dependent weights and that can be used for the purposes of height reconstruction. To smooth the reconstructed surface, we fit quadrics to the height data for each patch. The smoothed surface normal directions are updated ensuring compliance with Lambert's law. The processes of height recovery and surface normal adjustment are interleaved and iterated until a stable surface is obtained. We provide results on synthetic and real-world imagery.

[1]  Richard S. Varga,et al.  Matrix Iterative Analysis , 2000, The Mathematical Gazette.

[2]  Katsushi Ikeuchi,et al.  Numerical Shape from Shading and Occluding Boundaries , 1981, Artif. Intell..

[3]  Aaron F. Bobick,et al.  The direct computation of height from shading , 1991, Proceedings. 1991 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[4]  David A. Forsyth,et al.  Mutual illumination , 1989, Proceedings CVPR '89: IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[5]  Berthold K. P. Horn.  Height and gradient from shading , 1989, International Journal of Computer Vision.

[6]  Alfred M. Bruckstein,et al.  Shape from shading: Level set propagation and viscosity solutions , 1995, International Journal of Computer Vision.

[7]  Shree K. Nayar,et al.  Generalization of the Lambertian model and implications for machine vision , 1995, International Journal of Computer Vision.

[8]  E. Rouy,et al.  A viscosity solutions approach to shape-from-shading , 1992 .

[9]  Lingxiao Li,et al.  A line-integration based method for depth recovery from surface normals , 1988, Comput. Vis. Graph. Image Process..

[10]  Ping-Sing Tsai,et al.  Shape from Shading: A Survey , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[11]  Grahame B. Smith.  The Recovery of Surface Orientation from Image Irradiance , 1982 .

[12]  Berthold K. P. Horn.  Understanding Image Intensities , 1977, Artif. Intell..

[13]  Xavier Descombes,et al.  A Multiresolution Approach for Shape from Shading Coupling Deterministic and Stochastic Optimization , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[14]  Reinhard Klette,et al.  Theoretical Analysis of Finite Difference Algorithms for Linear Shape from Shading , 2001, CAIP.

[15]  Lawrence B. Wolff,et al.  Polarization-Based Material Classification from Specular Reflection , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[16]  Rama Chellappa,et al.  Estimation of Illuminant Direction, Albedo, and Shape from Shading , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[17]  P. Dupuis,et al.  Direct method for reconstructing shape from shading , 1991, Proceedings 1992 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[18]  H. Piaggio.  Differential Geometry of Curves and Surfaces , 1952, Nature.

[19]  Thomas M. Strat,et al.  A Numerical Method for Shape-From-Shading from a Single Image. , 1979 .

[20]  Jitendra Malik,et al.  Normalized cuts and image segmentation , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[21]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[22]  T. Poggio,et al.  Ill-Posed Problems and Regularization Analysis in Early Vision , 1984 .

[23]  K. Torrance,et al.  Theory for off-specular reflection from roughened surfaces , 1967 .

[24]  L. J. Krakauer.  COMPUTER ANALYSIS OF VISUAL PROPERTIES OF CURVED OBJECTS , 1971 .

[25]  Sang Uk Lee,et al.  Shape from shading using graph cuts , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[26]  Adrian G. Bors,et al.  Terrain Analysis Using Radar Shape-from-Shading , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[27]  Demetri Terzopoulos,et al.  The Role of Constraints and Discontinuities in Visible-Surface Reconstruction , 1983, IJCAI.

[28]  B K Horn,et al.  Calculating the reflectance map. , 1979, Applied optics.

[29]  Ikeuchi,et al.  Constructing a Depth Map from Images , 1983 .

[30]  Dr. M. G. Worster.  Methods of Mathematical Physics , 1947, Nature.

[31]  Berthold K. P. Horn,et al.  Shape from shading , 1989 .

[32]  Michael J. Brooks,et al.  Surface-Normals from Closed Paths , 1979, IJCAI.

[33]  J. Summers,et al.  “What you see is what you get” , 2005 .

[34]  Bruce Hendrickson,et al.  A Spectral Algorithm for Seriation and the Consecutive Ones Problem , 1999, SIAM J. Comput..

[35]  Andrea J. van Doorn,et al.  Surface shape and curvature scales , 1992, Image Vis. Comput..

[36]  Berthold K. P. Horn.  SHAPE FROM SHADING: A METHOD FOR OBTAINING THE SHAPE OF A SMOOTH OPAQUE OBJECT FROM ONE VIEW , 1970 .

[37]  Alex Pentland,et al.  A simple algorithm for shape from shading , 1992, Proceedings 1992 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[38]  Edwin R. Hancock,et al.  Lambertian Correction for Rough and Specular Surfaces , 2003, VVG.

[39]  Edwin R. Hancock,et al.  New Constraints on Data-Closeness and Needle Map Consistency for Shape-from-Shading , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[40]  T. Rindfleisch.  Photometric method for lunar topography. , 1966 .

[41]  László Lovász,et al.  Random Walks on Graphs: A Survey , 1993 .

[42]  Alex Pentland.  Linear shape from shading , 2004, International Journal of Computer Vision.

[43]  Michael J. Brooks,et al.  The variational approach to shape from shading , 1986, Comput. Vis. Graph. Image Process..

[44]  Rama Chellappa,et al.  A Method for Enforcing Integrability in Shape from Shading Algorithms , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

[45]  Robert J. Woodham,et al.  Analysing Images of Curved Surfaces , 1981, Artif. Intell..

[46]  Edwin R. Hancock,et al.  Needle map recovery using robust regularizers , 1999, Image Vis. Comput..

[47]  Dcructri l’cropoulos.  EFFICIENT MULTIRESOLUTION ALGORITHMS FOR COMPUTING LIGHTNESS, SHAPE-FROM-SHADING, AND OPTICAL FLOW , 1999 .

[48]  Edwin R. Hancock,et al.  A probabilistic framework for specular shape-from-shading , 2002, Object recognition supported by user interaction for service robots.