White Rose University Consortium logo
University of Leeds logo University of Sheffield logo York University logo

Shape-from-shading using the heat equation

Robles-Kelly, Antonio and Hancock, Edwin R. (orcid.org/0000-0003-4496-2028) (2007) Shape-from-shading using the heat equation. IEEE Transactions on Image Processing. pp. 7-21. ISSN 1057-7149

Text (hancocker13.pdf)

Download (3962Kb)


This paper offers two new directions to shape-from-shading, namely the use of the heat equation to smooth the field of surface normals and the recovery of surface height using a low-dimensional embedding. Turning our attention to the first of these contributions, we pose the problem of surface normal recovery as that of solving the steady state heat equation subject to the hard constraint that Lambert's law is satisfied. We perform our analysis on a plane perpendicular to the light source direction, where the z component of the surface normal is equal to the normalized image brightness. The x - y or azimuthal component of the surface normal is found by computing the gradient of a scalar field that evolves with time subject to the heat equation. We solve the heat equation for the scalar potential and, hence, recover the azimuthal component of the surface normal from the average image brightness, making use of a simple finite difference method. The second contribution is to pose the problem of recovering the surface height function as that of embedding the field of surface normals on a manifold so as to preserve the pattern of surface height differences and the lattice footprint of the surface normals. We experiment with the resulting method on a variety of real-world image data, where it produces qualitatively good reconstructed surfaces.

Item Type: Article
Copyright, Publisher and Additional Information: Copyright © 2007 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.
Institution: The University of York
Academic Units: The University of York > Computer Science (York)
Depositing User: Repository Officer
Date Deposited: 21 Feb 2007
Last Modified: 03 Apr 2016 11:31
Published Version: http://dx.doi.org/10.1109/TIP.2006.884945
Status: Published
Refereed: Yes
URI: http://eprints.whiterose.ac.uk/id/eprint/1984

Actions (repository staff only: login required)