Shape-From-Shading Using the Heat Equation

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 norma...

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Bibliographic Details
Published in:IEEE transactions on image processing 2007-01, Vol.16 (1), p.7-21
Main Authors: Robles-Kelly, A., Hancock, E.R.
Format: Article
Language:English
Subjects:
Algorithms
Applied sciences
Azimuthal component
Brightness
Computing time
Equations
Exact sciences and technology
Heat equations
Heat recovery
Image analysis
Image Enhancement - methods
Image Interpretation, Computer-Assisted - methods
Image processing
Imaging, Three-Dimensional - methods
Information Storage and Retrieval - methods
Information, signal and communications theory
Light sources
Lighting - methods
Mathematical analysis
Pattern recognition
Pattern Recognition, Automated - methods
Performance analysis
Photometry - methods
Preserves
Recovery
Scalars
Signal processing
Steady-state
Studies
Surface reconstruction
Telecommunications and information theory
Thermodynamics
Turning
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Summary: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
ISSN:1057-7149
1941-0042
DOI:10.1109/TIP.2006.884945