A Method for Enforcing Integrability in Shape from Shading Algorithms
IEEE Transactions on Pattern Analysis and Machine Intelligence
Computer Vision, Graphics, and Image Processing
Shape from shading
A viscosity solutions approach to shape-from-shading
SIAM Journal on Numerical Analysis
Existence and uniqueness for shape from shading around critical points: theory and an algorithm
International Journal of Robotics Research
Numerical methods for stochastic control problems in continuous time
Numerical methods for stochastic control problems in continuous time
Tracking level sets by level sets: a method for solving the shape from shading problem
Computer Vision and Image Understanding
Regular Article: Some Remarks on Shape from Shading
Advances in Applied Mathematics
Shape from shading: level set propagation and viscosity solutions
International Journal of Computer Vision
Computer Vision and Image Understanding
Extending viscosity solutions to Eikonal equations with discontinuous spatial dependence
Nonlinear Analysis: Theory, Methods & Applications
SIAM Journal on Numerical Analysis
Shape from Shading and Viscosity Solutions
ECCV '02 Proceedings of the 7th European Conference on Computer Vision-Part II
A Generic and Provably Convergent Shape-from-Shading Method for Orthographic and Pinhole Cameras
International Journal of Computer Vision
Shape-from-shading with discontinuous image brightness
Applied Numerical Mathematics - Numerical methods for viscosity solutions and applications
A fast marching formulation of perspective shape from shading under frontal illumination
Pattern Recognition Letters
Numerical methods for shape-from-shading: A new survey with benchmarks
Computer Vision and Image Understanding
Shape from shading using graph cuts
Pattern Recognition
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The height, u(x, y), of a continuous, Lambertian surface of known albedo (i.e., grayness) is related to u(x, y), information recoverable from a black and white flash photograph of the surface, by the partial differential equation \[\sqrt{u^2_x + u^2_y} - n\,{=}\,0.\] We review the notion of a unique viscosity solution for this equation when n is continuous and a recent unique extension of the viscosity solution when n is discontinuous. We prove convergence to this extension for a wide class of the numerical algorithms that converge when n is continuous. After discussing the properties of the extension and the order of error in the algorithms simulating the extension, we point out warning signs which, when observed in the numerical solution, usually indicate that the surface is not continuous or that the viscosity solution or its extension does not correspond to the actual surface. Finally, we discuss a method that, in some of these cases, allows us to correct the simulation and recover the actual surface again.