Improved methods of estimating shape from shading using the light source coordinate system
Artificial Intelligence
Computer Vision, Graphics, and Image Processing
Shape from shading
Height and gradient from shading
International Journal of Computer Vision
Uniqueness in shape from shading
International Journal of Computer Vision
Shape from shading as a partially well-constrained problem
CVGIP: Image Understanding
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
Tracking level sets by level sets: a method for solving the shape from shading problem
Computer Vision and Image Understanding
Shape from shading: level set propagation and viscosity solutions
International Journal of Computer Vision
Computer Vision and Image Understanding
Parametric Shape-from-Shading by Radial Basis Functions
IEEE Transactions on Pattern Analysis and Machine Intelligence
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In the theory of shape from shading, behaviours of the local solution around a critical point of the image play an important role. This paper shows that the second derivatives of the object surface can be locally determined at these image critical points. Closed form expressions of the surface second derivatives in terms of the second derivatives of the image brightness and of the reflectance map are shown. They are derived as follows: By differentiating the image irradiance equation twice at an image critical point, a set of polynomial equations is obtained that contains the second derivatives of the surface, of the image brightness and of the reflectance map. Regarding these equations as simultaneous equations for unknown surface second derivatives, they are algebraically solved and their explicit expressions are derived. Such a derivation is possible only at image critical points and is impossible at any other image point. The applicability of the derived expressions to noisy images is tested using synthetic images.