Abstract
The continuous Lambertian shape from shading is studied using a PDE approach in terms of Hamilton–Jacobi equations. The latter will then be characterized by a maximization problem. In this paper we show the convergence of discretization and propose to use the well-known Chambolle–Pock primal-dual algorithm to solve numerically the shape from shading problem. The saddle-point structure of the problem makes the Chambolle–Pock algorithm suitable to approximate solutions of the discretized problems.
Highlights
Shape from Shading (SfS) consists in reconstructing the 3D shape of an object from its given 2D image brightness
Where I(x1, x2) is the brightness greylevel measured in the image at point (x1, x2); R(n(x1, x2)) is the reflectance map and n(x1, x2) is the unit normal at point (x1, x2, u(x1, x2)) given by
The Chambolle–Pock algorithm is easy to implement on Matlab which allows working on images contrary to the augmented Lagrangian approach which was implemented using FreeFem++ to solve linear PDEs
Summary
Shape from Shading (SfS) consists in reconstructing the 3D shape of an object from its given 2D image brightness. The shape of a surface u(x1, x2) is related to the image brightness I(x1, x2) by the Horn image irradiance equation [27]: R(n(x1, x2)) = I(x1, x2),. Where I(x1, x2) is the brightness greylevel measured in the image at point (x1, x2); R(n(x1, x2)) is the reflectance map and n(x1, x2) is the unit normal at point (x1, x2, u(x1, x2)) given by. Hamilton–Jacobi equation, Shape-from-Shading, primal-dual algorithm, numerical analysis.
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