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<p><b>Abstract</b>—In this paper, we present a new method of shape from shading by using radial basis functions to parameterize the object depth. The radial basis functions are deformed by adjusting their centers, widths, and weights such that the intensity errors are minimized. The initial centers and widths are arranged hierarchically to speed up convergence and to stabilize the solution. Although the smoothness constraint is used, it can be eventually dropped out without causing instabilities in the solution. An important feature of our parametric shape-from-shading method is that it offers a unified framework for integration of multiple sensory information. We show that knowledge about surface depth and/or surface normals anywhere in the image can be easily incorporated into the shape from shading process. It is further demonstrated that even qualitative knowledge can be used in shape from shading to improve 3D reconstruction. Experimental comparisons of our method with several existing ones are made by using both synthetic and real images. Results show that our solution is more accurate than the others.</p>
Shape from shading, radial basis functions, hierarchical structure, depth constraints, normals constraints, qualitative knowledge, stochastic gradient method.

G. Wei and G. Hirzinger, "Parametric Shape-from-Shading by Radial Basis Functions," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 19, no. , pp. 353-365, 1997.
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