Integrating surface normal vectors using fast marching method
Jeffrey Ho,Jongwoo Lim,Ming-Hsuan Yang,David J. Kriegman +3 more
- 07 May 2006
- pp 239-250
TL;DR: In this article, the authors proposed a fast and efficient method for computing the depth values from surface normal vectors based on solving the Eikonal equation using Fast Marching Method, which is very easy to implement and demonstrate experimentally that, with insignificant loss in precision, their method is considerably faster than the usual optimization method that uses conjugate gradient to minimize an error function.
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Abstract: Integration of surface normal vectors is a vital component in many shape reconstruction algorithms that require integrating surface normals to produce their final outputs, the depth values. In this paper, we introduce a fast and efficient method for computing the depth values from surface normal vectors. The method is based on solving the Eikonal equation using Fast Marching Method. We introduce two ideas. First, while it is not possible to solve for the depths Z directly using Fast Marching Method, we solve the Eikonal equation for a function W of the form W = Z + λf. With appropriately chosen values for λ, we can ensure that the Eikonal equation for W can be solved using Fast Marching Method. Second, we solve for W in two stages with two different λ values, first in a small neighborhood of the given initial point with large λ, and then for the rest of the domain with a smaller λ. This step is needed because of the finite machine precision and rounding-off errors. The proposed method is very easy to implement, and we demonstrate experimentally that, with insignificant loss in precision, our method is considerably faster than the usual optimization method that uses conjugate gradient to minimize an error function.
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Figures
Fig. 6. From Left to Right: Reconstruction results for the function Z = sin(2π(x2 + y2)) + 3 using λ = 4, 30 and 100, respectively. 
Fig. 7. From Left to Right: Reconstruction results for Z = e−x 2−y2 +10 using λ = 0, 8 and 100, respectively. 
Fig. 4. From Left to Right: Plots of the mean, median and standard deviation of the relative errors of the reconstruction results for the Monkey Saddle with λ ranging from 0 to 100. 
Fig. 5. From Left to Right: Reconstruction results for the Monkey Saddle using λ = 0, 12 and 100, respectively. 
Fig. 1. Left: A function with two local minimums that cannot be recovered using Fast Marching Method. Right: We solve for W with two different λ values in two complementary regions. 
Fig. 8. From Left to Right: Three views of the reconstruction result of one individual in the Yale Face Database B.
Citations
Normal Integration: A Survey
TL;DR: In this article, the authors present a survey of the most important properties that one may expect from a normal integration method, based on a thorough study of two pioneering works by Horn and rooks (Comput Vis Graph Image Process 33(2): 174−208, 1986) and Frankot and Chellappa (IEEE Trans Pattern Anal Mach Intell 10(4): 439-451, 1988).
Patent
Various methods and apparatuses for achieving augmented reality
Nuno Moura e Silva Cruces,Ivan de Almeida Soares Franco,Nuno Ricardo Sequeira Cardoso,André Rui Soares Pereira de Almeida,João Frazão,Gonçalo Lopes +5 more
- 19 Apr 2011
TL;DR: In this article, a matching of the resolution of the image data and actual measured depth data as well as a linking formed between the image and depth data for each pixel representing the real-world object in the scene is described.
128
•Journal Article
Inspection of specular and partially specular surfaces
TL;DR: In this paper, the main principle of deflectometric surface acquisition is to use a highly controllable environment, where a well-defined pattern is presented via the specular reflecting surface.
Fast and accurate surface normal integration on non-rectangular domains
TL;DR: A classic approach for surface normal integration with modern computational techniques is uniting, using an iterative Krylov subspace solver as a core step in tackling the task and is shown on real-world photometric stereo datasets that the developed numerical framework is flexible enough to tackle modern computer vision applications.
•Posted Content
Normal Integration: A Survey
TL;DR: The most important properties that one may expect from a normal integration method are selected, based on a thorough study of two pioneering works by Horn and rooks and Frankot and Chellappa.
29
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