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Published byCarson Wheatcroft Modified about 1 year ago

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L1 sparse reconstruction of sharp point set surfaces HAIM AVRON, ANDREI SHARF, CHEN GREIF and DANIEL COHEN-OR

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Index 3d surface reconstruction Moving Least squares Moving away from least squares [l1 sparse recon] Reconstruction model Re-weighted l1 Results and discussions

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3D surface reconstruction

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Moving least squares Input: Dense set of sample points that lie near a closed surface F with approximate surface normals. [in practice the normals are obtained by local least squares fitting of a plane to the sample points in a neighborhood ] Output : Generate a surface passing near the sample points. How does one do that : Linear point function that represents the local shape of the surface near point s. Combine these by a weighted average to produce a 3D function {I}, the surface is the zero implicit surface set of I. How good is it ? How close Is the function I to the signed distance function.

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2D -> 1D

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Total variation The l1-sparsity paradigm has been applied successfully to image denoising and de- blurring using Total Variation (TV) methods [Rudin 1987; Rudin et al. 1992; Chan et al. 2001; Levin et al. 2007] Total variation utilizes the sparsity in variation of gradients in an image. Dij is the discrete gradient operator, u is the scalar value The corresponding term for gradient in a mesh is the normal of the simplex (triangle)

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Reconstruction model Error term : Smooth surfaces have smoothly varying normals Penalty function (error) defined on the normals Total curvature Quadratic ; instead use Pair wise normal difference l2 norm Pi and pj are adjacent points pairwise penalty

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Reweighted l1 Consists of solving a sequence of weighted l1 minimization problems. where the weights used for the next iteration are computed from the value of the current solution. Each iteration solves a convex optimization, The over all algorithm does not. [Enhancing Sparsity by Reweighteed l1 Minimiaztion, Candes 2008]

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Reweighted l1 What is the key difference between l1 and l0 ? Dependence on magnitude.

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Reweighted l1

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Geometric view error Minimize L2 –norm [sum of square errors] Minimize L1 norm [sum of differences] Minimize L0 norm [number of non zeros terms]

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2 steps

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Orientation reconstruction Orientation minimization consists of two terms Global l1 minimization of orientation (normal) distances. Constraining the solution to be close to the initial orientation.

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Orientation reconstruction ctd Orientation minimization consists of two terms Global l1 minimization of orientation (normal) distances. For a piece-wise smooth surface the set Is sparse … why ? Globally weighted l1 penalty function

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Orientation reconstruction ctd For a piece-wise smooth surface the set Is sparse Globally weighted l1 penalty function

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Orientation reconstruction Orientation minimization consists of two terms Global l1 minimization of orientation (normal) distances. Constraining the solution to be close to the initial orientation.

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Key idea !!!

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Geometric view error Minimize L2 –norm [sum of square errors] Minimize L1 norm [sum of differences] Minimize L0 norm [number of non zeros terms]

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Results and Discussions Advantages Global frame work Till now sharpness was a local concept Criticisms Slow In reality the convex optimization although there are readily available solutions is a slow process.

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Discussions A lot of room for improvement Can I express this as a different form ? Specially like the low rank and sparse error form we had before.

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