L1 sparse reconstruction of sharp point set surfaces
HAIM AVRON, ANDREI SHARF, CHEN GREIF and DANIEL COHEN-OR
Index
3d surface reconstruction Moving Least squares Moving away from least squares [l1 sparse recon]
Reconstruction model Re-weighted l1
Results and discussions
3D surface reconstruction
Moving least squaresInput: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.
2D -> 1D
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)
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
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]
Reweighted l1
What is the key difference between l1 and l0 ?
Dependence on magnitude.
Reweighted l1
Geometric view
error
Minimize L2 –norm [sum of square errors]
Minimize L1 norm [sum of differences]
Minimize L0 norm [number of non zeros terms]
2 steps
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.
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
Orientation reconstruction ctd
For a piece-wise smooth surface the set
Is sparse
Globally weighted l1 penalty function
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.
Key idea !!!
Geometric view
error
Minimize L2 –norm [sum of square errors]
Minimize L1 norm [sum of differences]
Minimize L0 norm [number of non zeros terms]
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.
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.
