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.