ℓ ₁ -Sparse reconstruction of sharp point set surfaces

Abstract

We introduce an 1 -sparse method for the reconstruction of a piecewise smooth point set surface. The technique is motivated by recent advancements in sparse signal reconstruction. The assumption underlying our work is that common objects, even geometrically complex ones, can typically be characterized by a rather small number of features. This, in turn, naturally lends itself to incorporating the powerful notion of sparsity into the model. The sparse reconstruction principle gives rise to a reconstructed point set surface that consists mainly of smooth modes, with the residual of the objective function strongly concentrated near sharp features. Our technique is capable of recovering orientation and positions of highly noisy point sets. The global nature of the optimization yields a sparse solution and avoids local minima. Using an interior-point log-barrier solver with a customized preconditioning scheme, the solver for the corresponding convex optimization problem is competitive and the results are of high quality.

Continuous projection for fast L 1 reconstruction

“…These methods are theoretically able to faithfully reconstruct a surface as long as there are less than 50 percent outliers (the breakdown point). Other methods perform a global minimization on the orientation of the input normals, using ideas from compressed sensing and sparse signal recovery [Avron et al 2010]. However, while these algorithms are of high quality, due to their global nature they are often extremely slow, do not scale well to large data or require special assumptions on the input data (e.g., data with a few planar elements).…”

“…Basic approaches use some form of local plane fitting [Hoppe et al 1992], but noisy point sets with outliers and possible sharp features require more robust normal estimations. Approaches range from inscribing empty balls [Dey and Sun 2006], smoothing and outlier removal [Huang et al 2009], global L1 norm optimization [Avron et al 2010] to randomized Hough transforms [Boulch and Marlet 2012]. Robust statistics-based methods have been shown to achieve superior results in the presence of outliers [Kalogerakis et al 2007;Li et al 2010;Zheng et al 2010;Oztireli et al 2009].…”

The possibilities of compressed-sensing-based Kirchhoff prestack migration

“…Although LSM suppresses the migration artifacts and helps focus the migrated image, this L2-norm minimization tends to produce a smeared version of the true reflectivity distribution (Avron et al, 2010;Yu et al, 2011). Thus, a major limitation of LSM is that S114 Aldawood et al…”

Voronoi-based feature curves extraction for sampled singular surfaces

“…Feature points along intersections of patches which have different normals are identified as feature points. Avron et al [15] introduce a two step reconstruction algorithm. In the first step, orientation is corrected as a feature aware L 1 -norm minimization.…”

“…We want to keep those points whose distance to the respective barycenters are small, but no two of them should be mutually close (steps [11][12][13][14][15]. This is achieved by sorting the points in ascending order of d p , and then selecting only those points that are at least d p distance away from the rest of the selected points so far.…”