3D Point Cloud Super-Resolution via Graph Total Variation on Surface Normals

Abstract

Point cloud is a collection of 3D coordinates that are discrete geometric samples of an object's 2D surfaces. Using a low-cost 3D scanner to acquire data means that point clouds are often in lower resolution than desired for rendering on high-resolution displays. Building on recent advances in graph signal processing, we design a local algorithm for 3D point cloud super-resolution (SR). First, we initialize new points at centroids of local triangles formed using the low-resolution point cloud, and connect all points using a k-nearestneighbor graph. Then, to establish a linear relationship between surface normals and 3D point coordinates, we perform bipartite graph approximation to divide all nodes into two disjoint sets, which are optimized alternately until convergence. For each node set, to promote piecewise smooth (PWS) 2D surfaces, we design a graph total variation (GTV) objective for nearby surface normals, under the constraint that coordinates of the original points are preserved. We pursue an augmented Lagrangian approach to tackle the optimization, and solve the unconstrained equivalent using the alternating method of multipliers (ADMM). Extensive experiments show that our proposed point cloud SR algorithm outperforms competing schemes objectively and subjectively for a large variety of point clouds.

Point Cloud Sampling via Graph Balancing and Gershgorin Disc Alignment

“…To mathematically define an objective for PC sub-sampling, we first assume that the sub-sampled PC will be subsequently superresolved from m 3D point samples back to n points, where m < n, via a simple PC super-resolution (SR) method (a simplified variant of [28]). We chose this SR method because the resulting SR problem is an unconstrained quadratic programming (QP) optimization ( 9), with a system of linear equations as solution.…”

“…For simplicity, we assume here that the original n 3D points are used to construct an n-node graph. In practice, to construct an n-node graph from m 3D point samples, n − m interior 3D points can be interpolated from m samples and then subsequently refined, as done in [28], [47]. In particular, we construct an undirected positive graph G p = (V p , E p ) composed of a node set V p of size n (each node represents a 3D point) and an edge set E p specified by (i, j, w i,j ), where i = j, i, j ∈ V and w i,j ∈ R + with no self-loops.…”

“…We review basic concepts in GCT, GSP and signed graph balancing in Section 3. We describe our PC SR algorithm, a simplified variant of [28], in Section 4. We present our PC sub-sampling strategy in Section 5.…”

Fractional Super-Resolution of Voxelized Point Clouds

“…Hamdi-Cherif et al [15] combined local descriptors by their similarities for PC SR, however this approach required several PCs, normals calculations, and assumed surface smoothness. Dinesh et al [16], [17] used Delaunay triangulation in the LR PC and optimize an L 1 -norm graph-total-variation (GTV) cost function for neighborhood surface normals. It promotes piecewise smoothness in reconstructed 2D surfaces, under the constraint that the LR coordinates are preserved.…”

Multiscale Anisotropic Harmonic Filters on non Euclidean domains

“…On the other hand, Signals (defined) on Graphs (SoGs) provide a powerful and effective model for data defined on the aforementioned supports [3]. Indeed, Graph Signal Processing (GSP) has found application in all the branches of signal processing on volumetric domains, for the purpose of dual description and filtering [4], enhancement and restoration [5], compression [6], and transmission [7].…”