Region Adaptive Graph Fourier Transform for 3D Point Clouds

Abstract

We introduce the Region Adaptive Graph Fourier Transform (RA-GFT) for compression of 3D point cloud attributes. We assume the points are organized by a family of nested partitions represented by a tree. The RA-GFT is a multiresolution transform, formed by combining spatially localized block transforms. At each resolution level, attributes are processed in clusters by a set of block transforms. Each block transform produces a single approximation (DC) coefficient, and various detail (AC) coefficients. The DC coefficients are promoted up the tree to the next (lower resolution) level, where the process can be repeated until reaching the root. Since clusters may have a different numbers of points, each block transform must incorporate the relative importance of each coefficient. For this, we introduce the Q-normalized graph Laplacian, and propose using its eigenvectors as the block transform. The RA-GFT outperforms the Region Adaptive Haar Transform (RAHT) by up to 2.5 dB, with a small complexity overhead.

Multi-Resolution Intra-Predictive Coding Of 3d Point Cloud Attributes

“…Several orthonormal transforms for 3DPC attributes can be described this way, including the block based graph Fourier transform [20], RAHT [4] and RAGFT [13]. Since RAGFT is a composition of spatially localized block transforms, there may be additional redundancy between transformed coefficients, similar to what is observed for the RAHT [12].…”

“…While forming transform coefficients (2), many transforms [4,13], either explicitely or implicitely, produce sets of point coordinates at various resolutions (e.g., by down-sampling), thus for each resolution , we have a point cloud (V , a ), where VL = V and aL = a.…”

“…In the RAGFT [13], each transform coefficient has an importance weight that depends on the size (number of points) of the region of the point cloud they represent. The ith point of the point cloud of resolution is denoted by v i, .…”

“…While this approach to construct a higher resolution point cloud is suitable for the RAHT, since it is formed by a composition of 2 × 2 orthogonal transforms, it fails to take into account the more complex geometry of the higher resolution point cloud. This becomes even more important when considering larger block transforms with more points used by the region adaptive graph Fourier transform (RAGFT) [13], as shown in Figure 1.…”

Spectral Folding And Two-Channel Filter-Banks On Arbitrary Graphs

“…These are major limitations since the graph structure is rarely bipartite (which dictates the down-sampling operator), whereas the graph variation operator (or graph shift) is determined by the application. To overcome these issues, we propose a new theory that can be applied to: 1) arbitrary graphs, 2) any vertex partition for down-sampling, and 3) positive semi-definite variation operators (see [15,16,17,18] for examples). The proposed filter-banks also satisfy (i), (ii), and either (iii) or (iv), as with BFBs.…”