Coding of Image Intra Prediction Residuals Using Symmetric Graphs

Abstract

The Discrete Cosine Transform (DCT) is widely deployed by modern image and video coding standards such as JPEG and H.26x. In most cases, the DCT is applied in a separable manner to rows and columns, which limits its ability to represent signals with diagonal orientation. As an alternative, non-separable transforms can represent signals with different orientations, but are significantly more computationally complex. To address this problem, in this paper we propose a set of non-separable Symmetry-Based Graph Fourier Transforms (SBGFTs), whose symmetric structures lead to a faster implementation. We study a practical image coding scenario that exploits the proposed SBGFTs, where for each intra predicted image residual block the optimal graph is chosen by solving a graph-based Rate-Distortion (R-D) problem. Experimental results indicate a coding efficiency higher than JPEG and JPEG2000.

Symmetry-Based Graph Fourier Transforms: Are They Optimal For Image Compression?

“…The work in [9] proposed a set of graphs with symmetry properties that lead to efficient representations of both image and residual blocks. The resulting SBGFTs were enclosed in place of the DCT in the intra coding phase of the HEVC standard, obtaining a significant performance improvement in terms of RD performance.…”

“…Also, different approaches propose to build graph-based separable transforms specifically for video coding [7,8]. In our recent work [9], we proposed to replace the DCT with a set of multiple Symmetry-Based Graph Fourier Transforms (SBGFTs) to encode the intra prediction residuals generated in the HECV intra coding. This approach is based on Graph Signal Processing (GSP) [10,11], in which the graphs are adopted as models that represent the correlation among data samples by determining edge weights between nodes.…”

“…In this paper, we want to significantly extend the experiments on the SBGFTs, in particular to further investigate their representation ability providing a comparison with two state-of-the-art multiple transforms sets. This analysis is detached from the HEVC coding framework, as opposed to [9] where SBGFTs were already shown to outperform the HEVC DCT-based coding scheme in a Rate-Distortion (RD) sense. In this work, we restrict the analysis to non-linear approximation to purely appraise their energy compaction properties, thus without introducing any bias due to the encoding method.…”

“…As a further contribution, the paper also introduces a preliminary analysis on the use of the SBGFTs in a practical coding system. In particular, starting from the results presented in [9], we investigate to what extent we may remove some graphs from the SBGFTs (thus reducing the cardinality of the transforms set), while still preserving high sparsity properties. This is especially important to preserve reasonable complexity at the encoder since choosing the optimal transform by brute force may be unfeasible.…”

2D Discrete Mirror Transform for Image Non-Linear Approximation

“…Then, these model-based approaches attempt to describe signal variations with a few transform coefficients. Popular transforms belonging to this category are the Fourier transform, the cosine transform, wavelets, and the more recent Curvelets [3], Contourlets [4], Bandelets [5] or Directionlets [6], in addition to the graph Fourier transforms [7] [8] that exploit the structure of some graphs in order to model the signal, e.g., [9] (actually, in some works the graph is constructed by learning graph Laplacians so that a particular class of input data forms graph signals with smooth variations on the resulting topology [10]; this case belongs to the second category).…”