Graph Transform Optimization With Application to Image Compression

Abstract

In this paper, we propose a new graph-based transform and illustrate its potential application to signal compression. Our approach relies on the careful design of a graph that optimizes the overall rate-distortion performance through an effective graph-based transform. We introduce a novel graph estimation algorithm, which uncovers the connectivities between the graph signal values by taking into consideration the coding of both the signal and the graph topology in rate-distortion terms. In particular, we introduce a novel coding solution for the graph by treating the edge weights as another graph signal that lies on the dual graph. Then, the cost of the graph description is introduced in the optimization problem by minimizing the sparsity of the coefficients of its graph Fourier transform (GFT) on the dual graph. In this way, we obtain a convex optimization problem whose solution defines an efficient transform coding strategy. The proposed technique is a general framework that can be applied to different types of signals, and we show two possible application fields, namely natural image coding and piecewise smooth image coding. The experimental results show that the proposed graph-based transform outperforms classical fixed transforms such as DCT for both natural and piecewise smooth images. In the case of depth map coding, the obtained results are even comparable to the state-of-the-art graph-based coding method, that are specifically designed for depth map images.

Learning Graphs From Data: A Signal Representation Perspective

“…Graph learning has been introduced only recently for this type of problems. A learning model based on signal smoothness, inspired by [39], [70], has been further extended in order to design a graph-based coding framework that takes into account the coding of the signal values as well as the cost of transmitting the graph in rate distortion terms [69]. In particular, the cost of coding the image signal is minimized by promoting its smoothness on the learned topology.…”

“…The transmission cost of the graph itself is further controlled by adding an additional term in the optimization problem which penalizes the sparsity of the graph Fourier coefficients of the edge weight signal. An illustrative example of the graph-based transform coding proposed in [69], as well as its application to image compression, is shown in Fig. 10.…”

“…For example, in image compression, the actual cost of coding the graph is at least equally important compared to the cost of coding the image signal. Such constraints are indicated by the application, and they should be incorporated in the graph learning framework (e.g., [69]) in order to make the learning framework more targeted to a specific application.…”

Sparse Graph Learning Under Laplacian-Related Constraints

“…If y m (t) is share price of mth stock on day t, we consider (as is conventional in such studies) x m (t) = ln(y m (t)/y m (t − 1)) as the time series to analyze, yielding n = 1258 and p = 97. These 97 stocks are classified into 11 sectors (according to the Global Industry Classification Standard) and we order the nodes to group them as information technology (nodes 1-12), health care (13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27), financials (28)(29)(30)(31)(32)(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44), real estate (45-46), consumer discretionary (47-56), industrials (57-68), communication services (69-76), consumer staples (77-87), energy (88-92), materials (93), utilities (94-97). For each m, x m (t) was centered and normalized to unit variance.…”

“…Graph Laplacian matrix has been extensively used for embedding, manifold learning, clustering and semi-supervised learning [2]- [4], [27], [42], [43]; see [9], [15] for further references to applications to web page categorization with graph information, etc., and [13] for graph-based transform coding where learning of the graph Laplacian plays a key role.…”