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In many state-of-the-art compression systems, signal transformation is an integral part of the encoding and decoding process, where transforms provide compact representations for the signals of interest. This paper introduces a class of transforms called graph-based transforms (GBTs) for video compression, and proposes two different techniques to design GBTs. In the first technique, we formulate an optimization problem to learn graphs from data and provide solutions for optimal separable and nonseparable GBT designs, called GL-GBTs. The optimality of the proposed GL-GBTs is also theoretically analyzed based on Gaussian-Markov random field (GMRF) models for intra and inter predicted block signals. The second technique develops edge-adaptive GBTs (EA-GBTs) in order to flexibly adapt transforms to block signals with image edges (discontinuities). The advantages of EA-GBTs are both theoretically and empirically demonstrated. Our experimental results demonstrate that the proposed transforms can significantly outperform the traditional Karhunen-Loeve transform (KLT).

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In this paper, we propose a new approach for image compression using graph-based biorthogonal wavelet filterbanks (referred to as graphBior filterbanks). These filterbanks, proposed in our previous work, operate on the graph representations of images, which are formed by linking nearby pixels with each other. The connectivity and the link weights are chosen so as to reflect the geometrical structure of the image. The filtering operations on these edge-aware image graphs avoid filtering across the image discontinuities, thus resulting in a significant reduction in the amount of energy in the high frequency bands. This reduces the bit-rate requirements for the wavelet coefficients, but at the cost of sending extra edge-information bits to the decoder. We discuss efficient ways of representing and encoding this edge information. Our experimental results, based on the SPIHT codec, demonstrate that the proposed approach achieves better R-D performance than the standard CDF9/7 filter on piecewise smooth images such as depth maps.

In this paper, we propose a graph-based lifting transform for intrapredicted video sequences. The transform can approximate the performance of a Graph Fourier Transform (GFT) for a given graph, but does not require computing eigenvectors. A predict-update bipartition is designed based on a Gaussian Markov Random Field (GMRF) model with the goal to minimize the energy in the prediction set. Additionally, a novel re-connection method is applied for multi-level graphs, leading to significant gain for the proposed bipartition method and for the conventional MaxCut based bipartition. Experiments on intra-predicted video sequences show that the proposed method, even considering the extra overhead for edge information, outperforms the Discrete Cosine Transform (DCT) and approximates the performance of the higher complexity GFT.