Graph-based transforms have been shown to be powerful tools in terms of image energy compaction. However, when the support increases to best capture signal dependencies, the computation of the basis functions becomes rapidly untractable. This problem is in particular compelling for high dimensional imaging data such as light fields. The use of local transforms with limited supports is a way to cope with this computational difficulty. Unfortunately, the locality of the support may not allow us to fully exploit long term signal dependencies present in both the spatial and angular dimensions in the case of light fields. This paper describes sampling and prediction schemes with local graph-based transforms enabling to efficiently compact the signal energy and exploit dependencies beyond the local graph support. The proposed approach is investigated and is shown to be very efficient in the context of spatio-angular transforms for quasi-lossless compression of light fields.
In this letter, we provide a theoretical analysis of optimal predictive transform coding based on the Gaussian Markov random field (GMRF) model. It is shown that the eigen-analysis of the precision matrix of the GMRF model is optimal in decorrelating the signal. The resulting graph transform degenerates to the well-known 2-D discrete cosine transform (DCT) for a particular 2-D first order GMRF, although it is not a unique optimal solution. Furthermore, we present an optimal scheme to perform predictive transform coding based on conditional probabilities of a GMRF model. Such an analysis can be applied to both motion prediction and intra-frame predictive coding, and may lead to improvements in coding efficiency in the future. In this paper a graph-based transform is proposed as an alternative to the discrete cosine transform. An image or video signal is represented as a graph signal, where the graph is generated so as not to cross an image edge in a local region, i.e., square block. Then, spectral representation of graph signal is used to form transform kernels by finding eigenvectors of Laplacian matrix of the graph. This method requires to include additional information, i.e., edge map or adjacency matrix, into a bitstream so that a decoder can regenerate the exactly same graph used at an encoder. The novelty of this paper includes finding the optimal adjacency matrix and compressing it using context-based adaptive binary arithmetic coding. Coding efficiency improvement can be achieved when an image block contains arbitrarily shaped edges by applying the transform not across the edges. The proposed transform is applied to coding depth maps used for view synthesis in a multi-view video coding system, and provides 14 bit rate savings on average. In applications such as social, energy, transportation, sensor, and neuronal networks, high-dimensional data naturally reside on the vertices of weighted graphs. The emerging field of signal processing on graphs merges algebraic and spectral graph theoretic concepts with computational harmonic analysis to process such signals on graphs. In this tutorial overview, we outline the main challenges of the area, discuss different ways to define graph spectral domains, which are the analogs to the classical frequency domain, and highlight the importance of incorporating the irregular structures of graph data domains when processing signals on graphs. We then review methods to generalize fundamental operations such as filtering, translation, modulation, dilation, and downsampling to the graph setting and survey the localized, multiscale transforms that have been proposed to efficiently extract information from high-dimensional data on graphs. We conclude with a brief discussion of open issues and possible extensions. Depth map compression is important for efficient network transmission of 3D visual data in texture-plus-depth format, where the observer can synthesize an image of a freely chosen viewpoint via depth-image-based rendering (DIBR) using received neighboring texture and depth maps as anchors. Unlike texture maps, depth maps exhibit unique characteristics like smooth interior surfaces and sharp edges that can be exploited for coding gain. In this paper, we propose a multi-resolution approach to depth map compression using previously proposed graph-based transform (GBT). The key idea is to treat smooth surfaces and sharp edges of large code blocks separately and encode them in different resolutions: encode edges in original high resolution (HR) to preserve sharpness, and encode smooth surfaces in low-pass-filtered and down-sampled low resolution (LR) to save coding bits. Because GBT does not filter across edges, it produces small or zero high-frequency components when coding smooth-surface depth maps and leads to a compact representation in the transform domain. By encoding down-sampled surface regions in LR GBT, we achieve representation compactness for a large block without the high computation complexity associated with an adaptive large-block GBT. At the decoder, encoded LR surfaces are up-sampled and interpolated while preserving encoded HR edges. Experimental results show that our proposed multi-resolution approach using GBT reduced bitrate by 68 compared to native H.264 intra with DCT encoding original HR depth maps, and by 55 compared to single-resolution GBT encoding small blocks. Piecewise smooth (PWS) images (e.g., depth maps or animation images) contain unique signal characteristics such as sharp object boundaries and slowly varying interior surfaces. Leveraging on recent advances in graph signal processing, in this paper, we propose to compress the PWS images using suitable graph Fourier transforms (GFTs) to minimize the total signal representation cost of each pixel block, considering both the sparsity of the signal’s transform coefficients and the compactness of transform description. Unlike fixed transforms, such as the discrete cosine transform, we can adapt GFT to a particular class of pixel blocks. In particular, we select one among a defined search space of GFTs to minimize total representation cost via our proposed algorithms, leveraging on graph optimization techniques, such as spectral clustering and minimum graph cuts. Furthermore, for practical implementation of GFT, we introduce two techniques to reduce computation complexity. First, at the encoder, we low-pass filter and downsample a high-resolution (HR) pixel block to obtain a low-resolution (LR) one, so that a LR-GFT can be employed. At the decoder, upsampling and interpolation are performed adaptively along HR boundaries coded using arithmetic edge coding, so that sharp object boundaries can be well preserved. Second, instead of computing GFT from a graph in real-time via eigen-decomposition, the most popular LR-GFTs are pre-computed and stored in a table for lookup during encoding and decoding. Using depth maps and computer-graphics images as examples of the PWS images, experimental results show that our proposed multiresolution-GFT scheme outperforms H.264 intra by 6.8 dB on average in peak signal-to-noise ratio at the same bit rate. In this work a new set of edge-adaptive transforms (EATs) is presented as an alternative to the standard DCTs used in image and video coding applications. These transforms avoid filtering across edges in each image block, thus, they avoid creating large high frequency coefficients. These transforms are then combined with the DCT in H.264 AVC and a transform mode selection algorithm is used to choose between DCT and EAT in an RD-optimized manner. These transforms are applied to coding depth maps used for view synthesis in a multi-view video coding system, and provides up to 29 bit rate reduction for a fixed quality in the synthesized views. Recent advent in graph signal processing (GSP) has led to the development of new graph-based transforms and wavelets for image video coding, where the underlying graph describes inter-pixel correlations. In this paper, we develop a new transform called signed graph Fourier transform (SGFT), where the underlying graph G contains negative edges that describe anti-correlations between pixel pairs. Specifically, we first construct a one-state Markov process that models both inter-pixel correlations and anti-correlations. We then derive the corresponding precision matrix, and show that the loopy graph Laplacian matrix Q of a graph G with a negative edge and two self-loops at its end nodes is approximately equivalent. This proves that the eigenvectors of Q — called SGFT — approximates the optimal Karhunen-Loeve Transform (KLT). We show the importance of the self-loops in G to ensure Q is positive semi-definite. We prove that the first eigenvector of Q is piecewise constant (PWC), and thus can well approximate a piecewise smooth (PWS) signal like a depth image. Experimental results show that a block-based coding scheme based on SGFT outperforms a previous scheme using graph transforms with only positive edges for several depth images.