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.
A subset of vertices @math is a @cite_1 for signals in @math if @math . It is also shown that @math is a for all signals @math , if and only if @math are linearly independent where @math is the @math smallest eigenvalue of @math and @math is a reduced eigenvector. The term reduced implies taking the rows of the eigenvectors corresponding to the indices of the sampling set @math @cite_35 . It can also be shown that for any minimum uniqueness set @math of size @math for signals in @math , there is always at least one node @math such that @math is a uniqueness set of size @math for signals in @math @cite_35 . This property will be useful for iteratively selecting the set of reference samples from the input light field data.
@cite_35 @cite_1
2891091350 2095414057
In this paper we propose a novel vertex based sampling method for k-bandlimited signals lying on arbitrary graphs, that has a reasonable computational complexity and results in low reconstruction error. Our goal is to find the smallest set of vertices that can guarantee a perfect reconstruction of any k-bandlimited signal on any connected graph. We propose to iteratively search for the vertices that yield the minimum reconstruction error, by minimizing the maximum eigenvalue of the error covariance matrix using a linear solver. We compare the performance of our method with state-of-the-art sampling strategies and random sampling on graphs. Experimental results show that our method successfully computes the smallest sample sets on arbitrary graphs without any parameter tuning. It provides a small reconstruction error, and is robust to noise. In this paper, we present two localized graph filtering based methods for interpolating graph signals defined on the vertices of arbitrary graphs from only a partial set of samples. The first method is an extension of previous work on reconstructing bandlimited graph signals from partially observed samples. The iterative graph filtering approach very closely approximates the solution proposed in the that work, while being computationally more efficient. As an alternative, we propose a regularization based framework in which we define the cost of reconstruction to be a combination of smoothness of the graph signal and the reconstruction error with respect to the known samples, and find solutions that minimize this cost. We provide both a closed form solution and a computationally efficient iterative solution of the optimization problem. The experimental results on the recommendation system datasets demonstrate effectiveness of the proposed methods.