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
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We consider the problem of estimating an unknown signal @math from noisy linear observations @math . In many practical instances, @math has a certain structure that can be captured by a structure inducing convex function @math . For example, @math norm can be used to encourage a sparse solution. To estimate @math with the aid of @math , we consider the well-known LASSO method and provide sharp characterization of its performance. We assume the entries of the measurement matrix @math and the noise vector @math have zero-mean normal distributions with variances @math and @math respectively. For the LASSO estimator @math , we attempt to calculate the Normalized Square Error (NSE) defined as @math as a function of the noise level @math , the number of observations @math and the structure of the signal. We show that, the structure of the signal @math and choice of the function @math enter the error formulae through the summary parameters @math and @math , which are defined as the Gaussian squared-distances to the subdifferential cone and to the @math -scaled subdifferential, respectively. The first LASSO estimator assumes a-priori knowledge of @math and is given by @math . We prove that its worst case NSE is achieved when @math and concentrates around @math . Secondly, we consider @math , for some @math . This time the NSE formula depends on the choice of @math and is given by @math . We then establish a mapping between this and the third estimator @math . Finally, for a number of important structured signal classes, we translate our abstract formulae to closed-form upper bounds on the NSE. We consider the problem of clustering a graph @math into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables @math , where @math is the incidence matrix of a graph @math , @math is the vector of unknown vertex variables (with a uniform prior), and @math is a noise vector with Bernoulli @math i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of @math is possible if and only the graph @math is connected, with a sharp threshold at the edge probability @math for Erdős-Renyi random graphs. The first goal of this paper is to determine how the edge probability @math needs to scale to allow exact recovery in the presence of noise. Defining the degree rate of the graph by @math , it is shown that exact recovery is possible if and only if @math . In other words, @math is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph @math , defining the degree rate as @math , where @math is the minimum degree of the graph, it is shown that the proposed method achieves the rate @math , where @math is the spectral gap of the graph @math . It is well known that the smallest eigenvalue of the adjacency matrix of a connected d-regular graph is at least − d and is strictly greater than − d if the graph is not bipartite. More generally, for any connected graph G = (V, E), consider the matrix Q = D + A where D is the diagonal matrix of degrees in the graph G and A is the adjacency matrix of G. Then Q is positive semidefinite, and the smallest eigenvalue of Q is 0 if and only if G is bipartite. We will study the separation of this eigenvalue from 0 in terms of the following measure of nonbipartiteness of G. For any S ⊆ V, we denote by emin(S) the minimum number of edges that need to be removed from the induced subgraph on S to make it bipartite. Also, we denote by cut(S) the set of edges with one end in S and the other in V − S. We define the parameter Ψ as ***image*** The parameter Ψ is a measure of the nonbipartiteness of the graph G. We will show that the smallest eigenvalue of Q is bounded above and below by functions of Ψ. For d-regular graphs, this characterizes the separation of the smallest eigenvalue of the adjacency matrix from −d. These results can be easily extended to weighted graphs. © 1994 Wiley Periodicals, Inc. A large set of signals can sometimes be described sparsely using a dictionary, that is, every element can be represented as a linear combination of few elements from the dictionary. Algorithms for various signal processing applications, including classification, denoising and signal separation, learn a dictionary from a given set of signals to be represented. Can we expect that the error in representing by such a dictionary a previously unseen signal from the same source will be of similar magnitude as those for the given examples? We assume signals are generated from a fixed distribution, and study these questions from a statistical learning theory perspective. We develop generalization bounds on the quality of the learned dictionary for two types of constraints on the coefficient selection, as measured by the expected L2 error in representation when the dictionary is used. For the case of l1 regularized coefficient selection we provide a generalization bound of the order of O(√np ln(mλ) m), where n is the dimension, p is the number of elements in the dictionary, λ is a bound on the l1 norm of the coefficient vector and m is the number of samples, which complements existing results. For the case of representing a new signal as a combination of at most k dictionary elements, we provide a bound of the order O(√np ln(mk) m) under an assumption on the closeness to orthogonality of the dictionary (low Babel function). We further show that this assumption holds for most dictionaries in high dimensions in a strong probabilistic sense. Our results also include bounds that converge as 1 m, not previously known for this problem. We provide similar results in a general setting using kernels with weak smoothness requirements.