Pseudo-inverses of difference matrices and their application to sparse signal approximation

Plonka, Gerlind; Hoffmann, Sebastian; Weickert, Joachim

doi:10.1016/j.laa.2016.03.033

Cited by 11 publications

(16 citation statements)

References 21 publications

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“…These formulations provide methods for the obtention of the Moore-Penrose inverse in the finite-dimensional case. We show that results appeared in [2,3,10,14], which were proved with different approaches, are particular cases of some of our formulations. In Example 3.4, we consider distance matrices D of weighted trees with all the weights being nonzero and with sum equal to zero.…”

Section: Introductionmentioning

confidence: 60%

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Moore-Penrose inverse of sums of operators and matrices

Morillas¹

2021

Preprint

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Under certain conditions, we prove that the Moore-Penrose inverse of a sum of operators is the sum of the Moore-Penrose inverses. From this, we derive characterizations and expressions for the Moore-Penrose inverse of an operator that are useful for its computation. We give formulations of them for finite matrices. We show that results appeared in the literature, and which were proved with different approaches, can be derived as particular cases of some of these formulations. We present an application to distance matrices of weighted trees.

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Section: Introductionmentioning

confidence: 60%

“…√ n e ∈ N(A), from Corollary 3.2(iii) we obtain [14,Theorem 2.1]. In [14], the authors first prove that A + αee t is nonsingular by showing that all its eigenvalues are nonzero.…”

Section: Formulations For Matricesmentioning

confidence: 99%

Moore-Penrose inverse of sums of operators and matrices

Morillas¹

2021

Preprint

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“…This has been used in [66] to relate linear PDE-based inpainting to sparsity concepts: Discrete Green's functions serve as atoms in a dictionary that gives a sparse representation of the inpainting solution. In the one-dimensional case, discrete analytic derivations are presented in [110].…”

Section: Relations To Other Methodsmentioning

confidence: 99%

2. Optimizing spatial and tonal data for PDE-based inpainting

Hoeltgen¹,

Mainberger²,

Hoffmann³

et al. 2016

Variational Methods

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Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. Apart from these specific contributions, we also give an extensive literature survey on PDE-based image compression methods. Summary and ConclusionsThe contributions of our work are twofold: On the one hand we gave a comprehensive survey on the state-of-the-art in PDE-based image compression.

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“…Here, L † is sparse with respect to a 2-hop localized neighborhood on the graph. While closed-form expressions for the higher-order MPPs need to be derived on a case by case basis, it has been shown that the columns of L †2 C describe 4th order polynomials [30], from which it then naturally follows that the columns of L †2 C S T C = L † C S † C are cubic polynomials which are sparse with respect to the operator L 2 C ; accordingly, L †2 C L C = L † C describe quadratic polynomials, following a twofold degree-reduction. In addition L †2 C is sparse with respect to S C L 2 C , which possesses an extra vanishing moment through S C .…”

Section: General Circulant Graphsmentioning

confidence: 99%

“…4 (c) further illustrates the shape of the perturbation given by the columns of P −1 α at α = 4π/N , which exhibits a small support followed by approximate sparsity. In particular, the matrix P −1 α represents the difference between the MPPs of the simple cycle in (a) and the extended circulant in (b), and accordingly perturbs the functions underlying the former around its discontinuities, resulting in the constrained analysis representation 30 , as depicted in (b). Further, in Fig.…”

Section: The Generalized Graph Laplacianmentioning

confidence: 99%

Analysis vs Synthesis - An Investigation of (Co)sparse Signal Models on Graphs

Kotzagiannidis

Davies

2018

2018 IEEE Global Conference on Signal and Information Processing (GlobalSIP)

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We consider the problem of characterizing the 'duality gap' between sparse synthesis-and cosparse analysis-driven signal models through the lens of spectral graph theory, in an effort to comprehend their precise equivalencies and discrepancies. By detecting and exploiting the inherent connectivity structure, and hence, distinct set of properties, of rank-deficient graph difference matrices such as the graph Laplacian, we are able to substantiate discrepancies between the cosparse analysis and sparse synthesis models, according to which the former constitutes a constrained and translated instance of the latter. In view of a general union of subspaces model, we conduct a study of the associated subspaces and their composition, which further facilitates the refinement of specialized uniqueness and recovery guarantees, and discover an underlying structured sparsity model based on the graph incidence matrix. Furthermore, for circulant graphs, we provide an exact characterization of underlying subspaces by deriving closed-form expressions as well as demonstrating transitional properties between equivalence and non-equivalence for a parametric generalization of the graph Laplacian.

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Pseudo-inverses of difference matrices and their application to sparse signal approximation

Cited by 11 publications

References 21 publications

Moore-Penrose inverse of sums of operators and matrices

Moore-Penrose inverse of sums of operators and matrices

2. Optimizing spatial and tonal data for PDE-based inpainting

Analysis vs Synthesis - An Investigation of (Co)sparse Signal Models on Graphs

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