Showing papers on "Circulant matrix published in 2013"

Deconvolving Images With Unknown Boundaries Using the Alternating Direction Method of Multipliers

Abstract: The alternating direction method of multipliers (ADMM) has recently sparked interest as a flexible and efficient optimization tool for inverse problems, namely, image deconvolution and reconstruction under non-smooth convex regularization. ADMM achieves state-of-the-art speed by adopting a divide and conquer strategy, wherein a hard problem is split into simpler, efficiently solvable sub-problems (e.g., using fast Fourier or wavelet transforms, or simple proximity operators). In deconvolution, one of these sub-problems involves a matrix inversion (i.e., solving a linear system), which can be done efficiently (in the discrete Fourier domain) if the observation operator is circulant, i.e., under periodic boundary conditions. This paper extends ADMM-based image deconvolution to the more realistic scenario of unknown boundary, where the observation operator is modeled as the composition of a convolution (with arbitrary boundary conditions) with a spatial mask that keeps only pixels that do not depend on the unknown boundary. The proposed approach also handles, at no extra cost, problems that combine the recovery of missing pixels (i.e., inpainting) with deconvolution. We show that the resulting algorithms inherit the convergence guarantees of ADMM and illustrate its performance on non-periodic deblurring (with and without inpainting of interior pixels) under total-variation and frame-based regularization.

On the Controllability Properties of Circulant Networks

Abstract: This paper examines the controllability of a group of first order agents, adopting a weighted consensus-type coordination protocol over a circulant network. Specifically, it is shown that a circulant network with Laplacian eigenvalues of maximum algebraic multiplicity q is controllable from q nodes. Our approach leverages on the Cauchy-Binet formula, which in conjunction with the Popov-Belevitch-Hautus test, leads to new insights on structural aspects of network controllability.

Multilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity

Abstract: We study the tensor structure of two operations: the transformation of a given multidimensional vector into a multilevel Toeplitz matrix and the convolution of two given multidimensional vectors. We show that the low-rank tensor structure of the input is preserved in the output and propose efficient algorithms for these operations in the newly introduced quantized tensor train (QTT) format. Consider a $d$-dimensional $2n \times\cdots\times 2n$-vector $\boldsymbol{x}$. If it is represented elementwise, the number of parameters is $(2n)^{d}$. However, if we assume that $\boldsymbol{x}$ is given in a QTT representation with ranks bounded by $p$, the number of parameters is reduced to $\mathcal{O}\left(dp^{2} \log n\right)$. Under this assumption we show how the multilevel Toeplitz matrix generated by $\boldsymbol{x}$ can be obtained in the QTT format with ranks bounded by $2p$ in $\mathcal{O}\left(dp^{2} \log n\right)$ operations. We also describe how the convolution $\boldsymbol{x}\star\boldsymbol{y}$ of $\...

Convolutional Compressed Sensing Using Deterministic Sequences

Abstract: In this paper, a new class of orthogonal circulant matrices built from deterministic sequences is proposed for convolution-based compressed sensing (CS). In contrast to random convolution, the coefficients of the underlying filter are given by the discrete Fourier transform of a deterministic sequence with good autocorrelation. Both uniform recovery and non-uniform recovery of sparse signals are investigated, based on the coherence parameter of the proposed sensing matrices. Many examples of the sequences are investigated, particularly the Frank-Zadoff-Chu (FZC) sequence, the m-sequence and the Golay sequence. A salient feature of the proposed sensing matrices is that they can not only handle sparse signals in the time domain, but also those in the frequency and/or or discrete-cosine transform (DCT) domain.

Accelerated Edge-Preserving Image Restoration Without Boundary Artifacts

Abstract: To reduce blur in noisy images, regularized image restoration methods have been proposed that use nonquadratic regularizers (like l1 regularization or total-variation) that suppress noise while preserving edges in the image. Most of these methods assume a circulant blur (periodic convolution with a blurring kernel) that can lead to wraparound artifacts along the boundaries of the image due to the implied periodicity of the circulant model. Using a noncirculant model could prevent these artifacts at the cost of increased computational complexity. In this paper, we propose to use a circulant blur model combined with a masking operator that prevents wraparound artifacts. The resulting model is noncirculant, so we propose an efficient algorithm using variable splitting and augmented Lagrangian (AL) strategies. Our variable splitting scheme, when combined with the AL framework and alternating minimization, leads to simple linear systems that can be solved noniteratively using fast Fourier transforms (FFTs), eliminating the need for more expensive conjugate gradient-type solvers. The proposed method can also efficiently tackle a variety of convex regularizers, including edge-preserving (e.g., total-variation) and sparsity promoting (e.g., l1-norm) regularizers. Simulation results show fast convergence of the proposed method, along with improved image quality at the boundaries where the circulant model is inaccurate.

Characterization of quantum circulant networks having perfect state transfer

Abstract: In this paper we answer the question of when circulant quantum spin networks with nearest-neighbor couplings can give perfect state transfer. The network is described by a circulant graph G, which is characterized by its circulant adjacency matrix A. Formally, we say that there exists a perfect state transfer (PST) between vertices $${a,b\in V(G)}$$ if |F(?) ab | = 1, for some positive real number ?, where F(t) = exp(i At). Saxena et al. (Int J Quantum Inf 5:417---430, 2007) proved that |F(?) aa | = 1 for some $${a\in V(G)}$$ and $${\tau\in \mathbb {R}^+}$$ if and only if all eigenvalues of G are integer (that is, the graph is integral). The integral circulant graph ICG n (D) has the vertex set Z n = {0, 1, 2, . . . , n ? 1} and vertices a and b are adjacent if $${\gcd(a-b,n)\in D}$$ , where $${D \subseteq \{d : d \mid n, \ 1 \leq d < n\}}$$ . These graphs are highly symmetric and have important applications in chemical graph theory. We show that ICG n (D) has PST if and only if $${n\in 4\mathbb {N}}$$ and $${D=\widetilde{D_3} \cup D_2\cup 2D_2\cup 4D_2\cup \{n/2^a\}}$$ , where $${\widetilde{D_3}=\{d\in D\ |\ n/d\in 8\mathbb {N}\}, D_2= \{d\in D\ |\ n/d\in 8\mathbb {N}+4\}{\setminus}\{n/4\}}$$ and $${a\in\{1,2\}}$$ . We have thus answered the question of complete characterization of perfect state transfer in integral circulant graphs raised in Angeles-Canul et al. (Quantum Inf Comput 10(3&4):0325---0342, 2010). Furthermore, we also calculate perfect quantum communication distance (distance between vertices where PST occurs) and describe the spectra of integral circulant graphs having PST. We conclude by giving a closed form expression calculating the number of integral circulant graphs of a given order having PST.

The power and Arnoldi methods in an algebra of circulants

Abstract: SUMMARY Circulant matrices play a central role in a recently proposed formulation of three-way data computations. In this setting, a three-way table corresponds to a matrix where each ‘scalar’ is a vector of parameters defining a circulant. This interpretation provides many generalizations of results from matrix or vector-space algebra. These results and algorithms are closely related to standard decoupling techniques on block-circulant matrices using the fast Fourier transform. We derive the power and Arnoldi methods in this algebra. In the course of our derivation, we define inner products, norms, and other notions. These extensions are straightforward in an algebraic sense, but the implications are dramatically different from the standard matrix case. For example, the number of eigenpairs may exceed the dimension of the matrix, although it is still polynomial in it. It is thus necessary to take an extra step and carefully select a smaller, canonical set of size equal to the dimension of the matrix from which all possible eigenpairs can be formed. Copyright © 2012 John Wiley & Sons, Ltd.

An SVD-free Approach to a Class of Structured Low Rank Matrix Optimization Problems with Application to System Identification

Abstract: —We present an algorithm to solve a structuredlow rank matrix optimization problem based on the nuclearnorm. We represent the desired structure by a linear map,termed mutation, that we characterize and use in our algorithm.Contrary to alternative techniques for structured low rankmatrices, the algorithm is SVD-free, which leads to improvedscalability. The idea relies on restating the nuclear norm via anequivalent variational reformulation involving explicit matrixfactors. We detail the procedure for a general class of problemsand then discuss its application to linear system identificationwith input and output missing data. A direct comparisonbetween alternative approaches highlights the advantage ofSVD-free computations. I. I NTRODUCTION Various rank minimization problems recently attractedrenewed interest in the technical literature. Applications in-clude recommendation systems, multi-task learning, systemidentification and realization techniques. In all these settings,general notions of model complexity can be convenientlyexpressed by the rank of an appropriate matrix. In particular,nuclear norm optimization methods for low-rank matrixapproximation have been discussed in several recent papers.In the system identification community the idea was firstproposed in [7], [6]. The nuclear norm is the largest convexlower bound of the rank function on the spectral unit ball[7]; this fact motivates the use of the nuclear norm in convexrelaxation of rank-based problems.In this paper, we focus on problems where we need to finda matrix that, in addition to being low-rank, is required tohave entries partitioned into known disjointed groups. Thissetting includes various type of structured matrices such asHankel, Toeplitz and circulant matrices. Generally speaking,it allows to deal with matrices that have reduced degreesof freedom. Our interest arises in particular from concate-nated block-Hankel matrices that appear in formulations forinput-output linear system identification problems with noisyand/or partially unobserved data [12].A. Main ContributionsExisting algorithms for nuclear-norm based system identi-fication typically involve at each iteration the singular valuedecomposition (SVD) of a structured matrix Y = B(x) 2

The Circulant Rational Covariance Extension Problem: The Complete Solution

Abstract: The rational covariance extension problem to determine a rational spectral density given a finite number of covariance lags can be seen as a matrix completion problem to construct an infinite-dimensional positive-definite Toeplitz matrix the northwest corner of which is given. The circulant rational covariance extension problem considered in this paper is a modification of this problem to partial stochastic realization of periodic stationary process, which are better represented on the discrete unit circle rather than on the discrete real line . The corresponding matrix completion problem then amounts to completing a finite-dimensional Toeplitz matrix that is circulant. Another important motivation for this problem is that it provides a natural approximation, involving only computations based on the fast Fourier transform, for the ordinary rational covariance extension problem, potentially leading to an efficient numerical procedure for the latter. The circulant rational covariance extension problem is an inverse problem with infinitely many solutions in general, each corresponding to a bilateral ARMA representation of the underlying periodic process. In this paper, we present a complete smooth parameterization of all solutions and convex optimization procedures for determining them. A procedure to determine which solution that best matches additional data in the form of logarithmic moments is also presented.

Circulant structures and graph signal processing

Abstract: Linear shift-invariant processing of graph signals rests on circulant graphs and filters. The spatial features of circulant structures also permit shift-varying operations such as sampling. Their spectral features-as described by their Graph Fourier Transform profiles-enable novel multiscale signal processing systems and methods. To extend the reach of circulant structures, we present a method to decompose an arbitrary graph or filter into a combination of circulant structures. Our decomposition is analogous to resolving a linear time-varying system into a bank of linear time-invariant systems. As an application, we perform multiscale decomposition on temperature data spanning the continental United States.

GGS-groups: Order of congruence quotients and Hausdorff dimension

Abstract: If G is a GGS-group defined over a p-adic tree, where p is an odd prime, we calculate the order of the congruence quotients $G_n=G/\Stab_G(n)$ for every n. If G is defined by the vector $e=(e_1,...,e_{p-1})\in\F_p^{p-1}$, the determination of the order of $G_n$ is split into three cases, according as e is non-symmetric, non-constant symmetric, or constant. The formulas that we obtain only depend on p, n, and the rank of the circulant matrix whose first row is e. As a consequence of these formulas, we also obtain the Hausdorff dimension of the closures of all GGS-groups over the p-adic tree.

On the norms of an r-circulant matrix with the generalized k-Horadam numbers

Abstract: In this paper, we present new upper and lower bounds for the spectral norm of an r-circulant matrix H = Cr(Hk,0, Hk,1, Hk,2, ... , Hk,n-1) whose entries are the generalized k-Horadam numbers. Furthermore, we obtain new formulas to calculate the eigenvalues and determinant of the matrix H. MSC: 11B39; 15A60; 15A15

Embeddings of circulant networks

Abstract: In this paper we solve the edge isoperimetric problem for circulant networks and consider the problem of embedding circulant networks into various graphs such as arbitrary trees, cycles, certain multicyclic graphs and ladders to yield the minimum wirelength.

Multiresolution graph signal processing via circulant structures

Abstract: We use circulant structures to present a new framework for multiresolution analysis and processing of graph signals. Among the essential features of circulant graphs is that they accommodate fundamental signal processing operations, such as linear shift-invariant filtering, downsampling, upsampling, and reconstruction-features that offer substantial advantage. We design two-channel, critically-sampled, perfect-reconstruction, orthogonal lattice-filter structures to process signals defined on circulant graphs. To extend our reach to noncirculant graphs, we present a method to decompose a connected, undirected graph into a combination of circulant graphs. To evaluate our proposed framework, we offer examples of synthetic and real-world graph signal data and their multiscale decompositions.

Critically-sampled perfect-reconstruction spline-wavelet filterbanks for graph signals

Abstract: Inspired by first-order spline wavelets in classical signal processing, we introduce two-channel (low-pass and high-pass), critically-sampled, perfect-reconstruction filterbanks for signals defined on circulant graphs, which accommodate linear shift-invariant filtering We then generalize to filters that process signals defined on noncirculant graphs We apply these filters, which can be tuned to approximate desired frequency responses, to signals defined on synthetic graphs and examine their performance

VLSI Architecture for Layered Decoding of QC-LDPC Codes With High Circulant Weight

Abstract: In this brief, we propose a high-throughput layered decoder architecture to support a broader family of quasicyclic low-density parity-check (QC-LDPC) codes, whose parity-check matrices are constructed from arrays of circulant submatrices. Each nonzero circulant submatrix is a superposition of K cyclic-shifted identity matrices, where the circulant weight K ≥ 1. We propose a novel layered decoder architecture to support QC-LDPC codes with any circulant weight. We present a block-serial decoding architecture which processes a layer of a parity check matrix block by block, where each block is a Z×Z circulant matrix with a circulant weight of K. In the case study, we demonstrate an LDPC decoder design for the China Mobile Multimedia Broadcasting (CMMB) standard, which was synthesized for a TSMC 65-nm CMOS technology. With a core area of 3.9 mm2, the CMMB LDPC decoder achieves a maximum throughput of 1.1 Gb/s with 15 iterations.

On the multivariate circulant rational covariance extension problem

Abstract: Partial stochastic realization of periodic processes from finite covariance data leads to the circulant rational covariance extension problem and bilateral ARMA models. In this paper we present a convex optimization-based theory for this problem that extends and modifies previous results by Carli, Ferrante, Pavon and Picci on the AR solution, which have been successfully applied to image processing of textures. We expect that our present results will provide an enhancement of these procedures.

A cycle decomposition and entropy production for circulant quantum markov semigroups

Abstract: We propose a definition of cycle representation for Quantum Markov Semigroups (QMS) and of Quantum Entropy Production Rate (QEPR) in terms of the ρ-adjoint. We introduce the class of circulant QMS, which admit non-equilibrium steady states but exhibit symmetries that allow us to compute explicitly the QEPR, gain a deeper insight into the notion of cycle decomposition and prove that quantum detailed balance holds if and only if the QEPR equals zero.

Graph Structured Data Viewed Through a Fourier Lens

Abstract: Graph-structured data appears in many modern applications like social networks, sensor networks, transportation networks and computer graphics. These applications are defined by an underlying graph (e.g. a social graph) with associated nodal attributes (e.g. number of ad-clicks by an individual). A simple model for such data is that of a graph signal--a function mapping every node to a scalar real value. Our aim is to develop signal processing tools for analysis of such signals de- fined over irregular graph-structured domains, analogous to classical Fourier and Wavelet analysis defined for regular structures like discrete-time sequences and two-dimensional grids.In this work, we start by reviewing the notion of a Graph Fourier Transform (GFT), which has been defined in the literature for graph signals. We examine the spatial and spectral features of circulant graphs, which accommodate linear shift-invariant operations. We describe fundamental operations such as shifting, sampling, graph-reconnection and linear filtering for signals on circulant graphs and derive associated sampling and graph-reconnection theorems. We also develop wavelet filter bank structures for multi resolution analysis of large-scale graphs.We present a method to decompose an arbitrary graph into a linear combination of circulant graphs. This helps extend fundamental operations such as sampling, filtering and multi resolution filter banks to general graphs. We present an application in the area of graph semi-supervised learning where some of the existing algorithms can be viewed as suitably designed filters defined in the GFT domain. We propose a wavelet regularized learning algorithm and evaluate the performance on some real-world datasets.

New extremal binary self-dual codes of length 68 from quadratic residue codes over f_2+uf_2+u^2f_2

Abstract: In this work, quadratic reside codes over the ring F2 +uF2 +u^2F2 with u^3 = u are considered. A duality and distance preserving Gray map from F2 + uF2 + u^2F2 to (F_2)^3 is defined. By using quadratic double circulant, quadratic bordered double circulant constructions and their extensions self- dual codes of different lengths are obtained. As Gray images of these codes and their extensions, a substantial number of new extremal self-dual binary codes are found. More precisely, thirty two new extremal binary self-dual codes of length 68, 363 Type I codes of parameters [72; 36; 12], a Type II [72; 36; 12] code and a Type II [96; 48; 16] code with new weight enumerators are obtained through these constructions. The results are tabulated.

Sparse block circulant matrices for compressed sensing

Abstract: An undetermined measurement matrix can capture sparse signals losslessly if the matrix satisfies the restricted isometry property (RIP) in compressed sensing (CS) framework. However, existing measurement matrices suffer from high computational burden because of their completely unstructured nature. In this study, the authors propose to construct a novel measurement matrix with a specific structure, called sparse block circulant matrix (SBCM), to reduce the computational burden. The RIP of the proposed SBCM is also guaranteed with overwhelming probability. The simulation results validate that SBCM reduces the computational burden significantly whereas keeps similar signal recovery accuracy as Gaussian random matrices.

Chaotic Image Encryption Algorithm Based on Circulant Operation

Abstract: A novel chaotic image encryption scheme based on the time-delay Lorenz system is presented in this paper with the description of Circulant matrix Making use of the chaotic sequence generated by the time-delay Lorenz system, the pixel permutation is carried out in diagonal and antidiagonal directions according to the first and second components Then, a pseudorandom chaotic sequence is generated again from time-delay Lorenz system using all components Modular operation is further employed for diffusion by blocks, in which the control parameter is generated depending on the plain-image Numerical experiments show that the proposed scheme possesses the properties of a large key space to resist brute-force attack, sensitive dependence on secret keys, uniform distribution of gray values in the cipher-image, and zero correlation between two adjacent cipher-image pixels Therefore, it can be adopted as an effective and fast image encryption algorithm

On Circulant Embedding for Gaussian Random Fields in R

Abstract: The high-dimensionality typically associated with discretized approximations to Gaussian random fields is a considerable hinderance to computationally efficient methods for their simulation. Many direct approaches require spectral decompositions of the associated covariance matrix and so are unable to complete the solving process in a timely fashion, if at all. However under certain conditions, we may construct block-circulant versions of the covariance matrix at hand thereby allowing access to fast-Fourier methods to perform the required operations with impressive speed. We demonstrate how circulant embedding and subsequent simulation can be performed directly in the R language. The approach is currently implemented in C for the R package RandomFields , and used in the recently released package lgcp . Motivated by applications dealing with spatial point processes we restrict attention to stationary Gaussian fields on R 2 , where sparsity of the covariance matrix cannot necessarily be assumed.

Comparative analysis of sensing matrices for compressed sensed thermal images

Abstract: In the conventional sampling process, in order to reconstruct the signal perfectly Nyquist-Shannon sampling theorem needs to be satisfied. Nyquist-Shannon theorem is a sufficient condition but not a necessary condition for perfect reconstruction. The field of compressive sensing provides a stricter sampling condition when the signal is known to be sparse or compressible. Compressive sensing contains three main problems: sparse representation, measurement matrix and reconstruction algorithm. This paper describes and implements 14 different sensing matrices for thermal image reconstruction using Basis Pursuit algorithm available in the YALL1 package. The sensing matrices include Gaussian random with and without orthogonal rows, Bernoulli random with bipolar entries and binary entries, Fourier with and without dc basis vector, Toeplitz with Gaussian and Bernoulli entries, Circulant with Gaussian and Bernoulli entries, Hadamard with and without dc basis vector, Normalised Hadamard with and without dc basis vector. Orthogonalization of the rows of the Gaussian sensing matrix and normalisation of Hadamard matrix greatly improves the speed of reconstruction. Semi-deterministic Toeplitz and Circulant matrices provide lower PSNR and require more iteration for reconstruction. The Fourier and Hadamard deterministic sensing matrices without dc basis vector worked well in preserving the object of interest, thus paving the way for object specific image reconstruction based on sensing matrices. The sparsifying basis used in this paper was Discrete Cosine Transform and Fourier Transform.

Harmonic analysis on Heisenberg–Clifford Lie supergroups

Abstract: We define a Fourier transform and a convolution product for functions and distributions on Heisenberg--Clifford Lie supergroups. The Fourier transform exchanges the convolution and a pointwise product, and is an intertwining operator for the left regular representation. We generalize various classical theorems, including the Paley--Wiener--Schwartz theorem, and define a convolution Banach algebra.

Every matrix is a product of Toeplitz matrices

Abstract: We show that every n-by-n matrix is generically a product of [n/2] + 1 Toeplitz matrices and always a product of at most 2n+5 Toeplitz matrices. The same result holds true if the word "Toeplitz" is replaced by "Hankel", and the generic bound [n/2] + 1 is sharp. We will see that these decompositions into Toeplitz or Hankel factors are unusual: We may not in general replace the subspace of Toeplitz or Hankel matrices by an arbitrary (2n-1)-dimensional subspace of n-by-n matrices. Furthermore such decompositions do not exist if we require the factors to be symmetric Toeplitz, persymmetric Hankel, or circulant matrices, even if we allow an infinite number of factors. Lastly, we discuss how the Toeplitz and Hankel decompositions of a generic matrix may be computed by either (i) solving a system of linear and quadratic equations if the number of factors is required to be [n/2] + 1, or (ii) Gaussian elimination in O(n^3) time if the number of factors is allowed to be 2n.