In this paper, we propose a patch-based image inpainting method using a low-rank Hankel structured matrix completion approach. The proposed method exploits the annihilation property between a ...shift-invariant filter and image data observed in many existing inpainting algorithms. In particular, by exploiting the commutative property of the convolution, the annihilation property results in a low-rank block Hankel structure data matrix, and the image inpainting problem becomes a low-rank structured matrix completion problem. The block Hankel structured matrices are obtained patch-by-patch to adapt to the local changes in the image statistics. To solve the structured low-rank matrix completion problem, we employ an alternating direction method of multipliers with factorization matrix initialization using the low-rank matrix fitting algorithm. As a side product of the matrix factorization, locally adaptive dictionaries can be also easily constructed. Despite the simplicity of the algorithm, the experimental results using irregularly subsampled images as well as various images with globally missing patterns showed that the proposed method outperforms existing state-of-the-art image inpainting methods.
This paper is a continuation of a recent investigation by Zhan and Dyachenko (2021) on the Hurwitz stability of monic matrix polynomials with algebraic techniques. By improving an inertia formula for ...matrix polynomials with respect to the imaginary axis, we show that, under some conditions, the quasi-stability of a monic matrix polynomial can be tested via the Hermitian nonnegative definiteness of two block Hankel matrices built from its matricial Markov parameters. Moreover, for the so-called doubly monic matrix polynomials, the quasi-stability criteria can be formulated in a much simpler form. In particular, the relationship between Hurwitz stable matrix polynomials and Stieltjes positive definite matrix sequences established in Zhan and Dyachenko (2021) is included as a special case.
We consider the nonlinear inverse problem of learning a transition operator A from partial observations at T different times, in the form of sparse observations of entries of the powers A,A 2 ,...,A ...T . We address the nonlinearity of this spatio-temporal transition operator recovery problem by a suitable embedding into block-Hankel matrices, transforming it to a low-rank matrix completion problem, even when A has full rank. For both a uniform and an adaptive random space-time sampling model, we quantify the recoverability of the transition operator via suitable measures of incoherence of these block-Hankel embedding matrices. For graph transition operators these measures of incoherence depend on the interplay between the dynamics and the graph topology. We develop a suitable non-convex iterative reweighted least squares (IRLS) algorithm, establish its quadratic local convergence, and show that, in optimal scenarios, no more than O ( rn log( nT )) space-time samples are sufficient to ensure accurate recovery of a rank- r operator A of size n × n . We provide an efficient implementation of the proposed IRLS algorithm with space complexity of order O ( rnT ) and periteration time complexity linear in n , and confirm in numerical experiments that for several graph transition operators, the theoretical findings accurately track empirical phase transitions.
Parallel MRI (pMRI) and compressed sensing MRI (CS-MRI) have been considered as two distinct reconstruction problems. Inspired by recent k-space interpolation methods, an annihilating filter-based ...low-rank Hankel matrix approach is proposed as a general framework for sparsity-driven k-space interpolation method which unifies pMRI and CS-MRI. Specifically, our framework is based on a novel observation that the transform domain sparsity in the primary space implies the low-rankness of weighted Hankel matrix in the reciprocal space. This converts pMRI and CS-MRI to a k-space interpolation problem using a structured matrix completion. Experimental results using in vivo data for single/multicoil imaging as well as dynamic imaging confirmed that the proposed method outperforms the state-of-the-art pMRI and CS-MRI.
Recently, coprime arrays have attracted lots of interest due to their ability of providing enhanced degrees-of-freedom and reduced mutual coupling effect compared to conventional uniform linear ...arrays. Benefitting from these excellent properties, coprime multiple-input multiple-output (MIMO) radar has been recently suggested for improving parameter identifiability and target detection. In this paper, we address the problem of joint direction-of-departure (DOD) and direction-of-arrival (DOA) estimation of coherent targets in coprime MIMO radar. First of all, we establish an extended virtual uniform rectangular array (URA) by performing array interpolation on the transmit and receive arrays of coprime MIMO radar. Subsequently, we derive a low-rank block Hankel matrix that is constructed using the correlation information of the virtual URA expected outputs, and utilize the matched filter outputs to recover the block Hankel matrix by solving a low-rank structured matrix completion problem. Finally, we estimate the DODs and DOAs of coherent targets by applying the modified matrix pencil method on the recovered low-rank matrix. We also derive the Cramér-Rao bounds with closed-form expressions for DOD and DOA estimation of coherent targets using coprime MIMO radar. The proposed algorithm can identify multiple coherent targets and achieve parameter automatic pairing. Numerical results demonstrate the superiority of the proposed algorithm over several existing approaches when handling coherent targets.
In this study, an automatic rescue system was proposed to monitor agricultural machinery operators using machine vision. The rescue system was developed to recognise the driver inattention status, ...that is, the distraction and fatigue by recognising the driver's actions. A Kinect sensor was used to collect image sequences of the operators, and the recognition system relied on the “player extraction” function of the Kinect sensor. A Hankel-based Kernel Mutual Subspace Method (KMSM) was developed to monitor tractor drivers and recognise driver inattention behaviours. To reduce the computational complexity for fulfilling the requirements of recognition, low-dimensional image vectors were used to generate low-dimensional block Hankel matrixes as representations for input action sequences. To evaluate the performance of the proposed KMSM, a driver action dataset was established that included 10 tractor drivers and 5 types of action that denote inattention. The drivers' inattention actions were classified into three danger levels, and the corresponding countermeasures for the actions at each danger level were similarly classified. Both offline and online experiments using similar subjects and different subjects were conducted to evaluate the designed inattention action recognition algorithm. In the offline experiment, the proposed Hankel-based KMSM achieved recognition rates of 91.18% and 86.18% when using similar and different subjects, respectively; and in the online experiment, the proposed method achieved 87.02 and 79.97% when using similar and different subjects, respectively. The average computation time of the Hankel-based KMSM was 0.07 s in the online experiment. Thus, the proposed Hankel-based KMSM method satisfies both the accuracy and the real-time requirements for a driver rescue system.
•A rescue system was proposed to monitor the tractor's driver inattention action.•Kernel Mutual subspace algorithm proposed using Hankel matrix for the action images.•Developed algorithm had the advantage of processing the action images in real time.•Offline recognition rate 91% and 86% for similar/different subjects, respectively.•Online recognition rate 87% and 79% for similar/different subjects, respectively.
Motivated by the Cadzow filtering in seismic data processing, this paper presents a fast SVD method for multilevel block Hankel matrices. A seismic data presented as a multidimensional array is first ...transformed into a two dimensional multilevel block Hankel (MBH) matrix. Then the Lanczos process is applied to reduce the MBH matrix into a bidiagonal or tridiagonal matrix. Finally, the SVD of the reduced matrix is computed using the twisted factorization method. To achieve high efficiency, we propose a novel fast MBH matrix-vector multiplication method for the Lanczos process. In comparison with existing fast Hankel matrix-vector multiplication methods, our method applies 1-D, instead of multidimensional, FFT and requires minimum storage. Moreover, a partial SVD is performed on the reduced matrix, since complete SVD is not required by the Caszow filtering. Our numerical experiments show that our fast MBH matrix-vector multiplication method significantly improves both the computational cost and storage requirement. Our fast MBH SVD algorithm is particularly efficient for large size multilevel block Hankel matrices.
The key step of syndrome-based decoding of Reed-Solomon codes up to half the minimum distance is to solve the so-called Key Equation. List decoding algorithms, capable of decoding beyond half the ...minimum distance, are based on interpolation and factorization of multivariate polynomials. This article provides a link between syndrome-based decoding approaches based on Key Equations and the interpolation-based list decoding algorithms of Guruswami and Sudan for Reed-Solomon codes. The original interpolation conditions of Guruswami and Sudan for Reed-Solomon codes are reformulated in terms of a set of Key Equations. These equations provide a structured homogeneous linear system of equations of Block-Hankel form, that can be solved by an adaption of the Fundamental Iterative Algorithm. For an ( n , k ) Reed-Solomon code, a multiplicity s and a list size l , our algorithm has time complexity O ( ls 4 n 2 ).
We propose a low rank structured matrix completion algorithm for image inpainting problems originated from scanning microscopy. The proposed method exploits the annihilation property observed in ...Gaussian Markov Random Field (GMRF) or partial differential equation (PDE)-based inpainting approaches. By utilizing the commutative property of the convolution, the annihilation property is embodied into rank-deficient block Hankel structure data matrices and the image inpainting problem is converted into low-rank structured matrix completion problem. To solve the structured low-rank matrix completion problem, an alternating direction method of multiplier (ADMM) method is used with factorization matrix initialization using the low rank matrix fitting (LMaFit) algorithm. Experimental results showed that the proposed method outperforms the existing state-of-the-art image inpainting methods.
Complex Valued, Multi-Layer Hopfield Neural Network Ramamurthy, Garimella; Swamy, Tata Jagannadha
2022 International Conference on Emerging Techniques in Computational Intelligence (ICETCI),
2022-Aug.-25
Conference Proceeding
Based on a Complex Valued Neural Network (CVNNs) with the neurons placed in multiple layers, several interesting associative memories are proposed and their dynamics is investigated. Specifically, ...the synaptic weight matrix was chosen to be Block Hermitian, Block Jacobi, Block Toeplitz and Block Hankel. It is reasoned that in the case of such structured multilayer CVNNs, an interesting Convergence theorem holds true which enables that utilization of such an Artificial Neural Network as an associative memory. It is expected that the CVNNs proposed in this research paper have biological significance.
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