Higher Order Singular Value Decomposition Research Articles

Motion guided Spatiotemporal Sparsity for high quality 4D-CBCT reconstruction

Conventional cone-beam computed tomography is often deteriorated by respiratory motion blur, which negatively affects target delineation. On the other side, the four dimensional cone-beam computed tomography (4D-CBCT) can be considered to describe tumor and organ motion. But for current on-board CBCT imaging system, the slow rotation speed limits the projection number at each phase, and the associated reconstructions are contaminated by noise and streak artifacts using the conventional algorithm. To address the problem, we propose a novel framework to reconstruct 4D-CBCT from the under-sampled measurements—Motion guided Spatiotemporal Sparsity (MgSS). In this algorithm, we try to divide the CBCT images at each phase into cubes (3D blocks) and track the cubes with estimated motion field vectors through phase, then apply regional spatiotemporal sparsity on the tracked cubes. Specifically, we recast the tracked cubes into four-dimensional matrix, and use the higher order singular value decomposition (HOSVD) technique to analyze the regional spatiotemporal sparsity. Subsequently, the blocky spatiotemporal sparsity is incorporated into a cost function for the image reconstruction. The phantom simulation and real patient data are used to evaluate this algorithm. Results show that the MgSS algorithm achieved improved 4D-CBCT image quality with less noise and artifacts compared to the conventional algorithms.

Validation of μ-volt T-wave alternans analysis using multiscale analysis-by-synthesis and higher-order SVD

Abstract Detection of microvolt T-wave alternans (TWA) and finding its clinical significance remains a challenging task. In this work, we propose a new TWA detection method based on multiscale analysis-by-synthesis followed by higher-order singular value decomposition (MAS–HOSVD). The multilead ECG (MECG) data, after R-peak detection, were represented as a third-order tensor. Then HOSVD was applied on the subband reconstructed T-wave tensor to determine a tensor containing T-wave information (TensorTinfo). The p-value, TWA ratio, and the probability of detection (PD) were computed, based on the information present in TensorTinfo, to validate the presence of TWA or not. The proposed method was evaluated with semi-synthetic and real signals by adding several types of noise at different SNR levels and was compared to a multilead and two single-lead schemes. Simulation results showed that the proposed method detected TWA accurately with the sensitivity of 95.2% for the alternans level of 10 μV. Also, the method detected TWA with an SNR at least lower by 10 dB than other methods. The MAS–HOSVD method was tested to be accurate in TWA analysis of MECG data. This method can be considered as an automatic tool for prior prognosticator of sudden cardiac death, consequently, can help reduce the mortality rate.

Fast computation of the compressive hyperspectral imaging by using alternating least squares methods

Hyperspectral imaging acquires up to several hundreds of narrow and adjacent spectral band images simultaneously. However, since the dimension of the hyperspectral imaging data, which typically forms a third order tensor, is increased in proportion to the size of spatial and the spectral information at the same time, the higher order singular value decomposition (HOSVD) is appropriate to reduce its dimension. One of the simplest and most accurate approaches for computing the HOSVD is higher order orthogonal iteration (HOOI), which computes the factor matrices from the unfolding matrices of the given tensor by using singular value decomposition alternatively until convergence is achieved. However, because of its expensive computational complexity, we propose a faster algorithm to compute the HOSVD even though the output shows no meaningful difference from that obtained by HOOI. Specifically instead of computing the factor matrix from the updated tensor in every iteration along each mode, we reuse the intermediate result after updating one factor matrix to modify the others in a single iteration. Numerical experiments reveal that the proposed algorithm computes the dimension-reduced hyperspectral imaging much faster than HOOI with fewer outer iterations. Moreover, the difference in accuracy between the proposed algorithm and HOOI is negligible.

Higher-order singular value decomposition-based discrete fractional random transform for simultaneous compression and encryption of video images

Existing image compression and encryption methods have several shortcomings: they have low reconstruction accuracy and are unsuitable for three-dimensional (3D) images. To overcome these limitations, this paper proposes a tensor-based approach adopting tensor compressive sensing and tensor discrete fractional random transform (TDFRT). The source video images are measured by three key-controlled sensing matrices. Subsequently, the resulting tensor image is further encrypted using 3D cat map and the proposed TDFRT, which is based on higher-order singular value decomposition. A multiway projection algorithm is designed to reconstruct the video images. The proposed algorithm can greatly reduce the data volume and improve the efficiency of the data transmission and key distribution. The simulation results validate the good compression performance, efficiency, and security of the proposed algorithm.

Uncertainty and Sensitivity Analysis of Bifurcation Loci Characterizing Nonlinear Landing-Gear Dynamics

A methodology is developed to efficiently, yet accurately, determine the uncertainty bounds of the bifurcation loci of nonlinear dynamic systems subjected to parametric variations. The chosen approach makes use of numerical continuation, higher-order singular-value decomposition, and surrogate modeling. The technique is demonstrated on a representative model of an aircraft undercarriage, where the effect of uncertainty in the structural parameters is propagated onto the prediction of the conditions for the onset of shimmy. Comparison with numerical integration results demonstrates the accuracy of the methodology.

A Novel Fast Tensor-Based Preconditioner for Image Restoration.

Image restoration is one of the main parts of image processing. Mathematically, this problem can be modeled as a large-scale structured ill-posed linear system. Ill-posedness of this problem results in low-convergence rate of iterative solvers. For speeding up the convergence, preconditioning usually is used. Despite the existing preconditioners for image restoration, which are constructed based on approximations of the blurring matrix, in this paper, we propose a novel preconditioner with a different viewpoint. Here, we show that image restoration problem can be modeled as a tensor contractive linear equation. This modeling enables us to propose a new preconditioner based on an approximation of the blurring tensor operator. Due to the particular structure of the blurring tensor for zero boundaries, we show that the truncated higher order singular value decomposition of the blurring tensor is obtained very fast and so could be used as a preconditioner. Experimental results confirm the efficiency of this new preconditioner in image restoration and its outperformance in comparison with the other well-known preconditioners.

Multi-linear sparse reconstruction for SAR imaging based on higher-order SVD

This paper focuses on the spotlight synthetic aperture radar (SAR) imaging for point scattering targets based on tensor modeling. In a real-world scenario, scatterers usually distribute in the block sparse pattern. Such a distribution feature has been scarcely utilized by the previous studies of SAR imaging. Our work takes advantage of this structure property of the target scene, constructing a multi-linear sparse reconstruction algorithm for SAR imaging. The multi-linear block sparsity is introduced into higher-order singular value decomposition (SVD) with a dictionary constructing procedure by this research. The simulation experiments for ideal point targets show the robustness of the proposed algorithm to the noise and sidelobe disturbance which always influence the imaging quality of the conventional methods. The computational resources requirement is further investigated in this paper. As a consequence of the algorithm complexity analysis, the present method possesses the superiority on resource consumption compared with the classic matching pursuit method. The imaging implementations for practical measured data also demonstrate the effectiveness of the algorithm developed in this paper.

Fast Algorithms for Higher-Order Singular Value Decomposition from Incomplete Data

Higher-order singular value decomposition (HOSVD) is an efficient way for data reduction and also eliciting intrinsic structure of multi-dimensional array data. It has been used in many applications, and some of them involve incomplete data. To obtain HOSVD of the data with missing values, one can first impute the missing entries through a certain tensor completion method and then perform HOSVD to the reconstructed data. However, the two-step procedure can be inefficient and does not make reliable decomposition. In this paper, we formulate an incomplete HOSVD problem and combine the two steps into solving a single optimization problem, which simultaneously achieves imputation of missing values and also tensor decomposition. We also present two algorithms for solving the problem based on block coordinate update. Global convergence of both algorithms is shown under mild assumptions. The convergence of the second algorithm implies that of the popular higher-order orthogonality iteration (HOOI) method, and thus we, for the first time, give global convergence of HOOI. In addition, we compare the proposed methods to state-of-the-art ones for solving incomplete HOSVD and also low-rank tensor completion problems and demonstrate the superior performance of our methods over other compared ones. Furthermore, we apply them to face recognition and MRI image reconstruction to show their practical performance.

Sparse approximations in complex domain based on BM3D modeling

In this paper the concept of sparsity for complex-valued variables is introduced in the following three types: directly in complex domain and for two real-valued pairs phase/amplitude and real/imaginary parts of complex variables. The nonlocal block-matching technique is used for sparsity implementation and filter design for each type of sparsity. These filters are complex domain generalizations of the Block Matching 3D collaborative (BM3D) filter based on the high-order singular value decomposition (HOSVD) in order to generate group-wise adaptive analysis/synthesis transforms. Complex domain denoising is developed and studied as a test-problem for comparison of the designed filters as well as the different types of sparsity modeling.

HOSVD based multidimensional parameter estimation for massive MIMO system from incomplete channel measurements

The dominant multipath components for massive multiple-input multiple-output systems can be described using geometry-based channel models with R-dimensional (R-D) parameters. These parameters are crucial for channel correlation acquisition, which is a prerequisite for many technical challenges. In this paper, we consider higher-order singular value decomposition based R-D channel modeling parameter estimation from incomplete measurements. Incomplete higher-order orthogonality iteration algorithm can be utilized to solve the problem, which simultaneously achieves tensor recovery and tensor decomposition. After obtaining the signal or noise subspace, the parameters of interest can be estimated by using subspace methods.

Noise removal from MR images via iterative regularization based on higher-order singular value decomposition

Despite the success of magnetic resonance imaging techniques in many applications, acquisition noise is still a limiting factor for the quality and hence the usefulness of the techniques. In this paper, a new algorithm for denoising magnetic resonance images based on higher-order singular value decomposition is proposed. The proposed algorithm first forms a single tensor from the noisy data. Next, higher-order singular value decomposition is applied on this tensor with respect to a set of learned orthogonal directional matrices over the corresponding tensor mode. Finally, soft thresholding is iteratively applied to the calculated coefficients, thereby suppressing the noise. The new algorithm is further enhanced with a post-process Wiener filtering. The proposed algorithm has two advantages over existing tensor denoising approaches: (1) It combines the noisy image slices into a single tensor, thereby exploiting non-local image similarity across slices and (2) it uses an iterative regularization framework to suppress the noise. Experiments are conducted on synthetic and real magnetic resonance images to compare the performance of the proposed algorithm to state-of-the-art denoising approaches. The comparison is made quantitatively in terms of peak of signal-to-noise ratio, structural similarity index and mean absolute difference, and qualitatively through visual comparisons. The results demonstrate the competitive performance of the proposed algorithm.

Virtual multidimensional structure‐based subspace estimation to improve parameter estimation accuracy in harmonic retrieval problems

For R-dimensional (R-D) harmonic retrieval problems with R ≥ 2, it has been demonstrated that higher-order singular value decomposition (HOSVD) of the measurement data can be used to improve the signal subspace estimation, leading to high parameter estimation accuracy. This, however, usually cannot be applied to the one-dimensional harmonic retrieval problem due to the absence of the multidimensional structure in the measurement data. To overcome this difficulty, the authors propose to construct a virtual multidimensional structure in the measurement data, upon which the HOSVD can be used to estimate the signal subspace. Moreover, the idea of constructing a virtual multidimensional structure is also introduced to the R-D harmonic retrieval problems. In conjunction with its inherent multidimensional structure, the virtual multidimensional structure is exploited to further enhance the signal subspace estimation. As examples, estimation of signal parameters via rotational invariance technique algorithms are employed to estimate the signal parameters. Finally, theoretical analysis and simulation results demonstrate the effectiveness and efficiency of the authors’ proposal.

Denoise diffusion-weighted images using higher-order singular value decomposition

Noise usually affects the reliability of quantitative analysis in diffusion-weighted (DW) magnetic resonance imaging (MRI), especially at high b-values and/or high spatial resolution. Higher-order singular value decomposition (HOSVD) has recently emerged as a simple, effective, and adaptive transform to exploit sparseness within multidimensional data. In particular, the patch-based HOSVD denoising has demonstrated superb performance when applied to T1-, T2-, and proton density-weighted MRI data. In this study, we aim to investigate the feasibility of denoising DW data using the HOSVD transform. With the low signal-to-noise ratio in typical DW data, the patch-based HOSVD denoising suffers from stripe artifacts in homogeneous regions because of the HOSVD bases learned from the noisy patches. To address this problem, we propose a novel denoising method. It first introduces a global HOSVD-based denoising as a prefiltering stage to guide the subsequent patch-based HOSVD denoising stage. The HOSVD bases from the patch groups in prefiltered images are then used to transform the noisy patch groups in original DW data. Experiments were performed using simulated and in vivo DW data. Results show that the proposed method significantly reduces stripe artifacts compared with conventional patch-based HOSVD denoising methods, and outperforms two state-of-the-art denoising methods in terms of denoising quality and diffusion parameters estimation.

Significance of Higher-Order Spectral Analysis in Infant Cry Classification

In this paper, higher-order spectral analysis is applied to infant cry signals for classification of normal infant cries from pathological infant cries. From the family of higher-order spectra, bispectrum is considered for the proposed task. Bispectrum is the Fourier transform of the third-order cumulant function. To extract features from the bispectrum, application of higher-order singular value decomposition theorem is proposed. Experimental results show the average classification accuracy of \({82.44} \pm {4.03}{ \%}\) and Matthew’s correlation coefficient (MCC) of 0.62 with proposed bispectrum features. In all of the experiments reported in this paper, support vector machine with radial basis function kernel is used as the pattern classifier. Performance of the proposed features is also compared with the state-of-the-art methods such as linear frequency cepstral coefficients, Mel frequency cepstral coefficients, perceptual linear prediction coefficients, linear prediction coefficients, linear prediction cepstral coefficients and perceptual linear prediction cepstral coefficients, and is found to be better than that given by these feature sets. The proposed bispectrum-based features are shown to be robust under signal degradation or noisy conditions at various SNR levels. Performance in the presence of noise is compared with the state-of-the-art spectral feature sets using MCC scores. In addition, effectiveness of cryunit segmentation in normal and pathological infant cry classification task is reported.

A novel unitary PARAFAC method for DOD and DOA estimation in bistatic MIMO radar

In this paper, a novel unitary parallel factor (U-PARAFAC) algorithm of estimating direction-of-departure (DOD) and direction-of-arrival (DOA) in bistatic multiple-input multiple-output (MIMO) radar is proposed. A real-valued tensor signal model is constructed by applying the traditional forward-backward averaging technique. Subsequently, the fact that the real-valued tensor follows a PARAFAC model is proved, thus the subspace-based high-order singular value decomposition (HOSVD) method can be avoided in the subsequent solving process. Furthermore, directly operating the real-valued loading factors instead of the signal subspace, traditional unitary ESPRIT (U-ESPRIT) method is firstly extended to the real-valued PARAFAC model. The new algorithm, which exploits the multidimensional structure and does not require the estimation of signal subspace, having good performance especially at low signal-to-noise ratio (SNR). More attractively, compared with classical tensor methods such as the PARAFAC algorithm and the unitary tensor-ESPRIT algorithm, the U-PARAFAC algorithm still performs well without sacrificing array aperture when targets are highly correlated or closely spaced. Additional angle pair-matching is not required. Simulation results verify the effectiveness of the proposed algorithm.

Tensor based multichannel reconstruction for breast tumours identification from DCE-MRIs.

A new methodology based on tensor algebra that uses a higher order singular value decomposition to perform three-dimensional voxel reconstruction from a series of temporal images obtained using dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) is proposed. Principal component analysis (PCA) is used to robustly extract the spatial and temporal image features and simultaneously de-noise the datasets. Tumour segmentation on enhanced scaled (ES) images performed using a fuzzy C-means (FCM) cluster algorithm is compared with that achieved using the proposed tensorial framework. The proposed algorithm explores the correlations between spatial and temporal features in the tumours. The multi-channel reconstruction enables improved breast tumour identification through enhanced de-noising and improved intensity consistency. The reconstructed tumours have clear and continuous boundaries; furthermore the reconstruction shows better voxel clustering in tumour regions of interest. A more homogenous intensity distribution is also observed, enabling improved image contrast between tumours and background, especially in places where fatty tissue is imaged. The fidelity of reconstruction is further evaluated on the basis of five new qualitative metrics. Results confirm the superiority of the tensorial approach. The proposed reconstruction metrics should also find future applications in the assessment of other reconstruction algorithms.

Low rank tensor recovery via iterative hard thresholding

We study extensions of compressive sensing and low rank matrix recovery (matrix completion) to the recovery of low rank tensors of higher order from a small number of linear measurements. While the theoretical understanding of low rank matrix recovery is already well-developed, only few contributions on the low rank tensor recovery problem are available so far. In this paper, we introduce versions of the iterative hard thresholding algorithm for several tensor decompositions, namely the higher order singular value decomposition (HOSVD), the tensor train format (TT), and the general hierarchical Tucker decomposition (HT). We provide a partial convergence result for these algorithms which is based on a variant of the restricted isometry property of the measurement operator adapted to the tensor decomposition at hand that induces a corresponding notion of tensor rank. We show that subgaussian measurement ensembles satisfy the tensor restricted isometry property with high probability under a certain almost optimal bound on the number of measurements which depends on the corresponding tensor format. These bounds are extended to partial Fourier maps combined with random sign flips of the tensor entries. Finally, we illustrate the performance of iterative hard thresholding methods for tensor recovery via numerical experiments where we consider recovery from Gaussian random measurements, tensor completion (recovery of missing entries), and Fourier measurements for third order tensors.

Multi-focus image fusion using HOSVD and edge intensity

The purpose of multi-focus image fusion is to integrate the partially focused images into one single image which is focused everywhere. To achieve this purpose, higher order singular value decomposition (HOSVD) and edge intensity (EDI) based multi-focus image fusion method is proposed. The main characteristics of the proposed method includes: 1. an effective and robust sharpness measure based on edge intensity is presented; 2. considering the fact that HOSVD is an effective data-driven decomposition technique and shows the outstanding ability in the representation of high-dimensional data, it is used to decompose multi-focus images; and 3. a multi-strategy fusion rule based on sigmoid function is used to fuse the decomposition coefficients. Furthermore, several experiments are conducted to verify the superiority of the proposed fusion framework in terms of visual and statistical analyses.

Multi-dimensional sinusoidal order estimation using angles between subspaces

Multi-dimensional harmonic retrieval (HR) in white noise is required in numerous applications such as channel estimation in wireless communications and imaging in multiple-input multiple-output radar. In this paper, we propose two R-dimensional (R-D) extensions of the subspace-based MUSIC model order selection scheme, for R≥2, to detect the number of multi-dimensional cisoids. The key idea in the algorithm development is to utilize the principle angles between multilinear signal subspaces via the truncated higher-order singular value decomposition. The first method is designed for multiple-snapshot scenarios. It considerably outperforms existing algorithms in terms of both detection accuracy and identifiability particularly when a large number of snapshots are available. However, its computational cost is relatively quite high. The second method is computationally much simpler and performs almost as well as the first one when the number of snapshots is small. Simulation results are conducted to demonstrate the performance of the proposed estimators.

Sparse phase imaging based on complex domain nonlocal BM3D techniques

The paper is addressed to 2D phase and amplitude estimation of complex-valued signals – that is, in particular, to estimation of modulo-2π interferometric phase images from periodic and noisy observations. These degradation mechanisms make phase image estimation a challenging problem. A sparse nonlocal data-adaptive imaging formalized in complex domain is used for phase and amplitude image reconstruction. Following the procedure of patch-based technique, the image is partitioned into small overlapping square patches. Block Matching Three Dimensional (BM3D) technique is developed for forming complex domain sparse spectral representations of complex-valued data. High Order Singular Value Decomposition (HOSVD) applied to BM3D groups enables the design of the orthonormal complex domain 3D transforms which are data adaptive and different for each BM3Ds group. An iterative version of the complex domain BM3D is designed from variational formulation of the problem. The convergence of this algorithm is shown. The effectiveness of the new sparse coding based algorithms is illustrated in simulation experiments where they demonstrate the state-of-the-art performance.