Number Of Non-zero Singular Values Research Articles

Dynamic speckle imaging with SVD compression

Dynamic speckle imaging (DSI) of areas with different speed of processes ongoing in industrial or biological objects relies on statistical processing of a large number of images of the speckle patterns formed on the objects surface under laser illumination. The DSI visualizes the speed spatial distribution as an activity map. We propose compression of the raw DSI data by applying singular value decomposition (SVD). A specific feature of speckle images for DSI is lack of a structure with areas of close intensity values. The gain from the direct SVD application may be modest in cases when a great number of non-zero singular values is needed to build an activity map comparable in quality to the ground truth map from bitmap images. For higher compression, we propose SVD to be applied to the 2D arrays containing the differences between the successive images. The SVD compression has been verified by using synthetic and experimental data.

2-D Seismic Data Reconstruction via Truncated Nuclear Norm Regularization

Rank-reduction (RR) methods have been widely applied for reconstructing seismic data. The popular convex relaxation formulation of RR is nuclear norm minimization (NNM), which is capitalized on its convexity. Consequently, global optimization can be effectively achieved with NNM method. However, NNM minimizes the summation of all singular values and is therefore not equivalent to the minimum of the rank function; in fact, the rank of the matrix is equal to the number of nonzero singular values. Thus, NNM is not a good approximation to the original RR problem, which affects the reconstruction performance. The truncated nuclear norm regularization (TNNR) method only minimizes the smallest $\min (m,n)-r$ singular values ( $m, n$ is the size of matrix and $r$ is the rank), yielding more accurate approximations to the rank function in real applications. In this article, we propose a new method for seismic reconstruction via TNNR based on the texture patch matrix, and further provide a strategy to estimate $r$ , which can efficiently reduce the cost of computation. Numerical experiments performed on both synthetic and real data indicate that the reconstruction quality based on our proposed algorithm is better than that based on the singular value thresholding algorithm and the low-rank factorization model (low-rank matrix fitting).

Signal denoising method for modal analysis of an offshore platform

The modal analysis of offshore structures is a key element of structural health monitoring based on vibration. A difficulty encountered by practitioners and researchers is that an accurate modal analysis is challenging in noisy environments. The objective of this work is to develop and implement a signal denoising method based on solving the inverse singular value problem of a measured (noisy) data matrix with prescribed entries to reconstruct a filtered data matrix. The measured (noisy) impulse response function (IRF) is used to build a square or nearly square Hankel matrix. The normalized rank determination indicator (RDI) of the Hankel matrix is introduced to determine the number of prescribed non-zero singular values in the prescribed entries inverse singular value problem (PEISVP). The reconstructed (filtered) matrix must maintain the original Hankel structure, and preserve the number of non-zero singular values. Once the filtered IRF has been obtained, the complex exponential (CE) method is applied for the modal analysis. To validate the proposed method, hereafter referred to as PEISVP-CE, we undertake the numerical simulation of a five-story shear building. Once successfully validated, we apply the PEISVP-CE method to an offshore field experiment of a jacket platform under the step relaxation. We find that the PEISVP-CE method is effective at eliminating noise and, therefore, appropriate for the modal analysis of offshore structures.

Accelerated exponential parameterization of T2 relaxation with model-driven low rank and sparsity priors (MORASA).

This work is to develop a novel image reconstruction method from highly undersampled multichannel acquisition to reduce the scan time of exponential parameterization of T2 relaxation. On top of the low-rank and joint-sparsity constraints, we propose to exploit the linear predictability of the T2 exponential decay to further improve the reconstruction of the T2-weighted images from undersampled acquisitions. Specifically, the exact rank prior (i.e., number of non-zero singular values) is adopted to enforce the spatiotemporal low rankness, while the mixed L2-L1 norm of the wavelet coefficients is used to promote joint sparsity, and the Hankel low-rank approximation is used to impose linear predictability, which integrates the exponential behavior of the temporal signal into the reconstruction process. An efficient algorithm is adopted to solve the reconstruction problem, where corresponding nonlinear filtering operations are performed to enforce corresponding priors in an iterative manner. Both simulated and in vivo datasets with multichannel acquisition were used to demonstrate the feasibility of the proposed method. Experimental results have shown that the newly introduced linear predictability prior improves the reconstruction quality of the T2-weighted images and benefits the subsequent T2 mapping by achieving high-speed, high-quality T2 mapping compared with the existing fast T2 mapping methods. This work proposes a novel fast T2 mapping method integrating the linear predictable property of the exponential decay into the reconstruction process. The proposed technique can effectively improve the reconstruction quality of the state-of-the-art fast imaging method exploiting image sparsity and spatiotemporal low rankness. Magn Reson Med 76:1865-1878, 2016. © 2016 International Society for Magnetic Resonance in Medicine.

Truncated SVD best rank choice through ROC curves for genomic annotation prediction

Motivations Correct interpretation of many biological experiments is currently based on consistency of biomolecular annotation databases. Such databases are very widespread and very useful for the scientific community, but, unfortunately, incomplete by definition. To support and quicken their time consuming curation procedure, and to improve their consistence, computational methods that supply a ranked list of predicted annotations are hence extremely useful. We depart from a previous work on the prediction of Gene Ontology (GO) annotations, based on the truncated Singular Value Decomposition (SVD) of the annotation matrix, where the truncation level k of the input matrix is a keypoint in obtaining both best biomolecular annotation predictions and best performance. Here we propose a method that chooses this truncation level by computing and evaluating the Area Under the Curve (AUC) of different Receiver Operating Characteristic (ROC) curves. Methods Let the matrix A(i,j), with m rows (genes) and n columns (annotation terms), represent all annotations of a specific controlled vocabulary for a given organism. The entry A(i,j)=1 if gene i is annotated to term j (or descendant), 0 otherwise. The annotation prediction is performed by computing a reduced rank approximation Ak of the matrix A, by means of the SVD. Ak contains real value entries related to the likelihood that gene i shall be annotated to term j. For a defined threshold t, if Ak(i,j)>t, gene i is predicted to be annotated to term j and, if A(i,j) 0 & Ak(i,j)<=t, an existing annotation is suggested as semantically inconsistent with the available data (AR). The method core is the truncation level k, which defines the size of the submatrix used by the algorithm to compute the SVD. For any considered truncation value, our greedy algorithm generates a ROC curve drawing the AR rate (1.0 - Sensitivity) vs the AP rate (1.0 - Specificity), and computes the ROC AUC. If p is the maximum rank of A, where p=min(m,n), and r<=p is the number of non-zero singular values along the diagonal of Sigma matrix, the best truncation value is in the [1;r] interval. To avoid performing the SVD and ROC analysis for every integer value in [1;r] we sample within this interval q values that could be used as adequate truncation values. To obtain the best sampling, we study the distribution of the AUC values for different truncation levels, for a sample dataset (i.e. organism Gallus gallus, GO Biological Process). First, we exclude fist and last 10% values, to avoid taking levels that, during SVD reconstruction, would consider too few or too many non-zero singular values of A. By analyzing gradient variations in AUCs distribution function, we sample q truncation values, inside the above range. We consider every qi as a new SVD truncation value, and compute the AUCqi of the corresponding ROCqi curve. Finally, we take min(AUCqi) as the best qi truncation value. Results For evaluation, we use old GO annotations of Gallus gallus and Bos taurus genes available on July 2009 in an old version of GO Annotation databases (http://geneontology.org). By analyzing Gallus gallus annotations between genes and Biological process (BP) (8,731 annotations; 275 genes; 610 BP terms), the algorithm suggests k=77 as the best truncation level for SVD. This level led to a ROC curve having AUC=40.27%, while the 2nd best value, 59, led to AUC=40.46%. From the 8,731 input annotations, with t=0.4, the SVD method with value 77 predicted 44 AP annotations. Out of these, 28 (63.63%) turned out to be present among the 27 month newer GO annotations in a more recent GO database version (Oct-2011); these 28 APs included 14 annotations (50%) with GO evidence different from IEA-ND. On the other hand, the 2nd best value, 59, led to worst results: same number of 44 APs, but just 14 of these (31.81%) were present among the newer GO annotations considered. Other truncation values, related to higher AUC values, led to even worst prediction results. Erratum. This is modified version replaced on 22 May 2012. The Editor guarantes the scientific integrety of the abstract content.

SENSOR VALIDATION IN DAMAGE LOCATING VECTOR METHOD FOR STRUCTURAL HEALTH MONITORING

The reliability of a structural health-monitoring system is very dependent on the quality of signals that are acquired and fed into the damage detection algorithm. Developed herein is an algorithm for sensor validation in the context of the damage locating vector (DLV) method for detecting damage in structures. Given the signals from ns sensors, ns combinatorial sets of signals from (ns - 1) sensors are formed. For each set, the change in the flexibility matrix relative to that of the reference or undamaged structure is computed and singular value decomposition is performed to estimate the number of nonzero singular values (NZV). The set that produces the smallest NZV is associated with healthy sensors whereas sensors that do not belong to that combination are suspected to be faulty. The performance of the proposed algorithm is illustrated using both simulated and experimental data obtained from a 3D modular truss structure monitored by sensors, some of which are faulty.

Extraction of fundamental mode shapes from superposition resonance of a plate

This work proposes a method that can extract a set of individual fundamental mode shapes from a set of measured superposition mode shapes, when a structure possesses two or more mode shapes with the same natural frequency. It is revealed that the number of nonzero singular values of a set of measured mode shapes indicates the geometric multiplicity resolved in measured mode shapes. Utilizing the sensitivity of singular values resolved in measured mode shapes, an updating algorithm is developed to extracta set of individual fundamental mode shapes. The performance of the proposed method is demonstrated using a numerical simulation for a selected class of structures.

Model order selection for the singular value decomposition and the discrete Karhunen–Loève transform using a Bayesian approach

Bayesian model order selection is considered in relation to the singular value decomposition (SVD) and the discrete Karhunen–Loève transform (DKLT). There are many applications of the SVD and DKLT where it is necessary to discard some of the small singular values that may represent corrupted signal information. Often this task is performed heuristically or in an ad hoc manner. The Bayesian approach to model order selection involves the determination of the evidence or the conditional posterior probability of the model structure given the data; this framework allows the relative probabilities of all possible candidate models to be compared explicitly. Applied to the SVD, the evidence formulation enables the number of nonzero singular values (and hence the effective rank) of a singular or ill-conditioned matrix to be determined analytically. For the DKLT, the evidence allows the determination of the optimal number of basis vectors to choose for the signal reconstruction. In addition, the Bayesian method allows prior information such as physical smoothness constraints to be incorporated directly into the problem specification. Derivations of the evidence formulae are included along with results that illustrate the usefulness of the method.