Standard Singular Value Decomposition Research Articles

Susceptibility of Tmax to Tracer Delay on Perfusion Analysis: Quantitative Evaluation of Various Deconvolution Algorithms Using Digital Phantoms

The time-to-maximum of the tissue residue function (T(max)) perfusion index has proven very predictive of infarct growth in large clinical trials, yet its dependency on simple tracer delays remains unknown. Here, we determine the dependency of computed tomography (CT) perfusion (CTP) T(max) estimates on tracer delay using a range of deconvolution techniques and digital phantoms. Digital phantom data sets simulating the tracer delay were created from CTP data of six healthy individuals, in which time frames of the left cerebral hemisphere were shifted forward and backward by up to ±5 seconds. These phantoms were postprocessed with three common singular value decomposition (SVD) deconvolution algorithms-standard SVD (sSVD), block-circulant SVD (bSVD), and delay-corrected SVD (dSVD)-with an arterial input function (AIF) obtained from the right middle cerebral artery (MCA). The T(max) values of the left hemisphere were compared among different tracer delays and algorithms by a region of interest-based analysis. The T(max) values by sSVD were positively correlated with 'positive shifts' but unchanged with 'negative shifts,' those by bSVD had an excellent positive linear correlation with both positive and negative shifts, and those by dSVD were relatively constant, although slightly increased with the positive shifts. The T(max) is a parameter highly dependent on tracer delays and deconvolution algorithm.

On the asymptotic behaviour of the solutions to the replicator equation

Selection systems and the corresponding replicator equations model the evolution of replicators with a high level of abstraction. In this paper, we apply novel methods of analysis of selection systems to the replicator equations. To be suitable for the suggested algorithm, the interaction matrix of the replicator equation should be transformed; in particular, the standard singular value decomposition allows us to rewrite the replicator equation in a convenient form. The original n-dimensional problem is reduced to the analysis of asymptotic behaviour of the solutions to the so-called escort system, which in some important cases can be of significantly smaller dimension than the original system. The Newton diagram methods are applied to study the asymptotic behaviour of the solutions to the escort system, when interaction matrix has Rank 1 or 2. A general replicator equation with the interaction matrix of Rank 1 is fully analysed; the conditions are provided when the asymptotic state is a polymorphic equilibrium. As an example of the system with the interaction matrix of Rank 2, we consider the problem from Adams & Sornborger (2007, Analysis of a certain class of replicator equations. J. Math. Biol., 54, 357–384), for which we show, for an arbitrary dimension of the system and under some suitable conditions, that generically one globally stable equilibrium exits on the 1-skeleton of the simplex.

Differences in CT Perfusion Maps Generated by Different Commercial Software: Quantitative Analysis by Using Identical Source Data of Acute Stroke Patients

To examine the variability in the qualitative and quantitative results of computed tomographic (CT) perfusion imaging generated from identical source data of stroke patients by using commercially available software programs provided by various CT manufacturers. Institutional review board approval and informed consent were obtained. CT perfusion imaging data of 10 stroke patients were postprocessed by using five commercial software packages, each of which had a different algorithm: singular-value decomposition (SVD), maximum slope (MS), inverse filter (IF), box modulation transfer function (bMTF), and by using custom-made original software with standard (sSVD) and block-circulant (bSVD) SVD methods. Areas showing abnormalities in cerebral blood flow (CBF), mean transit time (MTT), and cerebral blood volume (CBV) were compared with each other and with the final infarct areas. Differences among the ratios of quantitative values in the final infarct areas and those in the unaffected side were also examined. The areas with CBF or MTT abnormalities and the ratios of these values significantly varied among software, while those of CBV were stable. The areas with CBF or MTT abnormalities analyzed by using SVD or bMTF corresponded to those obtained with delay-sensitive sSVD, but overestimated the final infarct area. The values obtained from software by using MS or IF corresponded well with those obtained from the delay-insensitive bSVD and the final infarct area. Given the similarities between CBF and MTT, all software were separated in two groups (ie, sSVD and bSVD). The ratios of CBF or MTTs correlated well within both groups, but not across them. CT perfusion imaging maps were significantly different among commercial software even when using identical source data, presumably because of differences in tracer-delay sensitivity.

A New Scheme for the Tensor Representation

The paper presents a new scheme for the representation of tensors which is well-suited for high-order tensors. The construction is based on a hierarchy of tensor product subspaces spanned by orthonormal bases. The underlying binary tree structure makes it possible to apply standard Linear Algebra tools for performing arithmetical operations and for the computation of data-sparse approximations. In particular, a truncation algorithm can be implemented which is based on the standard matrix singular value decomposition (SVD) method.

Difference in Tracer Delay–induced Effect among Deconvolution Algorithms in CT Perfusion Analysis: Quantitative Evaluation with Digital Phantoms

Institutional review board approval and informed consent were obtained. The purpose was to evaluate the differences in tracer delay-induced effects of various deconvolution algorithms for computed tomographic (CT) perfusion imaging by using digital phantoms created from actual source data. Three methods of singular value decomposition (SVD) were evaluated. For standard SVD (sSVD), the delays induced significant errors in cerebral blood flow and mean transit time. In contrast, for block-circulant SVD (bSVD), these values remained virtually unchanged, whereas for delay-corrected SVD (dSVD), mild changes were observed. bSVD was superior to sSVD and dSVD for avoiding the tracer delay-induced effects in CT perfusion imaging.

Improved residue function and reduced flow dependence in MR perfusion using least-absolute-deviation regularization

Cerebral blood flow (CBF) estimates derived from singular value decomposition (SVD) of time intensity curves from Gadolinium bolus perfusion-weighted imaging are known to underestimate CBF, especially at high flow rates. We report the development of a model-independent delay-invariant deconvolution technique using least-absolute-deviation (LAD) regularization to improve the CBF estimation accuracy. Computer simulations were performed to compare the accuracy of CBF estimates derived from LAD, reformulated SVD (rSVD) and standard SVD (sSVD) techniques. Simulations were performed at image signal-to-noise ratios ranging from 20 to 400, cerebral blood volumes from 1% to 10%, and CBF from 2.5 mL/100 g/min to 176.5 mL/100 g/min to estimate the effect of these parameters on the accuracy of CBF estimation. The LAD method improved the CBF estimation accuracy by up to 32% in gray matter and 23% in white matter compared with rSVD and sSVD methods. LAD method also reduces the systematic bias of rSVD and sSVD methods to baseline SNR while producing more accurate and reproducible residue function calculation than either rSVD or sSVD method. Initial clinical implementation of the method on six representative clinical cases confirm the advantages of the LAD method over rSVD and sSVD methods.

Tracer Delay–Insensitive Algorithm Can Improve Reliability of CT Perfusion Imaging for Cerebrovascular Steno-Occlusive Disease: Comparison with Quantitative Single-Photon Emission CT

Reliability of CT perfusion (CTP) algorithms has not been fully validated. We investigated whether the cerebral blood flow (CBF) values obtained by using a dynamic CTP technique with a tracer delay-insensitive deconvolution algorithm are more accurate than those obtained by using CTP with delay-sensitive algorithms in unilateral cerebrovascular steno-occlusive disease, when compared with those generated by quantitative single-photon emission CT (SPECT). Using CTP and iodine-123-N-isopropyl-p-iodoamphetamine SPECT with an autoradiographic quantification technique, we examined 20 patients with suggested hemodynamic ischemia due to stenosis or occlusion of the unilateral internal carotid or middle cerebral artery. The algorithms used for CTP included delay-insensitive block-circulant singular value decomposition (SVD) (bSVD) and delay-sensitive standard SVD (sSVD) and box-modulation transfer function (bMTF). Absolute CBF values obtained by using CTP with bSVD were significantly lower than those obtained with SPECT, but the ratios to the nonaffected side were significantly correlated to the quantitative SPECT values with significant agreements, particularly when the arterial input function was obtained from the unaffected side. Contrastingly, CBF ratios with sSVD and bMTF were significantly underestimated, and no significant agreement was determined between CTP with sSVD or bMTF and SPECT, though there were substantial correlations between them in some parameters. With the CTP technique, the insensitivity of the deconvolution algorithm to the tracer-delay effect appears to be essential for estimating semiquantitative CBF values in patients with unilateral steno-occlusive lesions.

Spectral and entropic characterizations of Wigner functions: Applications to model vibrational systems

The Wigner function for the pure quantum states is used as an integral kernel of the non-Hermitian operator K, to which the standard singular value decomposition (SVD) is applied. It provides a set of the squared singular values treated as probabilities of the individual phase-space processes, the latter being described by eigenfunctions of KK(+) (for coordinate variables) and K(+)K (for momentum variables). Such a SVD representation is employed to obviate the well-known difficulties in the definition of the phase-space entropy measures in terms of the Wigner function that usually allows negative values. In particular, the new measures of nonclassicality are constructed in the form that automatically satisfies additivity for systems composed of noninteracting parts. Furthermore, the emphasis is given on the geometrical interpretation of the full entropy measure as the effective phase-space volume in the Wigner picture of quantum mechanics. The approach is exemplified by considering some generic vibrational systems. Specifically, for eigenstates of the harmonic oscillator and a superposition of coherent states, the singular value spectrum is evaluated analytically. Numerical computations are given for the nonlinear problems (the Morse and double well oscillators, and the Henon-Heiles system). We also discuss the difficulties in implementation of a similar technique for electronic problems.

Higher Order SVD Analysis for Dynamic Texture Synthesis

Videos representing flames, water, smoke, etc., are often defined as dynamic textures: "textures" because they are characterized by the redundant repetition of a pattern and "dynamic" because this repetition is also in time and not only in space. Dynamic textures have been modeled as linear dynamic systems by unfolding the video frames into column vectors and describing their trajectory as time evolves. After the projection of the vectors onto a lower dimensional space by a singular value decomposition (SVD), the trajectory is modeled using system identification techniques. Synthesis is obtained by driving the system with random noise. In this paper, we show that the standard SVD can be replaced by a higher order SVD (HOSVD), originally known as Tucker decomposition. HOSVD decomposes the dynamic texture as a multidimensional signal (tensor) without unfolding the video frames on column vectors. This is a more natural and flexible decomposition, since it permits us to perform dimension reduction in the spatial, temporal, and chromatic domain, while standard SVD allows for temporal reduction only. We show that for a comparable synthesis quality, the HOSVD approach requires, on average, five times less parameters than the standard SVD approach. The analysis part is more expensive, but the synthesis has the same cost as existing algorithms. Our technique is, thus, well suited to dynamic texture synthesis on devices limited by memory and computational power, such as PDAs or mobile phones.

Efficient Closed-Loop Schemes for MIMO-OFDM-Based WLANs

The single-input single-output (SISO) orthogonal frequency-division multiplexing (OFDM) systems for wireless local area networks (WLAN) defined by the IEEE 802.11a standard can support data rates up to 54 Mbps. In this paper, we consider deploying two transmit and two receive antennas to increase the data rate up to 108 Mbps. Applying our recent multiple-input multiple-output (MIMO) transceiver designs, that is, the geometric mean decomposition (GMD) and the uniform channel decomposition (UCD) schemes, we propose simple and efficient closed-loop MIMO-OFDM designs for much improved performance, compared to the standard singular value decomposition (SVD) based schemes as well as the open-loop V-BLAST (vertical Bell Labs layered space-time) based counterparts. In the explicit feedback mode, precoder feedback is needed for the proposed schemes. We show that the overhead of feedback can be made very moderate by using a vector quantization method. In the time-division duplex (TDD) mode where the channel reciprocity is exploited, our schemes turn out to be robust against the mismatch between the uplink and downlink channels. The advantages of our schemes are demonstrated via extensive numerical examples.

Is correction necessary when clinically determining quantitative cerebral perfusion parameters from multi-slice dynamic susceptibility contrast MR studies?**Preliminary results associated with this paper were presented at ISMRM 12th Scientific Meeting and Exhibition, Kyoto, Japan, 2004.

Several groups have modified the standard singular value decomposition (SVD) algorithm to produce delay-insensitive cerebral blood flow (CBF) estimates from dynamic susceptibility contrast (DSC) perfusion studies. However, new dependences of CBF estimates on bolus arrival times and slice position in multi-slice studies have been recently recognized. These conflicting findings can be reconciled by accounting for several experimental and algorithmic factors. Using simulation and clinical studies, the non-simultaneous measurement of arterial and tissue concentration curves (relative slice position) in a multi-slice study is shown to affect time-related perfusion parameters, e.g. arterial-tissue-delay measurements. However, the current clinical impact of relative slice position on amplitude-related perfusion parameters, e.g. CBF, can be expected to be small unless any of the following conditions are present individually or in combination: (a) high concentration curve signal-to-noise ratios, (b) small tissue mean transit times, (c) narrow arterial input functions or (d) low temporal resolution of the DSC image sequence. Recent improvements in magnetic resonance (MR) technology can easily be expected to lead to scenarios where these effects become increasingly important sources of inaccuracy for all perfusion parameter estimates. We show that using Fourier interpolated (high temporal resolution) residue functions reduces the systematic error of the perfusion parameters obtained from multi-slice studies.

An alternative viewpoint of the similarities and differences of SVD and FT deconvolution algorithms used for quantitative MR perfusion studies

Quantitative cerebral blood flow (CBF) values can be determined from residue function estimates obtained from magnetic resonance dynamic susceptibility contrast (DSC) perfusion studies using a variety of deconvolution approaches. However, there are significant differences between the CBF estimates obtained, differences that are not simply due to minor details of the implementation of the algorithms. The standard singular value decomposition (sSVD) shows a variation of CBF values with arterial-tissue delay (ATD) not present with the Fourier transform deconvolution algorithm. Fourier transform deconvolution and the newly suggested delay-invariant SVD algorithm implementations provide CBF estimates whose accuracy changes with tissue mean transit times (MTTs). Techniques combining sSVD with deliberate ATD manipulation have been proposed to compensate for this inaccuracy. Other studies indicate that CBF changes related to slice position in a multislice study, and other experimental factors, can be reduced using interpolative deconvolution algorithms. In this review, we use both time-domain and frequency-domain analysis to show the underlying theoretical relationships between these many approaches to obtain "the best" CBF estimate. This model allows us to better understand the similarities and differences, advantages and disadvantages between these variants of the deconvolution algorithms used in DSC perfusion studies.

A sequential algorithm of inverse heat conduction problems using singular value decomposition

This paper examines numerically and theoretically the application of truncated Singular Value Decomposition (SVD) in a sequential form. The Sequential SVD algorithm presents two tunable hyperparameters: the number of future temperature ( r ) and the rank of the truncated sensitivity matrix ( p ) . The regularization effect of both hyperparameters is consistent with the data filtering interpretation by truncated SVD (reported by Shenefelt [Internat. J. Heat Mass Transfer 45 (2002) 67]). This study reveals that the most suitable reduced rank is “one”. Under this assumption ( p = 1 ) , the sequential procedure proposed, presents several advantages with respect to the standard whole-domain procedure: The search of the optimum rank value is not required. The simplification of the model is the maximum that can be achieved. The unique tunable hyperparameter is the number of future temperatures, and a very simple algorithm is obtained. This algorithm has been compared to: Function Specification Method (FSM) proposed by Beck and the standard whole-domain SVD. In this comparative study, the parameters considered have been: the shape of the input, the noise level of measurement and the size of time step. In all cases, the FSM and sequential SVD algorithm give very similar results. In one case, the results obtained by the sequential SVD algorithm are clearly superior to the ones obtained by the whole-domain algorithm.

SVD technique for estimation of harmonic components in a power system: a statistical approach

The paper presents a statistical approach to the estimation of harmonic components in a power system. Mathematically it is based on singular value decomposition (SVD). Three different techniques are investigated: the standard averaged SVD, the total LS and double SVD. The results of numerical experiments illustrating the features of all these approaches are presented and discussed.

Singular value decomposition for identification of parameters in multicompartment analysis

Standard singular value decomposition is used in a practical approach to parameter identification in multicompartment analysis; a solution can be found even in not well-defined situations. The method is demonstrated on a simple 2-compartment glucose model and applied to an extended 7-compartment model.