Set Of Linear Measurements Research Articles

Extensions of Real Atomic Gauges for Complex Signal Recovery

Solving discrete linear inverse problems is one of the cornerstones of modern science and engineering. In abstract terms, these problems seek to recover an unknown vector from an incomplete set of linear measurements. When the object is a sparse convex combination of a known collection of atoms, the gauge associated to the convex hull of this collection, i.e., the atomic gauge, can be minimized subject to data consistency constraints to attempt to recover the original vector. In some practical applications, such as magnetic resonance imaging, the vector is complex-valued and it is the magnitude vector, i.e., the vector containing the magnitude of the components, that is a sparse convex combination of known real-valued atoms. To apply the atomic gauge to this setting, we propose extending the collection of real-valued atoms by considering their modulations by a collection of suitable phases. Furthermore, under minor assumptions, we provide computationally tractable expressions to evaluate both the gauge associated to the modulated set of atoms and its proximal map. Our results show the complexity of using the gauge associated to a collection of modulated atoms is comparable to that of using a collection of real-valued atoms.

Stable Analysis of Compressive Principal Component Pursuit

Compressive principal component pursuit (CPCP) recovers a target matrix that is a superposition of low-complexity structures from a small set of linear measurements. Pervious works mainly focus on the analysis of the existence and uniqueness. In this paper, we address its stability. We prove that the solution to the related convex programming of CPCP gives an estimate that is stable to small entry-wise noise. We also provide numerical simulation results to support our result. Numerical results show that the solution to the related convex program is stable to small entry-wise noise under board condition.

Exact Solution Analysis of Strongly Convex Programming for Principal Component Pursuit

In this paper, we address strongly convex programming for principal component analysis, which recovers a target matrix that is a superposition of low-complexity structures from a small set of linear measurements. In this paper, we firstly provide sufficient conditions under which the strongly convex models lead to the exact low-rank matrix recovery. Secondly, we also give suggestions that will guide us how to choose suitable parameters in practical algorithms. Finally, the proposed result is extended to the principal component pursuit with reduced linear measurements and we provide numerical experiments.

Principal Component Pursuit with Weighted Nuclear Norm

Principal Component Pursuit (PCP) recovers low-dimensional structures from a small set of linear measurements, such as low rank matrix and sparse matrix. Pervious works mainly focus on exact recovery without additional noise. However, in many applications the observed measurements are corrupted by an additional white Gaussian noise (AWGN). In this paper, we model the recovered matrix the sum a low-rank matrix, a sparse matrix and an AWGN. We propose a weighted PCP for the recovery matrix, which is solved by alternating direction method. Numerical results show that the reconstructions performance of weighted PCP outperforms the classical PCP in term of accuracy.

Three-Dimensional Assessment of Early Surgical Outcome in Repaired Unilateral Cleft Lip and Palate: Part 1. Nasal Changes

To evaluate three-dimensional nasal morphology following primary reconstruction in children with unilateral cleft lip and palate relative to contemporaneous noncleft data. Prospective, cross-sectional, controlled study. Glasgow Dental Hospital and School, Faculty of Medicine, Glasgow University. Two groups of 3-year-old children (21 with unilateral cleft lip and palate and 96 controls) with facial images taken using a three-dimensional, vision-based capture technique. Three-dimensional images of the face were reflected so the cleft was on the left side to create a homogeneous group for statistical analysis. Three-dimensional coordinates of anthropometric landmarks were extracted from facial images by a single operator. A set of linear measurements was used to compare cleft and control subjects on right and left sides, adjusting for sex differences. The mean nasal base width and the width of the nostril floor on right and left sides differed significantly between control and unilateral cleft lip and palate groups. The measurements were greater in children with unilateral cleft lip and palate. The differences in the mean nasal height and mean nasal projection between the groups were not statistically significant. Mean columellar lengths were different between the left and right sides in children with unilateral cleft lip and palate. There were significant nasal deformities following the surgical repair of unilateral cleft lip and palate.

Three-Dimensional Assessment of Early Surgical Outcome in Repaired Unilateral Cleft Lip and Palate: Part 2. Lip Changes

To evaluate three-dimensional lip morphology, following primary reconstruction in children with unilateral cleft lip and palate relative to contemporaneous noncleft data. Prospective, cross-sectional, controlled study. Glasgow Dental Hospital and School, University of Glasgow, U.K. Two groups of 3-year-old children (21 with unilateral cleft lip and palate and 96 controls) with facial images taken using a three-dimensional vision-based capture technique. Three-dimensional images of the face were reflected so the cleft was on the left side to create a homogeneous group for statistical analysis. Three-dimensional coordinates of anthropometric landmarks were extracted from facial images. Three-dimensional, generalized Procrustes superimposition was implemented and a set of linear measurements were used to compare cleft and control subjects for right and left sides, adjusting for sex differences. Crista philtri on both the cleft and noncleft sides were displaced laterally and posteriorly; there was also a statistically significant increase in philtrum width. No significant differences between cleft and control regarding the cutaneous height of the upper lip. The lip in the cleft patients was flatter than in the noncleft individuals, with less prominence of labialis superioris. Stereophotogrammetry allows detection of residual dysmorphology following cleft repair. There was significant increase of the philtrum width. The lip appeared flatter and more posterior displaced in unilateral cleft lip and palate patients compared with controls.

On Spectral Learning

In this paper, we study the problem of learning a matrix W from a set of linear measurements. Our formulation consists in solving an optimization problem which involves regularization with a spectral penalty term. That is, the penalty term is a function of the spectrum of the covariance of W. Instances of this problem in machine learning include multi-task learning, collaborative filtering and multi-view learning, among others. Our goal is to elucidate the form of the optimal solution of spectral learning. The theory of spectral learning relies on the von Neumann characterization of orthogonally invariant norms and their association with symmetric gauge functions. Using this tool we formulate a representer theorem for spectral regularization and specify it to several useful example, such as Schatten p-norms, trace norm and spectral norm, which should proved useful in applications.

Optimal reconstruction of signals from a set of linear measurements

The problem of statistically optimal reconstruction of a signal based on its arbitrary linear projection is considered. The general solution to the problem is obtained in the form of the optimal reverse filter. The equation defining the filter represents an extension of the Wiener-Hopf equation to the case when N observations are a noise-bearing projection of a continuous random signal onto the space R N .

A modified technique for evaluating apical base relationships

1. The ANB difference is not always an accurate method of establishing the actual amount of apical base divergence. In addition to the geometric shortcomings previously pointed out, the results of this study indicate that the ANB landmarks are subject to growth and treatment mechanics which cause corresponding variations in the measurements used. 2. The AXD method described in this study demonstrates a method of relating apical bases which eliminates two of the variables, nasion and point B. The AXD angle is a more critical evaluation and was found to correspond significantly with a linear measurement of the actual apical base difference. 3. This study and others have demonstrated conclusively that angular measurements cannot compensate for divergence of apical bases resulting from variations in vertical facial height. For this reason, a set of linear measurements was proposed which would offer an accurate method of evaluating the pre- and posttreatment changes taking place. 4. The linear measurements introduced in this study demonstrate an accurate method of assessing apical base relationships. The use of these measurements provides a critical evaluation of treatment and places proper emphasis on favorable growth and treatment mechanics. 5. The study of the pre- and posttreatment linear measurement concerning point A determined that this landmark remained remarkably constant during orthodontic treatment. It would appear from this study that treatment mechanics for Class II cases do not actively relocate point A but actually hold it in position while the middle face is growing forward. 6. The A-D' measurement is concerned with the actual distance, in millimeters, between the apical base landmarks. In the fifty cases studied, the mean pretreatment measurement of A-D' was reduced 15.8 per cent as the Class II malocclusion was corrected. The mean pretreatment ANB underwent a reduction of 43.1 per cent on these same cases. While the A-D' pre- and posttreatment change may not be as dramatic as the ANB evaluation, the major point to consider is the critical degree of accuracy which this measurement brings into an analysis.

Reconstructions of signals of a known class from a given set of linear measurements

This paper investigates the error in reconstructions of a signal based on a given finite set of linear measurements, and presents two schemes that, if there is available a priori knowledge of the class of signals of which the measured signal is a member, can achieve a reduction of this error beyond the best that could be done without such knowledge. The error measure used is the supremum over the class of the \mathcal{L}_2 distance between a signal and its reconstruction. The essence of the proposed reconstruction techniques is a coordinate transformation from the sampling subspace to a new reconstruction subspace known to be efficient for representation of signals of the given class. This study applies the theory of extremal subspaces and n widths of signal classes originated by Kolmogorov. Results are applied to the much studied class of time-concentrated band-limited signals. The measurement process is here assumed to be the convenient one of Nyquist rate time sampling. For this problem, plots of the error bounds and of several test functions and their reconstructions are presented, both for the proposed reconstructions, and for conventional cardinal-sampling-theorem reconstructions.

Eigenfunctions of Operator-Valued Analytic Functions

This paper is a sequel to [2], whose primary purposes are to clarify and generalize the concept introduced there of an eigenfunction of an inner function, and to answer questions raised there concerning the equivalence of several possible forms of the definition. A new definition, proposed here, leads to a complete characterization of the eigenfunctions of Potapov inner functions of normal operators, and the result is more satisfactory than [2, Theorem 3.4], although the latter is used strongly in the proof.Let be an inner function in the sense of Lax; i.e., is almost everywhere (a.e.) a unitary operator on a separable Hilbert space and belongs weakly to the Hardy class H2. An analytic function q (which will have to be a scalar inner function) was defined to be an eigenfunction of if the set of z in the disk {z: |z| ≦ 1} for which is invertible is a set of linear measure 0 on the circle {z: |z| = 1}.