Carlos Sing-Long

Unbiased Risk Estimates for Singular Value Thresholding and Spectral Estimators

In an increasing number of applications, it is of interest to recover an approximately low-rank data matrix from noisy observations. This paper develops an unbiased risk estimate-holding in a Gaussian model-for any spectral estimator obeying some mild regularity assumptions. In particular, we give an unbiased risk estimate formula for singular value thresholding (SVT), a popular estimation strategy that applies a soft-thresholding rule to the singular values of the noisy observations. Among other things, our formulas offer a principled and automated way of selecting regularization parameters in a variety of problems. In particular, we demonstrate the utility of the unbiased risk estimation for SVT-based denoising of real clinical cardiac MRI series data. We also give new results concerning the differentiability of certain matrix-valued functions.

Deterministic matrices matching the compressed sensing phase transitions of Gaussian random matrices

In compressed sensing, one takes samples of an N-dimensional vector using an matrix A, obtaining undersampled measurements . For random matrices with independent standard Gaussian entries, it is known that, when is k-sparse, there is a precisely determined phase transition: for a certain region in the (,)-phase diagram, convex optimization typically finds the sparsest solution, whereas outside that region, it typically fails. It has been shown empirically that the same property—with the same phase transition location—holds for a wide range of non-Gaussian random matrix ensembles. We report extensive experiments showing that the Gaussian phase transition also describes numerous deterministic matrices, including Spikes and Sines, Spikes and Noiselets, Paley Frames, Delsarte-Goethals Frames, Chirp Sensing Matrices, and Grassmannian Frames. Namely, for each of these deterministic matrices in turn, for a typical k-sparse object, we observe that convex optimization is successful over a region of the phase diagram that coincides with the region known for Gaussian random matrices. Our experiments considered coefficients constrained to for four different sets , and the results establish our finding for each of the four associated phase transitions.

Application of the fractional Fourier transform to image reconstruction in MRI

The classic paradigm for MRI requires a homogeneous B0 field in combination with linear encoding gradients. Distortions are produced when the B0 is not homogeneous, and several postprocessing techniques have been developed to correct them. Field homogeneity is difficult to achieve, particularly for short‐bore magnets and higher B0 fields. Nonlinear magnetic components can also arise from concomitant fields, particularly in low‐field imaging, or intentionally used for nonlinear encoding. In any of these situations, the second‐order component is key, because it constitutes the first step to approximate higher‐order fields. We propose to use the fractional Fourier transform for analyzing and reconstructing the objects magnetization under the presence of quadratic fields. The fractional fourier transform provides a precise theoretical framework for this. We show how it can be used for reconstruction and for gaining a better understanding of the quadratic field‐induced distortions, including examples of reconstruction for simulated and in vivo data. The obtained images have improved quality compared with standard Fourier reconstructions. The fractional fourier transform opens a new paradigm for understanding the MR signal generated by an object under a quadratic main field or nonlinear encoding. Magn Reson Med, 2012.

Single-photon sampling architecture for solid-state imaging sensors

Significance We propose a highly compressed readout architecture for arrays of imaging sensors capable of detecting individual photons. By exploiting sparseness properties of the input signal, our architecture can provide the same information content as conventional readout designs while using orders of magnitude fewer output channels. This is achieved using a unique interconnection topology based on group-testing theoretical considerations. Unlike existing designs, this promises a low-cost sensor with high fill factor and high photon sensitivity, potentially enabling increased spatial and temporal resolution in a number of imaging applications, including positron-emission tomography and light detection and ranging. Advances in solid-state technology have enabled the development of silicon photomultiplier sensor arrays capable of sensing individual photons. Combined with high-frequency time-to-digital converters (TDCs), this technology opens up the prospect of sensors capable of recording with high accuracy both the time and location of each detected photon. Such a capability could lead to significant improvements in imaging accuracy, especially for applications operating with low photon fluxes such as light detection and ranging and positron-emission tomography. The demands placed on on-chip readout circuitry impose stringent trade-offs between fill factor and spatiotemporal resolution, causing many contemporary designs to severely underuse the technology’s full potential. Concentrating on the low photon flux setting, this paper leverages results from group testing and proposes an architecture for a highly efficient readout of pixels using only a small number of TDCs. We provide optimized design instances for various sensor parameters and compute explicit upper and lower bounds on the number of TDCs required to uniquely decode a given maximum number of simultaneous photon arrivals. To illustrate the strength of the proposed architecture, we note a typical digitization of a 60 × 60 photodiode sensor using only 142 TDCs. The design guarantees registration and unique recovery of up to four simultaneous photon arrivals using a fast decoding algorithm. By contrast, a cross-strip design requires 120 TDCs and cannot uniquely decode any simultaneous photon arrivals. Among other realistic simulations of scintillation events in clinical positron-emission tomography, the above design is shown to recover the spatiotemporal location of 99.98% of all detected photons.

The fractional Fourier transform and quadratic field magnetic resonance imaging

The fractional Fourier transform (FrFT) is revisited in the framework of strongly continuous periodic semigroups to restate known results and to explore new properties of the FrFT. We then show how the FrFT can be used to reconstruct Magnetic Resonance (MR) images acquired under the presence of quadratic field inhomogeneity. Particularly, we prove that the order of the FrFT is a measure of the distortion in the reconstructed signal. Moreover, we give a dynamic interpretation to the order as time evolution of a function. We also introduce the notion of @r-@a space as an extension of the Fourier or k-space in MR, and we use it to study the distortions introduced in two common MR acquisition strategies. We formulate the reconstruction problem in the context of the FrFT and show how the semigroup theory allows us to find new reconstruction formulas for discrete sampled signals. Finally, the results are supplemented with numerical examples that show how it performs in a standard 1D MR signal reconstruction.

TRIO a Technique for Reconstruction Using Intensity Order: Application to Undersampled MRI

Long acquisition times are still a limitation for many applications of magnetic resonance imaging (MRI), specially in 3-D and dynamic imaging. Several undersampling reconstruction techniques have been proposed to overcome this problem. These techniques are based on acquiring less samples than specified by the Nyquist criterion and estimating the nonacquired data by using some sort of prior information. Most of these reconstruction methods use prior information based on estimations of the pixel intensities of the images and therefore they are prone to introduce spatial or temporal blurring. Instead of using the pixel intensities, we propose to use information that allows us to sort the pixels of an image from darkest to brightest. The set of order relations which sort the pixels of an image has been called intensity order. The intensity order of an image can be estimated from low-resolution images, adjacent slices in volumetric acquisitions, temporal correlation in dynamic sequences or from prior reconstructions. Our technique for reconstruction using intensity order (TRIO) consists of looking for an image that satisfies the intensity order and minimizes the discrepancy between the acquired and reconstructed data. Results show that TRIO can effectively reconstruct 2-D-cine cardiac MR images (under-sampling factor of 4), estimating correctly the temporal evolution of the objects. Furthermore, TRIO is used as a second stage reconstruction after reconstructing with other techniques, keyhole, sliding window and k-t BLAST, to estimate the order information. In all cases the images are improved by TRIO.

Level set segmentation with shape prior knowledge using intrinsic rotation, translation and scaling alignment

Level set-based algorithms have been extensively used for medical image segmentation. Despite their relative success, standard level set segmentations tend to fail when images are severely corrupted or in poorly defined regions. This problem has been tackled incorporating shape prior knowledge, i.e. restricting the evolving curve to a distribution of shapes pre-defined during a training process. Such shape restriction needs to incorporate invariance to translation, rotations and scaling. The common approach for this is to solve a registration problem during the curve evolution, i.e. finding optimal registration parameters. This procedure is slow and produces variable results depending on the order in which the registration parameters were optimized. To overcome this issue Cremers et al. (2006) proposed an intrinsic alignment formulation, which is a normalized coordinate system for each shape, thus avoiding the optimization step to account for the registration. Nevertheless, their proposed solution considered only scaling and translation, but not rotations which are critical for medical imaging applications. We added rotations to this intrinsic alignment, using eigenvalues and eigenvector matrices of the covariance matrix of each shape, and we incorporated them into the evolution equation, allowing us to use shape priors in complex segmentation problems. We tested our algorithm combined with a Chan-Vese functional in synthetic images and in 2D right ventricle MRI.

A nonlinear ordinary differential equation associated with the quantum sojourn time

We study a nonlinear ordinary differential equation on the half-line, with the Dirichlet boundary condition at the origin. This equation arises when studying the local maxima of the sojourn time for a free quantum particle whose states belong to an adequate subspace of the unit sphere of the corresponding Hilbert space. We establish several results concerning the existence and asymptotic behavior of the solutions.

3D axial and circumferential wall shear stress from 4D flow MRI data using a finite element method and a laplacian approach

To decompose the 3D wall shear stress (WSS) vector field into its axial (WSSA) and circumferential (WSSC) components using a Laplacian finite element approach.