Saturday, February 6, 2010 College of Design, North Building, Room 60 Justin Romberg, Georgia Tech Presenting at 9:15 AM FFTs on Spirals and Dynamic Updating for L1 Minimization We will discuss two algorithms related to compressive sampling for MRI. In the first part of the talk, we will show how samples along spirals in 3D frequency space can be arranged so that a fast, exact transform exists --- a ``spiral FFT''. This spiral FFT offers a significant speed increase over current interpolation-based non-uniform FFT algorithms. As both least-squares and compressive sensing reconstructions require many applications of this transform, this new algorithm is of general interest. In the second part of the talk, we will discuss recent progress on algorithms aimed at making compressive sampling ``dynamic''. We will show how the solutions to L1 optimization programs can be efficiently updated as 1) the signal we are measuring changes, and 2) new measurements are added, and stale ones are removed. The algorithms are based on homotopy methods, and are somewhat analogous to recursive least-squares in that they can be reduced to a series of low-rank updates. This is joint work with Chris Turnes and Salman Asif. Ted Trouard, University of Arizona Presenting at 10:10 AM Understanding Diffusion-Weighted Magnetic Resonance Imaging via Computational Modeling of the Diffusion of Water in Tissue Diffusion-weighted MRI provides a means to non-invasively measure diffusional motion of water in living tissue. The apparent diffusion coefficient (ADC) of water is a quantitative measure of this motion and has been shown to be sensitive to the tissue s cellular architecture and integrity. Dramatic changes in the ADC have been documented following ischemic stroke, where the ADC of water decreases 30-50% within the affected tissue. Changes in the ADC of water have also been observed in cancer following successful chemotherapy. Although these observations have been clinically useful, our understanding of the biophysical mechanisms responsible for them remains lacking. Measured ADC values are likely sensitive to many physical properties (e.g. cell size, membrane permeability, T2 relaxation time) as well as experimental parameters used to measure them (e.g. TE and diffusion time). We are implementing mathematical models of tissue water diffusion and simulations of DWMRI experiments to deepen our understanding of the biophysical mechanisms important to ADC measurements. Results from modeling will be discussed and compared to experimental measurements in humans and in cell cultures. 1 of 6
Luminita Vese, University of California, Los Angeles Presenting at 11:05 AM HARDI data denoising using vectorial total variation and logarithmic barrier This talk is devoted to two variational models for denoising HARDI (High Angular Resolution Diffusion Imaging) data arising in medical imaging. Diffusion imaging is a relatively new and powerful method to measure the three-dimensional profile of water diffusion in the body. These images can be used to reconstruct fiber directions and pathways in the living brain, providing detailed maps of fiber integrity and connectivity. HARDI data is a powerful new extension of diffusion imaging, which goes beyond the diffusion tensor imaging (DTI) model. Intensity data is given at every voxel and at any direction on the sphere. Unfortunately, HARDI data is usually highly contaminated with noise, depending on the b-value which is a tuning parameter preselected to collect the data. We propose here two variational methods to denoise HARDI data. The first one directly denoises the collected data S, while the second one denoises the so-called sadc (spherical Apparent Diffusion Coefficient), a field of radial functions derived from the data. These two quantities are related by an equation of the form S = S0 x exp(-b x sadc) (in the noise-free case). The theoretical analysis of the proposed models is presented, together with experimental results and comparisons for denoising synthetic and real HARDI data. This is joint work with Yunho Kim, Paul Thompson, Arthur Toga and Liang Zhan. Guowei Wei, Michigan State University Presenting at 12:00 PM Geometric evolution equations --- from image analysis to multiscale modeling The last twenty years have witnessed great interest in geometric evolution equations and their widely spread applications in image analysis, computer vision, material science, cellular membrane, and multiphase fluid flows. In this talk, we discuss the mathematical structure of a class of high order geometric evolution equations, some of the first that have ever proposed for image analysis. We show how such a structure can be utilized as a fundamental paradigm for mutliscale/multiphsyics modeling. Despite of tremendous challenges in mathematical analysis and numerical computation, this paradigm has promising applications to wide variety of fields, including fuel cells, nanofluidics, ion channels, viruses and biomolecular systems. 2 of 6
Oscar Bruno, Caltech University Presenting at 2:05 PM New Spectral Methods for General Domains with Applications to Medicine We present new methodologies for the numerical solution of Partial Differential Equations in general spatial domains. Based on a novel Fourier-Continuation (FC) method for the resolution of the Gibbs phenomenon, spectral surface-representation methods, fast high-order methods for evaluation of integral operators, and the well-known alternating direction approach, these methodologies give rise to fast and highly accurate frequency- and time-domain solvers for PDEs on general spatial domains - including challenging multiphysics problems in complex geometries often encountered in medical research and visualization - such as nonlinear acoustics (ultrasound), fluid-flow, diffusion, etc. Our differential solvers for time-dependent PDEs, for example, are based on use of the well known Alternating Direction Implicit (ADI) algorithm in conjunction with the FC method. Unlike previous alternating direction methods of order higher than one, which can only deliver unconditional stability for rectangular domains, the present spectral Fourier-Continuation Alternating-Direction (FC-AD) algorithm possesses the desirable property of unconditional stability for general domains; the computational time required by the algorithm to advance a solution by one time-step, in turn, grows in an essentially linear manner with the number of spatial discretization points used. A variety of applications of the FC-AD methodology to Parabolic, Elliptic and Hyperbolic PDEs demonstrate the unconditional stability and high-order convergence of the methods, as well the very significant improvements it can provide over the accuracy and speed resulting from other approaches. Wotao Yin, Rice University Presenting at 3:00 PM Compressive Sensing Algorithms for Fast and Accurate Imaging This talk covers some recent compressive sensing algorithms that are fast and reconstruct high quality images by taking advantages of parallel imaging, mutual information, image edge detection, and other optimization techniques. 3 of 6
Yoram Bresler, University of Illinois, Urbana Presenting at 4:00 PM Sampling theory and modeling in adaptive Cardiac MRI Sampling theory and modeling in adaptive Cardiac MRI Spatial and temporal resolution and image quality in dynamic MRI are severely limited by physical constraints on the rate of acquisition. Perhaps the most challenging example and important dynamic MRI application is cardiac MR (CMR) imaging. While CMRI is already a clinical tool, it does not offer sufficient temporal resolution, its spatial resolution is insufficient to discern small blocked arteries, and it requires long breath-holds that cannot be performed by infants or the sick. Similar challenges arise in functional brain imaging, in the imaging of the vocal tract during speech, and the imaging of joints during movement. Acquisition of additional dimensions of information -- such as spectral, or diffusion tensor, further exacerbates the problem. We describe an explicit model-based methodology enabling more than an order-of-magnitude reduction in the acquisition requirements in both single and multiple channel MRI, and providing guarantees on the quality of reconstruction subject to the modeling assumptions. Based on timesequential sampling theory, the approach uses the models to (i) design a minimum redundancy acquisition sequence; and (ii) reconstruct a movie (cine) of the object. By adapting the model to the imaged subject, both acquisition and reconstruction are adaptive. Phantom studies with known ground truth, and in-vivo CMR experiments demonstrate unprecedented spatial and temporal resolutions. Somantika Datta, Princeton University Presenting at 4:30 PM Deterministic compressed sensing for efficient image reconstruction The application of compressed sensing techniques for image reconstruction using deterministic sensing matrices will be discussed. Specifically, the sensing matrices used are constructed by either discrete chirps or second-order Reed-Muller sequences. Previous works by Applebaum et al. and Howard et al. used chirps and Reed-Muller sequences, respectively, for very sparse onedimensional signals and their experimental results are quite good. The speed and accuracy suffer when the degree of sparsity is not high, making it inapplicable for natural and medical images. We propose efficient reconstruction algorithms for images with deterministic compressed sensing. The steps of the reconstruction algorithms include: initial best approximation, a greedy algorithm for the nonzero locations, and a new approach in the least squares method. This is joint work with Kangyu Ni, Prasun Mahanti, Svetlana Roudenko and Douglas Cochran. 4 of 6
Sunday, February 7, 2010 College of Design, North Building, Room 60 Jim Pipe, Barrow Neurological Institute Presenting at 9:00 AM Reconstructing data from Non-Cartesian Trajectories MRI data are collected along a series of paths, or trajectories, through the Fourier domain of the desired image. Although most clinical images are collected in a Cartesian fashion, one row at a time, there are several advantages of employing Non-Cartesian trajectories for data sampling. The first half of this talk will discuss these reasons, and in particular the aspect of SNR and scan time. The second part will present some recent work by our lab in collecting and reconstructing data on a novel Non-Cartesian trajectory. Ed Walsh, Brown University Presenting at 9:55 AM Quantitative MR Imaging Using Parameter Assessment by Retrieval from Signal Encoding (PARSE) Image reconstruction methodology as typically implemented on clinical MRI scanners for use with Cartesian and other k-space trajectories (e.g. spiral) fails to account for signal evolution arising from causes other than gradient fields applied for spatial encoding. Consequences include geometric distortion and signal loss using echo planar imaging, or blurring with spiral trajectories in regions containing bulk susceptibility gradients. The PARSE method uses a nonlinear optimization process for generating an inverse solution to the MR signal equation to produce parametric maps of magnetization, signal decay rate (R2 and R2*) and frequency (field variation). By including frequency as an optimized parameter, the technique provides correction for geometric distortion. A rosette k-space trajectory is used to improve solution conditioning by virtue of its frequent sampling of low spatial frequency k-space, where most of the signal energy is contained. The discussion will include a description of the current signal model and optimization process and a discussion of the cost function and discrete evaluation model behavior. 5 of 6
Lingling Pu, University of Arizona Presenting at 11:00 AM Application of Compressive Sensing to Magnetic Resonance Imaging Compressive sensing (CS) is a new framework which illustrates that signals can be reconstructed from much fewer samples than what was previously suggested by the Nyquist theorem. CS has been applied to many areas of signal processing including medical imaging. In this talk, several applications of CS to MRI will be discussed. Techniques that exploit dependencies among sparse domain coefficients and utilize multiple sparse representations will be introduced. Results of a study using CS reconstructed T1-weighted images of the whole brain to investigate age-related decreases in gray matter volume via Voxel-based Morphometry (VBM) analysis will also be presented. Rodrigo Platte, Arizona State University Presenting at 11:30 AM Signal reconstruction with radial basis functions Radial basis functions are often used in the recovery of functions in multi-dimensions from scattered samples. In this talk we will explore convergence rates and stability of such methods in signal reconstruction. Of particular interest are the reconstruction and compression of images using multi-scale radial basis functions. 6 of 6