Sparse approximation

About: Sparse approximation is a research topic. Over the lifetime, 18037 publications have been published within this topic receiving 497739 citations. The topic is also known as: Sparse approximation.

k-t SPARSE: High frame rate dynamic MRI exploiting spatio-temporal sparsity

Abstract: M. Lustig, J. M. Santos, D. L. Donoho, J. M. Pauly Electrical Engineering, Stanford University, Stanford, CA, United States, Statistics, Stanford University, Stanford, CA, United States Introduction Recently rapid imaging methods that exploit the spatial sparsity of images using under-sampled randomly perturbed spirals and non-linear reconstruction have been proposed [1,2]. These methods were inspired by theoretical results in sparse signal recovery [1-5] showing that sparse or compressible signals can be recovered from randomly under-sampled frequency data. We propose a method for high frame-rate dynamic imaging based on similar ideas, now exploiting both spatial and temporal sparsity of dynamic MRI image sequences (dynamic scene). We randomly under-sample k-t space by random ordering of the phase encodes in time (Fig. 1). We reconstruct by minimizing the L1 norm of a transformed dynamic scene subject to data fidelity constraints. Unlike previously suggested linear methods [7, 8], our method does not require a known spatio-temporal structure nor a training set, only that the dynamic scene has a sparse representation. We demonstrate a 7-fold frame-rate acceleration both in simulated data and in vivo non-gated Cartesian balanced-SSFP cardiac MRI . Theory Dynamic MR images are highly redundant in space and time. By using linear transformations (such as wavelets, Fourier etc.), we can represent a dynamic scene using only a few sparse transform coefficients. Inadequate sampling of the spatial-frequency -temporal space (k-t space) results in aliasing in the spatial -temporal-frequency space (x-f space). The aliasing artifacts due to random under-sampling are incoherent as opposed to coherent artifacts in equispaced under sampling. More importantly the artifacts are incoherent in the sparse transform domain. By using the non-linear reconstruction scheme in [1-5] we can recover the sparse transform coefficients and as a consequence, recover the dynamic scene. We exploit sparsity by constraining our reconstruction to have a sparse representation and be consistent with the measured data by solving the constrained optimization problem: minimize ||Ψm||1 subject to: ||Fm – y||2 < e. Here m is the dynamic scene, Ψ transforms the scene into a sparse representation, F is randomized phase encode ordering Fourier matrix, y is the measured k-space data and e controls fidelity of the reconstruction to the measured data. e is usually set to the noise level. Methods For dynamic heart imaging, we propose using the wavelet transform in the spatial dimension and the Fourier transform in the temporal. Wavelets sparsify medical images [1] whereas the Fourier transform sparsifies smooth or periodic temporal behavior. Moreover, with random k-t sampling, aliasing is extremely incoherent in this particular transform domain. To validate our approach we considered a simulated dynamic scene with periodic heart-like motion. A random phase-encode ordered Cartesian acquisition (See Fig. 2) was simulated with a TR=4ms, 64 pixels, acquiring a total of 1024 phase encodes (4.096 sec). The data was reconstructed at a frame rate of 15FPS (a 4-fold acceleration factor) using the L1 reconstruction scheme implemented with non-linear conjugate gradients. The result was compared to a sliding window reconstruction (64 phase encodes in length). To further validate our method we considered a Cartesian balanced-SSFP dynamic heart scan (TR=4.4, TE=2.2, α=60°, res=2.5mm, slice=9mm). 1152 randomly ordered phase encodes (5sec) where collected and reconstructed using the L1 scheme at a 7-fold acceleration (25FPS). Result was compared to a sliding window (64 phase encodes) reconstruction. The experiment was performed on a 1.5T GE Signa scanner using a 5inch surface coil. Results and discussion Figs. 2 and 3 illustrate the simulated phantom and actual dynaic heart scan reconstructions. Note, that even at 4 to 7-fold acceleration, the proposed method is able to recover the motion, preserving the spatial frequencies and suppressing aliasing artifacts. This method can be easily extended to arbitrary trajectories and can also be easily integrated with other acceleration methods such as phase constrained partial k-space and SENSE [1]. In the current, Matlab implementation we are able to reconstruct a 64x64x64 scene in an hour. This can be improved by using newly proposed reconstruction techniques [5,6]. Previously proposed linear methods [7,8] exploit known or measured spatio-temporal structure. The advantage of the proposed method is that the signal need not have a known structure, only sparsity, which is a very realistic assumption in dynamic medical images [1,7,8]. Therefore, a training set is not required. References [1] Lustig et al. 13th ISMRM 2004:p605 [2] Lustig et al. ” Rapid MR Angiography ...” Accepted SCMR06’ [3] Candes et al. ”Robust Uncertainty principals". Manuscript. [4] Donoho D. “Compressesed Sensing”. Manuscript. [5] Candes et al. “Practical Signal Recovery from Random Projections” Manuscript. [6] M. Elad, "Why Simple Shrinkage is Still Relevant?" Manuscript. [7] Tsao et al.. Magn Reson Med. 2003 Nov;50(5):1031-42. [8] Madore et al. Magn Reson Med. 1999 Nov;42(5):813-28. Figure 2: Simulated dynamic data. (a) The transform domain of the cross section is truly sparse. (b) Ground truth crosssection. (c) L1 reconstruction from random phase encode ordering, 4-fold acceleration (d) Sliding window (64) reconstruction from random phase encode ordering.. Figure 1: (a) Sequential phase encode ordering. (b) Random Phase encode ordering. The k-t space is randomly sampled, which enables recovery of sparse spatio-temporal dynamic scenes using the L1 reconstruction.

Gradient-based algorithms with applications to signal-recovery problems.

Abstract: This chapter presents in a self-contained manner recent advances in the design and analysis of gradient-based schemes for specially structured smooth and nonsmooth minimization problems. We focus on the mathematical elements and ideas for building fast gradient-based methods and derive their complexity bounds. Throughout the chapter, the resulting schemes and results are illustrated and applied on a variety of problems arising in several specific key applications such as sparse approximation of signals, total variation-based image processing problems, and sensor location problems.

Wavelet-like bases for the fast solutions of second-kind integral equations

Abstract: A class of vector-space bases is introduced for the sparse representation of discretizations of integral operators An operator with a smooth, nonoscillatory kernel possessing a finite number of singularities in each row or column is represented in these bases as a sparse matrix, to high precision A method is presented that employs these bases for the numerical solution of second-kind integral equations in time bounded by $O(n\log ^2 n)$, where n is the number of points in the discretization Numerical results are given which demonstrate the effectiveness of the approach, and several generalizations and applications of the method are discussed

Kernel sparse representation for image classification and face recognition

Abstract: Recent research has shown the effectiveness of using sparse coding(Sc) to solve many computer vision problems. Motivated by the fact that kernel trick can capture the nonlinear similarity of features, which may reduce the feature quantization error and boost the sparse coding performance, we propose Kernel Sparse Representation(KSR). KSR is essentially the sparse coding technique in a high dimensional feature space mapped by implicit mapping function. We apply KSR to both image classification and face recognition. By incorporating KSR into Spatial Pyramid Matching(SPM), we propose KSRSPM for image classification. KSRSPM can further reduce the information loss in feature quantization step compared with Spatial Pyramid Matching using Sparse Coding(ScSPM). KSRSPM can be both regarded as the generalization of Efficient Match Kernel(EMK) and an extension of ScSPM. Compared with sparse coding, KSR can learn more discriminative sparse codes for face recognition. Extensive experimental results show that KSR outperforms sparse coding and EMK, and achieves state-of-the-art performance for image classification and face recognition on publicly available datasets.

Compressive light field photography using overcomplete dictionaries and optimized projections

Abstract: Light field photography has gained a significant research interest in the last two decades; today, commercial light field cameras are widely available. Nevertheless, most existing acquisition approaches either multiplex a low-resolution light field into a single 2D sensor image or require multiple photographs to be taken for acquiring a high-resolution light field. We propose a compressive light field camera architecture that allows for higher-resolution light fields to be recovered than previously possible from a single image. The proposed architecture comprises three key components: light field atoms as a sparse representation of natural light fields, an optical design that allows for capturing optimized 2D light field projections, and robust sparse reconstruction methods to recover a 4D light field from a single coded 2D projection. In addition, we demonstrate a variety of other applications for light field atoms and sparse coding, including 4D light field compression and denoising.