L+S matrix decomposition finds the low-rank (L) and sparse (S) components of a matrix M by solving the following
convex optimization problem: min‖L‖*L+S matrix decomposition finds the low-rank (L) and sparse (S) components of a matrix M by solving the following convex optimization problem: ‖L ‖* + λ‖S‖1, subject to M=L+S, where ‖L‖* is the nuclear-norm or sum of singular values of L and ‖S‖1 is the 11-norm| or sum of absolute values of S. This work presents the application of the L+S
decomposition to reconstruct incoherently undersampled dynamic MRI data as a superposition of a slowly or coherently changing background and sparse innovations. Feasibility of the method was tested in several accelerated dynamic MRI experiments including cardiac perfusion, time-resolved peripheral angiography and liver perfusion using Cartesian and radial sampling. The high acceleration and background separation enabled by L+S reconstruction promises to enhance spatial and temporal resolution and to enable background suppression without the need of subtraction or modeling.
In fluorescence microscopy, one can distinguish two kinds of imaging approaches, wide field and raster scan
microscopy, differing by their excitation and detection scheme. In both imaging modalities the acquisition is
independent of the information content of the image. Rather, the number of acquisitions N, is imposed by
the Nyquist-Shannon theorem. However, in practice, many biological images are compressible (or, equivalently
here, sparse), meaning that they depend on a number of degrees of freedom K that is smaller that their size N.
Recently, the mathematical theory of compressed sensing (CS) has shown how the sensing modality could take
advantage of the image sparsity to reconstruct images with no loss of information while largely reducing the number M of acquisition. Here we present a novel fluorescence microscope designed along the principles of CS. It uses a spatial light modulator (DMD) to create structured wide field excitation patterns and a sensitive point detector to measure the emitted fluorescence. On sparse fluorescent samples, we could achieve compression ratio N/M of up to 64, meaning that an image can be reconstructed with a number of measurements of only 1.5 % of its pixel number. Furthemore, we extend our CS acquisition scheme to an hyperspectral imaging system.
KEYWORDS: 3D image processing, Fourier transforms, Computing systems, Wavelets, Image processing, Electroluminescence, Analog electronics, Medical imaging, Computational mathematics, Video processing
In this paper, we present the first 3D discrete curvelet transform. This transform is an extension to the 2D transform described in Candes et al..1 The resulting curvelet frame preserves the important properties, such as parabolic scaling, tightness and sparse representation for singularities of codimension one. We describe three different implementations: in-core, out-of-core and MPI-based parallel implementations. Numerical results verify the desired properties of the 3D curvelets and demonstrate the efficiency of our implementations.
A widespread problem in the applied sciences is to recover an object of interest from a limited number of measurements. Recently, a series of exciting results have shown that it is possible to recover sparse (or approximately sparse) signals with high accuracy from a surprisingly small number of such measurements. The recovery procedure consists of solving a tractable convex program. Moreover, the procedure is robust to measurement error; adding a perturbation of size ε to the measurements will not induce a recovery error of more than a small constant times ε. In this paper, we will briefly overview these results, describe how stable recovery via convex optimization can be implemented in an efficient manner, and present some numerical results illustrating the practicality of the procedure.
Can we recover a signal f∈RN from a small number of linear measurements? A series of recent papers developed a collection of results showing that it is surprisingly possible to reconstruct certain types of signals accurately from limited measurements. In a nutshell, suppose that f is compressible in the sense that it is well-approximated by a linear combination of M vectors taken from a known basis Ψ. Then not knowing anything in advance about the signal, f can (very nearly) be recovered from about M log N generic nonadaptive measurements only. The recovery procedure is concrete and consists in solving a simple convex optimization program.
In this paper, we show that these ideas are of practical significance. Inspired by theoretical developments, we propose a series of practical recovery procedures and test them on a series of signals and images which are known to be well approximated in wavelet bases. We demonstrate that it is empirically possible to recover an object from about 3M-5M projections onto generically chosen vectors with an accuracy which is as good as that obtained by the ideal M-term wavelet approximation. We briefly discuss possible implications in the areas of data compression and medical imaging.
We outline digital implementations of two newly developed multiscale representation systems, namely, the ridgelet and curvelet transforms. We apply these digital transforms to the problem of restoring an image from noisy data and compare our results with those obtained via well established methods based on the thresholding of wavelet coefficients. We develop a methodology to combine wavelets together these new systems to perform noise removal by exploiting all these systems simultaneously. The results of the combined reconstruction exhibits clear advantages over any individual system alone. For example, the residual error contains essentially no visually intelligible structure: no structure is lost in the reconstruction.
The problem of recovering an input signal form noisy and linearly distorted data arises in many different areas of scientific investigation; e.g., noisy Radon inversion is a problem of special interest and considerable practical relevance in medical imaging. We will argue that traditional methods for solving inverse problems - damping of the singular value decomposition or cognate methods - behave poorly when the object to recover has edges. We apply a new system of representation, namely the curvelets in this setting. Curvelets provide near-optimal representations of otherwise smooth objects with discontinuities along smooth C2 edges. Inspired by some recent work on nonlinear estimation, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build a reconstruction based on the shrinkage of the noisy curvelet coefficients. This novel approach is shown to give a new theoretical understanding of the problem of edges in the Radon inversion problem.
Curvelets provide a new multiresolution representation with several features that set them apart from existing representations such as wavelets, multiwavelets, steerable pyramids, and so on. They are based on an anisotropic notion of scaling. The frame elements exhibit very high direction sensitivity and are highly anisotropic. In this paper we describe these properties and indicate why they may be important for both theory and applications.
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