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Register a new userA fast approach for overcomplete sparse decomposition based on smoothed L0 norm
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Added by
Massoud Babaie-Zadeh Dr
Publication date
2008
fields
Informatics Engineering
and research's language is
English
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In this paper, a fast algorithm for overcomplete sparse decomposition, called SL0, is proposed. The algorithm is essentially a method for obtaining sparse solutions of underdetermined systems of linear equations, and its applications include underdetermined Sparse Component Analysis (SCA), atomic decomposition on overcomplete dictionaries, compressed sensing, and decoding real field codes. Contrary to previous methods, which usually solve this problem by minimizing the L1 norm using Linear Programming (LP) techniques, our algorithm tries to directly minimize the L0 norm. It is experimentally shown that the proposed algorithm is about two to three orders of magnitude faster than the state-of-the-art interior-point LP solvers, while providing the same (or better) accuracy.
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In this paper, we propose a method to address the problem of source estimation for Sparse Component Analysis (SCA) in the presence of additive noise. Our method is a generalization of a recently proposed method (SL0), which has the advantage of directly minimizing the L0-norm instead of L1-norm, while being very fast. SL0 is based on minimization of the smoothed L0-norm subject to As=x. In order to better estimate the source vector for noisy mixtures, we suggest then to remove the constraint As=x, by relaxing exact equality to an approximation (we call our method Smoothed L0-norm Denoising or SL0DN). The final result can then be obtained by minimization of a proper linear combination of the smoothed L0-norm and a cost function for the approximation. Experimental results emphasize on the significant enhancement of the modified method in noisy cases.
Photoacoustic imaging (PAI) is a novel medical imaging modality that uses the advantages of the spatial resolution of ultrasound imaging and the high contrast of pure optical imaging. Analytical algorithms are usually employed to reconstruct the photoacoustic (PA) images as a result of their simple implementation. However, they provide a low accurate image. Model-based (MB) algorithms are used to improve the image quality and accuracy while a large number of transducers and data acquisition are needed. In this paper, we have combined the theory of compressed sensing (CS) with MB algorithms to reduce the number of transducer. Smoothed version of L0-norm (SL0) was proposed as the reconstruction method, and it was compared with simple iterative reconstruction (IR) and basis pursuit. The results show that S$ell_0$ provides a higher image quality in comparison with other methods while a low number of transducers were. Quantitative comparison demonstrates that, at the same condition, the SL0 leads to a peak-signal-to-noise ratio for about two times of the basis pursuit.
In this paper, a discrete LCT (DLCT) irrelevant to the sampling periods and without oversampling operation is developed. This DLCT is based on the well-known CM-CC-CM decomposition, that is, implemented by two discrete chirp multiplications (CMs) and one discrete chirp convolution (CC). This decomposition doesnt use any scaling operation which will change the sampling period or cause the interpolation error. Compared with previous works, DLCT calculated by direct summation and DLCT based on center discrete dilated Hermite functions (CDDHFs), the proposed method implemented by FFTs has much lower computational complexity. The relation between the proposed DLCT and the continuous LCT is also derived to approximate the samples of the continuous LCT. Simulation results show that the proposed method somewhat outperforms the CDDHFs-based method in the approximation accuracy. Besides, the proposed method has approximate additivity property with error as small as the CDDHFs-based method. Most importantly, the proposed method has perfect reversibility, which doesnt hold in many existing DLCTs. With this property, it is unnecessary to develop the inverse DLCT additionally because it can be replaced by the forward DLCT.
We offer a new approach to the information decomposition problem in information theory: given a target random variable co-distributed with multiple source variables, how can we decompose the mutual information into a sum of non-negative terms that quantify the contributions of each random variable, not only individually but also in combination? We derive our composition from cooperative game theory. It can be seen as assigning a fair share of the mutual information to each combination of the source variables. Our decomposition is based on a different lattice from the usual partial information decomposition (PID) approach, and as a consequence our decomposition has a smaller number of terms: it has analogs of the synergy and unique information terms, but lacks terms corresponding to redundancy. Because of this, it is able to obey equivalents of the axioms known as local positivity and identity, which cannot be simultaneously satisfied by a PID measure.
Sparse intersymbol-interference (ISI) channels are encountered in a variety of high-data-rate communication systems. Such channels have a large channel memory length, but only a small number of significant channel coefficients. In this paper, trellis-based equalization of sparse ISI channels is revisited. Due to the large channel memory length, the complexity of maximum-likelihood detection, e.g., by means of the Viterbi algorithm (VA), is normally prohibitive. In the first part of the paper, a unified framework based on factor graphs is presented for complexity reduction without loss of optimality. In this new context, two known reduced-complexity algorithms for sparse ISI channels are recapitulated: The multi-trellis VA (M-VA) and the parallel-trellis VA (P-VA). It is shown that the M-VA, although claimed, does not lead to a reduced computational complexity. The P-VA, on the other hand, leads to a significant complexity reduction, but can only be applied for a certain class of sparse channels. In the second part of the paper, a unified approach is investigated to tackle general sparse channels: It is shown that the use of a linear filter at the receiver renders the application of standard reduced-state trellis-based equalizer algorithms feasible, without significant loss of optimality. Numerical results verify the efficiency of the proposed receiver structure.
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