Dimensionality reduction via linear random projections are used in numerous applications including data streaming, information retrieval, data mining, and compressive sensing (CS). While CS has traditionally relied on normal random projections, corresponding to ℓ2distance preservation, a large body of work has emerged for applications where ℓ1approximate distances may be preferred. Dimensionality reduction in ℓ1use Cauchy random projections that multiply the original data matrix B ∈ 葷D×nwith a Cauchy random matrix R ∈ 葷n×k(k « min(n,D)), resulting in a projected matrix C ∈ 葷D×k. This paper focuses on developing signal reconstruction algorithms from Cauchy random projections, where the large suite of reconstruction algorithms developed in compressive sensing perform poorly due to the lack of finite second-order statistics in the projections. In particular, a set of regularized coordinate-descent Myriad regression based reconstruction algorithms are developed using, both l0and Lorentzian norms as sparsity inducing terms. The l0-regularized algorithm shows superior performance to other standard approaches. Simulations illustrate and compare accuracy of reconstruction.

In this paper, we address the compressive sensing signal reconstruction problem by solving an ℓ0-regularized Least Absolute Deviation (LAD) regression problem. A coordinate descent algorithm is developed to solve this ℓ0-LAD optimization problem leading to a two-stage operation for signal estimation and basis selection. In the first stage, an estimation of the sparse signal is found by a weighted median operator acting on a shifted-and-scaled version of the measurement samples with weights taken from the entries of the projection matrix. The resultant estimated value is then passed to the second stage that tries to identify whether the corresponding entry is relevant or not. This stage is achieved by a hard threshold operator with adaptable thresholding parameter that is suitably tuned as the algorithm progresses.

In this paper, a new framework for confocal microscopy based on the novel theory of compressive sensing is proposed. Unlike wide field microscopy or conventional parallel beam confocal imaging systems that use charge-coupled devices (CCD) as acquisition devices in addition to complex mechanical scanning system, the proposed compressive confocal microscopy is a kind of parallel beam confocal imaging system which exploits the rich theory of compressive sensing by using a single pixel detector and a digital micromirror device (DMD) to capture linear projections of the in-focus image. With the proposed system, confocal imaging of high optical sectioning ability can be achieved at sub-Nyquist sampling rates. Theoretical analysis, simulations and experimental results are shown to demonstrate the characteristics and potential of the proposed approach.

In this work, a new family of nonlinear filters based on support vector machine is presented. This new filter, called support vector machine filter (SVMF), is based on the general concept of binary filters and machine learning theory. Two applications that show the potential of these filters are designed. As a first application, the proposed filter is used as an impulsive noise image denoising. The second application presents a new edge detection structure using a different point of view from the traditional ones. The results obtained for the applications at hand show that the proposed filter outperforms center weighted median in the image denoising task and the traditional edge detectors. The proposed filter can be successfully applied for the processing of images corrupted with impulsive noise while maintaining the visual quality and a low reconstruction error.

In this paper, ultrawideband (UWB) channel estimation based on the novel theory of compressive sensing (CS) is developed. The proposed approach relies on the fact that transmitting an ultra-short UWB pulse through a multipath channel leads to a received UWB signal that can be approximated by a linear combination of a few atoms from a pre-defined dictionary, yielding thus a sparse representation of the received signal. The CS reconstruction capabilities are exploited to recover the composite pulse-multipath channel from a reduced set of random projections using the matching pursuit algorithm. This reconstructed signal is subsequently used as a referent template in a correlator based detector. Extensive simulations show that for different propagation scenarios and UWB communication environments, the CS detector outperforms traditional correlators using just 1/3 of the sampling rate leading thus to a reduced use of analog-to-digital resources in the channel estimation stage.

The emerging theory of compressed sensing (CS) has led to the remarkable result that signals having a sparse representation in some known basis can be represented (with high probability) by a small sample set, taken from random projections of the signal. Notably, this sample set can be smaller than that required by the ubiquitous Nyquist sampling theorem. Much like the generalized Nyquist sampling theorem dictates that the sampling rate can be further reduced for the representation of bandlimited signals, this paper points to similar results for the sampling density in CS. In particular, it is shown that if additional spectral information of the underlying sparse signals is known, colored random projections can be used in CS in order to further reduce the number of measurements needed. Such a priori information is often available in signal processing applications and communications. Algorithms to design colored random projection vectors are developed. Further, an adaptive CS sampling method is developed for applications where non-uniform spectral characteristics of the signal are expected but are not known a priori.

This paper proposes a method of normalization of cDNA microarray data. This approach uses all gene data to estimate the normalization parameters and it obtains these parameters using an algorithm based on least absolute deviation (LAD) regression. This method normalizes iteratively each microarray set after the estimation of the normalization parameters which uses a LAD regression algorithm. The normalization method has a robust performance since it assumes that the errors between arrays follow a Laplacian distribution, leading to the mean absolute error minimization as a performance criterion to be achieved. The proposed normalization method was evaluated using three performance measures and they show that LAD based normalization method minimizes the errors and provides a more consistent replicated data spread with respect to a least square based method

A class of robust weighted median (WM) sharpening algorithms is developed in this paper. Unlike traditional linear sharpening methods, weighted median sharpeners are shown to be less sensitive to background random noise or to image artifacts introduced by JPEG and other compression algorithms. These concepts are extended to include data dependent weights under the framework of permutation weighted medians leading to tunable sharpeners that, in essence, are insensitive to noise and compression artifacts. Permutation WM sharpeners are subsequently generalized to smoother/sharpener structures that can sharpen edges and image details while simultaneously filter out background random noise. A statistical analysis of the various algorithms is presented, theoretically validating the characteristics of the proposed sharpening structures. A number of experiments are shown for the sharpening of JPEG compressed images and sharpening of images with background film-grain noise. These algorithms can prove useful in the enhancement of compressed or noisy images posted on the World Wide Web (WWW) as well as in other applications where the underlying images are unavoidably acquired with noise.

A new approach to multichannel image compression is introduced where the intra- and cross-band correlations are jointly exploited in a surprisingly simple yet very effective manner. The key component of the algorithm is a bijection mapping of the original multichannel image into a virtual two-dimensional (2-D) scalar image. By optimally mapping the multichannel image set into a 2-D array and by subsequently applying a scalar image coding algorithm, the spatial correlation and the spectral correlation of the multichannel data set are jointly exploited. Based on the statistical characteristics of the multichannel data, the bijection mapping can be optimized to minimize the distortion introduced by the compression algorithm. The optimization reduces to the maximization of a function of the second-order statistics of the multichannel data. At high compression rates, the new algorithm outperforms traditional compression algorithms whenever the cross-band correlation is high and it yields comparable performance at low compression rates.

The output distribution formula for stack filters based on mirrored threshold decomposition is derived. This formula allows one to compute the cumulative distribution function of the output of a stack filter for a given input noise distribution. Mirrored threshold decomposition permits us to analyze the input-output characteristics of the stack filter in the binary domain. The sliding window operation of the stack filter is modeled by a deterministic finite automaton. The output distribution of the filter is obtained by interpreting the automaton as a Markov Chain whose transition probabilities depend on the probabilistic description of the binary input signal.

A generalized recursive weighted median (RWM) filter structure admitting negative weights is introduced. Much like the sample median is analogous to the sample mean, the proposed class of RWM filters is analogous to the class of infinite impulse response (IIR) linear filters. RWM filters provide advantages over linear IIR filters, offering near perfect "stop-band" characteristics and robustness against noise. A novel "recursive decoupling" adaptive optimization algorithm for the design of these RWM filters is also introduced. In the optimization algorithm, the previous outputs used to compute the recursive WM filter output are replaced by previous desired outputs. This structure avoids the feedback inherent in the recursive operation and therefore leads to a much simpler derivation of the gradient in the steepest descent algorithm used to update the filter coefficients.

A new approach to multispectral image compression is introduced where the intra- and cross-band correlations are jointly exploited in a surprisingly simple yet very effective manner. The proposed compression algorithm maps the multispectral image set into a virtual 2-dimensional array and applies a scalar image coding algorithm to the virtual array. Thus, the spatial correlation and the spectral correlation of the multispectral data set are jointly exploited. Based on the statistical characteristics of the multispectral data, the bijection mapping is optimized to minimize the distortion introduced by the compression algorithm. At high compression rates, the new algorithm outperforms traditional compression algorithms whenever the cross-band correlation is high and it yields comparable performance at low compression rates.