Yingsong Zhang

Improved Bounds for Subband-Adaptive Iterative Shrinkage/Thresholding Algorithms

This paper presents new methods for computing the step sizes of the subband-adaptive iterative shrinkage-thresholding algorithms proposed by Bayram & Selesnick and Vonesch & Unser. The method yields tighter wavelet-domain bounds of the system matrix, thus leading to improved convergence speeds. It is directly applicable to non-redundant wavelet bases, and we also adapt it for cases of redundant frames. It turns out that the simplest and most intuitive setting for the step sizes that ignores subband aliasing is often satisfactory in practice. We show that our methods can be used to advantage with reweighted least squares penalty functions as well as L1 penalties. We emphasize that the algorithms presented here are suitable for performing inverse filtering on very large datasets, including 3D data, since inversions are applied only to diagonal matrices and fast transforms are used to achieve all matrix-vector products.

Blind wavelet estimation using a zero-lag slice of the fourth-order statistics

This paper presents an iterative method for blind wavelet estimation using a zero-lag slice (ZLS) of the fourth-order cumulant of the seismic signal. The proposed method can estimate non-minimum phase wavelets since the higher order statistics carry the phase information. By just using the ZLS that can be estimated from the seismic data with high reliability, our method achieves good accuracy for wavelet estimation, even if only short seismic signals are available. Furthermore, the proposed method is not computationally intensive because it is performed either by applying simple 1D operations (frequency domain) or by solving a linear system (time domain) in each iteration with fast convergence, and only needs calculation of the ZLS once. Application results on synthetic and real field data sets show that our method behaves well in relatively low signal-to-noise ratios, and when the wavelet length is overspecified.

Image deconvolution using a Gaussian Scale Mixtures model to approximate the wavelet sparseness constraint

This paper proposes to use an extended Gaussian Scale Mixtures (GSM) model instead of the conventional l1 norm to approximate the sparseness constraint in the wavelet domain. We combine this new constraint with subband-dependent minimization to formulate an iterative algorithm on two shift-invariant wavelet transforms, the Shannon wavelet transform and dual-tree complex wavelet transform (DTCWT). This extented GSM model introduces spatially varying information into the deconvolution process and thus enables the algorithm to achieve better results with fewer iterations in our experiments.

A Bayesian wavelet-based multidimensional deconvolution with sub-band emphasis

This work proposes a new algorithm for wavelet-based multidimensional image deconvolution which employs subband-dependent minimization and the dual-tree complex wavelet transform in an iterative Bayesian framework. In addition, this algorithm employs a new prior instead of the popular ℓ1 norm, and is thus able to embed a learning scheme during the iteration which helps it to achieve better deconvolution results and faster convergence.

Restoration of images and 3D data to higher resolution by deconvolution with sparsity regularization

Image convolution is conventionally approximated by the LTI discrete model. It is well recognized that the higher the sampling rate, the better is the approximation. However sometimes images or 3D data are only available at a lower sampling rate due to physical constraints of the imaging system. In this paper, we model the under-sampled observation as the result of combining convolution and subsampling. Because the wavelet coefficients of piecewise smooth images tend to be sparse and well modelled by tree-like structures, we propose the L0 reweighted-L2 minimization (L0RL2 ) algorithm to solve this problem. This promotes model-based sparsity by minimizing the reweighted L2 norm, which approximates the L0 norm, and by enforcing a tree model over the weights. We test the algorithm on 3 examples: a simple ring, the cameraman image and a 3D microscope dataset; and show that good results can be obtained.

On the Reconstruction of Wavelet-Sparse Signals From Partial Fourier Information

The problem of reconstructing a wavelet-sparse signal from its partial Fourier information has received a lot of attention since the emergence of compressive sensing (CS). The latest theory within the CS framework analyzes the local coherence between the Fourier and wavelet bases, and recover the signal from frequencies randomly selected according to a variable density profile. Unlike these developments, we adopt a new approach that does not need to analyze the (local) coherence. We show that the problem can be tackled by recovering the wavelet coefficients from the finest to the coarse scale, and only a small set of frequencies are needed to recover the coefficients exactly. As long as the scaling function satisfies a mild condition, the reconstruction is exact. Moreover the frequency set can be deterministically pre-selected and does not need to change even if the wavelet basis changes.

FAST L0-based sparse signal recovery

This paper develops an algorithm for finding sparse signals from limited observations of a linear system. We assume an adaptive Gaussian model for sparse signals. This model results in a least square problem with an iteratively reweighted L2 penalty that approximates the L0-norm. We propose a fast algorithm to solve the problem within a continuation framework. In our examples, we show that the correct sparsity map and sparsity level are gradually learnt during the iterations even when the number of observations is reduced, or when observation noise is present. In addition, with the help of sophisticated interscale signal models, the algorithm is able to recover signals to a better accuracy and with reduced number of observations than typical L1-norm and reweighted L1 norm methods.

Sampling Streams of Pulses With Unknown Shapes

This paper develops a theory for sampling and perfectly reconstructing streams of short pulses of unknown shapes in the continuous-time domain. The single pulse is modelled as the delayed version of a wavelet sparse signal, which is normally not band-limited. As the delay can be an arbitrary real number, it is difficult to develop an exact sampling result for this type of signals. We manage to achieve the exact reconstruction of the pulses by using only the knowledge of the Fourier transform of the signal at specific frequencies. We further introduce a multichannel acquisition system that uses a new family of compact-support sampling kernels for extracting the Fourier information from the samples. The shape of the kernel is independent of the wavelet basis in which the pulse is sparse, and hence the same acquisition system can be used with pulses that are sparse on different wavelet bases. By exploiting the fact that pulses have short duration and that the sampling kernels have compact support, we finally propose a local and sequential algorithm to reconstruct streaming pulses from the samples.

The modulated E-spline with multiple subbands and its application to sampling wavelet-sparse signals

The theory of Finite Rate of Innovation (FRI) can be applied to sampling and reconstructing certain classes of parametric signals. The objective of this paper is to have a sub-Nyquist sampling scheme for continuous-time wavelet-sparse signals within the general framework of FRI theory. Though the signal has a parametric representation in the wavelet basis, it is not possible to recover the signal merely from its low-pass samples, which makes the problem different from the conventional FRI settings. The need for the Fourier coefficients at frequencies widely spread over the spectrum puts challenges on the design of the sampling kernel. This paper presents a new family of sampling kernels that are able to stably reproduce exponentials over a wide range of frequencies and gives numerical examples on applying the new kernel to sampling wavelet-sparse signals.