We attempt to recover an unknown function from noisy, sampled data. Using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coe cients. The shrinkage can be tuned to be nearly minimax over any member of a wide range of Triebeland Besov-type smoothness constraints, and asymptoti...

Starting from a practical implementation of Roth and Fisher’s algorithm to solve a Lasso-type problem, we propose and study the Active Set Iterative Shrinkage/Thresholding Algorithm (AS-ISTA). The convergence is proven by observing that the algorithm can be seen as a particular case of a coordinate gradient descent algorithm with a Gauss-Southwell-r rule. We provide experimental evidence that t...

We consider the problem of learning an unknown (overcomplete) basis from an unknown sparse linear combination. Introducing the “sparse coding neural gas” algorithm, we show how to employ a combination of the original neural gas algorithm and Oja’s rule in order to learn a simple sparse code that represents each training sample by a multiple of one basis vector. We generalise this algorithm usin...

This letter considers the problem of dictionary learning for sparse signal representation whose atoms have low mutual coherence. To learn such dictionaries, at each step, we first update the dictionary using the method of optimal directions (MOD) and then apply a dictionary rank shrinkage step to decrease its mutual coherence. In the rank shrinkage step, we first compute a rank 1 decomposition ...

form-based (DCT-based) coding algorithms (e.g., the A novel postprocessing method based on the optimal shiftJPEG standard for still image compression [1]). The invariant wavelet packet (SIWP) representation and wavelet enhancement of compressed images has been regarded as shrinkage is proposed to enhance compressed images. At the a filtering problem. Various linear/nonlinear spaceencoder, the o...

— This paper introduces a new rain removal model based on the shrinkage of the sparse codes for a single image. Recently, dictionary learning and sparse coding have been widely used for image restoration problems. These methods can also be applied to the rain removal by learning two types of rain and non-rain dictionaries and forcing the sparse codes of the rain dictionary to be zero vectors. H...

In Sparse Coding (SC), input vectors are reconstructed using a sparse linear combination of basis vectors. SC has become a popular method for extracting features from data. For a given input, SC minimizes a quadratic reconstruction error with an L1 penalty term on the code. The process is often too slow for applications such as real-time pattern recognition. We proposed two versions of a very f...

This paper presents an augmented Lagrangian (AL) based method for designing of overcomplete dictionaries for sparse representation with general lq-data fidelity term (q 6 2). In the proposed method, the dictionary is updated via a simple gradient descent method after each inner minimization step of the AL scheme. Besides, a modified Iterated Shrinkage/Thresholding Algorithm is employed to accel...

An approach for speckle reduction and feature enhancement under a framework of multiscale wavelet analysis is presented. The advantages of both soft thresholding and hard thresholding wavelet shrinkage techniques are utilized to eliminate noise and preserve the sharpness of salient features. We integrate a method of wavelet shrinkage with nonlinear processing to enhance contrast within structur...