A full-rank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easily-verifiable conditions under which optimally-sparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several well-known signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical

This paper proposes an adaptive threshold estimation method for image denoising in the wavelet domain based on the generalized Guassian distribution (GGD) modeling of subband coefficients. The proposed method called NormalShrink is computationally more efficient and adaptive because the parameters required for estimating the threshold depend on subband data. The threshold is computed by / y where and y are the standard deviation of the noise and the subband data of noisy image respectively. is the scale parameter, which depends upon the subband size and number of decompositions. Experimental results on several test images are compared with various denoising techniques like Wiener Filtering [2], BayesShrink [3] and SureShrink [4]. To benchmark against the best possible performance of a threshold estimate, the comparison also include Oracleshrink. Experimental results show that the proposed threshold removes noise significantly and remains within 4% of OracleShrink and outperforms SureShrink, BayesShrink and Wiener filtering most of the time.

We propose a vector/matrix extension of our denoising algorithm initially developed for grayscale images, in order to efficiently process multichannel (e.g., color) images. This work follows our recently published SURE-LET approach where the denoising algorithm is parameterized as a linear expansion of thresholds (LET) and optimized using Stein’s unbiased risk estimate (SURE). The proposed wavelet thresholding function is pointwise and depends on the coefficients of same location in the other channels, as well as on their parents in the coarser wavelet subband. A non-redundant, orthonormal, wavelet transform is first applied to the noisy data, followed by the (subband-dependent) vector-valued thresholding of individual multichannel wavelet coefficients which are finally brought back to the image domain by inverse wavelet transform. Extensive comparisons with the state-of-the-art multiresolution image denoising algorithms indicate that despite being non-redundant, our algorithm matches the quality of the best redundant approaches, while maintaining a high computational efficiency and a low CPU/memory consumption. An online Java demo illustrates these assertions.

Coupling the periodic time-invariance of the wavelet transform with the view of thresholding as a projection yields a simple, recursive, wavelet-based technique for denoising signals. Estimating a signal from a noise-corrupted observation is a fundamental problem of signal processing which has been addressed via many techniques. Previously, Coifman and Donoho introduced cycle spinning, a technique estimating the true signal as the linear average of individual estimates derived from wavelet-thresholded translated versions of the noisy signal. Here, it is demonstrated that such an average can be dramatically improved upon. The proposed algorithm recursively “cycle spins ” by repeatedly translating and denoising the input via basic wavelet denoising and then translating back; at each iteration, the output of the previous iteration is used as input. Exploiting the convergence properties of projections, the proposed algorithm can be regarded as a sequence of denoising projections that converge to the projection of the original noisy signal to a small subspace containing the true signal. It is proven that the algorithm is guaranteed to globally converge, and simulations on piecewise polynomial signals show marked improvement over both basic wavelet thresholding and standard cycle spinning.

Despite the success of wavelet decompositions in other areas of statistical signal and image processing, current wavelet-based image models are inadequate for modeling patterns in images, due to the presence of unknown transformations (e.g., translation, rotation, location of lighting source) inherent in most pattern observations. In this paper we introduce a hierarchical wavelet-based framework for modeling patterns in digital images. This framework takes advantage of the ecient image representations aorded by wavelets, while accounting for unknown pattern transformations. Given a trained model, we can use this framework to synthesize pattern observations. If the model parameters are unknown, we can infer them from labeled training data using TEMPLAR (Template Learning from Atomic Representations), a novel template learning algorithm with linear complexity. TEMPLAR employs minimum description length (MDL) complexity regularization to learn a template with a sparse representation in the wavelet domain. We discuss several applications, including template learning, pattern classication, and image registration.

Despite their simplicity, scalar threshold operators effec-tively remove additive white Gaussian noise from wavelet detail coefficients of many practical signals. This paper ex-plores the use of multivariate estimators that are almost as simple as scalar threshold operators. Şendur and Selesnick have recently shown the effectiveness of joint threshold es-timation of parent and child wavelet coefficients. This pa-per discusses analogous results in two situations. With a frame representation, a simple joint threshold estimator is derived and it is shown that its generalization is equivalent to a type of 1-regularized denoising. Then, for the case where multiple independent noisy observations are avail-able, the counter-intuitive results by Chang, Yu, and Vetterli on combining averaging and thresholding are explained as a fortuitous consequence of randomization. 1.

Wavelet transform technique has been used for image compression targeting high visual quality reconstructed images even with high compression ratio. A visual quality measure such as Picture Quality Scale (PQS), which correlates well with the subjective Mean Opinion Score (MOS) may be employed on the compressed image for the quantizer to select the optimum dynamic threshold. The use of optimum threshold permits the removal of redundant information, thus leading to better compression performance with acceptable picture quality. The Results obtained with the proposed approach of threshold selection is compared with the existing technique and the performance and it is found to be better in all of the cases of images or wavelets.

Image restoration from corrupted image is a classical problem in the field of image processing. Additive random noise can easily be removed using simple threshold methods with linear and nonlinear filtering techniques. De-noising of natural images corrupted by Gaussian noise using wavelet techniques is very effective because of its ability to capture the energy of a signal in few energy transform values. The wavelet de-noising scheme thresholds the wavelet coefficients arising from the standard discrete wavelet transform. In this paper, it is proposed to investigate the suitability of different wavelet bases and the decomposition levels on the performance of image de-noising algorithms in terms of peak signal-to- noise ratio.

This paper proposes a new method of medical image denoising based on a new nonlinear thresholding function in Nonsubsampled Contourlet Transform (NSCT) domain. In medical images, noise suppression is a particularly delicate and difficult task. A tradeoff between noise reduction and the preservation of actual image features has to be made in a way that enhances the diagnostically relevant image content. The contourlet transform is a new extension of the wavelet transform that provides a multi-resolution and multidirection analysis for two dimension images. The NSCT expansion is composed of basis images oriented at various directions in multiple scales, with flexible aspect ratios. Each coefficient of the NSCT transform is tuned by a polynomial function for denoising. We compared the results of the proposed method with other methods of image de-noising. Experimental results show that the proposed approach can obtain better visual results and higher PSNR values. Key Words: image denoising, NSCT, nonlinear thresholding function, PSNR.

Recovering a pattern or image from a collection of noisy and misaligned observations is a challenging problem that arises in image processing and pattern recognition. This paper presents an automatic, wavelet-based approach to this problem. Despite the success of wavelet decompositions in other areas of statistical signal and image processing, most wavelet-based image models are inadequate for modeling patterns in images, due to the presence of unknown transformations (e.g., translation, rotation, location of lighting source) inherent in pattern observations. In this paper we introduce a wavelet-based framework for modeling patterns in digital images. This framework takes advantage of the efficient image representations afforded by wavelets, while accounting for unknown translations and rotations. In order to learn the parameters of our model from training data, we introduce TEMPLAR (Template Learning from Atomic Representations), a novel template learning algorithm. The problem solved by TEMPLAR is the recovery a pattern template from a collection of noisy, randomly translated and rotated observations of the pattern. TEMPLAR employs minimum description length (MDL) complexity regularization to learn a template with a sparse representation in the wavelet domain. We discuss several applications, including template learning, pattern classification, and image registration.