Soft-thresholding is a sparse modeling method typically applied to wavelet denoising in statistical signal processing. It is also important in machine learning since it is an essential nature of the well-known LASSO (Least Absolute Shrinkage and Selection Operator). It is known that soft-thresholding, thus, LASSO suffers from a problem of dilemma between sparsity and generalization. This is caused by excessive shrinkage at a sparse representation. There are several methods for improving this problem in the field of signal processing and machine learning. In this paper, we considered to extend and analyze a method of scaling of soft-thresholding estimators. In a setting of non-parametric orthogonal regression problem including discrete wavelet transform, we introduced component-wise and data-dependent scaling that is indeed identical to non-negative garrote. We here considered a case where a parameter value of soft-thresholding is chosen from absolute values of the least squares estimates, by which the model selection problem reduces to the determination of the number of non-zero coefficient estimates. In this case, we firstly derived a risk and construct SURE (Stein's unbiased risk estimator) that can be used for determining the number of non-zero coefficient estimates. We also analyzed some properties of the risk curve and found that our scaling method with the derived SURE is possible to yield a model with low risk and high sparsity compared to a naive soft-thresholding method with SURE. This theoretical speculation was verified by a simple numerical experiment of wavelet denoising.

An improved multivariate wavelet denoising algorithm combined with subspace and principal component analysis is presented in this paper. The key element is deriving an optimal orthogonal matrix that can project the multivariate observation signal to a signal subspace from observation space. Univariate wavelet shrinkage operator is then applied to the projected signals channel-wise resulting in the improvement of the output SNR. Finally, principal component analysis is performed on the denoised signal in the observation space to further improve the denoising performance. Experimental results based on synthesized and real world ECG data verify the effectiveness of the proposed algorithm.

Space-variant approaches subject to local image characteristics are useful in practical image restoration because many natural images are nonstationary. Motivated by the success of denoising approaches in the wavelet domain, we propose a region-adaptive restoration approach which adopts a wavelet denoising technique in flat regions after an under-regularized constrained least squares restoration. Experimental results verify that this approach not only improves image quality in mean square error but also contributes to ringing reduction.