New deblocking artifact, or blocking artifact reduction, algorithms based on nonlinear adaptive soft-threshold anisotropic filter in wavelet are proposed. Our deblocking algorithm uses soft-threshold, adaptive wavelet direction, adaptive anisotropic filter, and estimation. The novelties of this paper are an adaptive soft-threshold for deblocking artifact and an optimal intersection of confidence intervals (OICI) method in deblocking artifact estimation. The soft-threshold values are adaptable to different thresholds of flat area, texture area, and blocking artifact. The OICI is a reconstruction technique of estimated deblocking artifact which improves acceptable quality level of estimated deblocking artifact and reduces execution time of deblocking artifact estimation compared to the other methods. Our adaptive OICI method outperforms other adaptive deblocking artifact methods. Our estimated deblocking artifact algorithms have up to 98% of MSE improvement, up to 89% of RMSE improvement, and up to 99% of MAE improvement. We also got up to 77.98% reduction of computational time of deblocking artifact estimations, compared to other methods. We have estimated shift and add algorithms by using Euler++(E++) and Runge-Kutta of order 4++ (RK4++) algorithms which iterate one step an ordinary differential equation integration method. Experimental results showed that our E++ and RK4++ algorithms could reduce computational time in terms of shift and add, and RK4++ algorithm is superior to E++ algorithm.

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.