Image restoration: Wavelet frame shrinkage, nonlinear evolution PDEs, and beyond

In the past few decades, mathematics based approaches have been widely adopted in various image restoration problems; the partial differential equation (PDE) based approach (e.g., the total variation model [L. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259-268] and its generalizations, nonlinear diffusions [P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intel., 12 (1990), pp. 629-639; F. Catte et al., SIAM J. Numer. Anal., 29 (1992), pp. 182-193], etc.) and wavelet frame based approach are some successful examples. These approaches were developed through different paths and generally provided understanding from different angles of the same problem. As shown in numerical simulations, implementations of the wavelet frame based approach and the PDE based approach quite often end up solving a similar numerical problem with similar numerical behaviors, even though different approaches have advantages in different applications. Since wavelet frame based and PDE based approaches have all been modeling the same types of problems with success, it is natural to ask whether the wavelet frame based approach is fundamentally connected with the PDE based approach when we trace them all the way back to their roots. A fundamental connection of a wavelet frame based approach with a total variation model and its generalizations was established in [J. Cai, B. Dong, S. Osher, and Z. Shen, J. Amer. Math. Soc., 25 (2012), pp. 1033-1089]. This connection gives the wavelet frame based approach a geometric explanation and, at the same time, it equips a PDE based approach with a time frequency analysis. Cai et al. showed that a special type of wavelet frame model using generic wavelet frame systems can be regarded as an approximation of a generic variational model (with the total variation model as a special case) in the discrete setting. A systematic convergence analysis, as the resolution of the image goes to infinity, which is the key step in linking the two approaches, is also given in Cai et al. Motivated by Cai et al. and [Q. Jiang, Appl. Numer. Math., 62 (2012), pp. 51-66], this paper establishes a fundamental connection between the wavelet frame based approach and nonlinear evolution PDEs, provides interpretations and analytical studies of such connections, and proposes new algorithms for image restoration based on the new understandings. Together with the results in [J. Cai et al., J. Amer. Math. Soc., 25 (2012), pp. 1033-1089], we now have a better picture of how the wavelet frame based approach can be used to interpret the general PDE based approach (e.g., the variational models or nonlinear evolution PDEs) and can be used as a new and useful tool in numerical analysis to discretize and solve various variational and PDE models. To be more precise, we shall establish the following: (1) The connections between wavelet frame shrinkage and nonlinear evolution PDEs provide new and inspiring interpretations of both approaches that enable us to derive new PDE models and (better) wavelet frame shrinkage algorithms for image restoration. (2) A generic nonlinear evolution PDE (of parabolic or hyperbolic type) can be approximated by wavelet frame shrinkage with properly chosen wavelet frame systems and carefully designed shrinkage functions. (3) The main idea of this work is beyond the scope of image restoration. Our analysis and discussions indicate that wavelet frame shrinkage is a new way of solving PDEs in general, which will provide a new insight that will enrich the existing theory and applications of numerical PDEs, as well as those of wavelet frames

Direct Signal Separation Via Extraction of Local Frequencies with Adaptive Time-Varying Parameters

In nature, real-world phenomena that can be formulated as signals (or in terms of time series) are often affected by a number of factors and appear as multi-component modes. The natural approach to understand and process such phenomena is to decompose, or even better, to separate the multi-component signals to their basic building blocks (called sub-signals or time-series components, or fundamental modes). Recently the synchro-squeezing transform (SST) and its variants have been developed for nonstationary signal separation. More recently, a direct method of the time-frequency approach, called signal separation operation (SSO), was introduced for multi-component signal separation. While both SST and SSO are mathematically rigorous on the instantaneous frequency (IF) estimation, SSO avoids the second step of the two-step SST method in signal separation, which depends heavily on the accuracy of the estimated IFs. In the present paper, we solve the signal separation problem by constructing an adaptive signal separation operator (ASSO) for more effective separation of the blind-source multi-component signal, via introducing a time-varying parameter that adapts to local IFs. A recovery scheme is also proposed to extract the signal components one by one, and the time-varying parameter is updated for each component. The proposed method is suitable for engineering implementation, being capable of separating complicated signals into their sub-signals and reconstructing the signal trend directly. Numerical experiments on synthetic and real-world signals are presented to demonstrate our improvement over the previous attempts

A Tree-Based Multiscale Regression Method

A tree-based method for regression is proposed. In a high dimensional feature space, the method has the ability to adapt to the lower intrinsic dimension of data if the data possess such a property so that reliable statistical estimates can be performed without being hindered by the “curse of dimensionality.” The method is also capable of producing a smoother estimate for a regression function than those from standard tree methods in the region where the function is smooth and also being more sensitive to discontinuities of the function than smoothing splines or other kernel methods. The estimation process in this method consists of three components: a random projection procedure that generates partitions of the feature space, a wavelet-like orthogonal system defined on a tree that allows for a thresholding estimation of the regression function based on that tree and, finally, an averaging process that averages a number of estimates from independently generated random projection trees

Synchro-Transient-Extracting Transform for the Analysis of Signals with Both Harmonic and Impulsive Components

Time-frequency analysis (TFA) techniques play an increasingly important role in the field of machine fault diagnosis attributing to their superiority in dealing with nonstationary signals. Synchroextracting transform (SET) and transient-extracting transform (TET) are two newly emerging techniques that can produce energy concentrated representation for nonstationary signals. However, SET and TET are only suitable for processing harmonic signals and impulsive signals, respectively. This poses a challenge for each of these two techniques when a signal contains both harmonic and impulsive components. In this paper, we propose a new TFA technique to solve this problem. The technique aims to combine the advantages of SET and TET to generate energy concentrated representations for both harmonic and impulsive components of the signal. Furthermore, we theoretically demonstrate that the proposed technique retains the signal reconstruction capability. The effectiveness of the proposed technique is verified using numerical and real-world signals