Sparsity-based denoising and its application in linear inverse problems

Generally, the presence of noise in geophysical measurements is inevitable and depending on the type and the level it affects the results of geophysical studies. So, denoising is an important part of the processing of geophysical data. On the other hand, geophysicists make inferences about the physical properties of the earth interior based on the indirect measurements (data) collected at or near the surface of the earth. So, an inverse problem must be solved in order to take estimates of the physical properties in the earth. The vast majority of inverse problems which arise in geophysics are ill-posed; in other words, they have not unique and stable solutions. Regularization tools are used to find a unique and stable solution for such problems. The regularization uses a priori information about the solution to make it stable and to suppress high-frequency oscillations generated by the noise. One of the common ways to perform the regularization is expanding the unknown model (i.e. solution) with respect to an orthonormal basis, separating the model coefficients from that of the noise, and finally recovering the model.
In singular value decomposition, the specific physical nature of the model under study is not considered when defining the basis. For homogeneous operators, such basis does not provide a parsimonious approximation of models which are smooth in some regions while having sharp local changes in others. This is due to the non-localized properties of the SVD basis vectors in space (time) domain.
Wavelet-vaguelette decomposition (WVD) was introduced as a first approach for adapting wavelet methods to the framework of ill-posed inverse problems. It is a linear projection method based on wavelet-like function systems which have similar properties as the singular value decompositions. WVD are compared to the SVD construct near the orthogonal basis where the vectors are well localized in space (time) and frequency, thus producing less Gibbs-phenomenon at discontinuities. This property and existence of fast algorithm to compute the basis make wavelets a suitable candidate for solving inverse problems.
Vaguelette-wavelet decomposition (VWD) is an alternative to WVD for solving ill-posed inverse problems. It is a linear projection method based on wavelet function systems. In VWD the noisy data are expanded in a wavelet series, generated wavelet coefficients are thresholded to obtain an estimate of the wavelet expansion of noise free data, and then the resulting coefficients are transformed back for smoothed data. Later on, the smoothed data are inverted for the desired model.
In this paper we discuss: 1. The performance of sparsifying transforms (e.g. wavelet transform) for the denoising problem and their application to solve other linear inverse problems including WVD and VWD. 2. Comparing nonlinear Amplitude-scale-invariant Bayes Estimator (ABE) and hard- and soft-shrinkage filters to estimate signal coefficients in sparse domain for different levels of noise. 3. Introducing an efficient method to estimate the standard deviation of noise which is an important task in the experiments with single realization. The obtained standard deviation is then used to determine the regularization parameter in both wavelet- and SVD- based inversion methods.
Finally, inversion of integration operator to find the variation rate of a function is used to show the performance of the introduced methods in comparison to the popular SVD method. The results indicate that a simple non-linear operation of weighting and thresholding of wavelet coefficients can consistently outperform classical linear inverse methods.