In this work we present two sparse deconvolution methods for nondestructive testing. The first method is a special matching pursuit (MP) algorithm in order to deconvolve the mixed data (signal and noise), and thus to remove the unwanted noise. The second method is based on the approximate Prony method (APM). Both methods employ the sparsity assumption about the measured ultrasonic signal as prior knowledge. The MP algorithm is used to derive a sparse representation of the measured data by a deconvolution and subtraction scheme. An orthogonal variant of the algorithm (OMP) is presented as well. The APM technique also relies on the assumption that the desired signals are sparse linear combinations of (reflections of) the transmitted pulse. For blind deconvolution, where the transducer impulse response is unknown, we offer a general Gaussian echo model whose parameters can be iteratively adjusted to the real measurements. Several test results show that the methods work well even for high noise levels. Fur

Low rank approximation has been extensively studied in the past. It is most suitable to reproduce rectangular like structures in the data. In this work we introduce a generalization using "shifted" rank-1 matrices to approximate A ∈ C M×N . These matrices are of the form S λ (uv * ) where u ∈ C M , v ∈ C N and λ ∈ Z N . The operator S λ circularly shifts the k-th column of uv * by λ k . These kind of shifts naturally appear in applications, where an object u is observed in N measurements at different positions indicated by the shift λ. The vector v gives the observation intensity. Exemplary, a seismic wave can be recorded at N sensors with different time of arrival λ; Or a car moves through a video changing its position in every frame. We present theoretical results as well as an efficient algorithm to calculate a shifted rank-1 approximation in O(N M log M). The benefit of the proposed method is demonstrated in numerical experiments. A comparison to other sparse approximation methods is given. Finally, we illustrate the utility of the extracted parameters for direct information extraction in several applications including video processing or non-destructive testing. Index Terms-low rank approximation, singular value decomposition, shift-invariant dictionary learning, column shifts, adaptive approximation

For a fast and accurate extraction of important information in seismic signals, a sparse representation based on physical parameters of the given data is crucial. In this paper we use the Asymmetric Gaussian Chirplet Model (AGCM) and establish a dictionary free variant of the Orthogonal Matching Pursuit (OMP), a Greedy algorithm for sparse approximation. The atoms of AGCM, so-called chirplets, display asymmetric oscillation-attenuation properties, which make the AGCM very suitable for sparse representation of absorption decay seismic signals. Unlike the Fourier transform which assumes that the seismic signals consist of plane waves, the AGCM assumes the seismic signal consists of non-stationary compressed plane waves, i.e., symmetric or asymmetric chirplets. Thus AGCM is a general model for seismic wave simulation, and its model parameters include envelope part and phase part. In this paper, we mainly concentrate on the parameters of envelope part such as envelope amplitude and arrival time. We will show numerical examples using the algorithm for seismic signal approximation and arrive-time detection. The results show a promising performance but may be improved considering also spatial correlations of seismic data.

An asymmetric chirplet transform, also called the asymmetric Gaussian chirplet model (AGCM), was recently introduced for fast and accurate extraction of important information from seismic signals. Unlike Fourier or wavelet transforms with fixed base functions, the AGCM is an adaptive sparse representation based on the physical parameters of the given data. The atoms of AGCM, so-called chirplets, display asymmetric oscillation-attenuation properties. The AGCM decomposes seismic signals into nonstationary compressed plane waves. The waves or atoms consist of two parts with seven physical parameters: the envelope and frequency parts. We have determined how to reconstruct the envelope part (e.g., envelope amplitude and arrival time) of the seismic data. We concentrate on the frequency part that involves three parameters: phase, frequency, and chirp rate. A Newton method with a step-size choice is established to deal with the highly oscillating frequency part of AGCM. The model parameters or coefficients in the transform domain may not only provide explicit physical interpretation (e.g., local phase of signal), but they can also be potentially used for feature extraction and other applications. Numerical results indicate excellent approximation results with good runtime performance. Physically reasonable parameter interpretation is demonstrated.

Mathematical methods of image inpainting involve the discretization of given continuous models. We present a method that avoids the standard pointwise discretization by modeling known variational approaches, in particular total variation (TV), using a finite dimensional spline space. Besides the analysis of the resulting model, we present a numerical implementation based on the alternating method of multipliers. We compare the results numerically with classical TV inpainting and give examples of applications.