Deconvolution by matching pursuit using spline wavelet packets dictionaries

We present an efficient method that restores signals from strongly noised blurred discrete data. The method can be characterized as a regularized matching pursuit (MP), where dictionaries consist of spline wavelet packets and their sampled convolutions with the blurring kernel. It combines ideas from spline theory, wavelet analysis and greedy algorithms. A unified computational engine, which enables to construct versatile libraries of spline wavelet packet dictionaries and efficient implementation of the algorithm, is the Spline Harmonic Analysis (SHA). SHA imposes harmonic analysis methodology onto spline spaces. It is especially applicable to convolution operations. The use of splines enables to map the discrete noised data into spaces of continuous functions, which approximate the sought-after solution in the proper smoothed class. The main distinction from the conventional MP is that different dictionaries are used to test the data and to approximate the solution. In addition, the oblique projections of data onto dictionary elements are used instead of orthogonal projections, which are used in the conventional MP. The slopes of the projections and the stopping rule for the algorithm are determined automatically. Experimental results exhibit a high efficient algorithm. The coherent structure of the signals, which were subjected to the strong blurring and immersed into deep noise, were extracted.