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ABSTRACT. We study the inversion of weighted Radon transforms in two dimensions

Wavelets are ill-suited to represent oscillatory patterns: rapid variations of intensity can only be described by the small scale wavelet coefficients, which are often quantized to zero, even at high bit rates. Our goal is to provide a fast numerical implementation of the best wavelet packet algorithm in order to demonstrate that an advantage can be gained by constructing a basis adapted to a target image. Emphasis is placed on developing algorithms that are computationally efficient. We developed a new fast two-dimensional (2-D) convolution decimation algorithm with factorized nonseparable 2-D filters. The algorithm is four times faster than a standard convolution-decimation. An extensive evaluation of the algorithm was performed on a large class of textured images. Because of its ability to reproduce textures so well, the wavelet packet coder significantly out performs one of the best wavelet coder on images such as Barbara and fingerprints, both visually and in term of PSNR.

We consider two theorems from the theory of compressive sensing. Mainly a theorem concerning uniform recovery of random sampling matrices, where the number of samples needed in order to recover an s-sparse signal from linear measurements (with high probability) is known to be m s(ln s) 3 ln N . We present new and improved constants together with what we consider to be a more explicit proof. A proof that also allows for a slightly larger class of m × N -matrices, by considering what we call low entropy. We also present an improved condition on the so-called restricted isometry constants, δs, ensuring sparse recovery via 1 -minimization. We show that δ2s < 4/ √ 41 is sufficient and that this can be improved further to almost allow for a sufficient condition of the type δ2s < 2/3.