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)3lnN. 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 is called effective sparsity. We also present a condition on the so-called restricted isometry constants, δs, ensuring sparse recovery via ℓ1-minimization. We show that is sufficient and that this can be improved further to almost allow for a sufficient condition of the type .

Compression is a relatively new introduced technique for seismic data operations. The main drive behind the use of data compression in seismic data is the very large size of seismic data acquired. Some of the most recent acquired marine seismic data sets exceed 10 Tbytes, and in fact there are currently seismic surveys planned with a volume of around 120 Tbytes. Thus, the need to compress these very large seismic data files is imperative. Nevertheless, seismic data are quite different from the typical images used in image processing and multimedia applications. Some of their major differences are the data dynamic range exceeding 100 dB in theory, very often it is data with extensive oscillatory nature, the x and y directions represent different physical meaning, and there is significant amount of coherent noise which is often present in seismic data. Up to now some of the algorithms used for seismic data compression were based on some form of wavelet or local cosine transform, while using a uniform or quasiuniform quantization scheme and they finally employ a Huffman coding scheme. Using this family of compression algorithms we achieve compression results which are acceptable to geophysicists, only at low to moderate compression ratios. For higher compression ratios or higher decibel quality, significant compression artifacts are introduced in the reconstructed images, even with high-dimensional transforms. The objective of this paper is to achieve higher compression ratio, than achieved with the wavelet/uniform quantization/Huffman coding family of compression schemes, with a comparable level of residual noise. The goal is to achieve above 40 dB in the decompressed seismic data sets. Several established compression algorithms are reviewed, and some new compression algorithms are introduced. All of these compression techniques are applied to a good representation of seismic data sets, and their results are documented in this paper. One of the conclusions is that adaptive multiscale local cosine transform with different windows sizes performs well on all the seismic data sets and outperforms the other methods from the SNR point of view. All the described methods cover wide range of different data sets. Each data set will have his own best performed method chosen from this collection. The results were performed on four different seismic data sets. Special emphasis was given to achieve faster processing speed which is another critical issue that is examined in the paper. Some of these algorithms are also suitable for multimedia type compression.

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.

In noisy environments, a constrained least-squares (CLS) approach is presented to restore images blurred by a Gaussian impulse response, where instead of choosing a global regularization parameter, each point in the signal has its own associated regularization parameter. These parameters are found by constraining the weighted standard deviation of the wavelet transform coefficients on the finest scale of the inverse signal by a function r which is a local measure of the intensity variations around each point of the blurred and noisy observed signal. Border ringing in the inverse solution is proposed decreased by manipulating its wavelet transform coefficients on the finest scales close to the borders. If the noise in the inverse solution is significant, wavelet transform techniques are also applied to denoise the solution. Examples are given for images, and the results are shown to outperform the optimum constrained least-squares solution using a global regularization parameter, both visually and in the mean squared error sense.

Summary form only given. Presents a new fast wavelet packet compression algorithm that encodes very efficiently textured images. This fast wavelet packet compression technique relies on four stages: 1. Very fast convolution and decimation of the image with factorized filters. 2. Selection of a best basis in a large library of waveforms. The best basis is that basis which is best adapted to the content of the image. 3. Scanning of the wavelet packet coefficients by increasing frequency. This organization yields sequences of coefficients with a rapid decay. 4. Successive embedded approximation quantization, and entropy coding of the coefficients. We implemented the wavelet packet coder and decoder, and actual bit streams were created for each experiment. Our implementation used the 7-9 biorthogonal filters. We present the results of the algorithm, using the test image 512 x 512 Barbara. In order to evaluate the performance of the algorithm, we compared our algorithm to the SPIHT wavelet coder of Said and Pearlman (1996).

We describe a new multi-layered representation technique for images. An image is encoded as the superposition of one main approximation, and a sequence of residuals. The strength of the multi-layered method comes from the fact that we use different bases to encode the main approximation and the residuals. The different bases complement each others: some of the basis functions will give reasonable account of the large trend of the data, while others will catch the local transients, or the oscillatory patterns. By selecting different bases, we allow different features to be discovered in the image.