Suboptimality of the Karhunen-Loève transform for transform coding

We examine the performance of the Karhunen-Loeve transform (KLT) for transform coding applications. The KLT has long been viewed as the best available block transform for a system that orthogonally transforms a vector source, scalar quantizes the components of the transformed vector using optimal bit allocation, and then inverse transforms the vector. This paper treats fixed-rate and variable-rate transform codes of non-Gaussian sources. The fixed-rate approach uses an optimal fixed-rate scalar quantizer to describe the transform coefficients; the variable-rate approach uses a uniform scalar quantizer followed by an optimal entropy code, and each quantized component is encoded separately. Earlier work shows that for the variable-rate case there exist sources on which the KLT is not unique and the optimal quantization and coding stage matched to a "worst" KLT yields performance as much as 1.5 dB worse than the optimal quantization and coding stage matched to a "best" KLT. In this paper, we strengthen that result to show that in both the fixed-rate and the variable-rate coding frameworks there exist sources for which the performance penalty for using a "worst" KLT can be made arbitrarily large. Further, we demonstrate in both frameworks that there exist sources for which even a best KLT gives suboptimal performance. Finally, we show that even for vector sources where the KLT yields independent coefficients, the KLT can be suboptimal for fixed-rate coding

Application of ensemble transform data assimilation methods for parameter estimation in reservoir modelling

Over the years data assimilation methods have been developed to obtain estimations of uncertain model parameters by taking into account a few observations of a model state. The most reliable methods of MCMC are computationally expensive. Sequential ensemble methods such as ensemble Kalman filers and particle filters provide a favourable alternative. However, Ensemble Kalman Filter has an assumption of Gaussianity. Ensemble Transform Particle Filter does not have this assumption and has proven to be highly beneficial for an initial condition estimation and a small number of parameter estimation in chaotic dynamical systems with non-Gaussian distributions. In this paper we employ Ensemble Transform Particle Filter (ETPF) and Ensemble Transform Kalman Filter (ETKF) for parameter estimation in nonlinear problems with 1, 5, and 2500 uncertain parameters and compare them to importance sampling (IS). We prove that the updated parameters obtained by ETPF lie within the range of an initial ensemble, which is not the case for ETKF. We examine the performance of ETPF and ETKF in a twin experiment setup and observe that for a small number of uncertain parameters (1 and 5) ETPF performs comparably to ETKF in terms of the mean estimation. For a large number of uncertain parameters (2500) ETKF is robust with respect to the initial ensemble while ETPF is sensitive due to sampling error. Moreover, for the high-dimensional test problem ETPF gives an increase in the root mean square error after data assimilation is performed. This is resolved by applying distance-based localization, which however deteriorates a posterior estimation of the leading mode by largely increasing the variance. A possible remedy is instead of applying localization to use only leading modes that are well estimated by ETPF, which demands a knowledge at which mode to truncate

Image processing algorithms employing two-dimensional Karhunen-Loeve Transform

In the fields of image processing and pattern recognition there is an important problem of acquiring, gathering, storing and processing large volumes of data. The most frequently used solution making these data reduced is a compression, which in many cases leads also to the speeding-up further computations. One of the most frequently employed approaches is an image handling by means of Principal Component Analysis and Karhunen-Loeve Transform, which are well known statistical tools used in many areas of applied science. Their main property is the possibility of reducing the volume of data required for its optimal representation while preserving its specific characteristics.The paper presents selected image processing algorithms such as compression, scrambling (coding) and information embedding (steganography) and their realizations employing the twodimensional Karhunen-Loeve Transform (2DKLT), which is superior to the standard, onedimensional KLT since it represents images respecting their spatial properties. The principles of KLT and 2DKLT as well as sample implementations and experiments performed on the standard benchmark datasets are presented. The results show that the 2DKLT employed in the above applications gives obvious advantages in comparison to certain standard algorithms, such as DCT, FFT and wavelets