Image denoising using regularized Butterworth wavelet frames

We present an efficient algorithm for image restoration from highly noised originals. The algorithm is based on diverse library of tight and semi-tight wavelet frames. Unlike majority of current denoising methods, which threshold the transform coefficients, our algorithm performs direct and inverse multiscale transforms using properly modified frame filters. No thresholding is applied. The processing is linear. The algorithm is fast and can be implemented in real time. It depends on one numerical parameter, which is estimated from the noise level.

Interpolatory frames in signal space

We present a new family of frames, which are generated by perfect reconstruction filter banks consisting of linear phase filters. The filter banks are designed on the base of the discrete interpolatory splines and are related to the Butterworth filters. Each filter bank comprises one interpolatory symmetric low-pass filter and two high-pass filters, one of which is also interpolatory and symmetric. The second high-pass filter may be symmetric or antisymmetric. These filter banks generate the analysis and synthesis scaling functions and pairs of framelets. We introduced the concept of semi-tight frame. While in the tight frame all the analysis waveforms coincide with their synthesis counterparts, in the semi-tight frame we can vary the second framelets making them different for the synthesis and the analysis cases. By this means we can switch the vanishing moments from the synthesis to the analysis framelets or to add smoothness to the synthesis framelet. We constructed dual pairs of frames, where all the waveforms are symmetric and all the framelets have the same number of vanishing moments. Although most of the designed filters are IIR, they allow fast implementation via recursive procedures. The waveforms are well localized in the time domain despite their infinite support. The frequency response of the designed filters are flat

Tight and sibling frames originated from discrete splines

We present a new family of frames, which are generated by perfect reconstruction filter banks. The filter banks are based on the discrete interpolatory splines and are related to Butterworth filters. Each filter bank comprises one interpolatory symmetric low-pass filter, one band-pass and one highpass filters. In the sibling frames case, all the filters are linear phase and generate symmetric scaling functions with analysis and synthesis pairs of framelets. In the tight frame case, all the analysis waveforms coincide with their synthesis counterparts. In the sibling frame, we can vary the framelets making them different for synthesis and analysis cases. This enables us to swap vanishing moments between the synthesis and the analysis framelets or to add smoothness to the synthesis framelet. We construct dual pairs of frames, where all the waveforms are symmetric and the framelets may have any number of vanishing moments. Although most of the designed filters are IIR, they allow fast implementation via recursive procedures. The waveforms are well localized in time domain despite their infinite support.

Splines computation by subdivision and sample rate conversion

We present algorithms for explicit computation of one- and multi-dimensional periodic splines of arbitrary order at triadic rational points and of splines of even order at dyadic rational points. These algorithms use the direct and the inverse Fast Fourier transform (FFT) and the implementation is as fast as the FFT. The algorithms are based on dyadic and triadic subdivision of splines. Interpolating and smoothing splines are used for sample rate conversion such as upsampling of discrete-time signals and digital images, which may be contaminated by noise. The performance of the spline based rate conversion is compared with the performance of the prolate spheroidal wave functions based rate conversion. Key words: Periodic splines, subdivision, interpolating splines, smoothing splines,sample rate conversion, prolate spheroidal wave functions

Modeling and simulation of Li-Ion conduction in POLY(Ethylene Oxide

Prof. Moshe Israeli passed away on Feburary 18, 2007. This paper is dedicated to his memory Polyethylene oxide (PEO) containing a lithium salt (e.g. LiI) serves as a solid polymer electrolyte (SPE) in thin-film batteries and its ionic conductivity is a key parameter of their performance. We model and simulate Li + ion conduction in a single PEO molecule. Our simplified stochastic model of ionic motion is based on an analogy between protein channels of biological membranes that conduct Na +, K +, and other ions, and the PEO helical chain that conducts Li + ions. In contrast with protein channels and salt solutions, the PEO is both the channel and the solvent for the lithium salt (e.g., LiI). The mobile ions are treated as charged spherical Brownian particles. We simulate Smoluchowski dynamics in channels with a radius of ca 0.1nm and study the effect of stretching and temperature on ion conductivity. We assume that each helix (molecule) forms a random angle with the axis between these electrodes and the polymeric film is composed of many uniformly distributed oriented boxes that include molecules with the same direction. We further assume that mechanical stretching aligns the molecular structures in each box along the axis of stretching (intra-box alignment). Our model thus predicts the PEO conductivity as a function of the stretching, the salt concentration and the temperature. The computed enhancement of the ionic conductivity in the stretch direction is in good agreement with experimental results. The simulation results are also in qualitative agreement with recent theoretical and experimental results.

An hybrid algorithm for data compression

We present an algorithm that compresses two-dimensional data arrays, which are piece-wise smooth in one direction and have oscillating events in the other direction. Seismic, hyper-spectral and fingerprints data have this mixed structure. The transform part of the compression process is an algorithm that combines wavelet and the local cosine transform (LCT) also called an hybrid transform. The quantization and the entropy coding parts of the compression were taken from the SPIHT codec. To efficiently apply the SPIHT codec to a mixed coefficients array, reordering of the LCT coefficients takes place. This algorithm outperforms other algorithms that are based only on the 2D wavelet transforms. Its compression capabilities are also demonstrated on multimedia images that have a fine texture. The wavelet part in the mixed transform of the hybrid algorithm utilizes the library of Butterworth wavelet transforms.

POLYMER GEOMETRY AND Li + CONDUCTION IN POLY(ETHYLENE OXIDE)

Prof. Moshe Israeli passed away on February 18, 2007. This paper is dedicated to his memory. We study the effect of molecular shape on Li + conduction in dilute and concentrated polymer electrolytes (LiI: P(EO)n(3 ≤ n ≤ 100)). We model the migration-diffusion of interacting Li + ions in the helical PEO molecule as Brownian motion in a field of electrical force. Our model demonstrates that ionic conductivity of the amorphous PE structure is increased by mechanical stretching due to the unraveling of loops in the polymer molecule and to increased order. The enhancement of the ionic conductivity in the stretch direction, observed in our Brownian simulations, is in agreement with experimental results. We find an up to 40-fold increase in the LiIP(EO)7 conductivity, which is also in agreement with experimental results. The good agreement with experiment lends much credibility to our physical model of conductivity.