ENO-wavelet transforms for piecewise smooth functions

We have designed an adaptive essentially nonoscillatory (ENO)-wavelet transform for approximating discontinuous functions without oscillations near the discontinuities. Our approach is to apply the main idea from ENO schemes for numerical shock capturing to standard wavelet transforms. The crucial point is that the wavelet coefficients are computed without differencing function values across jumps. However, we accomplish this in a different way than in the standard ENO schemes. Whereas in the standard ENO schemes the stencils are adaptively chosen, in the ENO-wavelet transforms we adaptively change the function and use the same uniform stencils. The ENO-wavelet transform retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies, obtaining maximum accuracy, maintained up to the discontinuities, and having a multiresolution framework and fast algorithms, all without any edge artifacts. We have obtained a rigorous approximation error bound which shows that the error in the ENO-wavelet approximation depends only on the size of the derivative of the function away from the discontinuities. We will show some numerical examples to illustrate this error estimate

Wavelets and Fast Numerical Algorithms

Wavelet based algorithms in numerical analysis are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in this new system of coordinates. However, due to the recursive definition of wavelets, their controllable localization in both space and wave number (time and frequency) domains, and the vanishing moments property, wavelet based algorithms exhibit new and important properties. For example, the multiresolution structure of the wavelet expansions brings about an efficient organization of transformations on a given scale and of interactions between different neighbouring scales. Moreover, wide classes of operators which naively would require a full (dense) matrix for their numerical description, have sparse representations in wavelet bases. For these operators sparse representations lead to fast numerical algorithms, and thus address a critical numerical issue. We note that wavelet based algorithms provide a systematic generalization of the Fast Multipole Method (FMM) and its descendents. These topics will be the subject of the lecture. Starting from the notion of multiresolution analysis, we will consider the so-called non-standard form (which achieves decoupling among the scales) and the associated fast numerical algorithms. Examples of non-standard forms of several basic operators (e.g. derivatives) will be computed explicitly.Comment: 32 pages, uuencoded tar-compressed LaTeX file. Uses epsf.sty (see `macros'

Multiscale Discriminant Saliency for Visual Attention

The bottom-up saliency, an early stage of humans' visual attention, can be considered as a binary classification problem between center and surround classes. Discriminant power of features for the classification is measured as mutual information between features and two classes distribution. The estimated discrepancy of two feature classes very much depends on considered scale levels; then, multi-scale structure and discriminant power are integrated by employing discrete wavelet features and Hidden markov tree (HMT). With wavelet coefficients and Hidden Markov Tree parameters, quad-tree like label structures are constructed and utilized in maximum a posterior probability (MAP) of hidden class variables at corresponding dyadic sub-squares. Then, saliency value for each dyadic square at each scale level is computed with discriminant power principle and the MAP. Finally, across multiple scales is integrated the final saliency map by an information maximization rule. Both standard quantitative tools such as NSS, LCC, AUC and qualitative assessments are used for evaluating the proposed multiscale discriminant saliency method (MDIS) against the well-know information-based saliency method AIM on its Bruce Database wity eye-tracking data. Simulation results are presented and analyzed to verify the validity of MDIS as well as point out its disadvantages for further research direction.Comment: 16 pages, ICCSA 2013 - BIOCA sessio

Phase Harmonic Correlations and Convolutional Neural Networks

A major issue in harmonic analysis is to capture the phase dependence of frequency representations, which carries important signal properties. It seems that convolutional neural networks have found a way. Over time-series and images, convolutional networks often learn a first layer of filters which are well localized in the frequency domain, with different phases. We show that a rectifier then acts as a filter on the phase of the resulting coefficients. It computes signal descriptors which are local in space, frequency and phase. The non-linear phase filter becomes a multiplicative operator over phase harmonics computed with a Fourier transform along the phase. We prove that it defines a bi-Lipschitz and invertible representation. The correlations of phase harmonics coefficients characterise coherent structures from their phase dependence across frequencies. For wavelet filters, we show numerically that signals having sparse wavelet coefficients can be recovered from few phase harmonic correlations, which provide a compressive representationComment: 26 pages, 8 figure