The curvelet transform for image denoising

We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a` trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement

Coding overcomplete representations of audio using the MCLT

We propose a system for audio coding using the modulated complex lapped transform (MCLT). In general, it is difficult to encode signals using overcomplete representations without avoiding a penalty in rate-distortion performance. We show that the penalty can be significantly reduced for MCLT-based representations, without the need for iterative methods of sparsity reduction. We achieve that via a magnitude-phase polar quantization and the use of magnitude and phase prediction. Compared to systems based on quantization of orthogonal representations such as the modulated lapped transform (MLT), the new system allows for reduced warbling artifacts and more precise computation of frequency-domain auditory masking functions

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'

State-of-the-Art and Trends in Scalable Video Compression with Wavelet Based Approaches

3noScalable Video Coding (SVC) differs form traditional single point approaches mainly because it allows to encode in a unique bit stream several working points corresponding to different quality, picture size and frame rate. This work describes the current state-of-the-art in SVC, focusing on wavelet based motion-compensated approaches (WSVC). It reviews individual components that have been designed to address the problem over the years and how such components are typically combined to achieve meaningful WSVC architectures. Coding schemes which mainly differ from the space-time order in which the wavelet transforms operate are here compared, discussing strengths and weaknesses of the resulting implementations. An evaluation of the achievable coding performances is provided considering the reference architectures studied and developed by ISO/MPEG in its exploration on WSVC. The paper also attempts to draw a list of major differences between wavelet based solutions and the SVC standard jointly targeted by ITU and ISO/MPEG. A major emphasis is devoted to a promising WSVC solution, named STP-tool, which presents architectural similarities with respect to the SVC standard. The paper ends drawing some evolution trends for WSVC systems and giving insights on video coding applications which could benefit by a wavelet based approach.partially_openpartially_openADAMI N; SIGNORONI. A; R. LEONARDIAdami, Nicola; Signoroni, Alberto; Leonardi, Riccard

Modeling techniques for quantum cascade lasers

Quantum cascade lasers are unipolar semiconductor lasers covering a wide range of the infrared and terahertz spectrum. Lasing action is achieved by using optical intersubband transitions between quantized states in specifically designed multiple-quantum-well heterostructures. A systematic improvement of quantum cascade lasers with respect to operating temperature, efficiency and spectral range requires detailed modeling of the underlying physical processes in these structures. Moreover, the quantum cascade laser constitutes a versatile model device for the development and improvement of simulation techniques in nano- and optoelectronics. This review provides a comprehensive survey and discussion of the modeling techniques used for the simulation of quantum cascade lasers. The main focus is on the modeling of carrier transport in the nanostructured gain medium, while the simulation of the optical cavity is covered at a more basic level. Specifically, the transfer matrix and finite difference methods for solving the one-dimensional Schr\"odinger equation and Schr\"odinger-Poisson system are discussed, providing the quantized states in the multiple-quantum-well active region. The modeling of the optical cavity is covered with a focus on basic waveguide resonator structures. Furthermore, various carrier transport simulation methods are discussed, ranging from basic empirical approaches to advanced self-consistent techniques. The methods include empirical rate equation and related Maxwell-Bloch equation approaches, self-consistent rate equation and ensemble Monte Carlo methods, as well as quantum transport approaches, in particular the density matrix and non-equilibrium Green's function (NEGF) formalism. The derived scattering rates and self-energies are generally valid for n-type devices based on one-dimensional quantum confinement, such as quantum well structures

Exploiting Prior Knowledge in Compressed Sensing Wireless ECG Systems

Recent results in telecardiology show that compressed sensing (CS) is a promising tool to lower energy consumption in wireless body area networks for electrocardiogram (ECG) monitoring. However, the performance of current CS-based algorithms, in terms of compression rate and reconstruction quality of the ECG, still falls short of the performance attained by state-of-the-art wavelet based algorithms. In this paper, we propose to exploit the structure of the wavelet representation of the ECG signal to boost the performance of CS-based methods for compression and reconstruction of ECG signals. More precisely, we incorporate prior information about the wavelet dependencies across scales into the reconstruction algorithms and exploit the high fraction of common support of the wavelet coefficients of consecutive ECG segments. Experimental results utilizing the MIT-BIH Arrhythmia Database show that significant performance gains, in terms of compression rate and reconstruction quality, can be obtained by the proposed algorithms compared to current CS-based methods.Comment: Accepted for publication at IEEE Journal of Biomedical and Health Informatic

Elliptical Monogenic Wavelets for the analysis and processing of color images

International audienceThis paper studies and gives new algorithms for image processing based on monogenic wavelets. Existing greyscale monogenic filterbanks are reviewed and we reveal a lack of discussion about the synthesis part. The monogenic synthesis is therefore defined from the idea of wavelet modulation, and an innovative filterbank is constructed by using the Radon transform. The color extension is then investigated. First, the elliptical Fourier atom model is proposed to generalize theanalytic signal representation for vector-valued signals. Then a color Riesz-transform is defined so as to construct color elliptical monogenic wavelets. Our Radon-based monogenic filterbank can be easily extended to color according to this definition. The proposed wavelet representation provides efficient analysis of local features in terms of shape and color, thanks to the concepts of amplitude, phase, orientation, and ellipse parameters. The synthesis from local features is deeply studied. We conclude the article by defining the color local frequency, proposing an estimation algorithm