Local Cosine Bases Research Articles

Beamlet migration using local cosine basis

We have developed the theoretical foundation and technical details of a migration method using a local-cosine-bases (LCB) beamlet propagator. A beamlet propagator for heterogeneous media based on local perturbation theory is derived, and a fast implementation method is constructed. The use of local background velocities and local perturbations results in a two-scale decomposition of beamlet propagators: a background propagator for large-scale structures and a local phase-screen correction for small-scale local perturbations. Because of its locally adaptive nature, the beamlet propagator can handle strong lateral velocity variations with improved accuracy. For high-efficiency migration, we use a table-driven method and apply some techniques of sparse matrix operations. Compared with the Fourier finite-dif-ference and generalized screen propagator methods, the image quality and computational efficiency are similar. In some cases, we see fewer migration artifacts around and inside salt bodies with our method. We attribute this to the better high-angle accuracy of beamlet propagators in strong-contrast media. Numerical tests using synthetic data sets of the SEG-EAGE 2D salt model, Marmousi model, and Sigsbee 2A model demonstrate its high accuracy and reasonable efficiency. Another special feature of LCB beamlet migration is the availability of information in the local wavenumber domain during migration, which can be used to correct acquisition aperture effect and for other processing. Compared to beamlet migration using the Gabor-Daubechies frame (GDF) propagator, LCB migration is much more efficient because LCB is an orthonormal basis, whereas GDF has redundancy (usually greater than two) in the decomposition.

A non-linear spatial hearing model based on bases pursuit algorithm

In the research on spatial hearing and realization of virtual auditory space, it is important to effectively model the head-related transfer functions (HRTFs) or head-related impulse responses (HRIRs). In our study, we managed to carry out adaptive non-linear approximation in the field of wavelet transformation. The results show that the HRIRs’ adaptive non-linear approximation model is a more effective data reduction model, is faster, and is 5 dB on average better than the traditional principal component analysis (PCA) (Karhunen-Loeve transform) model based on relative mean square error (MSE) criterion. Furthermore, we also discussed the best bases’ choice for the time-frequency representation of HRIRs, and the results show that local cosine bases are more propitious to HRIRs’ adaptive approximation than wavelet and wavelet packet base. However, the improved effect of local cosine bases is not distinct. Here, for the sake of modeling the HRIRs more truthfully, we consider choosing optimal time-frequency atoms from redundant dictionary to decompose this kind of signals of HRIRs and achieve better results than all the previous models.

Audio source separation with a signal-adaptive local cosine transform

Audio source separation is a very challenging problem, and many different approaches have been proposed in attempts to solve it. We consider the problem of separating sources from two-channel instantaneous audio mixtures. One approach to this is to transform the mixtures into the time-frequency domain to obtain approximately disjoint representations of the sources, and then separate the sources using time-frequency masking. We focus on demixing the sources by binary masking, and assume that the mixing parameters are known. In this paper, we investigate the application of cosine packet (CP) trees as a foundation for the transform. We determine an appropriate transform by applying a computationally efficient best basis algorithm to a set of possible local cosine bases organised in a tree structure. We develop a heuristically motivated cost function which maximises the energy of the transform coefficients associated with a particular source. Finally, we evaluate objectively our proposed transform method by comparing it against fixed-basis transforms such as the short-time Fourier transform (STFT) and modified discrete cosine transform (MDCT). Evaluation results indicate that our proposed transform method outperforms MDCT and is competitive with the STFT, and informal listening tests suggest that the proposed method exhibits less objectionable noise than the STFT.

MLC: a novel image coder based on multitree local cosine dictionaries

We build a multitree dictionary of local cosine bases and develop a novel image coder that extracts the best basis in such a dictionary with respect to a rate-distortion cost. This dictionary is much larger than standard quadtree-based dictionaries, and therefore, our algorithm needs more overhead bits to encode the basis selection. Despite this, we show that the resulting algorithm outperforms an existing quadtree local cosine image compression algorithm. We also show that our multitree local cosine coder outperforms both SPIHT and a multitree block-DCT method previously developed by us, on several textured images.

Adaptive biorthogonal local discrete cosine transform for interference excision in direct sequence spread spectrum communications

A novel time-frequency domain interference excision technique is proposed. The technique is based on adaptive biorthogonal local discrete cosine transform (BLDCT). It uses a redundant library of biorthogonal local discrete cosine bases and an efficient concave cost function to match the transform basis to the interfering signal. The main advantage of the algorithm over conventional transform domain excision algorithms is that the basis functions are not fixed but can be adapted to the time-frequency structure of the interfering signal. It is well suited to transform domain compression and suppression of various types of interference. Compared to the discrete wavelet transform (DWT) that provides logarithmic division of the frequency bands, the adaptive BLDCT can provide more flexible frequency resolution. Thus it is more insensitive to variations of jamming frequency. Simulation results demonstrate the improved bit error rate (BER) performance and the increased robustness of the receiver.

Interference excision in spread spectrum communications using adaptive biorthogonal local discrete cosine transform

A novel spread spectrum communication receiver employing adaptive biorthogonal local discrete cosine transform (BLDCT) is designed for stationary and nonstationary interference excision. The adaptive BLDCT-based excision algorithm uses a redundant library of biorthogonal local discrete cosine bases and an efficient concave cost function to match the transform basis to the interfering signal. The main advantage of the algorithm over conventional transform domain excision algorithms is that the basis functions are not fixed but can be adapted to the time-frequency structure of the interfering signal. It is well suited to transform domain compression and suppression of a variety of interference types. Compared with discrete wavelet transform (DWT) that provides logarithmic division of the frequency bands, the adaptive BLDCT can provide more flexible frequency resolution. Thus it is more insensitive to the jammer frequency. Simulation experiments demonstrate the improved bit error rate (BER) performance and the increased robustness of our receiver. Copyright © 2004 AEI.

A hybrid scheme for encoding audio signal using hidden Markov models of waveforms

This paper reports on recent results related to audiophonic signals encoding using time-scale and time–frequency transform. More precisely, nonlinear, structured approximations for tonal and transient components using local cosine and wavelet bases will be described, yielding expansions of audio signals in the form tonal + transient + residual. We describe a general formulation involving hidden Markov models, together with corresponding rate estimates. Estimators for the balance transient/tonal are also discussed.

Formulation of Localized Cosine Bases that Preserve Polynomial Modulated Sinusoids

We develop a class of compactly supported window functions Vm so that each polynomial p(x) of degree m - 1 as well as p(x) modulated with an integer frequency has a locally finite representation in terms of the local cosine basis ψ = {ψkj (x) = Vm(x - j /2 - 1/4 ) cos((2k + 1)π(x - j))}. Explicit formulations for Vm and its Fourier transform are derived. It is shown that these window functions are non-negative, have minimal support of length m, and maximal smoothness of order 2m - 2. Furthermore, we determine the exact Riesz bounds of ψ. Smoothness and stability for these bases are superior as compared to other local cosine bases with similar properties in the literature. Consequently, the bases ψ are particularly useful for applications in signal and image processing.

Determining Local Transientness of Audio Signals

We describe a new method for estimating the degree of "transientness" and "tonality" of a class of compound signals involving simultaneously transient and harmonic features. The key assumption is that both transient and tonal layers admit sparse expansions, respectively in wavelet and local cosine bases. The estimation is performed using particular form of entropy (or theoretical dimension) functions. We provide theoretical estimates on the behavior of the proposed estimators, as well as numerical simulations. Audio signal coding provides a natural field of application.

Approximation with brushlet systems

We consider an orthonormal basis for L 2( R) consisting of functions that are well localized in the spatial domain and have compact support in the frequency domain. The construction is based on smooth local cosine bases and is inspired by Meyer and Coifman's brushlets, which are local exponentials in the frequency domain. For brushlet bases associated with an exponential-type partition of the frequency axis, we show that the system constitutes an unconditional basis for L p ( R), 1< p<∞, B q s ( L p ( R)), 1< p, q<∞, s>0, and that the norm in these spaces can be expressed by the expansion coefficients. In L p ( R), we construct greedy brushlet-type bases and derive Jackson and Bernstein inequalities. Finally, we investigate a natural bivariate extension leading to ridgelet-type bases for L 2( R 2).

Locally stationary covariance and signal estimation with macrotiles

A macrotile estimation algorithm is introduced to estimate the covariance of locally stationary processes. A macrotile algorithm uses a penalized method to optimize the partition of the space in orthogonal subspaces, and the estimation is computed with a projection operator. It is implemented by searching for a best basis among a dictionary of orthogonal bases and by constructing an adaptive segmentation of this basis to estimate the covariance coefficients. The macrotile algorithm provides a consistent estimation of the covariance of locally stationary processes, using a dictionary of local cosine bases. This estimation is computed with a fast algorithm. Macrotile algorithms apply to other estimation problems such as the removal of additive noise in signals. This simpler problem is used as an intuitive guide to better understand the case of covariance estimation. Examples of removal of white noise from sounds illustrate the results.

Image compression with adaptive local cosines: a comparative study

The goal of this work is twofold. First, we demonstrate that an advantage can be gained by using local cosine bases over wavelets to encode images that contain periodic textures. We designed a coder that outperforms one of the best wavelet coders on a large number of images. The coder finds the optimal segmentation of the image in terms of local cosine bases. The coefficients are encoded using a scalar quantizer optimized for Laplacian distributions. This new coder constitutes the first concrete contribution of the paper. Second, we used our coder to perform an extensive comparison of several optimized bells in terms of rate-distortion and visual quality for a large collection of images. This study provides for the first time a rigorous evaluation in realistic conditions of these bells. Our experiments show that bells that are designed to reproduce exactly polynomials of degree 1 resulted in the worst performance in terms of the PSNR. However, a visual inspection of the compressed images indicates that these bells often provide reconstructed images with very few visual artifacts, even at low bit rates. The bell with the most narrow Fourier transform gave the best results in terms of the PSNR on most images. This bell tends however to create annoying visual artifacts in very smooth regions at low bit rate.

Best Bells for the Scattering By Perfectly Conducting Cylinders

Application of the local bell-modulated cosine bases of Coifman and Meyer to the problem of scattering by PEC cylinders leads to a sparse Method of Moments interaction matrix. To optimize the sparsity representation it is imperative to find bell functions with nice Fourier radiation properties. Here we pose the problem of bell optimization in terms of Chebyshev's inequality, resulting in a family of elliptic bells (depending on elliptic functions) parametrized by a real parameter. The effect of finite curvature being, as a first approximation, a deformation of the originally pythagorean bell into a non-pythagorean bell, we then extend the formulation to include general non-pythagorean bells. We propose a class of Kaiser-Bessel bells, well-known from signal and image processing theory for their nice radiation properties. We implement the method to scattering by elliptic and cardioid-like cylinders to illustrate the technique. Based on various specific indicators measuring sparsity, fastness and accuracy, we find that a pertinent Kaiser-Bessel bell performs generally better than the elliptic bells.

Minimax robust nonstationary signal estimation based on a p-point uncertainty model

We propose a time-varying Wiener filter for nonstationary signal estimation that is robust in a minimax sense. This robust Wiener filter optimizes worst-case performance within novel “ p-point” uncertainty classes of nonstationary random processes. Furthermore, it features constant performance within these uncertainty classes and requires less detailed prior knowledge than the ordinary time-varying Wiener filter. We also propose a time–frequency formulation that is intuitively appealing since signal subspaces are replaced by time–frequency regions, and an efficient on-line implementation using local cosine bases. Our theory is illustrated by numerical simulations and a real-data example.

Fast Spectral Projection Algorithms for Density-Matrix Computations

We present a fast algorithm for the construction of a spectral projector. This algorithm allows us to compute the density matrix, as used in, e.g., the Kohn–Sham iteration, and so obtain the electron density. We compute the spectral projector by constructing the matrix sign function through a simple polynomial recursion. We present several matrix representations for fast computation within this recursion, using bases with controlled space–spatial-frequency localization. In particular we consider wavelet and local cosine bases. Since spectral projectors appear in many contexts, we expect many additional applications of our approach.

On denoising and best signal representation

We propose a best basis algorithm for signal enhancement in white Gaussian noise. The best basis search is performed in families of orthonormal bases constructed with wavelet packets or local cosine bases. We base our search for the "best" basis on a criterion of minimal reconstruction error of the underlying signal. This approach is intuitively appealing, because the enhanced or estimated signal has an associated measure of performance, namely, the resulting mean-square error. Previous approaches in this framework have focused on obtaining the most "compact" signal representations, which consequently contribute to effective denoising. These approaches, however, do not possess the inherent measure of performance which our algorithm provides. We first propose an estimator of the mean-square error, based on a heuristic argument and subsequently compare the reconstruction performance based upon it to that based on the Stein (1981) unbiased risk estimator. We compare the two proposed estimators by providing both qualitative and quantitative analyses of the bias term. Having two estimators of the mean-square error, we incorporate these cost functions into the search for the "best" basis, and subsequently provide a substantiating example to demonstrate their performance.

Discrete- and continuous-time local cosine bases with multiple overlapping

Cosine-modulated filter banks (CMFBs) are filter banks whose impulse responses are obtained by modulating a window with cosines. Among their applications are video and audio compression and multitone modulation. Their continuous-time counterpart is known as local cosine bases. Even though there is an extended literature on the discrete-time case both for single and multiple overlapping, the continuous-time case has received less attention, and only the single overlapping case has been solved. This work gives a solution to the problem of continuous-time local cosine bases with multiple overlapping via a general theory that emphasizes the deep connection between discrete and continuous time. A sampling theorem for local cosine basis and an efficient algorithm to compute the expansion of a signal are also given.

Local cosine bases in two dimensions

We construct two-dimensional (2-D) local cosine bases in discrete time. Solutions are offered both for rectangular and nonrectangular lattices. In the case of nonrectangular lattices, the problem is solved by mapping it into a one-dimensional (1-D) equivalent problem.

Wavelet analysis in recruitment of loudness compensation

Wavelet-based multiband dynamic range compression is developed to compensate for a common hearing impairment known as recruitment of loudness. The algorithm combines standard compression with intensity-level dependent gain calculation. Complexity and performance are similar to traditional techniques. Further, the methods established can be applied to the more adaptive wavelet packets and local cosine bases which model the speech signal more closely. >