We study local deformations of time-frequency and time-scale representations, in the framework of the so-called reassignment methods, which aim at `deblurring' time- frequency representations. We focus on deformations generated by appropriate vector fields defined on time- frequency or time scale plane, and constructed on the basis of geometric and group-theoretical arguments. Such vector fields may be used as such for signal analysis (as quantities generalizing instantaneous frequency or group delay) in the framework of reassignment algorithms.

This paper discusses the utility of scale-angle continuous wavelet transform features for object classification. These features are used as input to two algorithms: character recognition and target recognition in FLIR images. The corresponding recognition algorithm is robust against noise and allows data reduction. A comparative study is made between two types of directional wavelets derived from the Mexican hat wavelet and the usual template matching.

The analysis of oriented features in images requires 2D directional wavelets, for instance in standard tasks such as edge detection or directional filtering. In addition we present here a new application, namely a technique for determining all the (statistical) symmetries of a given pattern with respect to rotations and dilations. Examples are Penrose tilings, mathematical quasicrystals or various quasiperiodic planar point sets or patterns.

We consider ill-posed inverse problems where inverting the distortion of signals and images in presence of additive noise is numerically unstable. The properties of linear and non-linear diagonal estimators in an orthogonal basis lead to general conditions to build nearly minimax optimal thresholding estimators. The deconvolution of bounded variation signals and images is studied in further details, with an application to the deblurring of satellite images. Besides their optimality properties, a competition set by the French spatial agency (CNES) showed that this type of algorithms gives the best numerical results among all competing algorithms.

We propose a hybrid approach to wavelet-based deconvolution that comprises Fourier-domain system inversion followed by wavelet-domain noise suppression. In contrast to conventional wavelet-based deconvolution approaches, the algorithm employs a regularized inverse filter, which allows it to operate even when the system in non-invertible. Using a mean-square-error (MSE) metric, we strike an optimal balance between Fourier-domain regularization (matched to the system) and wavelet-domain regularization (matched to the signal/image). Theoretical analysis reveals that the optimal balance is determined by the economics of the signal representation in the wavelet domain and the operator structure. The resulting algorithm is fast (O(Nlog²₂N) complexity for signals/images of N samples) and is well-suited to data with spatially-localized phenomena such as edges. In addition to enjoying asymptotically optimal rates of error decay for certain systems, the algorithm also achieves excellent performance at fixed data lengths. In simulations with real data, the algorithm outperforms the conventional time-invariant Wiener filter and other wavelet- based deconvolution algorithms in terms of both MSE performance and visual quality.

This paper presents a novel wavelet-based method for simultaneous image restoration and edge detection. The Bayesian framework developed here is general enough to treat a wide class of linear inverse problems involving (white or colored) Gaussian observation noises, but we focus on convolution operators. In our new approach, a signal prior is developed by modeling the signal/image wavelet coefficients as independent Gaussian mixture random variables. We specify a uniform (non-informative) distribution on the mixing parameters, which leads to an extremely simple iterative algorithm for joint MAP restoration and edge detection. This algorithm is similar to the popular EM algorithm in that it alternates between a state estimation step and a maximization step, yet it is much simpler in each step and has a very intuitive derivation. Moreover, we show that our algorithm converges monotonically to a local maximum of the posterior distribution. Experimental results show that this new method can perform better than wavelet-vaguelette type methods that are based on linear inverse filtering followed by wavelet coefficient denoising.

In this paper we introduce a Bayesian tomographic reconstruction technique employing a wavelet-based multiresolution prior model. While the image is modeled in the wavelet-domain, the actual tomographic reconstruction is performed in a fixed resolution pixel domain. In comparison to performing the reconstruction in the wavelet domain, the pixel based optimization facilitates enforcement of the positivity constraint and preserves the sparseness of the tomographic projection matrix. Thus our technique combines the advantages of multiresolution image modeling with those of performing the constrained optimization in the pixel domain. In addition to this reconstruction framework, we introduce a novel multiresolution prior model. This prior model attempts to capture the dependencies of wavelet coefficients across scales by using a Markov chain structure. Specifically, the model employs nonlinear predictors to locally estimate the prior distribution of wavelet coefficients from coarse scale information. We incorporate this prior into a coarse-to-fine scale tomographic reconstruction algorithm. Preliminary results indicate that this algorithm can potentially improve reconstruction quality over fix resolution Bayesian methods.

We present in this paper a new way to measure the information in a signal, based on noise modeling. We show that the use of such an entropy-related measure leads to good results for signal restoration.

This paper deals with multiwavelets and the different properties of approximation and smoothness that are associated with them. In particular, we focus on the important issue of the preservation of discrete time polynomial signals by multiwavelet based filter banks. We give here a precise definition of balancing for higher degree discrete time polynomial signals and link it to a very natural factorization of the lowpass refinement mask that is the counterpart of the well-known zeros at (pi) condition on the scaling function in the usual wavelet framework. This property of balancing proves them to be central to the issues of the preservation of smooth signals by the filter bank, the approximation power of the multiresolution analysis and the smoothness of the scaling functions and wavelets. Using these new results, we are able to construct a family of orthogonal multiwavelets with symmetries and compact support that is indexed by the order of balancing. We also give the minimum length orthogonal multiwavelets for any balancing order.

This paper describes a basic difference between multiwavelets and scalar wavelets that explains, without using zero moment properties, why certain complications arise in the implementation of discrete multiwavelet transforms. Assuming one wishes to avoid the use of prefilters in implementing the discrete multiwavelet transform, it is suggested that the behavior of the iterated filter bank associated with a multiwavelet basis of multiplicity r is more fully revealed by an expanded set of r² scaling functions (phi) _i,j. This paper also introduces new K-balanced orthogonal multiwavelet bases based on symmetric FIR filers. The nonlinear design equations arising in this work are solved using Groebner bases. The K-balanced multiwavelet bases based on even- length symmetric FIR filters are shown to be particularly well behaved, as illustrated by special relations they satisfy and by the examples constructed.

In the following, we give the complete solutions of all bivariate symmetric orthonormal scaling functions with filter size up to 6 X 6 using the standard dilation matrix 2I as well as construct a set of associated wavelets. In addition, we give two families of refinable functions which are not orthonormal, but upon using the same completion technique appear to form tight frames (i.e. they preserve the norm) at least experimentally.

Biorthogonal multiwavelets are generated from refinable function vectors by using multiresolution analyses. To obtain a biorthogonal multiwavelet, we need to construct a pair of primal and dual masks, from which two refinable function vectors are obtained so that a multiresolution analysis is formed to derive a biorthogonal multiwavelet. It is well known that the order of vanishing moments of a biorthogonal multiwavelet is one of the most desirable properties of a biorthogonal multiwavelet in various applications. To design a biorthogonal multiwavelet with high order of vanishing moments, we have to design a pair of primal and dual masks with high order of sum rules. In this paper, we shall study an important family of primal masks-- Hermite interpolatory masks. A general way for constructing Hermite interpolatory masks with increasing order of sum rules is presented. Such family of Hermite interpolants from the Hermite interpolatory masks includes the piecewise Hermite cubics as a special case. In particular, a C³ Hermite interpolant is constructed with support [-3,3] and multiplicity 2. Next, we shall present a construction by cosets (CBC) algorithm to construct biorthogonal multiwavelets with arbitrary order of vanishing moments. By employing the CBC algorithm, several examples of biorthogonal multiwavelets are provided to illustrate the general theory. In particular, a C¹ dual function vector of the well-known piecewise Hermite cubics is given.

This manuscript gives a construction of divergence-free multiwavelets which combines the Hardin-Marasovich (HM) construction with a recipe of Strela for increasing or decreasing regularity of biorthogonal wavelets. Strela's process preserves symmetry of the HM wavelets. This enables the divergence-free wavelets to be suitably adapted to the analysis of divergence-free vector fields whose boundary traces are tangent vectors.

In this paper, the symmetric periodic extension method is generalized for multiwavelet filter banks. It is described how the sequences should be extended for any symmetry type of multifilter of size two and the corresponding pre- /postfilter and how the filtered sequence should be downsampled. The present paper shows that there are restrictions for any symmetry type of multifilter relating to the size of the sequence and the type of pre-/postfilter. Examples are given. The symmetric periodic extension method is compared with the periodic extension method and the effects on the transformed sequence are discussed.

I describe a statistical model for natural photographic images, when decomposed in a multi-scale wavelet basis. In particular, I examine both the marginal and pairwise joint histograms of wavelet coefficients at adjacent spatial locations, orientations, and spatial scales. Although the histograms are highly non-Gaussian, they are nevertheless well described using fairly simple parameterized density models.

The more a priori knowledge we encode into a signal processing algorithm, the better performance we can expect. In this paper, we overview several approaches to capturing the structure of singularities (edges, ridges, etc.) in wavelet-based signal processing schemes. Leveraging results from approximation theory, we discuss nonlinear approximations on trees and point out that an optimal tree approximant exists and is easily computed. The optimal tree approximation inspires a new hierarchical interpretation of the wavelet decomposition and a tree-based wavelet denoising algorithm that suppresses spurious noise bumps.

This paper addresses the problem of modeling of Earth phenomena from remotely sensed images. In the field of remote sensing, the observed phenomena are most often multi- scale phenomena such as the waves in oceans, the spatial organization of street networks in cities. For extracting information, or processing this kind of images, a multiresolution approach and modeling is often most powerful than classical approaches. We intend to present some winning applications based on multiresolution models derived from the representation provided by multiresolution analysis and wavelet transforms (WT). After a short reminder of WT and an example of application to a remotely sensed image, three applications where modeling based on WT improves the quality of results are presented: vine detection, streets extraction in urban areas and improvement of the spatial resolution of remotely sensed images.

In this paper, we motivate and describe a scattered data interpolation scheme based on a hierarchical wavelet subfamily selection process named allocation. This interpolation method applies in any dimension, where it compares well to regularization techniques, especially in terms of stability, of adaptivity and of sparsity of the learned function representation. Adaptive convergence theorems are stated, and their proofs are outlined. We also describe a variant of this approach that can be incremental, and thus works as an online learning process.

In this paper, we propose a new algorithm for extracting a non smooth shape from its noisy observation. The key ideal is to project the noisy shape onto a set of orthogonal subspaces at different resolutions, and construct scale space representation gleaned from the locally smoothed shape. Using the curvature we proceed to filter the high resolution scale subspace by projecting it onto the scale which is in turn used for the reconstruction. Inspired by the conjugate mirror filter and the wavelet decomposition synergy, we propose a curvature based filter operating at different scales and with minimal knowledge about the noise statistics.

A Bayesian analysis based on the empirical wavelet coefficients is developed for the standard change-point problem. This analysis is considered first for the piecewise constant Haar wavelet basis, then extended to using smooth and orthogonal boundary wavelets. The analysis, though developed for use in the standard change-point model, can be applied to estimating a discontinuity in an otherwise smooth function by using only higher level coefficients in the computation of the posterior distribution, thereby effectively removing a `smooth' of the function. The procedure is illustrated with simulated data examples.

A well-suited approach to calculate the fractal dimension of digital images stems from the power spectrum of a fractional Brownian motion: the ratio between powers at different scales is related to the persistence parameter H and, thus, to the fractal dimension D equals 3 - H. The signal- dependent nature of the speckle noise, however, prevents from a correct estimation of fractal dimension from Synthetic Aperture Radar (SAR) images. Here, we propose and assess a novel method to obtain D based on the multiscale decomposition provided by the normalized Laplacian pyramid, which is a bandpass representation obtained by dividing the layers of an LP by its expanded baseband and is designed to force the noise to become signal-independent. Extensive experiments on synthetic fractal textures, both noise-free and noisy, corroborate the underlying assumptions and show the performances, in terms of both accuracy and confidence of estimation, of pyramid methods compared with the well- established method based on the wavelet transform. Preliminary results on true SAR images from ERS-1 look promising as well.

This paper deals with constructions of compactly supported biorthogonal wavelets from a pair of dual definable functions in L₂(R^s). In particular, an algorithmic method to construct wavelet systems and the corresponding dual systems from a given pair of dual refinable functions is given.

A filterbank decomposition can be seen as a series of projections onto several discrete wavelet subspaces. In this presentation, we analyze the projection onto one of them-- the low-pass one, since many signals tend to be low-pass. We prove a general but simple formula that allows the computation of the l²-error made by approximating the signal by its projection. This result provides a norm for evaluating the accuracy of a complete decimation/interpolation branch for arbitrary analysis and synthesis filters; such a norm could be useful for the joint design of an analysis and synthesis filter, especially in the non-orthonormal case. As an example, we use our framework to compare the efficiency of different wavelet filters, such as Daubechies' or splines. In particular, we prove that the error made by using a Daubechies' filter downsampled by 2 is of the same order as the error using an orthonormal spline filter downsampled by 6. This proof is valid asymptotically as the number of regularity factors tends to infinity, and for a signal that is essentially low- pass. This implies that splines bring an additional compression gain of at least 3 over Daubechies' filters, asymptotically.

We present a technique based on nilpotent matrices for building filter banks with FIR filters and perfect reconstruction. The general design method can be used to design bi-orthogonal filters with unequal filter lengths between analysis and synthesis. This is useful for audio or image coding applications. We can also explicitly control the overall system delay of causal filter banks. The design method is based on a factorization of the polyphase matrices into factors with nilpotent matrices. These factors guarantee mathematical perfect reconstruction of the filter bank, and lead to FIR filters for analysis and synthesis. Using matrices with nilpotency of higher order than 2 leads to FIR filter banks with unequal filter length for analysis and synthesis. The general theory is then applied to the design of cosine modulated filter banks. This leads to an efficient implementation, and it is shown that in this case the filters have to have the same length for analysis and synthesis.

This paper presents a comparison of Advanced Filter Banks (AFB) and Time-Interleaving for high-speed, high-resolution conversion between analog and digital signals using a parallel array of converters. The AFB is an unconventional class of filter bank that employs both analog and digital signal processing. The AFB improves the speed and resolution of the conversion compared to the standard Time-Interleaved array conversion technique. Gain and phase mismatch errors are analyzed for both the AFB and Time-Interleaving architectures. The filters in the AFB isolate the converters in the array from each other and attenuate the effects of mismatches. In four-channel example systems analyzed in this paper, gain and phase errors are attenuated by 21 dB more in the AFB (with 30 dB stopband attenuation) than in the Time- Interleaved system. The AFB is capable of analog-to-digital conversion with 14-bit resolution and 400 MHz sample rate.

In this paper, we study some properties of nonmaximally decimated multirate filterbanks precoders in blindly mitigating the intersymbol interference channel.

Transform coding and differential pulse code modulation (DPCM) are two of the most important techniques for signal compression. The Karhunen-Loeve transform (KLT) is the optimal unitary transform that maximizes the coding gain. For DPCM coder, the optimal linear predictor maximizes its coding gain. In this paper, we will provide a connection between KLT and DPCM. A new class of nonunitary transform called Prediction-based Lower triangular Transform (PLT) is introduced. We will show that PLT is a special case of vector prediction, when the predictor is a lower triangular matrix. PLT has the same coding gain as KLT but its design and computational cost is much lower. In addition, it has many other desired features that make it a good choice for signal compression.

The purpose of this paper is to demonstrate the optimality properties of principal component filter-banks for various noise reduction schemes. Optimization of filter-banks (FB's) for coding gain maximization has been carried out in the literature, and the optimized solutions have been observed to satisfy the principal component property, which has independently been studied. Here we show a strong connection between the optimality and the principal component property; which allows us to optimize FB's for many other objectives. Thus, we consider the noise-reduction scheme where a noisy signal is analyzed using a FB and the subband signals are processed either using a hard-threshold operation or a zeroth order Wiener filter. For these situations, we show that a principal FB is again optimal in the sense of minimizing the expected mean-square error.

This paper introduces a class of M-channel linear phase perfect reconstruction filter banks with integer coefficients. The construction of these filter banks is based on a VLSI-friendly lattice structure which employs the minimum number of delay elements and robustly enforces both linear phase and perfect reconstruction properties. The lattice coefficients are parameterized as a series of lifting steps, providing fast, efficient, in-place computational of the subband coefficients. Image coding experiments show that these novel FBs are very competitive with current state-of-the-art transforms such as the 8 X 8 DCT and the wavelet transform with 9/7-tap biorthogonal floating-point wavelet.

This paper develops new algorithms for adapted multiscale analysis and signal adaptive wavelet transforms. We construct our adaptive transforms with the lifting scheme, which decomposes the wavelet transform into prediction and update stages. We adapt the prediction stage to the signal structure and design the update stage to preserve the desirable properties of the wavelet transform. The resulting scale and spatially adaptive transforms are extended to the image estimation problem; our new image transforms show improved denoising performance over existing (non-adaptive) orthogonal transforms.

The discrete wavelet transform (DWT) has been touted as a very effective tool in many signal processing application, including compression, denoising and modulation. For example, the forthcoming JPEG 2000 image compression standard will be based on the DWT. However, in order for the DWT to achieve the popularity of other more established techniques (e.g., the DCT in compression) a substantial effort is necessary in order to solve some of the related implementation issues. Specific issues of interest include memory utilization, computation complexity and scalability. In this paper we concentrate on wavelet-based image compression and provide examples, based on our recent work, of how these implementation issues can be addressed in three different environments, namely, memory constrained applications, software-only encoding/decoding, and parallel computing engines. Specifically we will discuss (1) a low memory image coding algorithm that employs a line-based transform, (2) a technique to exploit the sparseness of non- zero wavelet coefficients in a software-only image decoder, and (3) parallel implementation techniques that take full advantage of lifting filterbank factorizations.

This paper describes a video coding algorithm that combines new ideas in motion estimation, wavelet filter design, and wavelet-based coding techniques. A motion compensation technique using image warping and overlapped block motion compensation is employed to reduce temporal redundancies in a given image sequence. This combined motion model has the advantage of representing more complex motion than simple block matching schemes. Spatial decorrelation of the motion compensated residual images is performed using an one- parametric family of biorthogonal IIR wavelet filters coupled with a highly efficient pre-coding scheme. Experimental results demonstrate substantial improvements in objective quality of 1.0 - 2.2 dB PSNR compared to the H.263+ standard. Especially at very low bit-rates where the reconstruction quality of block-based coders suffers from visually annoying blocking artifacts the proposed coding scheme produces a superior subjective quality.

Wilson bases consist of products of trigonometric functions with window functions which have good time-frequency localization, so that the basis functions themselves are well localized in time and frequency. Therefore, Wilson bases are well suited for time-frequency analysis. Daubechies, Jaffard and Journe have given conditions on the window function for which the resulting Wilson basis is orthonormal. In particular, they constructed an example where the basis functions have exponential decay in the time and the frequency domain. Here, we investigate biorthogonal Wilson bases with arbitrary shape. Necessary and sufficient conditions for the Riesz stability of these bases are given. Furthermore, we determine exact Riesz bounds and the dual bases.

We extend Schoenberg's B-splines to all fractional degrees (alpha) > -1/2. These splines are constructed using linear combinations of the integer shifts of the power functions x^(alpha ) (one-sided) or x^(alpha )_* (symmetric); in each case, they are (alpha) -Hoelder continuous for (alpha) > 0. They satisfy most of the properties of the traditional B-splines; in particular, the Biesz basis condition and the two-scale relation, which makes them suitable for the construction of new families of wavelet bases. What is especially interesting from a wavelet perspective is that the fractional B-splines have a fractional order of approximately ((alpha) + 1), while they reproduce the polynomials of degree [(alpha) ]. We show how they yield continuous-order generalization of the orthogonal Battle- Lemarie wavelets and of the semi-orthogonal B-spline wavelets. As (alpha) increases, these latter wavelets tend to be optimally localized in time and frequency in the sense specified by the uncertainty principle. The corresponding analysis wavelets also behave like fractional differentiators; they may therefore be used to whiten fractional Brownian motion processes.

In this work we display and multiresolution analysis scheme restricted on the interval [0,N]. This scheme is developed for the case of Hermite spline functions but it can be implemented in more general cases. Embedded in this scheme we construct semiorthogonal multiwavelets. Also we expose several methods and algorithms for signal processing applications.

The paper discusses the factorization of polyphase matrices corresponding to symmetric compactly supported biorthogonal multiwavelets into symmetric lifting factors. This factorization allows parameterization of symmetric compact supported biorthogonal multiwavelets, which can be used in conjunction with numerical optimization to construct symmetric compactly supported biorthogonal multiwavelets with prescribed properties.

Multiwavelets in one dimension have given good results for image compression. On the other hand, nonseparable bidimensional wavelets have certain advantages over the tensor product of 1D wavelets. In this work we give examples of multiwavelets that are bidimensional and nonseparable. They correspond to the dilation matrices D₁ equals [1 1;1 -1], a reflection followed by an expansion of (root)2, or to D₂ equals [1 1;1 1], a rotation followed by an expansion of (root)2, and possess suitable properties for image compression: short support and 2 or 3 vanishing moments. Multiwavelets are derived from multiscaling functions. Conditions on the matrix coefficients of the dilation equation are exploited to build examples of orthogonal, nonseparable, compactly supported, bidimensional multiscaling functions of accuracy 2 and 3. They are continuous and the joint spectral radius is estimated. To the author's knowledge no examples of bidimensional multiscaling functions with these characteristics had been found previously. Coefficients for the corresponding multiwavelets are given. Multiscaling functions and multiwavelets are plotted with a cascade algorithm.

Recently many new methods of image representation have been proposed, including wavelets, cosine packets, brushlets, edgelets, and ridgelets. Typically each of these is good for a specific class of features, but not good for others. We propose method of combining two image representations based on 2D wavelet transform and edgelet transform. The 2D wavelet transform is good at capturing point singularities, while the newly proposed edgelet transform is good at capturing linear singularities (edges). Both transforms have fast algorithms for digital images.

The purpose of this paper is to study signal denoising by thresholding coefficients of undecimated discrete wavelet packet transforms (UDWPT). The undecimated filterbank implementation of UDWPT is first considered, and the best basis selection algorithm that prunes the complete undecimated discrete wavelet packet binary tree is studied for the purpose of signal denoising. Distinct from the usual approach which selects the best subtree based on the original (unthresholded) transform coefficients, our selection is based on the thresholded coefficients, since we believe discarding the small coefficients permits to choose the best basis from the set of coefficients that will really contribute to the reconstructed signal. Another feature of the algorithm is the thresholding scheme. To threshold coefficients which are correlated differently from scale to scale and from band to band, a uniform threshold is not appropriate. Alternatively, two scale-band-dependent thresholding schemes are designed: a correlation-dependent model and a Monte Carlo simulation-based model. The cost function for the pruning algorithm is specifically designed for the purpose of signal denoising. We consider it profitable to split a band if more noise can be discarded by thresholding while signal components are preserved. So, higher SNR is desirable in the process of selection. Experiments conducted for 1D and 2D signals shows that the algorithm achieves good SNR performance while preserving high frequency details of signals.

We discover a new relationship between two seemingly different image modeling methodologies; the Besov space theory and the wavelet-domain statistical image models. Besov spaces characterize the set of real-world images through a deterministic characterization of the image smoothness, while statistical image models capture the probabilistic properties of images. By establishing a relationship between the Besov norm and the normalized likelihood function under an independent wavelet-domain generalized Gaussian model, we obtain a new interpretation of the Besov norm which provides a natural generalization of the theory for practical image processing. Base don this new interpretation of the Besov space, we propose a new image denoising algorithm based on projections onto the convex sets defined in the Besov space. After pointing out the limitations of Besov space, we propose possible generalizations using more accurate image models.

A nonlinear block-median pyramidal transform has been proposed. This transform is based on the iterative application of the median operation and linear Lagrange interpolation. The probability distribution of the transform coefficients has been analytically derived for i.i.d. input signals. The results of this statistical analysis are used for selecting the thresholds for denoising applications. Numerical simulation results are presented.

In this paper, we will present a systematic discussion for the optimum interpolatory approximation in a shift-invariant wavelet and/or scaling subspace. Firstly, we will present the optimum interpolation functions which minimize various worst case measure of approximation error among all the linear and the nonlinear approximations using the same sample values of the input signal. Secondly, we will show that the optimum interpolation functions are expressed as the parallel shifts of the finite number of one function. Finally, we will present the optimum interpolation function in wavelet and scaling subspace. These interpolation functions are optimum in the multi-resolution analysis which considers lower resolutions.

Pseudo orthogonal bases are a certain type of frames proposed in the engineering field, whose concept is equivalent to a tight frame with frame bound 1 in the frame terminology. This paper shows that pseudo orthogonal bases play an essential role in neural network learning. One of the most important issues in neural network learning is `what training data provides the optimal generalization capability?', which is referred to as active learning in the neural network community. We derive a necessary and sufficient condition of training data to provide the optimal generalization capability in the trigonometric polynomial space, where the concept of pseudo orthogonal bases is essential. By utilizing useful properties of pseudo orthogonal bases, we clarify the mechanism of achieving the optimal generalization.

Signal processing and imaging of biomedical phenomena pose significant challenges, with one dominant issue being that biological processes are usually time varying and non- stationary. Many traditional processing approaches are derived on assumptions of statistical stationarity and linear time-invariant propagation channels, which are not valid assumptions for many biomedical problems. In this paper, continuous wavelet transforms are shown to be appropriate tools for characterizing linear time-varying systems and propagation channels and for processing wideband signals in non-stationary Gaussian noise. Wideband processing of signals allows for the processing to be limited by the scattering object's acceleration versus the more common techniques where the processing is limited by the scattering object's velocity. It is shown that the continuous wavelet transform of the output signal with respect to the input signal provides a correct system characterization for time-varying channels and non- stationary signals. Finally, an approach to removing even the wideband limitation of acceleration is presented. Possible biomedical applications of this approach include bloodflow velocimetry and heart motion monitoring.

Functional magnetic resonance imaging is a recent technique that allows the measurement of brain metabolism (local concentration of deoxyhemoglobin using BOLD contrast) while subjects are performing a specific task. A block paradigm produces alternating sequences of images (e.g., rest versus motor task). In order to detect and localize areas of cerebral activation, one analyzes the data using paired differences at the voxel level. As an alternative to the traditional approach which uses Gaussian spatial filtering to reduce measurement noise, we propose to analyze the data using an orthogonal filterbank. This procedure is intended to simplify and eventually improve the statistical analysis. The system is designed to concentrate the signal into a fewer number of components thereby improving the signal-to- noise ratio. Thanks to the orthogonality property, we can test the filtered components independently on a voxel-by- voxel basis; this testing procedure is optimal for i.i.d. measurement noise. The number of components to test is also reduced because of down-sampling. This offers a straightforward approach to increasing the sensitivity of the analysis (lower detection threshold) while applying the standard Bonferroni correction for multiple statistical tests. We present experimental results to illustrate the procedure. In addition, we discuss filter design issues. In particular, we introduce a family of orthogonal filters which are such that any integer reduction m can be implemented as a succession of elementary reductions m₁ to m_p where m equals m₁...m_p is a prime number factorization of m.

Wavelet threshold algorithms replace wavelet coefficients with small magnitude by zero and keep or shrink the other coefficients. This is basically a local procedure, since wavelet coefficients characterize the local regularity of a function. Although a wavelet transform has decorrelating properties, structures in images, like edges, are never decorrelated completely, and these structures appear in the wavelet coefficients. We therefore introduce geometrical prior model for configurations of large wavelet coefficients and combine this with the local characterization of a classical threshold procedure into a Bayesian framework. The threshold procedure selects the large coefficients in the actual image. This observed configuration enters the prior model, which, by itself, only describes configurations, not coefficient values. In this way, we can compute for each coefficient the probability of being `sufficiently clean'.

We propose Wavelet ANOVA, a simple general-purpose statistical method for analysis of signals and images. We emphasize the application of the method to functional magnetic resonance imaging (fMRI). Wavelet ANOVA combines the false discovery rate approach to multiple comparisons with block wavelet thresholding and linear statistical models. We discuss the relationship of Wavelet ANOVA to a similar method of Ruttimann, et al. We illustrate the application of Wavelet ANOVA to analysis of an fMRI data set.

Registration of images subject to non-linear warping has numerous practical applications. We present an algorithm based on double multiresolution structure of warp and image spaces. Tuning a so-called scale parameter controls the coarseness of the grid by which the deformation is described and also the amount of implicit regularization. The application of our algorithm deals with undoing unidirectional non-linear geometrical distortion of echo- planar images (EPI) caused by local magnetic field inhomogeneities induced mainly by the subject presence. The unwarping is based on registering the EPI images with corresponding undistorted anatomical MRI images. We present evaluation of our method using a wavelet-based random Sobolev-type deformation generator as well as other experimental examples.

A new numerical wavelet transform, the discrete torus wavelet transform, is described and an application is given to the denoising of abdominal magnetic resonance imaging (MRI) data. The discrete tori wavelet transform is an undecimated wavelet transform which is computed using a discrete Fourier transform and multiplication instead of by direct convolution in the image domain. This approach leads to a decomposition of the image onto frames in the space of square summable functions on the discrete torus, l²(T²). The new transform was compared to the traditional decimated wavelet transform in its ability to denoise MRI data. By using denoised images as the basis for the computation of a nuclear magnetic resonance spin-spin relaxation-time map through least squares curve fitting, an error map was generated that was used to assess the performance of the denoising algorithms. The discrete torus wavelet transform outperformed the traditional wavelet transform in 88% of the T₂ error map denoising tests with phantoms and gynecologic MRI images.

The local Fourier dictionary contains a large number of localized complex exponential functions. Representations of a function using the dictionary elements locally inherit many nice properties of the conventional Fourier representation, such as translation invariance and orientation selectivity. In this paper, after giving an intuitive review of its construction, we describe an algorithm to recover location-dependent shifts of local features in signals for matching and registration, and propose a best local translation basis selected from the local Fourier basis. Then we will report our preliminary results on the statistical analysis of natural scene images using the local Fourier dictionary, whose purpose is to examine the importance of sparsity, statistical independence, and orientation selectivities in representation and modeling of such images.

Orthogonal frequency division multiplexing (OFDM) has recently become a popular technique for high-data-rate transmission over wireless channels. Due to the time- frequency dispersion caused by the channel, the performance of OFDM systems depends critically on the time-frequency localization of the pulse shaping filters. In this paper, we show how the recent duality and biorthogonality theory developed in the context of Weyl-Heisenberg frames can be used to devise simple and efficient design procedures for well-localized OFDM pulse shaping filters. We consider OFDM systems employing time-frequency guard regions and OFDM systems based on offset QAM. We propose FFT-based design methods for pulse shaping filters with arbitrary length and arbitrary overlapping factors. Finally, we present some design examples.

We analyze the relation between infinite-dimensional frame theory and finite-dimensional models for frames as they are used for numerical algorithms. Special emphasis in this paper is on perfect reconstruction oversampled filter banks, also known as shift-invariant frames. For certain finite- dimensional models it is shown that the corresponding finite dual frame provides indeed an approximation of the dual frame of the original infinite-dimensional dual frame. For filter banks on l² (Z) we derive error estimates for the approximation of the synthesis filter bank when the analysis filter bank satisfies certain decay conditions. We show how one has to design the finite-dimensional model to preserve important structural properties of filter banks, such as polyphase representation. Finally an efficient regularization method is presented to solve the ill-posed problem arising when approximating the dual frame on L²(R) via truncated Gram matrix.

Our goal in this paper is to provide a fast numerical implementation of the local trigonometric bases algorithm in order to demonstrate that an advantage can be gained by constructing a biorthogonal basis adapted to a target image. Different choices for the bells are proposed, and an extensive evaluation of the algorithm was performed on synthetic and seismic data. Because of its ability to reproduce textures so well, the coder performs very well, even at high bitrate.

We unify two widely used image processing kernels, a Gaussian's first derivative (Canny's step edge detector) and the modulated Gaussian, by introducing a nonorthogonal wavelet family: the sine-Gabor functions. This parametrized wavelet trades off frame rightness and time-frequency localization; generating a `snug' frame of Gaussian's first derivatives and a `loose' frame of modulated Gaussians with nearly optimum time-frequency localization (easily tightened using voices). We review the discretization of the wavelet transform and its efficient computation when using nonorthogonal wavelets. We show that there exists wavelets (including the sine-Gabor function) for which the discrete wavelet transform is not invertible, describe how to evade this problem by modifying the strategy for selecting discrete filters, and show for the sine-Gabor wavelet that desirable properties such as constant phase and time- frequency localization are preserved by the alternative filters.

We discuss the relation between lattice and ladder structures for two-channel filter banks. It is well-known that both lattice and ladder steps are powerful enough to generate all perfect reconstructing filter banks provided that the filter coefficients may take arbitrary values in a field. However, we will show that the two concepts differ in general. We relate the two concepts by looking at three properties of the coefficient ring. We discuss a number of incompleteness results of these parametrizations and point out some connections to open problems in group theory.