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.

A matrix filter is a linear and time-invariant operator on the space of vector-valued signals. Matrix filter bank is the generalization of filter bank. A perfect reconstruction matrix filter bank consists of an analysis matrix filter bank and a synthesis matrix filter bank. In the theory of filter design, generating a perfect reconstruction matrix filter bank from a given lowpass matrix filter is considered. Such a lowpass matrix filter is called a primary matrix filter. In this paper, we give a necessary and sufficient condition for a lowpass matrix filter being primary and discuss the relation between perfect reconstruction matrix filter bank and biorthogonal multiwavelet.

The measurement of shape, displacement and deformations is often performed using interferometric methods, featuring nm to mm sensitivities and very high spatial and temporal resolutions. We first give a brief overview of interferometric techniques. Emphasis is laid on the wide purposes of these techniques. Then, we present a novel method using wavelet analysis to process live interference patterns. Further developments of the method are then presented. Finally, through two practical examples, we intend to highlight the interest of fringe processing by wavelet transform.

We show how periodized wavelet packet transforms and periodized wavelet transforms can be implemented on a quantum computer. Surprisingly, we find that the implementation of wavelet packet transforms is less costly than the implementation of wavelet transforms on a quantum computer.

This paper deals with the problem of automatic recognition of music. Segments of digitized music are processed by means of a Continuous Wavelet Transform, properly chosen so as to match the spectral characteristics of the signal. In order to achieve a good time-scale representation of the signal components a novel wavelet has been designed suited to the musical signal features. particular care has been devoted towards an efficient implementation, which operates in the frequency domain, and includes proper segmentation and aliasing reduction techniques to make the analysis of long signals feasible. The method achieves very good performance in terms of both time and frequency selectivity, and can yield the estimate and the localization in time of both the fundamental frequency and the main harmonics of each tone. The analysis is used as a preprocessing step for a recognition algorithm, which we show to be almost independent on the instrument reproducing the sounds. Simulations are provided to demonstrate the effectiveness of the proposed method.

We report recognition results using scale-transform based cepstral features in a telephone based digit recognition task. The method is based on the use of scale-transform based features for speaker-independent applications, which are insensitive to linear-frequency scaling effects and therefore reduce inter-speaker variability due to differences in vocal-tract lengths. We have implemented a digit recognition task using the proposed scale-transform based features and have compared the recognition accuracy obtained when compared to using mel-cepstrum based front-end features.

In this paper, we present two filters that simulate the behavior of biological end-stopped cells. Both are zero-mean filters, and are well located in the spatial as well as frequency domains, that is, these filters are admissible wavelets. We refer to the two filters as ES1 and ES2. The ES1 filter responds to ends of linear structures which have a specific orientation, and the ES2 filter responds to line- segments which have a specific orientation, and which have a length within a specific range. We show sample results to demonstrate the behavior of the proposed wavelets, and we also discuss the scale-space behavior of these wavelets briefly.

With the advancement of multimedia technology and the internet, numerous applications have arisen which require the storage and retrieval of large image and video databases. A novel method (Eigenwavelet) was developed to retrieve images from a large heterogeneous image database upon a user-specified query. The queries are in the form of an image(s) that the user seeks to find similar matches to in the database. Using the queries, an efficient algorithm was developed which decomposed each image in the database using wavelet packet analysis. Along each node of the packet tree, Principal Component Analysis was applied to the database images after wavelet packet decomposition, and a set of eigenvectors were generated for each node of the packet tree. To search the image database, the query images are projected onto these eigenvectors (Eigenwavelet coefficients). A distance metric is computed between the projections of the queries and the projections of the images in the database onto the eigenwavelets. Those images with minimal distance (L1) are retrieved in response to a unique query set. Simulations with a heterogeneous image database suggest the Eigenwavelet method of image retrieval is a robust and computationally tractable method of retrieving images with a probability of detection >.8.

In this paper, the recovery of noisy process data or de- noising using wavelets is studied. In addition, the effectiveness of some wavelet-based modern algorithms of process data recovery is experimented. A novel approach in de-noising according to the WienerShrink method of wavelet coefficients thresholding is suggested. Finally, the results of computer simulation for a special case study and comparing the mean square errors of the algorithms are presented. Two Continuous Stirred Tank Reactors in series with an intermediate mixer, are considered as a case study. The results show the advantages of our method over the previous wavelet-based approaches.

In this paper, an efficient technique is presented for recognizing 2D occluded objects under orientation, location, and size transformations. The developed technique is based on analyzing the boundary of the object using the maxima lines of the continuous wavelet transform. The experimental results showed that this refined technique successfully classified the occluded objects.

In this paper, we propose an original decomposition scheme based on Meyer's wavelets. In opposition to a classical technique of wavelet packet analysis, the decomposition is an adaptative segmentation of the frequential axis which does not use a filters bank. This permits a higher flexibility in the band frequency definition. The decomposition computes all possible partitions from a sequential space: it does not only compute those that come from a dyadic decomposition. Our technique is applied on the electroencephalogram signal; here the purpose is to extract a best basis of frequential decomposition. This study is part of a multimodal functional cerebral imagery project.

Best-basis searching algorithm based on binary (in general, M-ary) segmentation was constructed by Coifman and Wickerhauser in 1992. However, there are several problems with the binary scheme. First, the binary segmentation is inflexible in grouping signals along the axis. Secondly, the binary-based segmentation method is very sensitive to time/space shifts of the original signal, such that the resulted best-basis will change a great deal if the signal is shifted by some samples. Thirdly, the reconstruction distortion after compression is relatively strong. In this paper, we design a new flexible segmentation algorithm with arbitrary time/space segmentation resolution which addresses the above-mentioned problems caused by the binary segmentation scheme. This new flexible segmentation algorithm is applied to 2D seismic data compression with two semi-adaptive schemes: Flexible 2D time-ALCT (Adapted Local Cosine Transform) and Flexible 2D space-ALCT. From our numerical tests on both synthetic signals and real seismic data using local cosine transform, the advantages of this new flexible segmentation technique over the binary searching scheme can be easily seen, from overcoming the constraint of dyadic segmentations, reducing time/space- shift sensitivity, less reconstruction distortions to superior performance in seismic data compression.

We use an automatic re-registering of frames taken from a surveillance video to improve signal-to-noise in regions of particular interest. A wavelet transform with automatic signal and noise estimation is used for this purpose. We describe the results obtained using a short surveillance video sequence.

Generalized Lapped Orthogonal Transform based image coder is used to compress 2D seismic data sets. Its performance is compared to the results using wavelet-based image coder. Both algorithms use the same state-of-the-art zerotree coding for consistency and fair comparison. Several parameters such as filter length and objective cost function are varied to find the best suited filter banks. It is found that for raw data, filter bank with long overlapping filters should be used for processing signals along the time direction whereas filter bank with short filters should be used for processing signal along the distance direction. This combination yields the best results.

In measuring the width of a laser beam, the scale of the object to be measured is much smaller than the measuring device. With a wavelet transform, on the other hand, the scale can be progressively reduced, since scale is part of the mathematics of wavelets. Wavelet signal processing (noise reduction, edge detection and multiscale feature detection) will be applied to laser beam diagnostics (focal point, divergence, and collimation).

In this paper, we introduce an object segmentation algorithm that can be used in object oriented video coding, computer vision, target tracking or video sequence analysis. The algorithm is based on two kinds of information from a video sequence: the color segmentation which provides the spatial domain information and the motion estimation which provides the temporal information. The DWT based MRF segmentation algorithm, which can avoid oversegmentation problems, is used for color segmentation. The quadtree motion estimation algorithm which employs the color segmentation results and DWT coefficients is used for motion segmentation. Finally, an object extracting procedure will be used to combine the spatial-temporal information and extract an object out.

In this paper, a multiwavelet based feature extraction method is presented and applied to classification of microcalcification clusters in mammograms. Multiwavelet is a natural generalization to scalar wavelet in which more than one scaling function and wavelet are used to further the design degrees of freedom. We extract energy and entropy features from different channels of multiwavelet. Using a real-valued genetic algorithm (GA), the best sets of features along with their optimal weights are found. The optimal weight vector is found such that within-class scatter is minimized and between-class scatter is maximized. For evaluating the individuals in GA, we use the area under Receiver Operating Characteristic (ROC) curve criterion such that the fittest individual has the largest value of area under ROC curve and the worst has the lowest value. To obtain the ROC curve, we use KNN classifier. Several multiwavelets with different features are employed. An area of 0.91 is obtained for Chui and Lian multiwavelet. A comparative study is conducted to show that the performance of multiwavelet is generally better than the packet wavelet in the present application.

The problem we are interested in is the restoration of nuclear medicine images acquired by a gamma camera. In a previous paper the authors have developed a wavelet based filtering method enabling to remove one of the major sources of error in nuclear medicine, namely Poisson noise. The purpose of this paper is to show how the restoration algorithm has been improved by introducing the point spread function as additional constraint in the restoration of the wavelet coefficients and choosing the regularization constraint in the object space. We describe a new restoration algorithm where filtering and deconvolution are coupled in a multiresolution frame. The performances are illustrated with simulated data and phantom images.

In this paper, the design of a video conference system is outlined. The proposed scheme allows the various attendees to access the multipoint video distribution center via channels of different nature and capacity without resorting to parallel bank of coders or multiple decoding-coding conversions. This approach allows to obtain a scalability of the user profile which is not present in DCT based video codecs, where the distribution of the encoded video to different attendees is performed by using a unique bit rate adjusted according to the capacity of the worst connection. In this contribution a video coder based on spatio-temporal multiresolution pyramid generated by a 3D separable Wavelet transform is proposed. Experimental results show the capability of the proposed method.

In a multimedia framework, digital image sequences (videos) are by far the most demanding as far as storage, search, browsing and retrieval requirements are concerned. In order to reduce the computational burden associated to video browsing and retrieval, a video sequence is usually decomposed into several scenes (shots) and each of them is characterized by means of some key frames. The proper selection of these key frames, i.e. the most representative frames in the scene, is of paramount importance for computational efficiency. In this contribution a novel key frame extraction technique based on the wavelet analysis is presented. Experimental results show the capability of the proposed algorithm to select key frames properly summarizing the shot.

Wavelet analysis of voltage sag completely depends on the choice of the wavelet basis. For better detection performance via wavelet analysis, the choice of the optimal wavelet basis must be provided within the constraints of the uncertainty principle which restricts arbitrary assignment of time-frequency resolution. In this paper, we describe local properties of the wavelet basis and voltage sag signal in terms of time duration and frequency bandwidth parameters. After comparison of the local properties of the wavelet basis and voltage sag signal, we suggest a set of performance indexes to measure the time-frequency resolution relation between the wavelet basis and the voltage sag signal. This procedure of determining the optimal wavelet basis can be extended to other possible applications of wavelets.

This paper presents an image coder that modifies the EZW coder and provides an improvement in its performance. The subband EZW image coder uses a uniform quantizer with a threshold (deadzone). Whereas, we know that the distribution/histogram of the wavelet tree subband coefficients, all except the lowest subband, tend to be Laplacian. To accommodate for this, we modify the refining procedure in EZW and use a non-uniform quantizer on the coefficients that better fits their distribution. The experimental results show that the new image coder performs better than EZW.

In this paper the use of permutations as a mean of spreading the spectrum of the frequency channels in a general multicarrier modulation scheme is presented. Our first objective is to demonstrate how signature waveforms suitable for multiple access communications can be generated by applying permutation transformations to orthogonal channels generated with filterbanks or DFT techniques. The second objective is to motivate the use of such techniques as a mean of achieving immunity to timing errors, which can cause severe Adjacent Channel Interference in multicarrier systems, due to large spectral overlap among adjacent channels. An efficient implementation of the interleaver implementing the considered permutations is also described.

The wavelet transform gives a compact multiscale representation of a digital image and provides a hierarchical structure which is well suited for post- processing. With its good localization property in both the spatial domain and the frequency domain, wavelet-based image compression has gained a huge success in the past several years. In this paper we present a wavelet-based image codec specially designed for an automatic target recognition (ATR) system. Due to the large amount of data size, a `good' image compression algorithm is both necessary and important as a pre-processing stage for an ATR system, especially in a `real-time' processing situation. We incorporate a constant false alarm rate (CFAR) detector into an embedded image compression algorithm to efficiently code target pixels exclusively in the bit stream. The new image codec, which is enhanced by the CFAR feature, clearly exhibits (potential) targets in the decompressed image. Another new feature of this codec is a wavelet-interpretation of the CFAR detector, a multiscale representation of the CFAR values in the wavelet domain.

One of the most remarkable properties of wavelet transform is its ability to separate data into different scale contents. For data that show self-similar characteristics in every scale, like fractal landscape, the wavelet spectrum also shows self-similarity. Nevertheless, the situation is not so clear for time dependent data, like seismic geology, solar flares, among others systems that are known to contain self-organized criticality. It is not obvious that these properties will be present in the wavelet spectrum in the form of self-similarity. In this work, we apply two gradient field computational operators R² yields R, the Complex Entropic Form and the Asymmetric Amplitude Fragmentation, as a mean to differentiate self-similarity from different sources.

We solve the transient heat conduction equation in an L- shaped region using domain decomposition and boundary integrals. The iterations need to be carried out only on the interfaces between the subdomains unlike finite difference or finite element methods where the iterations are to be performed in the entire domain. Numerically, the problem reduces to the multiplication of dense matrices by vectors of boundary or initial values. The DAUB4 wavelet transform is used to compress the matrices which leads to computational advantage without loss of accuracy. This procedure, capable of parallel implementation, is an extension of the work of Zarantonello and Elton for Laplace equation in overlapping circular discs.

In this paper, a new prediction based method, predictive depth coding, for lossy wavelet image compression is presented. It compresses a wavelet pyramid composition by predicting the number of significant bits in each wavelet coefficient quantized by the universal scalar quantization and then by coding the prediction error with arithmetic coding. The adaptively found linear prediction context covers spatial neighbors of the coefficient to be predicted and the corresponding coefficients on lower scale and in the different orientation pyramids. In addition to the number of significant bits, the sign and the bits of non-zero coefficients are coded. The compression method is tested with a standard set of images and the results are compared with SFQ, SPIHT, EZW and context based algorithms. Even though the algorithm is very simple and it does not require any extra memory, the compression results are relatively good.

In this paper we consider a new form of successive coefficient refinement which can be used in conjunction with embedded compression algorithms like Shapiro's EZW (Embedded Zerotree Wavelet) and Said & Pearlman's SPIHT (Set Partitioning in Hierarchical Trees). Using the conventional refinement process, the approximation of a coefficient that was earlier determined to be significantly is refined by transmitting one of two symbols--an `up' symbol if the actual coefficient value is in the top half of the current uncertainty interval or a `down' symbol if it is the bottom half. In the modified scheme developed here, we transmit one of 3 symbols instead--`up', `down', or `exact'. The new `exact' symbol tells the decoder that its current approximation of a wavelet coefficient is `exact' to the level of precision desired. By applying this scheme in earlier work to lossless embedded compression (also called lossy/lossless compression), we achieved significant reductions in encoder and decoder execution times with no adverse impact on compression efficiency. These excellent results for lossless systems have inspired us to adapt this refinement approach to lossy embedded compression. Unfortunately, the results we have achieved thus far for lossy compression are not as good.

The concept of the associative storage in a photorefractive material offers suitable methods to design a multichannel correlator for image identification. Wavelet transform is introduced to improve recognition accuracy of the system, which provides a sharper peak and lower sidelobes than the conventional correlation. The recognition accuracy of the system is significantly affected by the choice of wavelet function and its parameters. A neural network is proposed to optimize parameters of the wavelet filters to improve recognition performance of the system. The object function for optimization is to maximize the difference of correlation outputs among different categories and minimize the variation of correlation outputs in a same category. Simulation and experimental results are given to testify the effect of optimization. Its application in human face recognition is studied. The results show that it is attractive to use neural network to refine parameters of filters.

A scheme for extracting morphological information from images through a edge oriented wavelet decomposition is proposed. Multiresolution edge patterns extracted in the wavelet transform domain are represented by parameterized segments. Tracking these segments across consecutive frames of a sequence leads to a parametric cinematic characterization of the content of a video sequence, suited for high level syntactic processing. Parameterized images are visualizable by means of the inverse wavelet transform.

In this paper we proposed an algorithm using symbol- overlapping, redundant transforms for both, transmitter and receiver. We show that a combination of a low-delay cosine- modulated filter bank at the transmitter and an optimized filter bank at the receiver performs as good as the Discrete Multitone solution, but with much smaller latency time. For channel equalization Zero-Forcing and MMSE methods are used. Different methods of redundancy insertion used for a better channel equalization are discussed. Due to better stopband attenuation of the basis filters modulated filter banks show less sensitivity to Intersymbol and Intercarrier Interference and narrowband interferers.

Wavelets for the detection of masses in digitized mammograms are proposed in this study. Enhancement and segmentation of the image, based on texture, is performed using wavelets and Laws texture maps. Hexagonal wavelets are proposed in this study for improved detection and speed. In hexagonal sampling, orientations are partitioned into three bands of 60 degrees, equally covering the frequency domain such that no particular orientation is favored. The lifting scheme is used to construct the wavelets. This scheme has several advantages over the conventional Fourier approach. Most importantly the lifting scheme can easily be extended to uneven sampling, is not susceptible to boundary conditions, and is also computationally more efficient by a factor of two. One method uses wavelets to enhance the image before the Laws texture energy map is found. The second segments the image based on the wavelet coefficients for low texture. Both of the methods show promising results.

A novel method for simultaneous registration and segmentation is developed. The method is designed to register two similar images while a region with significant difference is adaptively segmented. This is achieved by minimization of a non-linear functional that models the statistical properties of the subtraction of the two images. Minimization is performed in the wavelet domain by a coarse- to-fine approach to yield a mapping that yields the registration and the boundary that yields the segmentation. The new method was applied to the registration of the left and the right lung regions in chest radiographs for extraction of lung nodules while the normal anatomic structures such as ribs are removed. A preliminary result shows that our method is very effective in reducing the number of false detections obtained with our computer-aided diagnosis scheme for detection of lung nodules in chest radiographs.