the shiftable complex directional pyramid part i theoretical aspects this paper presents an over complete multiscale decomposition by combining the laplacian pyramid and the complex directional filter bank dfb the filter bank is constructed in such a way that each complex directional filter is analytical using the dual tree structure of real fan filters necessary and sufficient conditions in order for the resulting multirate filter bank to be shift invariant in energy sense shiftability are derived in terms of the magnitude and phase responses of these filters their connection to 2d hilbert transform relationship is established the proposed transform possesses several desirable properties including multiresolution arbitrarily high directional resolution low redundant ratio and efficient implementation

eigenfaces vs fisherfaces recognition using class specific linear projection we develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression taking a pattern classification approach we consider each pixel in an image as a coordinate in a high dimensional space we take advantage of the observation that the images of a particular face under varying illumination but fixed pose lie in a 3d linear subspace of the high dimensional image space if the face is a lambertian surface without shadowing however since faces are not truly lambertian surfaces and do indeed produce self shadowing images will deviate from this linear subspace rather than explicitly modeling this deviation we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation our projection method is based on fisher s linear discriminant and produces well separated classes in a low dimensional subspace even under severe variation in lighting and facial expressions the eigenface technique another method based on linearly projecting the image space to a low dimensional subspace has similar computational requirements yet extensive experimental results demonstrate that the proposed fisherface method has error rates that are lower than those of the eigenface technique for tests on the harvard and yale face databases

feature selection in possibilistic modeling feature selection is becoming increasingly important for the reduction of computing complexity in this context conventional approaches have random performances because they can succeed for some contexts and fail for others possibilistic modeling is a powerful paradigm being able to handle data imperfection or redundancy and is not affected by data variability therefore in this paper we propose a new feature selection strategy for possibilitic modeling the proposed approach is based on two issues in order to extract relevant features the measure of feature importance as well as the possibility distribution uncertainty degree the importance of one feature can be considered under two aspects the first one is related to the scattering within one class and the second one reflects the feature power for class discrimination therefore we apply here shapley index paradigm which selects features who minimize the intra class distance and who maximize the inter class distance the previous process is refined using possibility distribution uncertainty degree in order to resolve some conflict problems between feature s importance values highlightswe present a feature selection strategy for possibilitic modeling we examine feature importance within one class and for classes s discrimination resolving conflict between feature s importance using possibility uncertainty degree proposed strategy is able to handle data imperfection and gives good performances proposed strategy gives better and reliable results on any data set

image clustering based on sparse patch alignment framework image clustering methods are efficient tools for applications such as content based image retrieval and image annotation recently graph based manifold learning methods have shown promising performance in extracting features for image clustering typical manifold learning methods adopt appropriate neighborhood size to construct the neighborhood graph which captures local geometry of data distribution because the density of data points distribution may be different in different regions of the manifold a fixed neighborhood size may be inappropriate in building the manifold in this paper we propose a novel algorithm named sparse patch alignment framework for the embedding of data lying in multiple manifolds specifically we assume that for each data point there exists a small neighborhood in which only the points that come from the same manifold lie approximately in a low dimensional affine subspace based on the patch alignment framework we propose an optimization strategy for constructing local patches which adopt sparse representation to select a few neighbors of each data point that span a low dimensional affine subspace passing near that point after that the whole alignment strategy is utilized to build the manifold experiments are conducted on four real world datasets and the results demonstrate the effectiveness of the proposed method

general tensor discriminant analysis and gabor features for gait recognition traditional image representations are not suited to conventional classification methods such as the linear discriminant analysis lda because of the undersample problem usp the dimensionality of the feature space is much higher than the number of training samples motivated by the successes of the two dimensional lda 2dlda for face recognition we develop a general tensor discriminant analysis gtda as a preprocessing step for lda the benefits of gtda compared with existing preprocessing methods such as the principal components analysis pca and 2dlda include the following 1 the usp is reduced in subsequent classification by for example lda 2 the discriminative information in the training tensors is preserved and 3 gtda provides stable recognition rates because the alternating projection optimization algorithm to obtain a solution of gtda converges whereas that of 2dlda does not we use human gait recognition to validate the proposed gtda the averaged gait images are utilized for gait representation given the popularity of gabor function based image decompositions for image understanding and object recognition we develop three different gabor function based image representations 1 gabord is the sum of gabor filter responses over directions 2 gabors is the sum of gabor filter responses over scales and 3 gaborsd is the sum of gabor filter responses over scales and directions the gabord gabors and gaborsd representations are applied to the problem of recognizing people from their averaged gait images a large number of experiments were carried out to evaluate the effectiveness recognition rate of gait recognition based on first obtaining a gabor gabord gabors or gaborsd image representation then using gdta to extract features and finally using lda for classification the proposed methods achieved good performance for gait recognition based on image sequences from the university of south florida usf humanid database experimental comparisons are made with nine state of the art classification methods in gait recognition

robustness of the inner outer factorization and of the spectral factorization for fir data the present paper analyzes the construction of outer functions the factorization of finite impulse response fir systems into a minimum phase system and an all pass part and the spectral factorization it investigates the behavior of these operations with respect to errors in the given data it shows that the error in the constructed outer function grows at least and at most proportional with the logarithm of the degree of the given fir system and proportional to the error in the given data for the other two operations it turns out that they show the same behavior with respect to errors in the given fir data as the construction of outer functions

allpass filter design and applications the problem of allpass filter design for phase approximation and equalization in the chebyshev sense is solved by using a generalized remez algorithm convergence to the unique optimum is guaranteed and is achieved rapidly in the actual implementation the well known numerical problems for higher degree filters are analyzed and solved by a simple approach the algorithm yields a solution to a variety of filter design problems such as the design of filters with a desired phase response e g a delay element the design of phase equalizers or the design of recursive filters with magnitude prescriptions using parallel allpass filters for the latter the algorithm can be modified to allow arbitrary tolerance schemes for the magnitude response

multiresolution monogenic signal analysis using the riesz laplace wavelet transform the monogenic signal is the natural 2d counterpart of the 1d analytic signal we propose to transpose the concept to the wavelet domain by considering a complexified version of the riesz transform which has the remarkable property of mapping a real valued primary wavelet basis of l2 r2 into a complex one the riesz operator is also steerable in the sense that it give access to the hilbert transform of the signal along any orientation having set those foundations we specify a primary polyharmonic spline wavelet basis of l2 r2 that involves a single mexican hat like mother wavelet laplacian of a b spline the important point is that our primary wavelets are quasi isotropic they behave like multiscale versions of the fractional laplace operator from which they are derived which ensures steerability we propose to pair these real valued basis functions with their complex riesz counterparts to specify a multiresolution monogenic signal analysis this yields a representation where each wavelet index is associated with a local orientation an amplitude and a phase we give a corresponding wavelet domain method for estimating the underlying instantaneous frequency we also provide a mechanism for improving the shift and rotation invariance of the wavelet decomposition and show how to implement the transform efficiently using perfect reconstruction filterbanks we illustrate the specific feature extraction capabilities of the representation and present novel examples of wavelet domain processing in particular a robust tensor based analysis of directional image patterns the demodulation of interferograms and the reconstruction of digital holograms

rational coefficient dual tree complex wavelet transform design and implementation the dual tree complex wavelet transform cwt has recently received significant interest in the wavelet community owing primarily to its directional selective and near shift invariant properties it has been shown that with two separate maximally decimated and dyadic decompositions where filters are offset by a half sample the resulting cwt wavelet bases form an approximate hilbert transform pair in this paper we present the design implementation and applications of several families of orthogonal as well as biorthogonal rational coefficient wavelet filters that satisfy the hilbert transform pair condition and meet other desirable properties such as high coding gain good directional sensitivity and sufficient degree of regularity the wavelet filters presented here which confirm to selesnick s and kingsbury s design schemes are designed and implemented directly in the lattice and lifting domain using vlsi friendly dyadic coefficients we confirm the fact that rational coefficient constraint does not impose a significant loss in terms of energy compaction wavelet smoothness time invariance or directionality we also propose the time reversal relationships between the two cwt filter pairs in lattice and lifting domain thereby facilitating both the design and implementation process in the end we present several applications and evaluations to illustrate the performance of the proposed designs

nonlinear analysis of irregular animal vocalizations animal vocalizations range from almost periodic vocal fold vibration to completely atonal turbulent noise between these two extremes a variety of nonlinear dynamics such as limit cycles subharmonics biphonation and chaotic episodes have been recently observed these observations imply possible functional roles of nonlinear dynamics in animal acoustic communication nonlinear dynamics may also provide insight into the degree to which detailed features of vocalizations are under close neural control as opposed to more directly reflecting biomechanical properties of the vibrating vocal folds themselves so far nonlinear dynamical structures of animal voices have been mainly studied with spectrograms in this study the deterministic versus stochastic dvs prediction technique was used to quantify the amount of nonlinearity in three animal vocalizations macaque screams piglet screams and dog barks results showed that in vocalizations with pronounced harmonic components adult macaque screams certain piglet screams and dog barks deterministic nonlinear prediction was clearly more powerful than stochastic linear prediction the difference termed low dimensional nonlinearity measure lnm indicates the presence of a low dimensional attractor in highly irregular signals such as juvenile macaque screams piglet screams and some dog barks the detectable amount of nonlinearity was comparatively small analyzing 120 samples of dog barks it was further shown that the harmonic to noise ratio hnr was positively correlated with lnm it is concluded that nonlinear analysis is primarily useful in animal vocalizations with strong harmonic components including subharmonics and biphonation or low dimensional chaos

spectral factorization for polynomial spectral densities impact of dimension this correspondence investigates the continuity behavior of the spectral factorization mapping for trigonometric polynomials it is clear that this factorization mapping is continuous on the space of all trigonometric polynomials of a fixed degree n which means that a small perturbation in the given spectrum yields always a bounded error in the spectral factor the correspondence derives a lower bound on the continuity constant of the spectral factorization mapping which shows that the error in the spectral factor grows at least proportional with the logarithm of the degree n of the given spectrum

estimation of subspace arrangements with applications in modeling and segmenting mixed data recently many scientific and engineering applications have involved the challenging task of analyzing large amounts of unsorted high dimensional data that have very complicated structures from both geometric and statistical points of view such unsorted data are considered mixed as different parts of the data have significantly different structures which cannot be described by a single model in this paper we propose to use subspace arrangements a union of multiple subspaces for modeling mixed data each subspace in the arrangement is used to model just a homogeneous subset of the data thus multiple subspaces together can capture the heterogeneous structures within the data set in this paper we give a comprehensive introduction to a new approach for the estimation of subspace arrangements this is known as generalized principal component analysis gpca in particular we provide a comprehensive summary of important algebraic properties and statistical facts that are crucial for making the inference of subspace arrangements both efficient and robust even when the given data are corrupted by noise or contaminated with outliers this new method in many ways improves and generalizes extant methods for modeling or clustering mixed data there have been successful applications of this new method to many real world problems in computer vision image processing and system identification in this paper we will examine several of those representative applications this paper is intended to be expository in nature however in order that this may serve as a more complete reference for both theoreticians and practitioners we take the liberty of filling in several gaps between the theory and the practice in the existing literature

a novel scheme for the design of approximate hilbert transform pairs of orthonormal wavelet bases in designing the hilbert transform pairs of orthonormal wavelet bases several authors have shown that the requirements of the equal magnitude responses and the half sample phase offset are the necessary and sufficient conditions on the scaling filters of conjugate quadrature filter cqf banks in this paper we first show that the scaling filters of cqf banks with equal magnitude responses cannot approach the half sample phase offset in the high frequency range properly then a new design scheme making a tradeoff between the requirements of the equal magnitude responses and the half sample phase offset is presented the design scheme can make the scaling filter pairs approximately satisfy the requirement of the half sample phase offset over the full frequency range while the magnitude responses of each pair remain nearly equal as a result the corresponding orthonormal wavelet bases can approximate to hilbert transform pairs better some orthogonal wavelet bases we designed demonstrate the superiority of our scheme

graph embedding and extensions a general framework for dimensionality reduction a large family of algorithms supervised or unsupervised stemming from statistics or geometry theory has been designed to provide different solutions to the problem of dimensionality reduction despite the different motivations of these algorithms we present in this paper a general formulation known as graph embedding to unify them within a common framework in graph embedding each algorithm can be considered as the direct graph embedding or its linear kernel tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided furthermore the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms by utilizing this framework as a tool we propose a new supervised dimensionality reduction algorithm called marginal fisher analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class while the penalty graph connects the marginal points and characterizes the interclass separability we show that mfa effectively overcomes the limitations of the traditional linear discriminant analysis algorithm due to data distribution assumptions and available projection directions real face recognition experiments show the superiority of our proposed mfa in comparison to lda also for corresponding kernel and tensor extensions

the design and use of steerable filters the authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters allowing one to adaptively steer a filter to any orientation and to determine analytically the filter output as a function of orientation steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation the authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase angularly adaptive filtering edge detection and shape from shading one can also build a self similar steerable pyramid representation the same concepts can be generalized to the design of 3 d steerable filters

wavelets and recursive filter banks it is shown that infinite impulse response iir filters lead to more general wavelets of infinite support than finite impulse response fir filters a complete constructive method that yields all orthogonal two channel filter banks where the filters have rational transfer functions is given and it is shown how these can be used to generate orthonormal wavelet bases a family of orthonormal wavelets that have a maximum number of disappearing moments is shown to be generated by the halfband butterworth filters when there is an odd number of zeros at pi it is shown that closed forms for the filters are available without need for factorization a still larger class of orthonormal wavelet bases having the same moment properties and containing the daubechies and butterworth filters as the limiting cases is presented it is shown that it is possible to have both linear phase and orthogonality in the infinite impulse response case and a constructive method is given it is also shown how compactly supported bases may be orthogonalized and bases for the spline function spaces are constructed

a no reference image sharpness metric based on structural information using sparse representation as a ubiquitous image distortion blur casts non trivial influence on image visual quality many image sharpness assessment methods have been proposed in the views of edge information gradient map frequency spectrum or other natural image statistics features in this paper we propose a no reference image sharpness metric based on structural information using sparse representation sr we observe that the dictionary atoms learned by sr algorithm convey clear structural information considering the distinct sensibility of human visual system hvs to different structures we use the learned dictionary to encode the patches of the blurry image to embed the locality of the representation a multi scale spatial max pooling scheme is incorporated the final sharpness score is given by an efficient linear support vector regression svr model we evaluate our approach on three public databases i e live ii tid2008 and csiq the experiments demonstrate that the proposed method achieves competitive performance compared with the state of the art blind image sharpness assessment algorithms

a fast dbscan clustering algorithm by accelerating neighbor searching using groups method density based clustering methods are proposed for clustering spatial databases with noise density based spatial clustering of applications with noise dbscan can discover clusters of arbitrary shape and also handles outliers effectively dbscan obtains clusters by finding the number of points within the specified distance from a given point it involves computing distances from given point to all other points in the dataset the conventional index based methods construct a hierarchical structure over the dataset to speed up the neighbor search operations the hierarchical index structures fail to scale for datasets of dimensionality above 20 in this paper we propose a novel graph based index structure method groups that accelerates the neighbor search operations and also scalable for high dimensional datasets experimental results show that the proposed method improves the speed of dbscan by a factor of about 1 5 2 2 on benchmark datasets the performance of dbscan degrades considerably with noise due to unnecessary distance computations introduced by noise points while the proposed method is robust to noise by pruning out noise points early and eliminating the unnecessary distance computations the cluster results produced by our method are exactly similar to that of dbscan but executed at a much faster pace a graph based index structure is built for speeding up neighbor search operations no additional inputs are required to build the index structure proposed method is scalable for high dimensional datasets handles noise effectively to improve the performance of dbscan

max min distance analysis by using sequential sdp relaxation for dimension reduction we propose a new criterion for discriminative dimension reduction max min distance analysis mmda given a data set with c classes represented by homoscedastic gaussians mmda maximizes the minimum pairwise distance of these c classes in the selected low dimensional subspace thus unlike fisher s linear discriminant analysis flda and other popular discriminative dimension reduction criteria mmda duly considers the separation of all class pairs to deal with general case of data distribution we also extend mmda to kernel mmda kmmda dimension reduction via mmda kmmda leads to a nonsmooth max min optimization problem with orthonormal constraints we develop a sequential convex relaxation algorithm to solve it approximately to evaluate the effectiveness of the proposed criterion and the corresponding algorithm we conduct classification and data visualization experiments on both synthetic data and real data sets experimental results demonstrate the effectiveness of mmda kmmda associated with the proposed optimization algorithm

geometric mean for subspace selection subspace selection approaches are powerful tools in pattern classification and data visualization one of the most important subspace approaches is the linear dimensionality reduction step in the fisher s linear discriminant analysis flda which has been successfully employed in many fields such as biometrics bioinformatics and multimedia information management however the linear dimensionality reduction step in flda has a critical drawback for a classification task with c classes if the dimension of the projected subspace is strictly lower than c 1 the projection to a subspace tends to merge those classes which are close together in the original feature space if separate classes are sampled from gaussian distributions all with identical covariance matrices then the linear dimensionality reduction step in flda maximizes the mean value of the kullback leibler kl divergences between different classes based on this viewpoint the geometric mean for subspace selection is studied in this paper three criteria are analyzed 1 maximization of the geometric mean of the kl divergences 2 maximization of the geometric mean of the normalized kl divergences and 3 the combination of 1 and 2 preliminary experimental results based on synthetic data uci machine learning repository and handwriting digits show that the third criterion is a potential discriminative subspace selection method which significantly reduces the class separation problem in comparing with the linear dimensionality reduction step in flda and its several representative extensions

low rank matrix factorization with multiple hypergraph regularizer this paper presents a novel low rank matrix factorization method named multihmmf which incorporates multiple hypergraph manifold regularization to the low rank matrix factorization in order to effectively exploit high order information among the data samples the hypergraph is introduced to model the local structure of the intrinsic manifold specifically multiple hypergraph regularization terms are separately constructed to consider the local invariance the optimal intrinsic manifold is constructed by linearly combining multiple hypergraph manifolds then the regularization term is incorporated into a truncated singular value decomposition framework resulting in a unified objective function so that matrix factorization is changed into an optimization problem alternating optimization is used to solve the optimization problem with the result that the low dimensional representation of data space is obtained the experimental results of image clustering demonstrate that the proposed method outperforms state of the art data representation methods we propose a novel low rank matrix factorization method we incorporate multiple hypergraph manifold regularization to the matrix factorization we adopt alternating optimization to solve the optimization problem

surface geodesic pattern for 3d deformable texture matching this paper presents a surface geodesic pattern sgp representation for matching textured 3d deformable surfaces sgp encodes the local variations of the surface texture derivatives to extract local information from distinctive textural relationships contained in a geodesic neighborhood thus sgp derives its strength from the fusion of surface texture and shape information at the data level in a way that is invariant to non rigid deformations we also propose gabor topography wavelet gtw for direct feature extraction from the range data both features are combined using a multi view sparse representation to achieve higher discrimination capability while matching non rigid 3d surfaces the performance of the proposed method is evaluated extensively on the bosphorus face database the frgc v2 face database and the polyu contact free hand database and the results are compared to state of the art methods experimental results show the effectiveness and superiority of the proposed method in recognizing objects under non rigid surface deformations

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

formulas for orthogonal iir wavelet filters explicit solutions are given for the rational function p z for two classes of iir orthogonal two band wavelet bases for which the scaling filter is maximally flat p z denotes the rational transfer function h z h 1 z where h z is the lowpass scaling filter the first is the class of solutions that are intermediate between the daubechies 1992 and the butterworth wavelets it is found that the daubechies the butterworth and the intermediate solutions are unified by a single formula the second is the class of scaling filters realizable as a parallel sum of two allpass filters a particular case of which yields the class of symmetric iir orthogonal wavelet bases for this class a closed form solution is provided by the solution to an older problem in group delay approximation by digital allpole filters

protein synthesis is associated with high speed dynamics and broad band stability of functional hubs in the brain l 1 11c leucine pet can be used to measure in vivo protein synthesis in the brain however the relationship between regional protein synthesis and on going neural dynamics is unclear we use a graph theoretical approach to examine the relationship between cerebral protein synthesis rcps and both static and dynamical measures of functional connectivity measured using resting state functional mri r fmri our graph theoretical analysis demonstrates a significant positive relationship between protein turnover and static measures of functional connectivity we compared these results to simple measures of metabolism in the cortex using 18f fdg pet whilst some relationships between 18f fdg binding and graph theoretical measures was present there remained a significant relationship between protein turnover and graph theoretical measures which were more robustly explained by l 1 11c leucine than 18f fdg pet this relationship was stronger in dynamics at a faster temporal resolution relative to dynamics measured over a longer epoch using a dynamic connectivity approach we also demonstrate that broad band dynamic measures of functional connectivity fc are inversely correlated with protein turnover suggesting greater stability of fc in highly interconnected hub regions is supported by protein synthesis overall we demonstrate that cerebral protein synthesis has a strong relationship independent of tissue metabolism to neural dynamics at the macroscopic scale

twist tensor total variation regularized reweighted nuclear norm based tensor completion for video missing area recovery abstract this paper focuses on recovering multi dimensional signal called tensor which is corrupted by random missing areas the performance of the conventional tensor completion techniques deteriorate when the tensor multi rank is large and or large missing areas moreover these techniques are weak in preserving the edges in the signals like images videos this paper proposes an efficient method to overcome these problems by simultaneously combining novel twist tensor total variation norm to exploit spatio temporal correlation and tensor singular value decomposition t svd based reweighted nuclear norm to improve low multi rank tensor recovery the twist tensor total variation norm takes care of edges in the recovered data and aids the recovery of missing areas by utilising the similarities in the adjacent samples the reweighted nuclear norm handles corrupted large rank tensors by sparsity enhancement via reweighting its singular values the effectiveness of the proposed method is established by applying to video completion problem and experimental results reveal that the algorithm outperforms its counterparts

multimodal graph based reranking for web image search this paper introduces a web image search reranking approach that explores multiple modalities in a graph based learning scheme different from the conventional methods that usually adopt a single modality or integrate multiple modalities into a long feature vector our approach can effectively integrate the learning of relevance scores weights of modalities and the distance metric and its scaling for each modality into a unified scheme in this way the effects of different modalities can be adaptively modulated and better reranking performance can be achieved we conduct experiments on a large dataset that contains more than 1000 queries and 1 million images to evaluate our approach experimental results demonstrate that the proposed reranking approach is more robust than using each individual modality and it also performs better than many existing methods

non negative patch alignment framework in this paper we present a non negative patch alignment framework npaf to unify popular non negative matrix factorization nmf related dimension reduction algorithms it offers a new viewpoint to better understand the common property of different nmf algorithms although multiplicative update rule mur can solve npaf and is easy to implement it converges slowly thus we propose a fast gradient descent fgd to overcome the aforementioned problem fgd uses the newton method to search the optimal step size and thus converges faster than mur experiments on synthetic and real world datasets confirm the efficiency of fgd compared with mur for optimizing npaf based on npaf we develop non negative discriminative locality alignment ndla experiments on face image and handwritten datasets suggest the effectiveness of ndla in classification tasks and its robustness to image occlusions compared with representative nmf related dimension reduction algorithms

principal manifolds and nonlinear dimensionality reduction via tangent space alignment we present a new algorithm for manifold learning and nonlinear dimensionality reduction based on a set of unorganized data points sampled with noise from a parameterized manifold the local geometry of the manifold is learned by constructing an approximation for the tangent space at each data point and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold we also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases we illustrate our algorithm using curves and surfaces both in two dimensional three dimensional 2d 3d euclidean spaces and in higher dimensional euclidean spaces we also address several theoretical and algorithmic issues for further research and improvements

a new framework for complex wavelet transforms although the discrete wavelet transform dwt is a powerful tool for signal and image processing it has three serious disadvantages shift sensitivity poor directionality and lack of phase information to overcome these disadvantages we introduce two stage mapping based complex wavelet transforms that consist of a mapping onto a complex function space followed by a dwt of the complex mapping unlike other popular transforms that also mitigate dwt shortcomings the decoupled implementation of our transforms has two important advantages first the controllable redundancy of the mapping stage offers a balance between degree of shift sensitivity and transform redundancy this allows us to create a directional non redundant complex wavelet transform with potential benefits for image coding systems to the best of our knowledge no other complex wavelet transform is simultaneously directional and non redundant the second advantage of our approach is the flexibility to use any dwt in the transform implementation as an example we can exploit this flexibility to create the complex double density dwt cddwt a shift insensitive directional complex wavelet transform with a low redundancy of 3 sup m 1 2 sup m 1 in m dimensions to the best of our knowledge no other transform achieves all these properties at a lower redundancy

the design of approximate hilbert transform pairs of wavelet bases several authors have demonstrated that significant improvements can be obtained in wavelet based signal processing by utilizing a pair of wavelet transforms where the wavelets form a hilbert transform pair this paper describes design procedures based on spectral factorization for the design of pairs of dyadic wavelet bases where the two wavelets form an approximate hilbert transform pair both orthogonal and biorthogonal fir solutions are presented as well as iir solutions in each case the solution depends on an allpass filter having a flat delay response the design procedure allows for an arbitrary number of vanishing wavelet moments to be specified a matlab program for the procedure is given and examples are also given to illustrate the results