projective fourier analysis for patterns abstract identifying psl 2 c as a projective group for patterns in the conformal camera model the projective harmonic analysis on its double covering group sl 2 c is presented in the noncompact and compact pictures the pictures used to study different aspects of irreducible unitary representations of semisimple lie groups bypassing technicalities of representation theory but stressing the motivation and similarities with euclidean fourier analysis each constructed picture of the projective fourier analysis includes the fourier transform plancherel s theorem and convolution property projectively covariant characteristics of the analysis in the noncompact picture allow rendering any of image projective transformations of a pattern after removing conformal distortions by using only one projective fourier transform of the original pattern what is demonstrated in a computer simulation the convolution properties in both pictures must by used to develop algorithms for projectively invariant matching of patterns work in progress on fast algorithms for computing with projective fourier transforms and for rendering image projective transformations is discussed efficient computations of the convolutions would follow from the both fast projective fourier transforms and their inverses

weight adjustment rule of neural networks for computing discrete 2 d gabor transforms image processing it is demonstrated that the weight adjustment rule used in the neural network for computing the 2 d gabor transform proposed by j daugman 1988 can be shown to be equivalent to the jacobi iteration scheme for solving simultaneous linear equations it is shown that faster convergence of the algorithm can be achieved by using gauss seidel iteration successive overrelaxation conjugate gradient algorithms and multigrid methods

complex multipole beam approach to acoustic scattering problems a novel approach to reducing the matrix size associated with the boundary element moment method solution of the problem of acoustic scattering from arbitrarily shaped closed bodies is presented in this paper the key step in this approach is to represent the scattered field in terms of a series of beams produced by multipole sources located in a complex space on the scatterer boundary the fields generated by these multipole sources resemble the gabor basis functions by utilizing the properties of the gabor series guidelines for selecting the orders as well as locations of the multipole sources are developed the present approach not only reduces the number of unknowns but also generates a generalized impedance matrix with a banded structure the accuracy of the proposed method is verified by using internal accuracy checks and by comparing the numerical results with the analytic solutions for spherical scatterers

a new implementation of the mellin transform and its application to radar classification of ships a modified mellin transform for digital implementation is developed and applied to range radar profiles of naval vessels the scale invariance property of the mellin transform provides a means for extracting features from the profiles which are insensitive to the aspect angle of the radar past implementations of the mellin transform based on the fft have required exponential sampling interpolation and the computation of a correction term all of which introduce errors into the transform in addition exponential sampling results in a factor of ln n increase in the number of data points an alternate implementation developed in the paper utilizes a direct expansion of the mellin integral definition this direct mellin transform dmt eliminates the implementation problems associated with the fft approach and does not increase the number of samples a scale and translation invariant transform is developed from a modification of the dmt the mdmt applied to the fft of the radar profiles results in the desired insensitivity without having the low pass filtering characteristic that exists in other fourier mellin implementations

complete discrete 2 d gabor transforms by neural networks for image analysis and compression a three layered neural network is described for transforming two dimensional discrete signals into generalized nonorthogonal 2 d gabor representations for image analysis segmentation and compression these transforms are conjoint spatial spectral representations which provide a complete image description in terms of locally windowed 2 d spectral coordinates embedded within global 2 d spatial coordinates in the present neural network approach based on interlaminar interactions involving two layers with fixed weights and one layer with adjustable weights the network finds coefficients for complete conjoint 2 d gabor transforms without restrictive conditions in wavelet expansions based on a biologically inspired log polar ensemble of dilations rotations and translations of a single underlying 2 d gabor wavelet template image compression is illustrated with ratios up to 20 1 also demonstrated is image segmentation based on the clustering of coefficients in the complete 2 d gabor transform

efficient computation of gabor filter based multiresolution responses abstract multiresolution representation in machine vision systems provides a means to detect analyze and interpret the information content of an image at multiple resolutions the pyramidal representation of images suggested by burt and crowley and the hierarchical construction of orientation and velocity tuned filters developed by fleet and jepson are some of the examples for multiresolution representation of image details in recent years gabor filter based methods have been suggested by several researchers for machine vision applications such as edge detection texture classification and optical flow estimation these applications require generating a family of gabor filters tuned to several resolutions and orientations and computing their responses the computation of gabor filter responses at multiple resolutions and orientations is a very computationally intensive task two efficient schemes for computing gabor filter based multiresolution responses are proposed the first scheme namely recursive filtering method generates a family of gabor filters and their responses starting from a high resolution and proceeding towards the lowest resolution in steps of an octave the second approach namely successive sampling method generates the same responses in the reverse order in both these methods the symmetric asymmetric and wavelet nature of gabor filters are exploited in order to speed up the computation the applicability and usefulness of the schemes are established through experimental results special purpose architectures are proposed to implement the proposed techniques in hardware in order to further speed up the computation

hamiltonian systems near relative equilibria abstract we give explicit differential equations for the dynamics of hamiltonian systems near relative equilibria these split the dynamics into motion along the group orbit and motion inside a slice transversal to the group orbit the form of the differential equations that is inherited from the symplectic structure and symmetry properties of the hamiltonian system is analysed and the effects of time reversing symmetries are included the results will be applicable to the stability and bifurcation theories of relative equilibria of hamiltonian systems

mellin transform methods applied to integral evaluation taylor series and asymptotic approximations mellin transform methods have been used to obtain taylor series or asymptotic approximations to definite integrals whose integrands consist of the product of two generalized hypergeometric functions this approach has proved extremely useful in many applied physics problems including the analysis of electromagnetic propagation in a turbulent medium the main result in this article is a generalization of the method to integrands containing the product of an arbitrary number of hypergeometric functions providing a very powerful and general technique for integral evaluation with many applications in physics

the canonical coordinates method for pattern recognition ii isomorphisms with affine transformations abstract the canonical coordinates method for pattern recognition is extended conditions are examined for the existence of an isomorphism between a given deformation group and some affine transformation new invariants for many non linear deformations are derived several examples with real images are given

the canonical coordinates method for pattern deformation theoretical and computational considerations a method for the analysis of deformed patterns is presented and analyzed the image is transformed into a new set of coordinates in which the deformation has a particular simple form a number of deformations are considered the practical implementation of the method is discussed similar aspects of biological vision are also considered

design of steerable wavelets to detect multifold junctions we propose a framework for the detection of junctions in images although the detection of edges and key points is a well examined and described area the multiscale detection of junction centers especially for odd orders poses a challenge in pattern analysis the goal of this paper is to build optimal junction detectors based on 2d steerable wavelets that are polar separable in the fourier domain the approaches we develop are general and can be used for the detection of arbitrary symmetric and asymmetric junctions the backbone of our construction is a multiscale pyramid with a radial wavelet function where the directional components are represented by circular harmonics and encoded in a shaping matrix we are able to detect m fold junctions in different scales and orientations we provide experimental results on both simulated and real data to demonstrate the effectiveness of the algorithm

rotation invariant texture classification using a complete space frequency model a method of rotation invariant texture classification based on a complete space frequency model is introduced a polar analytic form of a two dimensional 2 d gabor wavelet is developed and a multiresolution family of these wavelets is used to compute information conserving microfeatures from these microfeatures a micromodel which characterizes spatially localized amplitude frequency and directional behavior of the texture is formed the essential characteristics of a texture sample its macrofeatures are derived from the estimated selected parameters of the micromodel classification of texture samples is based on the macromodel derived from a rotation invariant subset of macrofeatures in experiments comparatively high correct classification rates were obtained using large sample sets

what is the goal of sensory coding a number of recent attempts have been made to describe early sensory coding in terms of a general information processing strategy in this paper two strategies are contrasted both strategies take advantage of the redundancy in the environment to produce more effective representations the first is described as a compact coding scheme a compact code performs a transform that allows the input to be represented with a reduced number of vectors cells with minimal rms error this approach has recently become popular in the neural network literature and is related to a process called principal components analysis pca a number of recent papers have suggested that the optimal compact code for representing natural scenes will have units with receptive field profiles much like those found in the retina and primary visual cortex however in this paper it is proposed that compact coding schemes are insufficient to account for the receptive field properties of cells in the mammalian visual pathway in contrast it is proposed that the visual system is near to optimal in representing natural scenes only if optimality is defined in terms of sparse distributed coding in a sparse distributed code all cells in the code have an equal response probability across the class of images but have a low response probability for any single image in such a code the dimensionality is not reduced rather the redundancy of the input is transformed into the redundancy of the firing pattern of cells it is proposed that the signature for a sparse code is found in the fourth moment of the response distribution i e the kurtosis in measurements with 55 calibrated natural scenes the kurtosis was found to peak when the bandwidths of the visual code matched those of cells in the mammalian visual cortex codes resembling wavelet transforms are proposed to be effective because the response histograms of such codes are sparse i e show high kurtosis when presented with natural scenes it is proposed that the structure of the image that allows sparse coding is found in the phase spectrum of the image it is suggested that natural scenes to a first approximation can be considered as a sum of self similar local functions the inverse of a wavelet possible reasons for why sensory systems would evolve toward sparse coding are presented

rotation recovery from spherical images without correspondences this paper addresses the problem of rotation estimation directly from images defined on the sphere and without correspondence the method is particularly useful for the alignment of large rotations and has potential impact on 3d shape alignment the foundation of the method lies in the fact that the spherical harmonic coefficients undergo a unitary mapping when the original image is rotated the correlation between two images is a function of rotations and we show that it has an so 3 fourier transform equal to the pointwise product of spherical harmonic coefficients of the original images the resolution of the rotation space depends on the bandwidth we choose for the harmonic expansion and the rotation estimate is found through a direct search in this 3d discretized space a refinement of the rotation estimate can be obtained from the conservation of harmonic coefficients in the rotational shift theorem a novel decoupling of the shift theorem with respect to the euler angles is presented and exploited in an iterative scheme to refine the initial rotation estimates experiments show the suitability of the method for large rotations and the dependence of the method on bandwidth and the choice of the spherical harmonic coefficients

on the shiftability of dual tree complex wavelet transforms the dual tree complex wavelet transform dt bbcwt is known to exhibit better shift invariance than the conventional discrete wavelet transform we propose an amplitude phase representation of the dt ibbc wt which among other things offers a direct explanation for the improvement in the shift invariance the representation is based on the shifting action of the group of fractional hilbert transform fht operators which extends the notion of arbitrary phase shifts from sinusoids to finite energy signals wavelets in particular in particular we characterize the shift ability of the dt bbcwt in terms of the shifting property of the fhts at the heart of the representation are certain fundamental invariances of the fht group namely that of translation dilation and norm which play a decisive role in establishing the key properties of the transform it turns out that these fundamental invariances are exclusive to this group next by introducing a generalization of the bedrosian theorem for the fht operator we derive an explicitly understanding of the shifting action of the fht for the particular family of wavelets obtained through the modulation of lowpass functions e g the shannon and gabor wavelet this in effect links the corresponding dual tree transform with the framework of windowed fourier analysis finally we extend these ideas to the multidimensional setting by introducing a directional extension of the fht the fractional directional hilbert transform in particular we derive a signal representation involving the superposition of direction selective wavelets with appropriate phase shifts which helps explain the improved shift invariance of the transform along certain preferential directions

ranging through gabor logons a consistent hierarchical approach in this work the correspondence problem in stereo vision is handled by matching two sets of dense feature vectors inspired by biological evidence these feature vectors are generated by a correlation between a bank of gabor sensors and the intensity image the sensors consist of two dimensional gabor filters at various scales spatial frequencies and orientations which bear close resemblance to the receptive field profiles of simple v1 cells in visual cortex a hierarchical stochastic relaxation method is then used to obtain the dense stereo disparities unlike traditional hierarchical methods for stereo feature based hierarchical processing yields consistent disparities to avoid false matchings due to static occlusion a dual matching based on the imaging geometry is used

continuation theorems for boundary value problems abstract this paper gives constructive proofs of some important continuation theorems for boundary value problems via the homotopy method this leads to a global convergent method of finding solutions

transforms associated to square integrable group representations i general results let g be a locally compact group which need not be unimodular let x u x x g be an irreducible unitary representation of g in a hilbert space h u assume that u is square integrable i e that there exists in h u at least one nonzero vector g such that u x g g 2 dx we give here a reasonably self contained analysis of the correspondence associating to every vector f h u the function u x g f on g discussing its isometry characterization of the range inversion and simplest interpolation properties this correspondence underlies many properties of generalized coherent states

a constructive proof of the poincare birkhoff theorem in this paper with the use of the homotopy method a constructive proof of the poincare birkhoff theorem is given this approach provides a global method for finding fixed points of area preserving maps and periodic solutions of duffing equations

video orbits of the projective group a simple approach to featureless estimation of parameters we present direct featureless methods for estimating the eight parameters of an exact projective homographic coordinate transformation to register pairs of images together with the application of seamlessly combining a plurality of images of the same scene resulting in a single image or new image sequence of greater resolution or spatial extent the approach is exact for two cases of static scenes 1 images taken from the same location of an arbitrary three dimensional 3 d scene with a camera that is free to pan tilt rotate about its optical axis and zoom or 2 images of a flat scene taken from arbitrary locations the featureless projective approach generalizes interframe camera motion estimation methods that have previously used a camera model which lacks the degrees of freedom to exactly characterize such phenomena as camera pan and tilt and or which have relied upon finding points of correspondence between the image frames the featureless projective approach which operates directly on the image pixels is shown to be superior in accuracy and the ability to enhance the resolution the proposed methods work well on image data collected from both good quality and poor quality video under a wide variety of conditions sunny cloudy day night these new fully automatic methods are also shown to be robust to deviations from the assumptions of static scene and no parallax

form invariant linear filtering theory and applications the form invariant filters are by definition those shift variant filters such that their output for any given input turns out to be linearly scaled implying that its form does not change whenever the input is linearly scaled in this paper the most general classes of 1 d and 2 d linear form invariant filters are derived and their properties are discussed together with their implementation techniques two main implementation approaches are considered one based on the mellin transform the other on a combination of coordinate mappings and shift invariant filtering the paper also discusses the related works of other authors covering quite different fields such as optical pattern recognition image restoration and image reconstruction from projections radar signal processing etc it is shown that the mathematics of form invariant filtering provides a common framework if not a powerful unified approach to the many signal processing techniques spread in the above mentioned works and apparently different application areas the paper ends with a processing example showing the usefulness of form invariant filtering in a pattern recognition problem that is in the area where the most promising applications of such a filtering are foreseen

global bifurcation of polygonal relative equilibria for masses vortices and dnls oscillators given a regular polygonal arrangement of identical objects turning around a central object masses vortices or dnls oscillators this paper studies the global bifurcation of relative equilibria in function of a natural parameter central mass central circulation or amplitude of the oscillation the symmetries of the problem are used in order to find the irreducible representations the linearization and with the help of a degree theory the symmetries of the bifurcated solutions

lyapunov exponents for the n 3 discrete self trapping equation abstract lyapunov characteristic exponents for the n 3 discrete self trapping equation are numerically computed and the connection between the vanishing ones and the existence of independent constants of motion is exhibited the results obtained by this approach are in agreement with previous ones about the long term behaviour of the system

homoclinic chaos in the discrete self trapping trimer we study the discrete self trapping dst equation with three degrees of freedom by taking the dst dimer as the underlying unperturbed system we treat the coupling to the additional oscillator as a small perturbation using the generalized melnikov method we prove the existence of homoclinic chaos in the dst trimer dynamics

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

bifurcation structures and dominant modes near relative equilibria in the one dimensional discrete nonlinear schrodinger equation abstract we investigate the bifurcation structure of a family of relative equilibria of a ring of seven oscillators described by the discrete nonlinear schrodinger equation dnlse when the period of these orbits and a suitable defect act as bifurcation parameters we find a reduced hamiltonian that gives substantial insight into the dynamics of this system the convexity of this hamiltonian at given nonresonant equilibria supports the stability of nearby quasiperiodic solutions we show that the local loss of convexity in the reduced hamiltonian is determined by the hessian of its integrable part in the family of relative equilibria under study stable quasiperiodic solutions are studied by considering the power spectral densities of a set of suitable fast and slow actions whose origin is suggested by the averaging principle we also show that the return times form an optimal embedding to characterize the system dynamics we show that the power spectral density of a suitable interference signal arising from a ring of bose einstein condensates and described by the dnlse has a single prominent peak at the breather like relative equilibria

wide band proportional bandwidth wigner ville analysis the wigner ville w v distribution is a time frequency representation that yields a highly accurate estimate of instantaneous frequency it is related to the narrow band ambiguity function by an internal transform and it can be used in a variety of detection and estimation problems a spectrogram constructed with constant bandwidth filters can be obtained by convolving two w v distributions or by forming a magnitude squared narrow band cross ambiguity function the wide band ambiguity function represents the doppler effect with dilation or compression rather than with frequency shift as in the narrow band approximation the q distributions a modified w v representation that is related to the wide band ambiguity function by an integral transform is defined a spectrogram constructed with proportional bandwidth or constant q filters can be obtained by a convolution like operation involving two q distributions or by forming a magnitude squared wide band cross ambiguity function the q distribution is thus a wide band version of the w v distribution properties of the q distribution indicate that it may prove useful for detection and parameter estimation as well as for tomographic measurement of wideband scattering functions with relatively few transmitted waveforms

image representations using multiscale differential operators differential operators have been widely used for multiscale geometric descriptions of images the efficient computation of these differential operators is always desirable moreover it has not been clear whether such representations are invertible for certain applications it is usually required that such representations should be invertible so that one can facilitate the processing of information in the transform domain and then recover it in this paper such problems are studied we consider multiscale differential representations of images using different types of operators such as the directional derivative operators and laplacian operators in particular we provide a general approach to represent images by their multiscale and multidirectional derivative components for practical implementation efficient pyramid like algorithms are derived using the spline technique for both the the composition and reconstruction of images it is shown that using these representations various meaningful geometric information of images can be extracted at multiple scales therefore these representations can be used for edge based image processing purposes furthermore the intrinsic relationships of the proposed representations with the compact wavelet models and some classical multiscale approaches are also elucidated in the paper

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

a multi expert approach for robust face detection both detection accuracy and speed are of major concerns in developing a robust face detection system for real world applications to this end we propose a robust face detection approach by combining multiple experts in both cascade and parallel manner we design three detection experts which employ different feature representation schemes of local images 2d haar wavelet gradient direction and gabor filter the three features are classified using the same classification model namely a polynomial neural network pnn on reduced feature subspace the detection experts are used in multiple stages with simple ones in proceeding stages and complex ones in succeeding stages for improving detection speed meanwhile the output of each expert is combined with the outputs of its preceding experts to improve detection accuracy the effectiveness of the multi expert approach has been demonstrated in experiments on a large number of images the obtained detection results are superior to the best individual expert and state of the art approaches while the detection speed is fast

texture classification using logical operators in this paper a new algorithm for texture classification based on logical operators is presented operators constructed from logical building blocks are convolved with texture images an optimal set of six operators are selected based on their texture discrimination ability the responses are then converted to standard deviation matrices computed over a sliding window zonal sampling features are computed from these matrices a feature selection process is applied and the new set of features are used for texture classification classification of several natural and synthetic texture images are presented demonstrating the excellent performance of the logical operator method the computational superiority and classification accuracy of the algorithm is demonstrated by comparison with other popular methods experiments with different classifiers and feature normalization are also presented the euclidean distance classifier is found to perform best with this algorithm the algorithm involves only convolutions and simple arithmetic in the various stages which allows faster implementations the algorithm is applicable to different types of classification problems which is demonstrated by segmentation of remote sensing images compressed and reconstructed images and industrial images

reinterpreting the application of gabor filters as a manipulation of the margin in linear support vector machines linear filters are ubiquitously used as a preprocessing step for many classification tasks in computer vision in particular applying gabor filters followed by a classification stage such as a support vector machine svm is now common practice in computer vision applications like face identity and expression recognition a fundamental problem occurs however with respect to the high dimensionality of the concatenated gabor filter responses in terms of memory requirements and computational efficiency during training and testing in this paper we demonstrate how the preprocessing step of applying a bank of linear filters can be reinterpreted as manipulating the type of margin being maximized within the linear svm this new interpretation leads to sizable memory and computational advantages with respect to existing approaches the reinterpreted formulation turns out to be independent of the number of filters thereby allowing the examination of the feature spaces derived from arbitrarily large number of linear filters a hitherto untestable prospect further this new interpretation of filter banks gives new insights other than the often cited biological motivations into why the preprocessing of images with filter banks like gabor filters improves classification performance

geometric fourier analysis of the conformal camera for active vision suppose one intends to design an active vision system that should perform some artificial intelligence functions for instance it should recognize a planar object or a three dimensional object containing a piece of a planar surface in a dynamic scene ideally such a system should be built upon some data model representing visual inputs and algorithms storing processing and analyzing visual information that are well adapted to image transformations produced by different perspectives between planar objects and the imaging system in spite of its importance this problem remained unsolved until recently in this article building on the author s work projective fourier analysis for patterns is constructed in the framework of geometric fourier analysis on groups and homogeneous spaces it is done by identifying in the conformal camera the group mathbf sl 2 mathbb c which gives image projective transformations by acting through linear fractional mappings on the image plane homogeneous under th

facial expression recognition in perceptual color space this paper introduces a tensor perceptual color framework tpcf for facial expression recognition fer which is based on information contained in color facial images the tpcf enables multilinear image analysis in different color spaces and demonstrates that color components provide additional information for robust fer using this framework the components in either rgb ycbcr cielab or cieluv space of color images are unfolded to 2 d tensors based on multilinear algebra and tensor concepts from which the features are extracted by log gabor filters the mutual information quotient method is employed for feature selection these features are classified using a multiclass linear discriminant analysis classifier the effectiveness of color information on fer using low resolution and facial expression images with illumination variations is assessed for performance evaluation experimental results demonstrate that color information has significant potential to improve emotion recognition performance due to the complementary characteristics of image textures furthermore the perceptual color spaces cielab and cieluv are better overall for fer than other color spaces by providing more efficient and robust performance for fer using facial images with illumination variation

recognition of distorted patterns by invariance kernels abstract a novel method for the recognition of deformed patterns is presented it is shown that for planar deformations generated by lie transformation groups there exist kernels such that the associated integral transforms are invariant under the deformations many examples of deformations and the corresponding kernels are given

a parametric texture model based on joint statistics of complex wavelet coefficients we present a universal statistical model for texture images in the context of an overcomplete complex wavelet transform the model is parameterized by a set of statistics computed on pairs of coefficients corresponding to basis functions at adjacent spatial locations orientations and scales we develop an efficient algorithm for synthesizing random images subject to these constraints by iteratively projecting onto the set of images satisfying each constraint and we use this to test the perceptual validity of the model in particular we demonstrate the necessity of subgroups of the parameter set by showing examples of texture synthesis that fail when those parameters are removed from the set we also demonstrate the power of our model by successfully synthesizing examples drawn from a diverse collection of artificial and natural textures

generalizations of the poincare birkhoff theorem in this article we prove some generalizations of the classical poincar birkhoff theorem on area preserving homeomorphisms of the annulus which satisfy a boundary twist condition the work of g d birkhoff on this theorem and its applications can be found in b1 b2 and chapter v of b3 a more modern treatment can be found in b n we prove a theorem for the open annulus a s1 x 0 1 since as we will see the theorem for the closed annulus can easily be obtained from this result for the open annulus however it is not immediately obvious what should be the analogue of the twist condition it turns out that the most general hypothesis and the most natural from the point of view of our proof involves the notion of positively and negatively returning disks for some lift of f to the covering space a r x 0 1 more precisely if f a a is a lift of f a a we will say that the e is a positively returning disk for f if there is an open disk u c a such that f u n u 0 and fp u n u k 0 forsome n k 0 here u k denotes the set x k t i x t e u thus u is disjoint from its image but under iteration by f returns to a positive translate of itself a negatively returning disk is defined similarly but with k 0 such that fl u n u 0 in section 2 we prove the following

adaptive smoothing respecting feature directions accurate extraction of the image feature directions is essential to image smoothing and other image processing tasks we show that the gradient based feature direction extraction method can be very erroneous the gradient is too local and it cannot detect oscillations we have developed two new methods the hessian method an approach using higher order differentiations and the gabor 1946 method an approach using space frequency analysis

computationally attractive real gabor transforms we present a gabor transform for real discrete signals and present a computationally attractive method for computing the transform for the critically sampled case we derive a biorthogonal function which is very localized in the time domain therefore truncation of this biorthogonal function allows us to compute approximate expansion coefficients with significantly reduced computational requirements further truncation does not degrade the numerical stability of the transform we present a tight upper bound on the reconstruction error incurred due to use of a truncated biorthogonal function and summarize computational savings for example the expense of transforming a length 2048 signal using length 16 blocks is reduced by a factor of 26 over similar fft based methods with at most 0 04 squared error in the reconstruction

gabor filter based edge detection abstract it is common practice to utilize evidence from biological and psychological vision experiments to develop computational models for low level feature extraction the receptive profiles of simple cells in mammalian visual systems have been found to closely resemble gabor filters daugman proved that gabor filters achieve joint minimal joint uncertainty these results led researchers to develop computational models based on gabor filters for several low level vision applications such as edge detection texture classification optical flow estimation and data compression in this paper the performance of a gabor odd filter based edge detector is investigated using the measures proposed by canny based on this performance analysis a design criterion for one dimensional 1d gabor filter based edge detector is derived it is shown that this design criterion also holds good for a two dimensional 2d gabor filter based edge detector experimental results are presented to demonstrate the performance of the gabor filter based edge detector

discrete nonlinear schrodinger equation with defects we investigate the dynamical properties of the one dimensional discrete nonlinear schrodinger equation dnls with periodic boundary conditions and with an arbitrary distribution of on site defects we study the propagation of a traveling plane wave with momentum k the dynamics in fourier space mainly involves two localized states with momenta k corresponding to a transmitted and a reflected wave within a two mode ansatz in fourier space the dynamics of the system maps on a nonrigid pendulum hamiltonian the several analytically predicted and numerically confirmed regimes include states with a vanishing time average of the rotational states implying complete reflections and refocusing of the incident wave oscillations around fixed points corresponding to quasi stationary states and above a critical value of the nonlinearity self trapped states with the wave traveling almost undisturbed through the impurity we generalize this treatment to the case of several traveling waves and time dependent defects the validity of the two mode ansatz and the continuum limit of the dnls are discussed

handprinted character recognition based on spatial topology distance measurement in this work we present a self organization matching approach to accomplish the recognition of handprinted characters drawn with thick strokes this approach is used to flex the unknown handprinted character toward matching its object characters gradually the extracted character features used in the self organization matching are center loci orientation and major axes of ellipses which fit the inked area of the patterns simulations provide encouraging results using the proposed method

analyzing looming motion components from their spatiotemporal spectral signature this paper addresses the use of spatio temporal transform methods applied to the analysis of dynamic image sequences and the characterization of image motion the image motion including a divergent component resulting from a looming camera component is analyzed in the spatio temporal mellin transform mt domain resulting in the separation of the spectrum into two parts a structural term corresponding to the spatial mt of the static image and a kinematic term depending on time to collision a motion support we examine potential applications of this property for the recovery of image motion from integral image brightness measurements and the computation of time to collision using spatio temporal mt analysis

wave propagation and sampling theory part ii sampling theory and complex waves morlet et al 1982 this issue showed the advantages of using complex values for both waves and characteristics of the media we simulated the theoretical tools we present here using the goupillaud kunetz algorithm now we present sampling methods for complex signals or traces corresponding to received waves and sampling methods for complex characterization of multilayered or heterogeneous media regarding the complex signals we present a twodimecsional 2 d method of sampling in the time frequency domain using a special or extended gabor expansion on a set of basic wavelets adapted to phase preservation such a 2 d expansion permits us to handle in a proper manner instantaneous frequency spectra we show the differences between wavelet resolution and sampling grid resolution we also show the importance of phase preservation in high resolution seismic regarding the media we show how analytical studies of wave propagation in periodic structured layers could help when trying to characterize the physical properties of the layers and their large scale granularity as a result of complex deconvolution analytical studies of wave propagation in periodic structures are well known in solid state physics and lead to the so called bloch waves the introduction of complex waves leads to replacing the classical wave equation by a schriidinger equation finally we show that complex wave equations gabor expansion and bloch waves are three different ways of introducing the tools of quantum mechanics in highresolution seismic gabor 1946 kittel 1976 morlet 1975 and conversely the goupillaud kunetz algorithm and an extended gabor expansion may be of some use in solid state physics

wave propagation and sampling theory part i complex signal and scattering in multilayered media from experimental studies in digital processing of seismic reflection data geophysicists know that a seismic signal does vary in amplitude shape frequency and phase versus propagation time to enhance the resolution of the seismic reflection method we must investigate these variations in more detail we present quantitative results of theoretical studies on propagation of plane waves for normal incidence through perfectly elastic multilayered media as wavelet shapes we use zero phase cosine wavelets modulated by a gaussian envelope and the corresponding complex wavelets a finite set of such wavelets for an appropriate sampling of the frequency domain may be taken as the basic wavelets for a gabor expansion of any signal or trace in a two dimensional 2 d domain time and frequency we can then compute the wave propagation using complex functions and thereby obtain quantitative results including energy and phase of the propagating signals these results appear as complex 2 d functions of time and frequency i e as instantaneous frequency spectra choosing a constant sampling rate on the logarithmic scale in the frequency domain leads to an appropriate sampling method for phase preservation of the complex signals or traces for this purpose we developed a gabor expansion involving basic wavelets with a constant time duration mean period ratio for layered media as found in sedimentary basins

a joint frequency position domain structure identification of nonlinear discrete time systems by neural networks a new technique is proposed to identify the structure and the parameters of nonlinear discrete time system models the structure is represented in a frequency position domain of gabor basis functions gbfs a simplification to the gbf is also presented where the spatial gaussian envelope of gbf is replaced with a triangular one a modification to the gbf has also been introduced in order to suppress the effects of noise on the procedure a three layered neural network augmented with nonuniform sampling is described for solving the system identification problem

gaussian networks for direct adaptive control a direct adaptive tracking control architecture is proposed and evaluated for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture uses a network of gaussian radial basis functions to adaptively compensate for the plant nonlinearities under mild assumptions about the degree of smoothness exhibit by the nonlinear functions the algorithm is proven to be globally stable with tracking errors converging to a neighborhood of zero a constructive procedure is detailed which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy a stable weight adjustment mechanism is determined using lyapunov theory the network construction and performance of the resulting controller are illustrated through simulations with example systems

frequency analysis for multi dimensional systems global dynamics and diffusion abstract frequency analysis is a new method for analyzing the stability of orbits in a conservative dynamical system it was first devised in order to study the stability of the solar system j laskar icarus 88 1990 266 291 and then applied to the 2d standard mapping laskar et al physica d 56 1992 253 269 it is a powerful method for analyzing weakly chaotic motion in hamiltonian systems or symplectic maps for regular motions it yields an analytical representation of the solutions the analysis of the regularity of the frequency map with respect to the action space and of its variations with respect to time gives rise to two criteria for the regularity of the motion which are valid for multi dimensional systems for a 4d symplectic map plotting the frequency map in the frequency plane provides a clear representation of the global dynamics and reveals that high order resonances are of great importance in understanding the diffusion of non regular orbits through the invariant tori in particular it appears in several examples that diffusion along the resonance lines arnold diffusion is of less importance than diffusion across the resonances lines which can lead to large diffusion due to the phenomenon of overlap of higher order resonances chaotic layers many fine features of the dynamics are also revealed by frequency analysis which would require more theoretical study for a better understanding

acceleration of the frame algorithm shows how polynomial acceleration techniques which have been developed for the solution of large linear systems can be employed to improve and accelerate the frame algorithm these methods permit a reduction in the number of necessary iterations by an order of magnitude when the frame algorithm is slow the author gives several examples from the theory of irregular sampling from wavelet theory and from gabor theory where these methods are probably mandatory for efficient reconstruction

the generalized gabor scheme of image representation in biological and machine vision a scheme suitable for visual information representation in a combined frequency position space is investigated through image decomposition into a finite set of two dimensional gabor elementary functions gef the scheme is generalized to account for the position dependent gabor sampling rate oversampling logarithmic frequency scaling and phase quantization characteristic of the visual system comparison of reconstructed signal highlights the advantages of the generalized gabor scheme in coding typical bandlimited images it is shown that there exists a tradeoff between the number of frequency components used per position and the number of such clusters sampling rate utilized along the spatial coordinate

hamiltonian hopf bifurcations in the discrete nonlinear schrodinger trimer oscillatory instabilities quasi periodic solutions and a new type of self trapping transition oscillatory instabilities in hamiltonian anharmonic lattices are known to appear through hamiltonian hopf bifurcations of certain time periodic solutions of multibreather type here we analyse the basic mechanisms for this scenario by considering the simplest possible model system of this kind where they appear the three site discrete nonlinear schrodinger model with periodic boundary conditions the stationary solution having equal amplitude and opposite phases on two sites and zero amplitude on the third is known to be unstable for an interval of intermediate amplitudes we numerically analyse the nature of the two bifurcations leading to this instability and find them to be of two different types close to the lower amplitude threshold stable two frequency quasi periodic solutions exist surrounding the unstable stationary solution and the dynamics remains trapped around the latter so that in particular the amplitude of the originally unexcited site remains small by contrast close to the higher amplitude threshold all two frequency quasi periodic solutions are detached from the unstable stationary solution and the resulting dynamics is of population inversion type involving also the originally unexcited site

detection of arnold diffusion in hamiltonian systems we detect diffusion along resonances in a quasi integrable system at small values of the perturbing parameter the diffusion coefficient goes to zero as the perturbation parameter goes to zero faster than a power law typical of chirikov diffusion and is compatible with an exponential law expected in the nekhoroshev theorem

invariant pattern recognition using the rfm descriptor a pattern descriptor invariant to rotation scaling translation rst and robust to additive noise is proposed by using the radon fourier and mellin transforms the radon transform converts the rst transformations applied on a pattern image into transformations in the radial and angular coordinates of the pattern s radon image these beneficial properties of the radon transform make it an useful intermediate representation for the extraction of invariant features from pattern images for the purpose of indexing matching in this paper invariance to rst is obtained by applying the 1d fourier mellin and discrete fourier transforms on the radial and angular coordinates of the pattern s radon image respectively the implementation of the proposed descriptor is reasonably fast and correct based mainly on the fusion of the radon and fourier transforms and on a modification of the mellin transform theoretical arguments validate the robustness of the proposed descriptor to additive noise and empirical evidence on both occlusion deformation and noisy datasets shows its effectiveness

two different approaches for iris recognition using gabor filters and multiscale zero crossing representation abstract importance of biometric user identification is increasing everyday one of the most promising techniques is the one based on the human iris the authors in this work describe different approaches to develop this biometric technique based on the works carried out by daugman the authors have worked using gabor filters and hamming distance but in addition they have also worked in zero crossing representation of the dyadic wavelet transform applied to two different iris signatures one based on a single virtual circle of the iris the other one based on an annular region also other metrics have been applied to be compared with the results obtained with the hamming distance in this work euclidean distance and d z will be shown the last proposed approach is translation rotation and scale invariant results will show a classification success up to 99 6 achieving an equal error rate down to 0 12 and the possibility of having null false acceptance rates with very low false rejection rates

augmented dt cwt feature based classification using regularized neighborhood projection discriminant analysis for face recognition this paper presents a novel face recognition method which integrates the augmented dual tree complex wavelet transform adt cwt representation of face images and regularized neighborhood projection discriminant analysis rnpda method adt cwt first derives desirable facial features characterized by spatial frequency spatial locality and orientation selectivity to cope with the variations due to illumination and facial expression changes different from dt cwt which does not consider the structural characteristics of the face images our representation method not only considers the statistical property of the input features but also adopts an eigenmask to emphasize those important facial feature points the dimensionality of the derivation of adt cwt feature is further reduced by using rnpda which directly obtain a set of optimal eigenvectors with a simple regression framework and thus can overcome the small sample size problem of npda extensive experiments have been made to compare the recognition performance of the proposed method with some popular dimensionality reduction methods on the feret database the extended yaleb database and the cmu pie database the results verify the effectiveness of the proposed method

an fft based technique for translation rotation and scale invariant image registration this correspondence discusses an extension of the well known phase correlation technique to cover translation rotation and scaling fourier scaling properties and fourier rotational properties are used to find scale and rotational movement the phase correlation technique determines the translational movement this method shows excellent robustness against random noise

analysis of multichannel narrow band filters for image texture segmentation a model for texture analysis and segmentation using multiple oriented channel filters is analyzed in the general framework several different arguments are applied leading to the conclusion that the two dimensional gabor filters possess strong optimality properties for this task properties of the multiple channel segmentation approach are analyzed in particular perturbations of textures from an ideal model are found to have important effects on the segmentation that can usually be ameliorated by simply preceding the segmentation process by a logarithmic operation and using a low pass postfilter prior to making region assignments the difficult problems of space variant textures and multiple component textures are also considered local spatial frequency estimation approaches are suggested that use the responses as constraints in estimating the locally emergent texture frequencies complex texture aggregates containing multiple shared frequency components can be analyzed if the textures are distinct and few in number

a boundary perturbation interior point homotopy method for solving fixed point problems abstract in this paper a boundary perturbation interior point homotopy method is proposed to give a constructive proof of the general brouwer fixed point theorem and thus solve fixed point problems in a class of nonconvex sets compared with the previous results by using the newly proposed method initial points can be chosen in the whole space of r n which may improve greatly the computational efficiency of reduced predictor corrector algorithms resulted from that method some numerical examples are given to illustrate the results of this paper

boundedness for solutions of nonlinear hill s equations with periodic forcing terms via moser s twist theorem we consider the nonlinear hill s equation with periodic forcing term x x2n 1 a1 ea t x p t n 1 where a t and p t are continuous and 1 periodic a1 and are positive constants and e is a small parameter the purpose of this paper is to prove that all solutions of the above mentioned equation are bounded for t r and that there are infinitely many quasi periodic solutions and an infinity of periodic solutions of minimal period m for each positive integer m

on the numerical detection of the effective stability of chaotic motions in quasi integrable systems we describe and compare two recent tools for detecting the geometry of resonances of a dynamical system in order to get information about the long term stability of chaotic solutions one of these tools is the so called fast lyapunov indicator fli celest mech dyn astr 67 1997 41 while the other is a recently introduced spectral fourier analysis of chaotic motions discrete contin dyn syst b 1 2001 1 for the first tool we provide new analytical estimates which explain why the fli is a sensitive means of discriminating between resonant and non resonant regular orbits thus providing a method to detect the geometry of resonances of a quasi integrable system the second tool based on a recent theoretical result can test directly whether a chaotic motion is in the nekhoroshev stability regime so that it practically cannot diffuse in the phase space or on the contrary if it is in the chirikov diffusive regime using these two methods we determine the value of the critical parameter at which the transition from the nekhoroshev to the chirikov regime occurs in a quasi integrable model hamiltonian system and standard four dimensional map

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

gabor feature based classification using the enhanced fisher linear discriminant model for face recognition this paper introduces a novel gabor fisher 1936 classifier gfc for face recognition the gfc method which is robust to changes in illumination and facial expression applies the enhanced fisher linear discriminant model efm to an augmented gabor feature vector derived from the gabor wavelet representation of face images the novelty of this paper comes from 1 the derivation of an augmented gabor feature vector whose dimensionality is further reduced using the efm by considering both data compression and recognition generalization performance 2 the development of a gabor fisher classifier for multi class problems and 3 extensive performance evaluation studies in particular we performed comparative studies of different similarity measures applied to various classifiers we also performed comparative experimental studies of various face recognition schemes including our novel gfc method the gabor wavelet method the eigenfaces method the fisherfaces method the efm method the combination of gabor and the eigenfaces method and the combination of gabor and the fisherfaces method the feasibility of the new gfc method has been successfully tested on face recognition using 600 feret frontal face images corresponding to 200 subjects which were acquired under variable illumination and facial expressions the novel gfc method achieves 100 accuracy on face recognition using only 62 features

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

continuation of periodic orbits in conservative and hamiltonian systems we introduce and justify a computational scheme for the continuation of periodic orbits in systems with one or more first integrals and in particular in hamiltonian systems having several independent symmetries our method is based on a generalization of the concept of a normal periodic orbit as introduced by sepulchre and mackay nonlinearity 10 1997 679 we illustrate the continuation method on some integrable hamiltonian systems with two degrees of freedom and briefly discuss some further applications

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed