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

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

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

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

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

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

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

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

redefined fuzzy hv submodules and many valued implications this paper considers a relationship between fuzzy sets and algebraic hyperstructures a generalization of ideas presented by davvaz b davvaz fuzzy h v submodules fuzzy sets syst 117 2001 477 484 and bhakat and das s k bhakat p das email protected q fuzzy subgroup fuzzy sets syst 80 1996 359 368 is presented in fact the main purpose of this paper is to introduce and study a new sort of fuzzy h v submodule of an h v module called email protected q fuzzy h v submodules these fuzzy h v submodules are characterized by their level h v submodules the concept of a fuzzy h v submodule with thresholds is introduced and some interesting properties are investigated

a kernel problem with fuzzy visibility this paper proposes a new visibility concept fuzzy visibility reflecting that the visibility property varies according to the condition of application areas the concept of visibility is defined on the basis of the fuzzy theory for two points p and q p is visible from q if the visibility degree between them is no less than 0 1 if there exists a point from which all points in a polygon are visible then it is said that the visibility value of the polygon is the kernel problem is constructing the region such that all points inside simple polygon can be visible from any point of the region an optimal algorithm is given for the kernel problem with o n time complexity for a given simple polygon we present o n algorithms for computing the maximum and minimum visibility values of the polygon

a globally optimal bilinear programming approach to the design of approximate hilbert pairs of orthonormal wavelet bases it is understood that the hilbert transform pairs of orthonormal wavelet bases can only be realized approximately by the scaling filters of conjugate quadrature filter cqf banks in this paper the approximate fir realization of the hilbert transform pairs is formulated as an optimization problem in the sense of the lp p 1 2 or infinite norm minimization on the approximate error of the magnitude and phase conditions of the scaling filters the orthogonality and regularity conditions of the cqf bank pairs are taken as the constraints of such an optimization problem whereafter the branch and bound technique is employed to obtain the globally optimal solution of the resulting bilinear program optimization problem since the orthogonality and regularity conditions are explicitly taken as the constraints of our optimization problem the attained solution is an approximate hilbert transform pair satisfying these conditions exactly some orthogonal wavelet bases designed herein demonstrate that our design scheme is superior to those that have been reported in the literature moreover the designed orthogonal wavelet bases show that minimizing the l 1 norm of the approximate error should be advocated for obtaining better approximated hilbert pairs

characterization and sampled data design of dual tree filter banks for hilbert transform pairs of wavelet bases characterization and design of dual tree filter banks for forming hilbert transform pairs of wavelet bases are studied the characterization extends the existing results for quadrature mirror filter banks to general prefect reconstruction filter banks that satisfy only a mild technical assumption regarding the ratio of determinants of the two filter banks we establish equivalent relationships of hilbert transform pairs on scaling filters wavelet filters or scaling functions the design of scaling filters of a dual filter bank is formulated as a sampled data hinfin optimization problem the wavelet filters are then determined using the relationship on the determinants of the filter banks we convert the sampled data problem into an equivalent discrete time hinfin control problem which can be solved by standard hinfin control theory an analytical solution to the sampled data design problem is obtained for a special case the sampled data design approach usually gives infinite impulse response filter in the case where the primal filter bank is of finite impulse response fir we may truncate the impulse responses to get fir approximations they also lead to approximate hilbert transform pairs design examples are presented

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

combined complex ridgelet shrinkage and total variation minimization a new algorithm for the characterization of engineering surface topographies with line singularities is proposed it is based on thresholding complex ridgelet coefficients combined with total variation tv minimization the discrete ridgelet transform is designed by first using a discrete radon transform based on the nonequispaced fast fourier transform nfft and then applying a dual tree complex wavelet transform dt cwt the nfft based approach of the radon transform completely avoids linear interpolations of the cartesian to polar grid and requires only o n2 log n arithmetic operations for n by n arrays while its inverse preserves the good reconstruction quality of the filtered backprojection the dt cwt in the second step of the ridgelet transform provides approximate shift invariance on the projections of the radon transform after hard thresholding the ridgelet coefficients they are restored using tv minimization to eliminate the pseudo gibbs artifacts near the discontinuities numerical experiments demonstrate the remarkable ability of the methodology to extract line scratches

hilbert transform pairs of biorthogonal wavelet bases the forming of hilbert transform pairs of biorthogonal wavelet bases of two band filter banks is studied in this paper we first derive necessary and sufficient conditions on the scaling filters that render two hilbert transform pairs one decomposition pair and one reconstruction pair we show that the hilbert transform pairs are achieved if and only if the decomposition scaling filter of one filter bank is half sample delayed from that of the other filter bank and the reconstruction scaling filter of the former is half sample advanced from that of the latter hilbert transform pairs of wavelet bases are also characterized by equivalent relationships on the wavelet filters and the scaling functions associated with the two filter banks an illustrative example is provided

the phaselet transform an integral redundancy nearly shift invariant wavelet transform this paper introduces an approximately shift invariant redundant dyadic wavelet transform the phaselet transform that includes the popular dual tree complex wavelet transform of kingsbury see phil r soc london a sept 1999 as a special case the main idea is to use a finite set of wavelets that are related to each other in a special way and hence called phaselets to achieve approximate shift redundancy the bigger the set the better the approximation a sufficient condition on the associated scaling filters to achieve this is that they are fractional shifts of each other algorithms for the design of phaselets with a fixed number vanishing moments is presented building on the work of selesnick see ieee trans signal processing for the design of wavelet pairs for kingsbury s dual tree complex wavelet transform construction of two dimensional 2 d directional bases from tensor products of one dimensional 1 d phaselets is also described phaselets as a new approach to redundant wavelet transforms and their construction are both novel and should be interesting to the reader independent of the approximate shift invariance property that this paper argues they possess

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