the monogenic signal this paper introduces a two dimensional 2 d generalization of the analytic signal this novel approach is based on the riesz transform which is used instead of the hilbert transform the combination of a 2 d signal with the riesz transformed one yields a sophisticated 2 d analytic signal the monogenic signal the approach is derived analytically from irrotational and solenoidal vector fields based on local amplitude and local phase an appropriate local signal representation that preserves the split of identity i e the invariance equivariance property of signal decomposition is presented this is one of the central properties of the one dimensional 1 d analytic signal that decomposes a signal into structural and energetic information we show that further properties of the analytic signal concerning symmetry energy allpass transfer function and orthogonality are also preserved and we compare this with the behavior of other approaches for a 2 d analytic signal as a central topic of this paper a geometric phase interpretation that is based on the relation between the 1 d analytic signal and the 2 d monogenic signal established by the radon 1986 transform is introduced possible applications of this relationship are sketched and references to other applications of the monogenic signal are given

multidimensional directional filter banks and surfacelets in 1992 bamberger and smith proposed the directional filter bank dfb for an efficient directional decomposition of 2 d signals due to the nonseparable nature of the system extending the dfb to higher dimensions while still retaining its attractive features is a challenging and previously unsolved problem we propose a new family of filter banks named ndfb that can achieve the directional decomposition of arbitrary n dimensional nges2 signals with a simple and efficient tree structured construction in 3 d the ideal passbands of the proposed ndfb are rectangular based pyramids radiating out from the origin at different orientations and tiling the entire frequency space the proposed ndfb achieves perfect reconstruction via an iterated filter bank with a redundancy factor of n in n d the angular resolution of the proposed ndfb can be iteratively refined by invoking more levels of decomposition through a simple expansion rule by combining the ndfb with a new multiscale pyramid we propose the surfacelet transform which can be used to efficiently capture and represent surface like singularities in multidimensional data

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

subsystem eth motivated by the qualitative picture of canonical typicality we propose a refined formulation of the eigenstate thermalization hypothesis eth for chaotic quantum systems the new formulation which we refer to as subsystem eth is in terms of the reduced density matrix of subsystems this strong form of eth outlines the set of observables defined within the subsystem for which it guarantees eigenstate thermalization we discuss the limits when the size of the subsystem is small or comparable to its complement in the latter case we outline the way to calculate the leading volume proportional contribution to the von neumann and renyi entanglment entropies finally we provide numerical evidence for the proposal in the case of a one dimensional ising spin chain

characterization of color images with multiscale monogenic maxima can we build a feature based analysis that fully characterizes images the literature answers with edge based reconstruction methods inspired by marr s paradigm but limited to the greyscale case this paper studies the color case a new sparse representation is carried out with the monogenic concept and the mallat zhong wavelet maxima method our monogenic maxima provide efficient contour shape and color characterization as a sparse set of local features including amplitude phase orientation and ellipse parameters this rich description takes the wavelet maxima representation further towards the wide topic of keypoint analysis we propose a reconstruction process that retrieves the image from its monogenic maxima while known works all rely on constrained optimization implying an iterative use of the filterbank we propose to interpolate the data in the feature domain by exploiting the visual knowledge from the feature set this direct retrieval is accurate enough so that no iteration is required the main question is finally answered with comparative experiments it is shown that a reasonably small amount of features is sufficiently informative for visually appealing image retrieval the features appear numerically stable to rotation and can be intuitively simplified to perform image regularization

approach to thermal equilibrium of macroscopic quantum systems we consider an isolated macroscopic quantum system let mathcal h be a microcanonical energy shell i e a subspace of the system s hilbert space spanned by the finitely many energy eigenstates with energies between e and e ensuremath delta e the thermal equilibrium macrostate at energy e corresponds to a subspace mathcal h eq of mathcal h such that text dim text mathcal h eq text dim text mathcal h is close to 1 we say that a system with state vector ensuremath psi mathcal h is in thermal equilibrium if ensuremath psi is close to mathcal h eq we show that for typical hamiltonians with given eigenvalues all initial state vectors ensuremath psi 0 evolve in such a way that ensuremath psi t is in thermal equilibrium for most times t this result is closely related to von neumann s quantum ergodic theorem of 1929

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

image denoising using scale mixtures of gaussians in the wavelet domain we describe a method for removing noise from digital images based on a statistical model of the coefficients of an overcomplete multiscale oriented basis neighborhoods of coefficients at adjacent positions and scales are modeled as the product of two independent random variables a gaussian vector and a hidden positive scalar multiplier the latter modulates the local variance of the coefficients in the neighborhood and is thus able to account for the empirically observed correlation between the coefficient amplitudes under this model the bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimates over all possible values of the hidden multiplier variable we demonstrate through simulations with images contaminated by additive white gaussian noise that the performance of this method substantially surpasses that of previously published methods both visually and in terms of mean squared error

representation and compression of multidimensional piecewise functions using surflets we study the representation approximation and compression of functions in m dimensions that consist of constant or smooth regions separated by smooth m 1 dimensional discontinuities examples include images containing edges video sequences of moving objects and seismic data containing geological horizons for both function classes we derive the optimal asymptotic approximation and compression rates based on kolmogorov metric entropy for piecewise constant functions we develop a multiresolution predictive coder that achieves the optimal rate distortion performance for piecewise smooth functions our coder has near optimal rate distortion performance our coder for piecewise constant functions employs surflets a new multiscale geometric tiling consisting of m dimensional piecewise constant atoms containing polynomial discontinuities our coder for piecewise smooth functions uses surfprints which wed surflets to wavelets for piecewise smooth approximation both of these schemes achieve the optimal asymptotic approximation performance key features of our algorithms are that they carefully control the potential growth in surflet parameters at higher smoothness and do not require explicit estimation of the discontinuity we also extend our results to the corresponding discrete function spaces for sampled data we provide asymptotic performance results for both discrete function spaces and relate this asymptotic performance to the sampling rate and smoothness orders of the underlying functions and discontinuities for approximation of discrete data we propose a new scale adaptive dictionary that contains few elements at coarse and fine scales but many elements at medium scales simulation results on synthetic signals provide a comparison between surflet based coders and previously studied approximation schemes based on wedgelets and wavelets

wavelet steerability and the higher order riesz transform our main goal in this paper is to set the foundations of a general continuous domain framework for designing steerable reversible signal transformations a k a frames in multiple dimensions d 2 to that end we introduce a self reversible nth order extension of the riesz transform we prove that this generalized transform has the following remarkable properties shift invariance scale invariance inner product preservation and steerability the pleasing consequence is that the transform maps any primary wavelet frame or basis of l 2 bbr d into another steerable wavelet frame while preserving the frame bounds the concept provides a functional counterpart to simoncelli s steerable pyramid whose construction was primarily based on filterbank design the proposed mechanism allows for the specification of wavelets with any order of steerability in any number of dimensions it also yields a perfect reconstruction filterbank algorithm we illustrate the method with the design of a novel family of multidimensional riesz laplace wavelets that essentially behave like the n th order partial derivatives of an isotropic gaussian kernel

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

a fast multilevel algorithm for wavelet regularized image restoration we present a multilevel extension of the popular ldquothresholded landweberrdquo algorithm for wavelet regularized image restoration that yields an order of magnitude speed improvement over the standard fixed scale implementation the method is generic and targeted towards large scale linear inverse problems such as 3 d deconvolution microscopy the algorithm is derived within the framework of bound optimization the key idea is to successively update the coefficients in the various wavelet channels using fixed subband adapted iteration parameters step sizes and threshold levels the optimization problem is solved efficiently via a proper chaining of basic iteration modules the higher level description of the algorithm is similar to that of a multigrid solver for pdes but there is one fundamental difference the latter iterates though a sequence of multiresolution versions of the original problem while in our case we cycle through the wavelet subspaces corresponding to the difference between successive approximations this strategy is motivated by the special structure of the problem and the preconditioning properties of the wavelet representation we establish that the solution of the restoration problem corresponds to a fixed point of our multilevel optimizer we also provide experimental evidence that the improvement in convergence rate is essentially determined by the unconstrained linear part of the algorithm irrespective of the type of wavelet finally we illustrate the technique with some image deconvolution examples including some real 3 d fluorescence microscopy data

complex wavelet bases steerability and the marr like pyramid our aim in this paper is to tighten the link between wavelets some classical image processing operators and david marr s theory of early vision the cornerstone of our approach is a new complex wavelet basis that behaves like a smoothed version of the gradient laplace operator starting from first principles we show that a single generator wavelet can be defined analytically and that it yields a semi orthogonal complex basis of l 2 r2 irrespective of the dilation matrix used we also provide an efficient fft based filterbank implementation we then propose a slightly redundant version of the transform that is nearly translation invariant and that is optimized for better steerability gaussian like smoothing kernel we call it the marr like wavelet pyramid because it essentially replicates the processing steps in marr s theory of early vision we use it to derive a primal wavelet sketch which is a compact description of the image by a multiscale subsampled edge map finally we provide an efficient iterative algorithm for the reconstruction of an image from its primal wavelet sketch

the monogenic wavelet transform this paper extends the 1 d analytic wavelet transform to the 2 d monogenic wavelet transform the transformation requires care in its specification to ensure suitable transform coefficients are calculated and it is constructed so that the wavelet transform may be considered as both local and monogenic this is consistent with defining the transform as a real wavelet transform of a monogenic signal in analogy with the analytic wavelet transform classes of monogenic wavelets are proposed with suitable local properties it is shown that the monogenic wavelet annihilates anti monogenic signals that the monogenic wavelet transform is phase shift covariant and that the transform magnitude is phase shift invariant a simple form for the magnitude and orientation of the isotropic transform coefficients of a unidirectional signal when observed in a rotated frame of reference is derived the monogenic wavelet ridges of local plane waves are given

3d solid texture classification using locally oriented wavelet transforms many image acquisition techniques used in biomedical imaging material analysis and structural geology are capable of acquiring 3d solid images computational analysis of these images is complex but necessary since it is difficult for humans to visualize and quantify their detailed 3d content one of the most common methods to analyze 3d data is to characterize the volumetric texture patterns texture analysis generally consists of encoding the local organization of image scales and directions which can be extremely diverse in 3d current state of the art techniques face many challenges when working with 3d solid texture where most approaches are not able to consistently characterize both scale and directional information 3d riesz wavelets can deal with both properties one key property of riesz filterbanks is steerability which can be used to locally align the filters and compare textures with arbitrary local orientations this paper proposes and compares three novel local alignment criteria for higher order 3d riesz wavelet transforms the estimations of local texture orientations are based on higher order extensions of regularized structure tensors an experimental evaluation of the proposed methods for the classification of synthetic 3d solid textures with alterations such as rotations and noise demonstrated the importance of local directional information for robust and accurate solid texture recognition these alignment methods achieved an accuracy of 0 95 in the rotated data three times more than the unaligned riesz descriptor that achieved 0 32 the accuracy obtained is better than all other techniques that are published and tested on the same database

from quantum dynamics to the canonical distribution general picture and a rigorous example derivation of the canonical or boltzmann distribution based only on quantum dynamics is discussed consider a closed system which consists of a mutually interacting subsystem and a heat bath and assume that the whole system is initially in a pure state which can be far from equilibrium with small energy fluctuation under the hypothesis of equal weights for eigenstates we derive the canonical distribution in the sense that at sufficiently large and typical time the instantaneous quantum mechanical expectation value of an arbitrary operator of the subsystem is almost equal to the desired canonical expectation value we present a class of examples in which the above derivation can be rigorously established without any unproven hypotheses

on the explicit constructions of certain unitary t designs unitary t designs are good finite subsets of the unitary group ug d that approximate the whole unitary group ug d well unitary t designs have been applied in randomized benchmarking tomography quantum cryptography and many other areas of quantum information science if a unitary t design itself is a group then it is called a unitary t group although it is known that unitary t designs in ug d exist for any t and d the unitary t groups do not exist for t geq 4 if d geq 3 as it is shown by guralnick tiep 2005 and bannai navarro rizo tiep bnrt 2018 explicit constructions of exact unitary t designs in ug d are not easy in general in particular explicit constructions of unitary 4 designs in ug 4 have been an open problem in quantum information theory we prove that some exact unitary t 1 designs in the unitary group ug d are constructed from unitary t groups in ug d that satisfy certain specific conditions based on this result we specifically construct exact unitary 3 designs in ug 3 from the unitary 2 group sl 3 2 in ug 3 and also unitary 4 designs in ug 4 from the unitary 3 group sp 4 3 in ug 4 numerically we also discuss some related problems and a few applications to physics problems

a fast wavelet based reconstruction method for magnetic resonance imaging in this work we exploit the fact that wavelets can represent magnetic resonance images well with relatively few coefficients we use this property to improve magnetic resonance imaging mri reconstructions from undersampled data with arbitrary k space trajectories reconstruction is posed as an optimization problem that could be solved with the iterative shrinkage thresholding algorithm ista which unfortunately converges slowly to make the approach more practical we propose a variant that combines recent improvements in convex optimization and that can be tuned to a given specific k space trajectory we present a mathematical analysis that explains the performance of the algorithms using simulated and in vivo data we show that our nonlinear method is fast as it accelerates ista by almost two orders of magnitude we also show that it remains competitive with tv regularization in terms of image quality

normal typicality and von neumann s quantum ergodic theorem we discuss the content and significance of john von neumann s quantum ergodic theorem qet of 1929 a strong result arising from the mere mathematical structure of quantum mechanics the qet is a precise formulation of what we call normal typicality i e the statement that for typical large systems every initial wave function 0 from an energy shell is normal it evolves in such a way that t t is for most t macroscopically equivalent to the micro canonical density matrix the qet has been mostly forgotten after it was criticized as a dynamically vacuous statement in several papers in the 1950s however we point out that this criticism does not apply to the actual qet a correct statement of which does not appear in these papers but to a different indeed weaker statement furthermore we formulate a stronger statement of normal typicality based on the observation that the bound on the deviations from the average specified by von neumann is unnecessarily coarse and a much tighter and more relevant bound actually follows from his proof

a nonparametric boundary detection technique applied to 3d inverted surveys of the kevitsa ni cu pge deposit abstractgeophysical inversion can produce 3d models of the subsurface s physical properties the smoothness of property variations in these models makes it challenging to automatically find boundaries of homogeneous regions where these boundaries may have implications for petrophysical transition and are significant for geologic interpretation we have developed a new boundary detection technique that nonparametrically identifies and subtracts homogeneous regions from the 3d model leaving exposed edges the method is based on kernel density estimation of local property variations in which the number of modes in the kernel density estimates i e the local mode cardinality mc is used to identify edge voxels within the model two edge detection operators were developed one using local values exclusively and the other incorporating the spatial distribution of local values into the local mc which is more sensitive to local variations to assist in the geologic interpretation continuous boundary surf

three dimensional subband coding of video we describe and show the results of video coding based on a three dimensional 3 d spatio temporal subband decomposition the results include a 1 mbps coder based on a new adaptive differential pulse code modulation scheme adpcm and adaptive bit allocation this rate is useful for video storage on cd rom coding results are also shown for a 384 kbps rate that are based on adpcm for the lowest frequency band and a new form of vector quantization geometric vector quantization gvq for the data in the higher frequency bands gvq takes advantage of the inherent structure and sparseness of the data in the higher bands results are also shown for a 128 kbps coder that is based on an unbalanced tree structured vector quantizer utsvq for the lowest frequency band and gvq for the higher frequency bands the results are competitive with traditional video coding techniques and provide the motivation for investigating the 3 d subband framework for different coding schemes and various applications

steerable wavelet frames based on the riesz transform we consider an extension of the 1 d concept of analytical wavelet to n d which is by construction compatible with rotations this extension called a monogenic wavelet yields a decomposition of the wavelet coefficients into amplitude phase and phase direction the monogenic wavelet is based on the hypercomplex monogenic signal which is defined using riesz transforms and perfectly isotropic wavelets frames employing the new concept of clifford frames we can show that the monogenic wavelet generates a wavelet frame furthermore this approach yields wavelet frames that are steerable with respect to direction applications to descreening and contrast enhancement illustrate the versatility of this approach to image analysis and reconstruction

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

directed networks different link formation mechanisms causing degree distribution distinction abstract within undirected networks scientists have shown much interest in presenting power law features for instance barabasi and albert 1999 claimed that a common property of many large networks is that vertex connectivity follows scale free power law distribution and in another study barabasi et al 2002 showed power law evolution in the social network of scientific collaboration at the same time jiang et al 2011 discussed deviation from power law distribution others indicated that size effect bagrow et al 2008 information filtering mechanism mossa et al 2002 and birth and death process shi et al 2005 could account for this deviation within directed networks many authors have considered that outlinks follow a similar mechanism of creation as inlinks faloutsos et al 1999 krapivsky et al 2001 tanimoto 2009 with link creation rate being the linear function of node degree resulting in a power law shape for both indegree and outdegree distribution some other authors have made an assumption that directed networks such as scientific collaboration or citation behave as undirected resulting in a power law degree distribution accordingly barabasi et al 2002 at the same time we claim 1 outlinks feature different degree distributions than inlinks where different link formation mechanisms cause the distribution distinctions 2 in outdegree distribution distinction holds for different levels of system decomposition therefore this distribution distinction is a property of directed networks first we emphasize in outlink formation mechanisms as causal factors for distinction between indegree and outdegree distributions where this distinction has already been noticed in barker et al 2010 and baxter et al 2006 within a sample network of oss projects as well as java software corpus as a network second we analyze whether this distribution distinction holds for different levels of system decomposition open source software oss project project dependency within a cluster package package dependency within a project and class class dependency within a package we conclude that indegree and outdegree dependencies do not lead to similar type of degree distributions implying that indegree dependencies follow overall power law distribution or power law with flat top or exponential cut off in some cases while outdegree dependencies do not follow heavy tailed distribution

elliptical monogenic wavelets for the analysis and processing of color images this paper studies and gives new algorithms for image processing based on monogenic wavelets existing greyscale monogenic filterbanks are reviewed and we reveal a lack of discussion about the synthesis part the monogenic synthesis is therefore defined from the idea of wavelet modulation and an innovative filterbank is constructed by using the radon transform the color extension is then investigated first the elliptical fourier atom model is proposed to generalize the analytic signal representation for vector valued signals then a color riesz transform is defined so as to construct color elliptical monogenic wavelets our radon based monogenic filterbank can be easily extended to color according to this definition the proposed wavelet representation provides efficient analysis of local features in terms of shape and color thanks to the concepts of amplitude phase orientation and ellipse parameters the synthesis from local features is deeply studied we conclude the paper by defining the color local frequency proposing an estimation algorithm

3d steerable wavelets in practice we introduce a systematic and practical design for steerable wavelet frames in 3d our steerable wavelets are obtained by applying a 3d version of the generalized riesz transform to a primary isotropic wavelet frame the novel transform is self reversible tight frame and its elementary constituents riesz wavelets can be efficiently rotated in any 3d direction by forming appropriate linear combinations moreover the basis functions at a given location can be linearly combined to design custom and adaptive steerable wavelets the features of the proposed method are illustrated with the processing and analysis of 3d biomedical data in particular we show how those wavelets can be used to characterize directional patterns and to detect edges by means of a 3d monogenic analysis we also propose a new inverse problem formalism along with an optimization algorithm for reconstructing 3d images from a sparse set of wavelet domain edges the scheme results in high quality image reconstructions which demonstrate the feature reduction ability of the steerable wavelets as well as their potential for solving inverse problems

convergence to equilibrium under a random hamiltonian we analyze equilibration times of subsystems of a larger system under a random total hamiltonian in which the basis of the hamiltonian is drawn from the haar measure we obtain that the time of equilibration is of the order of the inverse of the arithmetic average of the bohr frequencies to compute the average over a random basis we compute the inverse of a matrix of overlaps of operators which permute four systems we first obtain results on such a matrix for a representation of an arbitrary finite group and then apply it to the particular representation of the permutation group under consideration

steerable pyramids and tight wavelet frames in l_ 2 bbr d we present a functional framework for the design of tight steerable wavelet frames in any number of dimensions the 2 d version of the method can be viewed as a generalization of simoncelli s steerable pyramid that gives access to a larger palette of steerable wavelets via a suitable parametrization the backbone of our construction is a primal isotropic wavelet frame that provides the multiresolution decomposition of the signal the steerable wavelets are obtained by applying a one to many mapping n th order generalized riesz transform to the primal ones the shaping of the steerable wavelets is controlled by an m m unitary matrix where m is the number of wavelet channels that can be selected arbitrarily this allows for a much wider range of solutions than the traditional equiangular configuration steerable pyramid we give a complete functional description of these generalized wavelet transforms and derive their steering equations we describe some concrete examples of transforms including some built around a mallat type multiresolution analysis of l2 rd and provide a fast fourier transform based decomposition algorithm we also propose a principal component based method for signal adapted wavelet design finally we present some illustrative examples together with a comparison of the denoising performance of various brands of steerable transforms the results are in favor of an optimized wavelet design equalized principal component analysis which consistently performs best

nonseparable two and three dimensional wavelets we present two and three dimensional nonseparable wavelets they are obtained from discrete time bases by iterating filter banks we consider three sampling lattices quincunx separable by two in two dimensions and fco the design methods are based either on cascade structures or on the mcclellan transformation in the quincunx case we give a few design examples in particular the first example of an orthonormal 2 d wavelet basis with symmetries is constructed

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