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

a steerable complex wavelet construction and its application to image denoising this work addresses the design of a novel complex steerable wavelet construction the generation of transform space feature measurements associated with corner and edge presence and orientation properties and the application of these measurements directly to image denoising the decomposition uses pairs of bandpass filters that display symmetry and antisymmetry about a steerable axis of orientation while the angular characterization of the bandpass filters is similar to those previously described the radial characteristic is new as is the manner of constructing the interpolation functions for steering the complex filters have been engineered into a multirate system providing a synthesis and analysis subband filtering system with good reconstruction properties although the performance of our proposed denoising strategy is currently below that of recently reported state of the art techniques in denoising it does compare favorably with wavelet coring approaches employing global thresholds and with an oracle shrinkage technique and presents a very promising avenue for exploring structure based denoising in the wavelet domain

fluctuation slow down of the death of brownian particles in the case of movable traps the influence of trap diffusion on the fluctuation slow down of death of brownian particles discovered earlier in the case of stationary traps is analysed it is shown that fluctuation slow down also takes place with movable traps if the diffusion is slow enough

general affine surface areas abstract two families of general affine surface areas are introduced basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for l affine surface areas are established

extrapolative spatial models for detecting perceptual boundaries in colour images we present a novel approach for automatically building image specific extrapolative spatial models of non boundary local energy that can be used to perform local statistical tests to detect perceptual boundaries the non boundary model consists of statistics of local energy that are spatially extrapolated by a non boundary confidence map and a scale adaptive normalised filtering algorithm we exploit the flexibility of steerable filters to both extract oriented local energy and to provide local statistics of the energy distribution in the orientation domain to compute the non boundary confidence map finally we apply our local thresholding technique separately to the three channels of colour images and adopt a max operator to combine the results we provide a qualitative and quantitative comparison on real images from a hand segmented natural image database against the best combination of the most widely cited colour edge detectors and automatic global thresholding methods

the lp minkowski problem and the minkowski problem in centroaffine geometry abstract the l p minkowski problem introduced by lutwak is solved for p n 1 in the smooth category the relevant monge ampere equation 0 1 is solved for all p 1 the same equation for p 1 is also studied and solved for p n 1 1 when p n 1 the equation is interpreted as a minkowski problem in centroaffine geometry a kazdan warner type obstruction for this problem is obtained

adaptive image denoising using scale and space consistency this paper proposes a new method for image denoising with edge preservation based on image multiresolution decomposition by a redundant wavelet transform in our approach edges are implicitly located and preserved in the wavelet domain whilst image noise is filtered out at each resolution level the image edges are estimated by gradient magnitudes obtained from the wavelet coefficients which are modeled probabilistically and a shrinkage function is assembled based on the model obtained joint use of space and scale consistency is applied for better preservation of edges the shrinkage functions are combined to preserve edges that appear simultaneously at several resolutions and geometric constraints are applied to preserve edges that are not isolated the proposed technique produces a filtered version of the original image where homogeneous regions appear separated by well defined edges possible applications include image presegmentation and image denoising

the orlicz mean zonoid operator abstract in this paper we introduce an orlicz mean zonoid operator z k n 2 k n which has zhang s mean zonoid operator as a special case we show that the new operator which actually maps k n 2 to k 0 n is commutative gl n covariant and continuous we also establish an affine isoperimetric inequality for the orlicz mean zonoid

the planar orlicz minkowski problem in the l1 sense abstract in this paper we solve l p minkowski problem for l 1 data and all p 0 and orlicz minkowski problem with two nonlinear terms in l 1 sense a byproduct is the blaschke santalo inequality which was previously established for only constant data and now is shown to hold for l 1 data

a tiling of phase space through self convolution a novel tiling for fourier half space is constructed using erlang functions this tiling is constructed by shifts in discrete time and self convolutions in the fourier domain we show that a subframe of this complex wavelet family can be used for the detection and characterization of events step and peaks in one dimensional 1 d discrete time signals

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

new families of fourier eigenfunctions for steerable filtering a new diadic family of eigenfunctions of the 2 d fourier transform has been discovered specifically new wavelets are derived by steering the elongated hermite gauss filters with respect to rotations thus obtaining a natural generalization of the laguerre gauss harmonics interestingly these functions are also proportional to their 2 d fourier transform their analytical expression is provided in a compact and treatable form by means of a new ad hoc matrix notation in which the cases of even and odd orders of the hermite polynomials are unified moreover these functions can be efficiently implemented by means of a recursive formula that is derived in this paper the proposed filters are applied to the problem of gradient estimation to improve the theoretical canny tradeoff of position accuracy versus noise rejection that occurs in edge detection experimental results show considerable improvements in using the new wavelets over both isotropic gaussian derivatives and other elongated steerable filters more recently introduced finally being the proposed wavelets a set of fourier eigenfunctions they can be of interest in other fields of science such as optics and quantum mechanics

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

primitive features by steering quadrature and scale the impulse response of neurons in the visual cortex of the mammalian brain has been known for some time how to make use of these as filters has led to many hypotheses the response of a single filter is ambiguous because the result depends on stimulus type contrast position orientation and scale we show that a set of quadrature filters at sparse positions can be constructed so that it is possible to disambiguate the 2d responses of the individual filters detecting edges is not the goal of the present work rather we seek to detect relevant edges thus we make the assumption that at the scale of interest a local image patch consists predominantly of an edge or a bar when this patch is processed by five or seven oriented filters one can compute the exact orientation and centroid position of the feature when the set of filters is applied at two different scales it is possible to distinguish edges from ridges and to identify the polarity intensity and width it is also possible to find corners and blobs these computations are stable under image shifts in position and orientation and can be made to subpixel resolution

image denoising by averaging of piecewise constant simulations of image partitions this paper investigates the problem of image denoising when the image is corrupted by additive white gaussian noise we herein propose a spatial adaptive denoising method which is based on an averaging process performed on a set of markov chain monte carlo simulations of region partition maps constrained to be spatially piecewise uniform i e constant in the grey level value sense for each estimated constant value regions for the estimation of these region partition maps we have adopted the unsupervised markovian framework in which parameters are automatically estimated in the least square sense this sequential averaging allows to obtain under our image model an approximation of the image to be recovered in the minimal mean square sense error the experiments reported in this paper demonstrate that the discussed method performs competitively and sometimes better than the best existing state of the art wavelet based denoising methods in benchmark tests

steerable weighted median filters a filter is steerable if transformed i e rotated scaled etc versions of its impulse response can be expressed as linear combinations of a fixed set of basis functions steerability is important for numerous image processing applications however it is a property presently shared only by a specific class of linear filters on the other hand several classes of nonlinear filters such as weighted median filters wmfs may offer certain advantages over linear filters such as robustness and edge preserving capabilities in this paper the concept of steerability is extended to encompass wmfs it will be shown that in general a steerable wmf design technique needs to be capable of handling negative weights although methods that allow the design of wmfs admitting negative weights have already been proposed such methods do not necessarily produce filters that are steerable as opposed to the approach presented in this work experimental results illustrate the applicability of steerable wmfs in two applications namely edge detection and orientation analysis

reunion of vicious walkers results from epsilon expansion the anomalous exponent eta p for the decay of the reunion probability of p vicious walkers each of length n in d 2 epsilon dimensions is shown to come from the multiplicative renormalization constant of a p directed polymer partition function using renormalization group rg we evaluate eta p to o epsilon 2 the survival probability exponent is eta p 2 for p 2 our rg is exact and eta p stops at o epsilon for d 2 the log corrections are also determined the number of walkers that are sure to reunite is 2 and has no epsilon expansion

on the _ minkowski problem a volume normalized formulation of the lp minkowski problem is presented this formulation has the advantage that a solution is possible for all p 1 including the degenerate case where the index p is equal to the dimension of the ambient space a new approach to the lp minkowski problem is presented which solves the volume normalized formulation for even data and all p 1

on the uniqueness of lp minkowski problems the constant p curvature case in r3 abstract we study the c 4 smooth convex bodies k r n 1 satisfying k x u x 1 p where x s n k is the gauss curvature of k u is the support function of k and p is a constant in the case of n 2 either when p 1 0 or when p 0 1 in addition to a pinching condition we show that k must be the unit ball this partially answers a conjecture of lutwak yang and zhang about the uniqueness of the l p minkowski problem in r 3 moreover we give an explicit pinching constant depending only on p when p 0 1

dual orlicz brunn minkowski theory abstract in this paper a dual orlicz brunn minkowski theory is presented an orlicz radial sum and dual orlicz mixed volumes are introduced the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality are established the variational formula for the volume with respect to the orlicz radial sum is proved the equivalence between the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality is demonstrated orlicz intersection bodies are defined and the orlicz busemann petty problem is posed

a new ellipsoid associated with convex bodies it is shown that corresponding to each convex body there is an ellipsoid that is in a sense dual to the legendre ellipsoid of classical mechanics sharp affine isoperimetric inequalities are obtained between the volume of the convex body and that of its corresponding new ellipsoid these inequalities provide exact bounds for the isotropic constant associated with the new ellipsoid among other things this leads to a new approach to establishing ball s maximal shadows conjecture for symmetric convex bodies corresponding to each origin symmetric convex or more general subset of euclidean n space r there is a unique ellipsoid with the following property the moment of inertia of the ellipsoid and the moment of inertia of the convex set is the same about every 1 dimensional subspace of r this ellipsoid is called the legendre ellipsoid of the convex set the legendre ellipsoid and its polar the binet ellipsoid are well known concepts from classical mechanics see milman and pajor mpa1 mpa2 lindenstrauss and milman lim and leichtweis le for some historical references it has slowly come to be recognized that along side the brunn minkowski theory there is a dual theory the nature of the duality between this dual brunn minkowski theory and the brunn minkowski theory is subtle and not yet understood it is easily seen that the legendre and binet ellipsoid is an object of this dual brunnminkowski theory this observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical legendre ellipsoid in the brunn minkowski theory it is the aim of this paper to demonstrate the existence of precisely this dual object in retrospect one may well wonder why research supported in part by nsf grant dms 9803261 typeset by ams tex 1 2 a new ellipsoid associated with convex bodies the new ellipsoid presented in this note was not discovered long ago the simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of l2 curvature in hand however the brunn minkowski theory was only recently extended to incorporate the new notion of lp curvature see l2 l3 a positive definite n n real symmetric matrix a generates an ellipsoid a in r defined by a x r x ax 1 where x ax denotes the standard inner product of x and ax in r associated with a star shaped about the origin set k r is its legedre ellipsoid 2k which is generated by the matrix mij k where mij k n 2 v k k ei x ej x dx with e1 en denoting the standard basis for r and v k denoting the ndimensional volume of k we will associate a new ellipsoid 2k with each convex body k r one approach to defining 2k without introducing new notation is to first define it for polytopes and then use approximation with respect to the hausdorff metric to extend the definition to all convex bodies suppose p r is a polytope that contains the origin in its interior let u1 un denote the outer unit normals to the faces of p let a1 an denote the areas i e n 1 dimensional volumes of the corresponding faces and let h1 hn denote the distances from the origin to the corresponding faces the ellipsoid 2p is generated by the matrix mij p where mij p 1 v p n l 1 al hl ei ul ej ul an alternate definition of the operator 2 will be given after additional notation is introduced the easily established affine nature of the operator 2 is formally stated in lemma 1 if k r is star shaped about the origin then for each gl n

rotationally symmetric solutions to the l p minkowski problem abstract in this paper we study the l p minkowski problem for p n 1 which corresponds to the critical exponent in the blaschke santalo inequality we first obtain volume estimates for general solutions then establish a priori estimates for rotationally symmetric solutions by using a kazdan warner type obstruction finally we give sufficient conditions for the existence of rotationally symmetric solutions by a blow up analysis we also include an existence result for the l p minkowski problem which corresponds to the super critical case of the blaschke santalo inequality

orlicz centroid bodies the sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body from below by the volume of the star body itself is the busemann petty centroid inequality a decade ago the lp analogue of the classical busemannpetty centroid inequality was proved here the definition of the centroid body is extended to an orlicz centroid body of a star body and the corresponding analogue of the busemann petty centroid inequality is established for convex bodies the centroids of the intersections of an origin symmetric body with half spaces form the surface of a convex body this centroid body is a concept that dates back at least to dupin the classical affine isoperimetric inequality that relates the volume of a convex body with that of its centroid body was conjectured by blaschke and established in a landmark work by petty 53 petty s inequality became known as the busemann petty centroid inequality because in establishing his inequality petty not only made critical use of busemann s random simplex inequality but as petty stated he reinterpreted it see e g the books by gardner 12 leichtweiss 25 schneider 55 and thompson 58 for reference the concept of a centroid body had a natural extension in what became known as the lp brunn minkowski theory and its dual this theory had its origins in the early 1960s when firey see e g schneider 55 introduced his concept of lp compositions of convex bodies three decades later in 34 and 35 these firey minkowski lp combinations were shown to lead to an embryonic lp brunn minkowski theory this theory and its dual has witnessed a rapid growth see e g 1 9 17 23 26 32 34 44 46 48 54 56 57 59 62 the lp analogues of centroid bodies became a central focus within the lp brunn minkowski theory and its dual and establishing the lpanalogue of the busemann petty centroid inequality became a major goal this was accomplished by the authors of the present paper in 37 with an independent approach presented by campi and gronchi 3 the lp centroid bodies quickly became objects of interest in asymptotic mathematics subject classification mr 52a40 research supported in part by nsf grant dms 0706859 proof copy 1 not for distribution 2 erwin lutwak deane yang and gaoyong zhang geometric analysis see e g 10 11 24 49 50 51 52 and even the theory of stable distributions see e g 48 using concepts introduced by ludwig 29 haberl and schuster 21 were led to establish asymmetric versions of the lp busemann petty centroid inequality that for bodies that are not origin symmetric are stronger than the lp busemann petty centroid inequality presented in 37 and 3 the asymmetric inequalities obtained by haberl and schuster are ideally suited for non symmetric bodies this can be seen by looking at the haberl schuster version of the lp analogue of the classical blaschke santalo inequality that was presented in 46 while for origin symmetric bodies the lp extension of 46 does recover the original blaschke santalo inequality as p for arbitrary bodies only the haberl schuster version does so the works of haberl and schuster 21 see also 22 ludwig and reitzner 32 and ludwig 31 have demonstrated the clear need to move beyond the lp brunn minkowski theory to what we are calling an orlicz brunn minkowski theory this need is not only motivated by compelling geometric considerations such as those presented in ludwig and reitzner 32 but also by the desire to obtain sobolev bounds see 22 of a far more general nature this paper is the second in a series intended to develop a few of the elements of an orlicz brunn minkowski theory and its dual here we define the orlicz centroid body establish some of its basic properties and most importantly establish what we call the orlicz busemannpetty centroid inequality for orlicz centroid bodies we consider convex r 0 such that 0 0 this means that must be decreasing on 0 and increasing on 0 we require that one of these is happening strictly so i e is either strictly decreasing on 0 or strictly increasing on 0 the class of such will be denoted by c if k is a star body see section 1 for precise definitions with respect to the origin in rn with volume k and c then we define the orlicz centroid body k of k as the convex body whose support function at x rn is given by h k x inf 0 1 k k x y dy 1 where x y denotes the standard inner product of x and y in rn and the integration is with respect to lebesgue measure in rn when p t t p with p 1 then

survival probability of a diffusing particle in the presence of poisson distributed mobile traps the problem of a diffusing particle moving among diffusing traps is analyzed in general space dimension d we consider the case where the traps are initially randomly distributed in space with uniform density ensuremath rho and derive upper and lower bounds for the probability q t averaged over all particle and trap trajectories that the particle survives up to time t we show that for 1l dl 2 the bounds converge asymptotically to give q t ensuremath sim mathrm exp ensuremath ensuremath lambda d t d 2 for 1l dl2 where ensuremath lambda d 2 ensuremath pi d mathrm sin ensuremath pi d 2 4 ensuremath pi d d 2 ensuremath rho and d is the diffusion constant of the traps and that q t ensuremath sim mathrm exp ensuremath 4 ensuremath pi ensuremath rho dt mathrm ln t for d 2 for dg2 bounds can still be derived but they no longer converge for large t for 1l dl 2 these asymptotic forms are independent of the diffusion constant of the particle the results are compared with simulation results obtained using a new algorithm v mehra and p grassberger phys rev e 65 050101 2002 which is described in detail deviations from the predicted asymptotic forms are found to be large even for very small values of q t indicating slowly decaying corrections whose form is consistent with the bounds we also present results in d 1 for the case where the trap densities on either side of the particle are different for this case we can still obtain exact bounds but they no longer converge

the logarithmic minkowski problem for polytopes abstract the logarithmic minkowski problem asks for necessary and sufficient conditions for a finite borel measure on the unit sphere so that it is the cone volume measure of a convex body this problem was solved recently by boroczky lutwak yang and zhang for even measures boroczky et al 2013 8 this paper solves the case of discrete measures whose supports are in general position

bivariate shrinkage functions for wavelet based denoising exploiting interscale dependency most simple nonlinear thresholding rules for wavelet based denoising assume that the wavelet coefficients are independent however wavelet coefficients of natural images have significant dependencies we only consider the dependencies between the coefficients and their parents in detail for this purpose new non gaussian bivariate distributions are proposed and corresponding nonlinear threshold functions shrinkage functions are derived from the models using bayesian estimation theory the new shrinkage functions do not assume the independence of wavelet coefficients we show three image denoising examples in order to show the performance of these new bivariate shrinkage rules in the second example a simple subband dependent data driven image denoising system is described and compared with effective data driven techniques in the literature namely visushrink sureshrink bayesshrink and hidden markov models in the third example the same idea is applied to the dual tree complex wavelet coefficients

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

bayesian wavelet based image denoising using the gauss hermite expansion the probability density functions pdfs of the wavelet coefficients play a key role in many wavelet based image processing algorithms such as denoising the conventional pdfs usually have a limited number of parameters that are calculated from the first few moments only consequently such pdfs cannot be made to fit very well with the empirical pdf of the wavelet coefficients of an image as a result the shrinkage function utilizing any of these density functions provides a substandard denoising performance in order for the probabilistic model of the image wavelet coefficients to be able to incorporate an appropriate number of parameters that are dependent on the higher order moments a pdf using a series expansion in terms of the hermite polynomials that are orthogonal with respect to the standard gaussian weight function is introduced a modification in the series function is introduced so that only a finite number of terms can be used to model the image wavelet coefficients ensuring at the same time the resulting pdf to be non negative it is shown that the proposed pdf matches the empirical one better than some of the standard ones such as the generalized gaussian or bessel k form pdf a bayesian image denoising technique is then proposed wherein the new pdf is exploited to statistically model the subband as well as the local neighboring image wavelet coefficients experimental results on several test images demonstrate that the proposed denoising method both in the subband adaptive and locally adaptive conditions provides a performance better than that of most of the methods that use pdfs with limited number of parameters

the orlicz brunn minkowski inequality abstract the orlicz brunn minkowski theory originated with the work of lutwak yang and zhang in 2010 in this paper we first introduce the orlicz addition of convex bodies containing the origin in their interiors and then extend the l p brunn minkowski inequality to the orlicz brunn minkowski inequality furthermore we extend the l p minkowski mixed volume inequality to the orlicz mixed volume inequality by using the orlicz brunn minkowski inequality

orlicz john ellipsoids abstract the orlicz john ellipsoids which are in the framework of the booming orlicz brunn minkowski theory are introduced for the first time it turns out that they are generalizations of the classical john ellipsoid and the evolved l p john ellipsoids the analog of ball s volume ratio inequality is established for the new orlicz john ellipsoids the connection between the isotropy of measures and the characterization of orlicz john ellipsoids is demonstrated

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

lp john ellipsoids it is shown that the classical john ellipsoid the petty ellipsoid and a recently discovered dual of the legendre ellipsoid are all special cases p infty p 1 and p 2 of a family of l p ellipsoids which can be associated with a fixed convex body this insight allows for a unified view of alternative approaches to and extensions of some basic results in convex geometry

existence of solutions to the orlicz minkowski problem abstract in this paper the orlicz minkowski problem a generalization of the classical minkowski problem is studied using the variational method we obtain a new existence result of solutions to this problem for general measures

adaptive wavelet thresholding for image denoising and compression the first part of this paper proposes an adaptive data driven threshold for image denoising via wavelet soft thresholding the threshold is derived in a bayesian framework and the prior used on the wavelet coefficients is the generalized gaussian distribution ggd widely used in image processing applications the proposed threshold is simple and closed form and it is adaptive to each subband because it depends on data driven estimates of the parameters experimental results show that the proposed method called bayesshrink is typically within 5 of the mse of the best soft thresholding benchmark with the image assumed known it also outperforms sureshrink donoho and johnstone 1994 1995 donoho 1995 most of the time the second part of the paper attempts to further validate claims that lossy compression can be used for denoising the bayesshrink threshold can aid in the parameter selection of a coder designed with the intention of denoising and thus achieving simultaneous denoising and compression specifically the zero zone in the quantization step of compression is analogous to the threshold value in the thresholding function the remaining coder design parameters are chosen based on a criterion derived from rissanen s minimum description length mdl principle experiments show that this compression method does indeed remove noise significantly especially for large noise power however it introduces quantization noise and should be used only if bitrate were an additional concern to denoising

shiftable multiscale transforms one of the major drawbacks of orthogonal wavelet transforms is their lack of translation invariance the content of wavelet subbands is unstable under translations of the input signal wavelet transforms are also unstable with respect to dilations of the input signal and in two dimensions rotations of the input signal the authors formalize these problems by defining a type of translation invariance called shiftability in the spatial domain shiftability corresponds to a lack of aliasing thus the conditions under which the property holds are specified by the sampling theorem shiftability may also be applied in the context of other domains particularly orientation and scale jointly shiftable transforms that are simultaneously shiftable in more than one domain are explored two examples of jointly shiftable transforms are designed and implemented a 1 d transform that is jointly shiftable in position and scale and a 2 d transform that is jointly shiftable in position and orientation the usefulness of these image representations for scale space analysis stereo disparity measurement and image enhancement is demonstrated

wavelet based statistical signal processing using hidden markov models wavelet based statistical signal processing techniques such as denoising and detection typically model the wavelet coefficients as independent or jointly gaussian these models are unrealistic for many real world signals we develop a new framework for statistical signal processing based on wavelet domain hidden markov models hmms that concisely models the statistical dependencies and non gaussian statistics encountered in real world signals wavelet domain hmms are designed with the intrinsic properties of the wavelet transform in mind and provide powerful yet tractable probabilistic signal models efficient expectation maximization algorithms are developed for fitting the hmms to observational signal data the new framework is suitable for a wide range of applications including signal estimation detection classification prediction and even synthesis to demonstrate the utility of wavelet domain hmms we develop novel algorithms for signal denoising classification and detection

deformable kernels for early vision early vision algorithms often have a first stage of linear filtering that extracts from the image information at multiple scales of resolution and multiple orientations a common difficulty in the design and implementation of such schemes is that one feels compelled to discretize coarsely the space of scales and orientations in order to reduce computation and storage costs a technique is presented that allows 1 computing the best approximation of a given family using linear combinations of a small number of basis functions and 2 describing all finite dimensional families i e the families of filters for which a finite dimensional representation is possible with no error the technique is based on singular value decomposition and may be applied to generating filters in arbitrary dimensions and subject to arbitrary deformations the relevant functional analysis results are reviewed and precise conditions for the decomposition to be feasible are stated experimental results are presented that demonstrate the applicability of the technique to generating multiorientation multi scale 2d edge detection kernels the implementation issues are also discussed

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

fuzzy density weight based support vector regression for image denoising support vector machine svm is a popular machine learning technique and its variant least squares support vector regression ls svr is effective for image denoising however conventional ls svr does not fully consider the sampling distribution of noisy images which may degrade the performance of the algorithm in this paper we propose a new fuzzy density weight svr fdw svr denoising algorithm which assigns fuzzy priority to each sample according to its density weight fdw is designed to estimate the joint probability density function via the fuzzy theory based on the pixel density and neighborhood density extensive experimental results show that fdw svr is superior to those state of the art denoising techniques in light of both subjective and objective evaluations

trapping of a random walk by diffusing traps we present a systematic analytical approach to the trapping of a random walk by a finite density rho of diffusing traps in arbitrary dimension d we confirm the phenomenologically predicted e c d rho t d 2 time decay of the survival probability and compute the dimension dependent constant c d to leading order within an eps 2 d expansion

orlicz projection bodies abstract minkowski s projection bodies have evolved into l p projection bodies and their asymmetric analogs these all turn out to be part of a far larger class of orlicz projection bodies the analog of the classical petty projection inequality is established for the new orlicz projection bodies

trapping of random walks on the line several features of the trapping of random walks on a one dimensional lattice are analyzed the results of this investigation are as follows 1 the correction term to the known asymptotic form for the survival probability ton steps is o 2n 1 3 where ln 1 c andc is the trap concentration 2 the short time form for the survival probability is found to be exp a c n1 2 wherea c is given in eq 21 3 the mean square displacement of a surviving random walker is found to go liken2 3for largen 4 when the distribution of trap free regions is changed so that very large regions are much rarer than for ideally random trap placement the asymptotic survival probability changes its dependence onn one such model is studied

nonuniqueness of solutions to the l p minkowski problem abstract in this paper we show by example that there is no uniform estimate for the l p minkowski problem as a result we obtain the nonuniqueness of solutions to the problem

efficient steerable circular harmonic functions two dimensional 2d filtering and pattern matching applications often involve processing of images using a set sf which includes several transformed e g rotated scaled warped versions of the same 2d function f depending on the application f may correspond to a filter kernel or a pattern in general image processing using all functions in sf poses a computational burden steerability offers a solution to the problem specifically each function in sf can be exactly expressed or closely approximated by a linear combination of a few basis or steering functions in such cases processing images by employing steering functions and obtaining linear combinations of the associated responses may be preferable over utilizing all functions in sf however employing the steering functions is still a computationally expensive process this paper presents a method to significantly improve the computational efficiency of circular harmonic functions which are the steering functions when rotation is the transformation of interest the method is presented mainly in the context of image filtering yet 2d pattern matching is also discussed the advantages of the proposed technique in relation to other techniques are presented and its efficiency is demonstrated in terms of execution time and number of operations an extension of the technique to more general cases where sf includes rotated but also otherwise transformed versions of f is also discussed highlightsvery efficient circular harmonic function chf steerable filter technique presented chf implementation several times faster than direct convolution and fft conditionally technique provides exact results as direct convolution comparisons presented for pattern matching orientation analysis and denoising