3 d shape estimation of dna molecules from stereo cryo electron micro graphs using a projection steerable snake we introduce a three dimensional 3 d parametric active contour algorithm for the shape estimation of dna molecules from stereo cryo electron micrographs we estimate the shape by matching the projections of a 3 d global shape model with the micrographs we choose the global model as a 3 d filament with a b spline skeleton and a specified radial profile the active contour algorithm iteratively updates the b spline coefficients which requires us to evaluate the projections and match them with the micrographs at every iteration since the evaluation of the projections of the global model is computationally expensive we propose a fast algorithm based on locally approximating it by elongated blob like templates we introduce the concept of projection steerability and derive a projection steerable elongated template since the two dimensional projections of such a blob at any 3 d orientation can be expressed as a linear combination of a few basis functions matching the projections of such a 3 d template involves evaluating a weighted sum of inner products between the basis functions and the micrographs the weights are simple functions of the 3 d orientation and the inner products are evaluated efficiently by separable filtering we choose an internal energy term that penalizes the average curvature magnitude since the exact length of the dna molecule is known a priori we introduce a constraint energy term that forces the curve to have this specified length the sum of these energies along with the image energy derived from the matching process is minimized using the conjugate gradients algorithm we validate the algorithm using real as well as simulated data and show that it performs well

design of multiparameter steerable functions using cascade basis reduction an efficient method of computing the least squares optimal basis functions to steer any function locally is presented the method combines the lie group theoretic and the singular value decomposition approaches its efficiency is demonstrated with the design of basis functions to steer a gabor function under the four parameter linear transformation group

on the radius of injectivity of null hypersurfaces we investigate the regularity of past future boundaries of points in regular einstein vacuum spacetimes we provide conditions expressed in terms of a space like foliation and which imply in particular uniform bounds for the curvature tensor sufficient to ensure the local nondegeneracy of these boundaries more precisely we provide a uniform lower bound on the radius of injectivity of the null boundaries of the causal past future sets such lower bounds are essential in understanding the causal structure and the related propagation properties of solutions to the einstein equations they are particularly important in construction of an effective kirchoff sobolev type parametrix for solutions of wave equations on such parametrices are used by the authors in a forthcoming paper to prove a large data break down criterion for solutions of the einstein vacuum equations

fast algorithms for low level vision a recursive filtering structure is proposed that drastically reduces the computational effort required for smoothing performing the first and second directional derivatives and carrying out the laplacian of an image these operations are done with a fixed number of multiplications and additions per output point independently of the size of the neighborhood considered the key to the approach is first the use of an exponentially based filter family and second the use of the recursive filtering applications to edge detection problems and multiresolution techniques are considered and an edge detector allowing the extraction of zero crossings of an image with only 14 operations per output element at any resolution is proposed various experimental results are shown

a generalized representation formula for systems of tensor wave equations in this paper we generalize the kirchhoff sobolev parametrix of klainerman and rodnianski hyperbolic equ 4 3 401 433 2007 to systems of tensor wave equations with additional first order terms we also present a different derivation which better highlights that such representation formulas are supported entirely on past null cones this generalization of hyperbolic equ 4 3 401 433 2007 is a key component for extending klainerman and rodnianski s breakdown criterion result for einstein vacuum spacetimes in j amer math soc 23 2 345 382 2009 to einstein maxwell and einstein yang mills spacetimes

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

a geometric perspective on the method of descent we derive a first order representation formula for the tensorial wave equation box mathbf g phi i f i in globally hyperbolic lorentzian spacetimes mathcal m 2 1 mathbf g by giving a geometric formulation of the method of descent which is applicable for any dimension

spatially variant convolution with scaled b splines we present an efficient algorithm to compute multidimensional spatially variant convolutions or inner products between n dimensional signals and b splines or their derivatives of any order and arbitrary sizes the multidimensional b splines are computed as tensor products of 1 d b splines and the input signal is expressed in a b spline basis the convolution is then computed by using an adequate combination of integration and scaled finite differences as to have for moderate and large scale values a computational complexity that does not depend on the scaling factor to show in practice the benefit of using our spatially variant convolution approach we present an adaptive noise filter that adjusts the kernel size to the local image characteristics and a high sensitivity local ridge detector

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

maximum likelihood orientation estimation of 1 d patterns in laguerre gauss subspaces a method for measuring the orientation of linear 1 d patterns based on a local expansion with laguerre gauss circular harmonic lg ch functions is presented it lies on the property that the polar separable lg ch functions span the same space as the 2 d cartesian separable hermite gauss 2 d hg functions exploiting the simple steerability of the lg ch functions and the peculiar block linear relationship among the two expansion coefficients sets maximum likelihood ml estimates of orientation and cross section parameters of 1 d patterns are obtained projecting them in a proper subspace of the 2 d hg family it is shown in this paper that the conditional ml solution derived by elimination of the cross section parameters surprisingly yields the same asymptotic accuracy as the ml solution for known cross section parameters the accuracy of the conditional ml estimator is compared to the one of state of art solutions on a theoretical basis and via simulation trials a thorough proof of the key relationship between the lg ch and the 2 d hg expansions is also provided

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

spaces of surface group representations let gamma g denote the fundamental group of a closed surface of genus g ge 2 we show that every geometric representation of gamma g into the group of orientation preserving homeomorphisms of the circle is rigid meaning that its deformations form a single semi conjugacy class as a consequence we give a new lower bound on the number of topological components of the space of representations of gamma g into mathrm homeo s 1 precisely for each nontrivial divisor k of 2g 2 there are at least k 2g 1 components containing representations with euler number frac 2g 2 k our methods apply to representations of surface groups into finite covers of mathrm psl 2 mathbb r and into mathrm diff s 1 as well in which case we recover theorems of w goldman and j bowden the key technique is an investigation of stability phenomena for rotation numbers of products of circle homeomorphisms using techniques of calegari walker this is a new approach to studying deformation classes of group actions on the circle and may be of independent interest

on the breakdown criterion in general relativity we give a geometric criterion for the breakdown of an einstein vacuum space time foliated by a constant mean curvature or maximal foli ation more precisely we show that the foliated space time can be extended as long as the the second fundamental form and the first derivatives of the logarithm of the lapse of the foliation remain uniformly bounded we make no restrictions on the size of the initial data

a computational approach for corner and vertex detection corners and vertexes are strong and useful features in computer vision for scene analysis stereo matching and motion analysis here we deal with the development of a computational approach to these important features we consider first a corner model and study analytically its behavior once it has been smoothed using the well known gaussian filter this allows us to clarify the behavior of some well knowncornerness measure based approaches used to detect these points of interest most of these classical approaches appear to detect points that do not correspond to the exact position of the corner a new scale space based approach that combines useful properties from the laplacian and beaudet s measure beaudet 1978 is then proposed in order to correct and detect exactly the corner position an extension of this approach is then developed to solve the problem of trihedral vertex characterization and detection in particular it is shown that a trihedral vertex has two elliptic maxima on extremal contrast surfaces if the contrast is sufficient and this allows us to classify trihedral vertexes in 2 classes vertex and vertex as corner the corner detection approach developed is applied to accurately detect trihedral vertexes using an additional test in order to make a distinction between trihedral vertexes and corners many experiments have been carried out using noisy synthetic data and real images containing corners and vertexes most of the promising results obtained are used to illustrate the experimental section of this paper

the fourier argand representation an optimal basis of steerable patterns computing the convolution between a 2d signal and a corresponding filter with variable orientations is a basic problem that arises in various tasks ranging from low level image processing e g ridge edge detection to high level computer vision e g pattern recognition through decades of research there still lacks an efficient method for solving this problem in this paper we investigate this problem from the perspective of approximation by considering the following problem what is the optimal basis for approximating all rotated versions of a given bivariate function surprisingly solely minimising the l 2 approximation error leads to a rotation covariant linear expansion which we name fourier argand representation this representation presents two major advantages 1 rotation covariance of the basis which implies a strong steerability rotating by an angle alpha corresponds to multiplying each basis function by a complex scalar e ik alpha 2 optimality of the fourier argand basis which ensures a few number of basis functions suffice to accurately approximate complicated patterns and highly direction selective filters we show the relation between the fourier argand representation and the radon transform leading to an efficient implementation of the decomposition for digital filters we also show how to retrieve accurate orientation of local structures patterns using a fast frequency estimation algorithm

global minimum for a finsler elastica minimal path approach in this paper we propose a novel curvature penalized minimal path model via an orientation lifted finsler metric and the euler elastica curve the original minimal path model computes the globally minimal geodesic by solving an eikonal partial differential equation pde essentially this first order model is unable to penalize curvature which is related to the path rigidity property in the classical active contour models to solve this problem we present an eikonal pde based finsler elastica minimal path approach to address the curvature penalized geodesic energy minimization problem we were successful at adding the curvature penalization to the classical geodesic energy caselles et al in int j comput vis 22 1 61 79 1997 cohen and kimmel in int j comput vis 24 1 57 78 1997 the basic idea of this work is to interpret the euler elastica bending energy via a novel finsler elastica metric that embeds a curvature penalty this metric is non riemannian anisotropic and asymmetric and is defined over an orientation lifted space by adding to the image domain the orientation as an extra space dimension based on this orientation lifting the proposed minimal path model can benefit from both the curvature and orientation of the paths thanks to the fast marching method the global minimum of the curvature penalized geodesic energy can be computed efficiently we introduce two anisotropic image data driven speed functions that are computed by steerable filters based on these orientation dependent speed functions we can apply the proposed finsler elastica minimal path model to the applications of closed contour detection perceptual grouping and tubular structure extraction numerical experiments on both synthetic and real images show that these applications of the proposed model indeed obtain promising results

steerable wedge filters for local orientation analysis steerable filters have been used to analyze local orientation patterns in imagery such filters are typically based on directional derivatives whose symmetry produces orientation responses that are periodic with period spl pi independent of image structure we present a more general set of steerable filters that alleviate this problem

corner detection and classification using anisotropic directional derivative representations this paper proposes a corner detector and classifier using anisotropic directional derivative andd representations the andd representation at a pixel is a function of the oriented angle and characterizes the local directional grayscale variation around the pixel the proposed corner detector fuses the ideas of the contour and intensity based detection it consists of three cascaded blocks first the edge map of an image is obtained by the canny detector and from which contours are extracted and patched next the andd representation at each pixel on contours is calculated and normalized by its maximal magnitude the area surrounded by the normalized andd representation forms a new corner measure finally the nonmaximum suppression and thresholding are operated on each contour to find corners in terms of the corner measure moreover a corner classifier based on the peak number of the andd representation is given experiments are made to evaluate the proposed detector and classifier the proposed detector is competitive with the two recent state of the art corner detectors the he yung detector and cpda detector in detection capability and attains higher repeatability under affine transforms the proposed classifier can discriminate effectively simple corners y type corners and higher order corners

consistency analysis on orientation features for fast and accurate palmprint identification a typical palmprint identification system identifies a query palmprint by matching it with the templates in the database and comparing the similarity score with a pre defined threshold however continual database expansion would make conventional brute force search method inefficient for large scale palmprint identification in this paper we propose a consistent orientation pattern cop hashing method to enforce fast search using the orientation and response features extracted by steerable filter we first give an analysis on the consistency of orientation features and then introduce a method to construct cop using the consistent features because the cop is very stable across the samples of the same subject the cop hashing method can find the target template quickly and thus can lead to early termination of the searching process experiments have been carried out on several real and synthetic databases and the results show that the cop hashing method can accelerate the identification process by more than an order of magnitude and can achieve comparable identification accuracy while compared with the state of the art approaches

complex derivative filters steerable filters are a valuable tool for various low level vision tasks in this paper we argue for the use of complex analysis in the context of 2 d steerable filters in particular we recommend the use of complex partial derivatives as a computational basis complex derivatives have a major advantage in comparison to real derivatives they show a canonical rotation behavior namely a rotation affects the derivative just by a multiplication with a complex unit number so the complex derivatives can be steered in a more elegant way and above that they are less expensive to compute we present several analytical formulas for common and new filter kernels in terms of complex derivatives further we relate the complex derivatives of a gaussian with the gauss laguerre transform and show that the gauss laguerre functions provide an optimal signal representation for local and smooth images we discuss various finite difference schemes for the realization of the derivatives and use them in practice in a first experiment we use a newly introduced filter kernel for anisotropic blurring the complex formalism offers an elegant way to locally adapt the shape and orientation of the kernel second we use the proposed filters as matched filters to detect vessels in retinal images

analysis of superimposed oriented patterns estimation of local orientation in images may be posed as the problem of finding the minimum gray level variance axis in a local neighborhood in bivariate images the solution is given by the eigenvector corresponding to the smaller eigenvalue of a 2times2 tensor for an ideal single orientation the tensor is rank deficient i e the smaller eigenvalue vanishes a large minimal eigenvalue signals the presence of more than one local orientation what may be caused by non opaque additive or opaque occluding objects crossings bifurcations or corners we describe a framework for estimating such superimposed orientations our analysis is based on the eigensystem analysis of suitably extended tensors for both additive and occluding superpositions unlike in the single orientation case the eigensystem analysis does not directly yield the orientations rather it provides so called mixed orientation parameters mops we therefore show how to decompose the mops into the individual orientations we also show how to use tensor invariants to increase efficiency and derive a new feature for describing local neighborhoods which is invariant to rigid transformations applications are e g in texture analysis directional filtering and interpolation feature extraction for corners and crossings tracking and signal separation

counterexamples to local existence for semi linear wave equations in this paper we study how much regularity of initial data is needed to ensure existence of a local solution to a semi linear wave equation we give counterexamples to local existence for the typical model equations the counterexamples we construct are sharp i e one does have a local solution if the data has slightly more regularity

a multiresolution approach for texture synthesis using the circular harmonic functions in this paper an unsupervised model based texture reproduction technique is described in accordance with the julesz s 1962 conjecture the statistical properties of the prototype up to the second order are copied in order to generate a synthetic texture perceptually indistinguishable from the given sample however this task is accomplished using a hybrid approach which operates partially in the spatial domain and partially in a multiresolution domain the latter employed is the circular harmonic function chf domain since it has been proven to be well suited for mimicking the behavior of the human visual system hvs this approach allows for a wide range of textures typologies obtaining synthetic textures that better match the prototype with respect to the ones obtained using techniques based on the julesz s conjecture operating only in the spatial domain and to dramatically reduce the computational complexity of similar methods operating only in the multiresolution domain

a high performance edge detector based on fuzzy inference rules edge detection is an important topic in computer vision and image processing in this paper a novel edge detector based on fuzzy if then inference rules and edge continuity is proposed the fuzzy if then rule system is designed to model edge continuity criteria the maximum entropy principle is used in the parameter adjusting process we also discuss the related issues in designing fuzzy edge detectors we compare it with the popular edge detectors sobel and canny edge detectors the proposed fuzzy edge detector does not need parameter setting as canny edge detector does and it can preserve an appropriate detection in details it is very robust to noise and can work well under high level noise situations while other edge detectors cannot the detector efficiently extracts edges in images corrupted by noise without requiring the filtering process the experimental results demonstrate the superiority of the proposed method to existing ones

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

efficient energies and algorithms for parametric snakes parametric active contour models are one of the preferred approaches for image segmentation because of their computational efficiency and simplicity however they have a few drawbacks which limit their performance in this paper we identify some of these problems and propose efficient solutions to get around them the widely used gradient magnitude based energy is parameter dependent its use will negatively affect the parametrization of the curve and consequently its stiffness hence we introduce a new edge based energy that is independent of the parameterization it is also more robust since it takes into account the gradient direction as well we express this energy term as a surface integral thus unifying it naturally with the region based schemes the unified framework enables the user to tune the image energy to the application at hand we show that parametric snakes can guarantee low curvature curves but only if they are described in the curvilinear abscissa since normal curve evolution do not ensure constant arc length we propose a new internal energy term that will force this configuration the curve evolution can sometimes give rise to closed loops in the contour which will adversely interfere with the optimization algorithm we propose a curve evolution scheme that prevents this condition

causal geometry of einstein vacuum spacetimes with finite curvature flux one of the central difficulties of settling the l2 bounded curvature conjecture for the einstein vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity in this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of enstein vacuum spacetimes can be controlled in terms of initial data and the total curvature flux through the hypersurface

efficient shape priors for spline based snakes parametric active contours are an attractive approach for image segmentation thanks to their computational efficiency they are driven by application dependent energies that reflect the prior knowledge on the object to be segmented we propose an energy involving shape priors acting in a regularization like manner thereby the shape of the snake is orthogonally projected onto the space that spans the affine transformations of a given shape prior the formulation of the curves is continuous which provides computational benefits when compared with landmark based discrete methods we show that this approach improves the robustness and quality of spline based segmentation algorithms while its computational overhead is negligible an interactive and ready to use implementation of the proposed algorithm is available and was successfully tested on real data in order to segment drosophila flies and yeast cells in microscopic images

combining local filtering and multiscale analysis for edge ridge and curvilinear objects detection this paper presents a general method for detecting curvilinear structures like filaments or edges in noisy images this method relies on a novel technique the feature adapted beamlet transform fabt which is the main contribution of this paper it combines the well known beamlet transform bt introduced by donoho with local filtering techniques in order to improve both detection performance and accuracy of the bt moreover as the desired feature detector is chosen to belong to the class of steerable filters our transform requires only o n log n operations where n n 2 is the number of pixels besides providing a fast implementation of the fabt on discrete grids we present a statistically controlled method for curvilinear objects detection to extract significant objects we propose an algorithm in four steps 1 compute the fabt 2 normalize beamlet coefficients 3 select meaningful beamlets thanks to a fast energy based minimization and 4 link beamlets together in order to get a list of objects we present an evaluation on both synthetic and real data and demonstrate substantial improvements of our method over classical feature detectors

equivariant holomorphic filters for contour denoising and rapid object detection it is well known that linear filters are not powerful enough for many low level image processing tasks however it is also very difficult to design robust nonlinear filters that respond exclusively to features of interest and that are at the same time equivariant with respect to translation and rotation this paper proposes a new class of rotation equivariant nonlinear filters that is based on the principle of group integration these filters become efficiently computable by an iterative scheme based on repeated differentiation of products and summations of intermediate results the relations of the proposed approach to volterra filters and steerable filters are shown in the context of detection problems the filter may be interpreted as some kind of generalized hough transform the experiments show that the new filter can be used for enhancing noisy contours and rapid object detection in microscopical images in the detection context our experiments show that the proposed filter is definitely superior to alternative approaches when high localization accuracy is required

boundary harmonic coordinates on manifolds with boundary in low regularity in this paper we prove the existence of h 2 regular coordinates on riemannian 3 manifolds with boundary assuming only l 2 bounds on the ricci curvature l 4 bounds on the second fundamental form of the boundary and a positive lower bound on the volume radius the proof follows by extending the theory of cheeger gromov convergence to include manifolds with boundary in the above low regularity setting the main tools are boundary harmonic coordinates together with elliptic estimates and a geometric trace estimate and a rigidity argument using manifold doubling assuming higher regularity of the ricci curvature we also prove corresponding higher regularity estimates for the coordinates

a nonlinear derivative scheme applied to edge detection this paper presents a nonlinear derivative approach to addressing the problem of discrete edge detection this edge detection scheme is based on the nonlinear combination of two polarized derivatives its main property is a favorable signal to noise ratio snr at a very low computation cost and without any regularization a 2d extension of the method is presented and the benefits of the 2d localization are discussed the performance of the localization and snr are compared to that obtained using classical edge detection schemes tests of the regularized versions and a theoretical estimation of the snr improvement complete this work

accurate image rotation using hermite expansions in this paper we propose an approach for the accurate rotation of a digital image using hermite expansions this exploits the fact that if a 2 d continuous bandlimited hermite expansion is rotated the resulting function can be expressed as a hermite expansion with the same bandlimit furthermore the hermite coefficients of the initial 2 d expansion and the rotated expansion are mapped through an invertible linear relationship two efficient methods to compute the mapping between hermite coefficients during rotation are proposed we also propose a method for connecting the hermite expansion and a discrete image using this method we can obtain the hermite expansion from a discrete image and vice versa combining these techniques we propose new methods for the rotation of discrete images we assess the accuracy of our methods and compare them with an existing fft based method implementing three shears we find that the method proposed here consistently has better accuracy than the fft based method

derivative kernels numerics and applications a generalized framework for numerical differentiation nd is proposed for constructing a finite impulse response fir filter in closed form the framework regulates the frequency response of nd filters for arbitrary derivative order and cutoff frequency selected parameters relying on interpolating power polynomials and maximally flat design techniques compared with the state of the art solutions such as gaussian kernels the proposed nd filter is sharply localized in the fourier domain with ripple free artifacts here we construct 2d maxflat kernels for image directional differentiation to calculate image differentials for arbitrary derivative order cutoff level and steering angle the resulted kernel library renders a new solution capable of delivering discrete approximation of gradients hessian and higher order tensors in numerous applications we tested the utility of this library on three different imaging applications with main focus on the unsharp masking the reported results highlight the high efficiency of the 2d maxflat kernel and its versatility with respect to robustness and parameter control accuracy

local propagation of impulsive gravitationalwaves in this paper we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves we construct such spacetimes as solutions to the characteristic initial value problem of the einstein vacuum equations with a data curvature delta singularity we show that in the resulting spacetime the delta singularity propagates along a characteristic hypersurface while away from that hypersurface the spacetime remains smooth unlike the known explicit examples of impulsive gravitational spacetimes this work in particular provides the first construction of an impulsive gravitational wave of compact extent and does not require any symmetry assumptions the arguments in the present paper also extend to the problem of existence and uniqueness of solutions to a larger class of nonregular characteristic data 2015 wiley periodicals inc

on optimal linear filtering for edge detection in this paper we revisit the analytical expressions of the three canny s 1983 criteria for edge detection quality good detection good localization and low multiplicity of false detections our work differs from canny s work on two essential points here the criteria are given for discrete sampled signals i e for the real implemented filters instead of a single step edge as input signal we use pulses of various width the proximity of other edges affects the quality of the detection process this is taken into account in the new expressions of these criteria we derive optimal filters for each of the criteria and for any combination of them in particular we define an original filter which maximizes detection and localization and a simple approximation of the optimal filter for the simultaneous maximization of the three criteria the upper bounds of the criteria are computed which allow users to measure the absolute and relative performance of any filter exponential deriche 1987 and first derivative of gaussian filters are evaluated our criteria can also be used to compute the optimal value of the scale parameter of a given filter when the resolution of the detection is fixed

hermite snakes with control of tangents we introduce a new model of parametric contours defined in a continuous fashion our curve model relies on hermite spline interpolation and can easily generate curves with sharp discontinuities it also grants direct access to the tangent at each location with these two features the hermite snake distinguishes itself from classical spline snake models and allows one to address certain bioimaging problems in a more efficient way more precisely the hermite snake construction allows introducing sharp corners in the snake curve and designing directional energy functionals relying on local orientation information in the input image using the formalism of spline theory the model is shown to meet practical requirements such as invariance to affine transformations and good approximation properties finally the dependence on initial conditions and the robustness to the noise is studied on synthetic data in order to validate our hermite snake model and its usefulness is illustrated on real biological images acquired using brightfield phase contrast differential interference contrast and scanning electron microscopy

efficient deformable filter banks this article describes efficient schemes for the computation of a large number of differently scaled oriented filtered versions of an image we generalize the well known steerable scalable deformable filter bank structure by imposing x y separability on the basis filters the resulting systems designed by an iterative projections technique achieve substantial reduction of the computational cost to reduce the memory requirement we adopt a multirate implementation due to the inner sampling rate alteration the resulting structure is not shift invariant we introduce a design criterion for multirate deformable structures that jointly controls the approximation error and the shift variance

hessian based norm regularization for image restoration with biomedical applications we present nonquadratic hessian based regularization methods that can be effectively used for image restoration problems in a variational framework motivated by the great success of the total variation tv functional we extend it to also include second order differential operators specifically we derive second order regularizers that involve matrix norms of the hessian operator the definition of these functionals is based on an alternative interpretation of tv that relies on mixed norms of directional derivatives we show that the resulting regularizers retain some of the most favorable properties of tv i e convexity homogeneity rotation and translation invariance while dealing effectively with the staircase effect we further develop an efficient minimization scheme for the corresponding objective functions the proposed algorithm is of the iteratively reweighted least square type and results from a majorization minimization approach it relies on a problem specific preconditioned conjugate gradient method which makes the overall minimization scheme very attractive since it can be applied effectively to large images in a reasonable computational time we validate the overall proposed regularization framework through deblurring experiments under additive gaussian noise on standard and biomedical images

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

design of steerable filters for feature detection using canny like criteria we propose a general approach for the design of 2d feature detectors from a class of steerable functions based on the optimization of a canny like criterion in contrast with previous computational designs our approach is truly 2d and provides filters that have closed form expressions it also yields operators that have a better orientation selectivity than the classical gradient or hessian based detectors we illustrate the method with the design of operators for edge and ridge detection we present some experimental results that demonstrate the performance improvement of these new feature detectors we propose computationally efficient local optimization algorithms for the estimation of feature orientation we also introduce the notion of shape adaptable feature detection and use it for the detection of image corners

scale invariant filtering design and analysis for edge detection existing edge detection filters work well on straight edges but make significant errors near sharp corners by producing rounded corners this is due to the fact that the edge maps produced by these filters are scale variant we enhance canny s optimality criteria to incorporate detection performance near corners as an explicit design objective the resulting optimal filter termed bessel integral filter can be derived analytically and exhibits superior performance over recent alternatives both in terms of numerical accuracy and experimental fidelity a noise free localization index is also derived here to account for the detection accuracy of discontinuities forming sharp corners in the absence of noise we prove here that edges detected by the filters which are not optimal with respect to this noise free localization index are scale variant however the bessel integral filter proposed here is optimal with respect to the noise free localization index and therefore it is a scale invariant filter

a new orientation adaptive interpolation method we propose an isophote oriented orientation adaptive interpolation method the proposed method employs an interpolation kernel that adapts to the local orientation of isophotes and the pixel values are obtained through an oriented bilinear interpolation we show that by doing so the curvature of the interpolated isophotes is reduced and thus zigzagging artifacts are largely suppressed analysis and experiments show that images interpolated using the proposed method are visually pleasing and almost artifact free

analysis of multiple orientations estimation of local orientations in multivariate signals is an important problem in image processing and computer vision this general problem formulation also covers optical flow estimation which can be regarded as orientation estimation in space time volumes modelling a signal using only a single orientation however is often too restrictive since occlusions and transparencies occur frequently thus necessitating the modelling and analysis of multiple orientations we therefore develop a unifying mathematical model for multiple orientations beyond describing an arbitrary number of orientations in scalar and vector valued image data such as color image sequences it allows the unified treatment of additively and occludingly superimposed oriented structures as well as of combinations of these based on this model we describe estimation schemes for an arbitrary number of additively or occludingly superimposed orientations in images we confirm the performance of our framework on both synthetic and real image data

equations d ondes quasilineaires et estimations de strichartz dans cet article notre but est de resoudre des equations d ondes quasilineaires pour des donnees initiales moins regulieres que ce qu imposent les methodes d energie ceci impose de demontrer des estimees de type strichartz pour des operateurs d ondes a coefficients seulement lipschitziens

construction of cauchy data of vacuum einstein field equations evolving to black holes we show the existence of complete asymptotically at cauchy initial data for the vacuum einstein eld equations free of trapped surfaces whose future development must admit a trapped surface moreover the datum is exactly a constant time slice in minkowski space time inside and exactly a constant time slice in kerr space time outside the proof makes use of the full strength of christodoulou s work on the dynamical formation of black holes and corvino schoen s work on the construction of initial data sets

characteristic initial data and wavefront singularities in general relativity the structure of singularities caustics self intersections of wavefronts null hypersurfaces and wavefront families null coordinates in arbitrary space times is discussed in detail and illustrated by explicit examples of stable wavefront singularities in minkowski space it is shown how characteristic initial data determine the caustics and the self intersections of the characteristics of einstein s field equations

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

on the formation of trapped surfaces in a recent important breakthrough d christodoulou has solved a long standing problem of general relativity of evolutionary formation of trapped surfaces in the einstein vacuum space times he has identified an open set of regular initial conditions on an outgoing null hypersurface both finite and at past null infinity leading to a formation a trapped surface in the corresponding vacuum space time to the future of the initial outgoing hypersurface and another incoming null hypersurface with the prescribed minkowskian data in this paper we give a simpler proof for a finite problem by enlarging the admissible set of initial conditions and consistent with this relaxing the corresponding propagation estimates just enough that a trapped surface still forms we also reduce the number of derivatives needed in the argument from two derivatives of the curvature to just one more importantly the proof which can be easily localized with respect to angular sectors has the potential for further developments

noise robust color edge detector using gradient matrix and anisotropic gaussian directional derivative matrix in this paper a noise robust color edge detector using gradient matrix and anisotropic gaussian directional derivative andd matrix is proposed in order to alleviate the conflict between high edge resolution and noise robustness in the color canny detector where the isotropic gaussian kernels and gradient matrix of the r g b components are used the andds of the three components of a color image are arranged into the andd matrix from its singular value decomposition svd the andd based color edge strength map cesm is constructed and is relevant to the pixelwise optimal fusion of the andds of the three components in the sense of color edge enhancement the merits and defects of the andd based cesm and gradient based cesm are contrasted to show their complementarity in color edge detection the two cesms are fused to develop a new color edge detector it is compared with the color canny detector and two recent color edge detectors the results show that the proposed detector attains better detection performance for noiseless and noisy color images corrupted by white gaussian noise or impulsive noise of small percentage a color edge detector using gradient matrix and andd matrix is proposed the svd of the andd matrix is used to determine cesm and cesd of a color image the new detector has both high edge resolution and noise robustness it attains better performance than the two existing color edge detectors

rough solutions of the einstein vacuum equations this is the first in a series of papers in which we initiate the study of very rough solutions to the initial value problem for the einstein vacuum equations expressed relative to wave coordinates by very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and sobolev inequalities following kl ro we develop new analytic methods based on strichartz type inequalities which result in a gain of half a derivative relative to the classical result our methods blend paradifferential techniques with a geometric approach to the derivation of decay estimates the latter allows us to take full advantage of the specific structure of the einstein equations

template free wavelet based detection of local symmetries our goal is to detect and group different kinds of local symmetries in images in a scale and rotation invariant way we propose an efficient wavelet based method to determine the order of local symmetry at each location our algorithm relies on circular harmonic wavelets which are used to generate steerable wavelet channels corresponding to different symmetry orders to give a measure of local symmetry we use the f test to examine the distribution of the energy across different channels we provide experimental results on synthetic images biological micrographs and electron microscopy images to demonstrate the performance of the algorithm

tubular structure segmentation based on minimal path method and anisotropic enhancement we present a new interactive method for tubular structure extraction the main application and motivation for this work is vessel tracking in 2d and 3d images the basic tools are minimal paths solved using the fast marching algorithm this allows interactive tools for the physician by clicking on a small number of points in order to obtain a minimal path between two points or a set of paths in the case of a tree structure our method is based on a variant of the minimal path method that models the vessel as a centerline and surface this is done by adding one dimension for the local radius around the centerline the crucial step of our method is the definition of the local metrics to minimize we have chosen to exploit the tubular structure of the vessels one wants to extract to built an anisotropic metric the designed metric is well oriented along the direction of the vessel admits higher velocity on the centerline and provides a good estimate of the vessel radius based on the optimally oriented flux this measure is required to be robust against the disturbance introduced by noise or adjacent structures with intensity similar to the target vessel we obtain promising results on noisy synthetic and real 2d and 3d images and we present a clinical validation

on optimal infinite impulse response edge detection filters the authors outline the design of an optimal computationally efficient infinite impulse response edge detection filter the optimal filter is computed based on canny s high signal to noise ratio good localization criteria and a criterion on the spurious response of the filter to noise an expression for the width of the filter which is appropriate for infinite length filters is incorporated directly in the expression for spurious responses the three criteria are maximized using the variational method and nonlinear constrained optimization the optimal filter parameters are tabulated for various values of the filter performance criteria a complete methodology for implementing the optimal filter using approximating recursive digital filtering is presented the approximating recursive digital filter is separable into two linear filters operating in two orthogonal directions the implementation is very simple and computationally efficient has a constant time of execution for different sizes of the operator and is readily amenable to real time hardware implementation

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

sharp local well posedness results for the nonlinear wave equation this article is concerned with local well posedness of the cauchy problem for second order quasilinear hyperbolic equations with rough initial data the new results obtained here are sharp in low dimension

the bounded l 2 curvature conjecture this is the main paper in a sequence in which we give a complete proof of the bounded l 2 curvature conjecture more precisely we show that the time of existence of a classical solution to the einstein vacuum equations depends only on the l 2 norm of the curvature and a lower bound on the volume radius of the corresponding initial data set we note that though the result is not optimal with respect to the scaling of the einstein equations it is nevertheless critical with respect to its causal geometry indeed l 2 bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries we note also that while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than 1 1 based on strichartz estimates were obtained in bahouri and chemin am j math 121 1337 1777 1999 imrn 21 1141 1178 1999 tataru am j math 122 349 376 2000 jams 15 2 419 442 2002 klainerman and rodnianski duke math j 117 1 1 124 2003 and optimized in klainerman and rodnianski ann math 161 1143 1193 2005 smith and tataru ann math 162 291 366 2005 the result we present here is the first in which the full structure of the quasilinear hyperbolic system not just its principal part plays a crucial role to achieve our goals we recast the einstein vacuum equations as a quasilinear so 3 1 valued yang mills theory and introduce a coulomb type gauge condition in which the equations exhibit a specific new type of null structure compatible with the quasilinear covariant nature of the equations to prove the conjecture we formulate and establish bilinear and trilinear estimates on rough backgrounds which allow us to make use of that crucial structure these require a careful construction and control of parametrices including l 2 error bounds which is carried out in szeftel parametrix for wave equations on a rough background i regularity of the phase at initial time arxiv 1204 1768 2012 parametrix for wave equations on a rough background ii construction of the parametrix and control at initial time arxiv 1204 1769 2012 parametrix for wave equations on a rough background iii space time regularity of the phase arxiv 1204 1770 2012 parametrix for wave equations on a rough background iv control of the error term arxiv 1204 1771 2012 as well as a proof of sharp strichartz estimates for the wave equation on a rough background which is carried out in szeftel sharp strichartz estimates for the wave equation on a rough background arxiv 1301 0112 2013 it is at this level that our problem is critical indeed any known notion of a parametrix relies in an essential way on the eikonal equation and our space time possesses barely the minimal regularity needed to make sense of its solutions

design and implementation of multisteerable matched filters image analysis problems such as feature tracking edge detection image enhancement or texture analysis require the detection of multi oriented patterns which can appear at arbitrary orientations direct rotated matched filtering for feature detection is computationally expensive but can be sped up with steerable filters so far steerable filter approaches were limited to only one direction many important low level image features are however characterized by more than a single orientation we therefore present here a framework for efficiently detecting specific multi oriented patterns with arbitrary orientations in grayscale images the core idea is to construct multisteerable filters by appropriate combinations of single steerable filters we exploit that steerable filters are closed under addition and multiplication this allows to derive a design guide for multisteerable filters by means of multivariate polynomials furthermore we describe an efficient implementation scheme and discuss the use of weighting functions to reduce angular oscillations applications in camera calibration junction analysis of images from plant roots and the discrimination of l t and x junctions demonstrate the potential of this approach