formality for the nilpotent cone and a derived springer correspondence abstract recall that the springer correspondence relates representations of the weyl group to perverse sheaves on the nilpotent cone we explain how to extend this to an equivalence between the triangulated category generated by the springer perverse sheaf and the derived category of differential graded modules over a dg ring related to the weyl group

feature guided gaussian mixture model with semi supervised em and local geometric constraint for retinal image registration abstract retinal image registration which can be formulated by matching two sets of sparse features extracted from two observed retinal images is a crucial step in the diagnosis and treatment of various eye diseases existing methods suffer from missing true correspondences or do not fully consider local appearance information which causes difficulty in matching low quality retinal images due to insufficient reliable features in addition the relationships between retinal image pairs are usually modeled by linear transformation such as affine transformation which cannot generate accurate alignments in large viewpoint changes due to the nonplanar eyeball surface to address these issues a feature guided gaussian mixture model gmm is proposed for the non rigid registration of retinal images we formulate the problem as an estimation of a feature guided mixture of densities a gmm is fitted to one point set in which the centers of the gaussian densities characterized by spatial positions associated with local appearance descriptors are constrained to coincide with the other point set the problem is solved under a maximum likelihood framework and semi supervised expectation maximization is used to iteratively estimate the feature correspondence and spatial transformation which is initialized by a set of confidential feature matches obtained previously non rigid transformation is specified in a reproducing kernel hilbert space and a local geometric constraint is imposed to establish the transformation estimation for obtaining a meaningful solution a fast implementation based on sparse approximation is also provided and reduces the time complexity from cubic to quadratic moreover we use the edge map which can extract more reliable features as a uniform representation of retinal images experimental results on publicly available retinal images show that our approach is robust in different registration tasks and outperforms several competing approaches especially when data is severely degraded

retinal artery vein classification via topology estimation we propose a novel graph theoretic framework for distinguishing arteries from veins in a fundus image we make use of the underlying vessel topology to better classify small and midsized vessels we extend our previously proposed tree topology estimation framework by incorporating expert domain specific features to construct a simple yet powerful global likelihood model we efficiently maximize this model by iteratively exploring the space of possible solutions consistent with the projected vessels we tested our method on four retinal datasets and achieved classification accuracies of 91 0 93 5 91 7 and 90 9 outperforming existing methods our results show the effectiveness of our approach which is capable of analyzing the entire vasculature including peripheral vessels in wide field of view fundus photographs this topology based method is a potentially important tool for diagnosing diseases with retinal vascular manifestation

three dimensional motion tracking of coronary arteries in biplane cineangiograms a three dimensional 3 d method for tracking the coronary arteries through a temporal sequence of biplane x ray angiography images is presented a 3 d centerline model of the coronary vasculature is reconstructed from a biplane image pair at one time frame and its motion is tracked using a coarse to fine hierarchy of motion models three dimensional constraints on the length of the arteries and on the spatial regularity of the motion field are used to overcome limitations of classical two dimensional vessel tracking methods such as tracking vessels through projective occlusions this algorithm was clinically validated in five patients by tracking the motion of the left coronary tree over one cardiac cycle the root mean square reprojection errors were found to be submillimeter in 93 54 58 of the image pairs the performance of the tracking algorithm was quantified in three dimensions using a deforming vascular phantom rms 3 d distance errors were computed between centerline models tracked in the x ray images and gold standard centerline models of the phantom generated from a gated 3 d magnetic resonance image acquisition the mean error was 0 69 spl plusmn 0 06 mm over eight temporal phases and four different biplane orientations

edge detection with embedded confidence computing the weighted average of the pixel values in a window is a basic module in many computer vision operators the process is reformulated in a linear vector space and the role of the different subspaces is emphasized within this framework wellknown artifacts of the gradient based edge detectors such as large spurious responses can be explained quantitatively it is also shown that template matching with a template derived from the input data is meaningful since it provides an independent measure of confidence in the presence of the employed edge model the widely used three step edge detection procedure gradient estimation non maxima suppression hysteresis thresholding is generalized to include the information provided by the confidence measure the additional amount of computation is minimal and experiments with several standard test images show the ability of the new procedure to detect weak edges

a computational method for segmenting topological point sets and application to image analysis we propose a computational method for segmenting topological subdimensional point sets in scalar images of arbitrary spatial dimensions the technique is based on calculating the homotopy class defined by the gradient vector in a subdimensional neighborhood around every image point this neighborhood is defined as the linear envelope spawned over a given subdimensional vector frame in the simplest case where the rank of this frame is maximal we obtain a technique for localizing the critical points we consider in particular the important case of frames formed by an arbitrary number of the first largest by absolute value principal directions of the hessian the method then segments positive and and negative ridges as well as other types of critical surfaces of different dimensionalities the signs of the eigenvalues associated to the principal directions provide a natural labeling of the critical subsets the result in general is a constructive definition of a hierarchy of point sets of different dimensionalities linked by inclusion relations because of its explicit computational nature the method gives a fast way to segment height ridges or edges in different applications the defined topological point sets are connected manifolds and therefore our method provides a tool for geometrical grouping using only local measurements we have demonstrated the grouping properties of our construction by presenting two different cases where an extra image coordinate is introduced

retinal vessel centerline extraction using multiscale matched filters confidence and edge measures motivated by the goals of improving detection of low contrast and narrow vessels and eliminating false detections at nonvascular structures a new technique is presented for extracting vessels in retinal images the core of the technique is a new likelihood ratio test that combines matched filter responses confidence measures and vessel boundary measures matched filter responses are derived in scale space to extract vessels of widely varying widths a vessel confidence measure is defined as a projection of a vector formed from a normalized pixel neighborhood onto a normalized ideal vessel profile vessel boundary measures and associated confidences are computed at potential vessel boundaries combined these responses form a six dimensional measurement vector at each pixel a training technique is used to develop a mapping of this vector to a likelihood ratio that measures the vesselness at each pixel results comparing this vesselness measure to matched filters alone and to measures based on the hessian of intensities show substantial improvements both qualitatively and quantitatively the hessian can be used in place of the matched filter to obtain similar but less substantial improvements or to steer the matched filter by preselecting kernel orientations finally the new vesselness likelihood ratio is embedded into a vessel tracing framework resulting in an efficient and effective vessel centerline extraction algorithm

on discontinuous first order implicit boundary value problems t 0 1 2 under various boundary conditions cf e g 1 5 8 13 29 31 32 thereis little known about the existence of solutions of 1 2 even in the casewhen f is continuous unless it is possible to solve the equation 1 2 forthe highest derivative and then to apply standard existence theory such asleray schauder theory one approach taken for instance in 8 16 is toreduce equation 1 2 to a differential inclusion of the formu

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

on singular functional nonsmooth and implicit phi laplacian initial and boundary value problems in this paper we apply fixed point results for mappings in partially ordered spaces to derive existence results for extremal solutions of phi laplacian initial and boundary value problems the considered problems can be singular functional discontinuous nonlocal and implicit concrete examples are also solved

the dual bootstrap iterative closest point algorithm with application to retinal image registration motivated by the problem of retinal image registration this paper introduces and analyzes a new registration algorithm called dual bootstrap iterative closest point dual bootstrap icp the approach is to start from one or more initial low order estimates that are only accurate in small image regions called bootstrap regions in each bootstrap region the algorithm iteratively 1 refines the transformation estimate using constraints only from within the bootstrap region 2 expands the bootstrap region and 3 tests to see if a higher order transformation model can be used stopping when the region expands to cover the overlap between images steps 1 and 3 the bootstrap steps are governed by the covariance matrix of the estimated transformation estimation refinement step 2 uses a novel robust version of the icp algorithm in registering retinal image pairs dual bootstrap icp is initialized by automatically matching individual vascular landmarks and it aligns images based on detected blood vessel centerlines the resulting quadratic transformations are accurate to less than a pixel on tests involving approximately 6000 image pairs it successfully registered 99 5 of the pairs containing at least one common landmark and 100 of the pairs containing at least one common landmark and at least 35 image overlap

statistical edge detection learning and evaluating edge cues we formulate edge detection as statistical inference this statistical edge detection is data driven unlike standard methods for edge detection which are model based for any set of edge detection filters implementing local edge cues we use presegmented images to learn the probability distributions of filter responses conditioned on whether they are evaluated on or off an edge edge detection is formulated as a discrimination task specified by a likelihood ratio test on the filter responses this approach emphasizes the necessity of modeling the image background the off edges we represent the conditional probability distributions nonparametrically and illustrate them on two different data sets of 100 sowerby and 50 south florida images multiple edges cues including chrominance and multiple scale are combined by using their joint distributions hence this cue combination is optimal in the statistical sense we evaluate the effectiveness of different visual cues using the chernoff information and receiver operator characteristic roc curves this shows that our approach gives quantitatively better results than the canny edge detector when the image background contains significant clutter in addition it enables us to determine the effectiveness of different edge cues and gives quantitative measures for the advantages of multilevel processing for the use of chrominance and for the relative effectiveness of different detectors furthermore we show that we can learn these conditional distributions on one data set and adapt them to the other with only slight degradation of performance without knowing the ground truth on the second data set this shows that our results are not purely domain specific we apply the same approach to the spatial grouping of edge cues and obtain analogies to nonmaximal suppression and hysteresis

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

extremal solutions for nonlinear functional ϕ laplacian impulsive equations abstract in this paper we deal with some nonlinear laplacian second order boundary value problems with impulses subject to a general type of boundary value conditions which cover dirichlet and periodic boundary data as particular cases our approach is that of upper and lower solutions together with growth restrictions of nagumo s type the impulsive functions depend implicitly on the different considered variables and the boundary value conditions are nonlinear in both cases functional dependence on the solution is allowed

koszul duality for kac moody groups and characters of tilting modules we establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic p in terms of p kazhdan lusztig polynomials for p h the coxeter number using results of andersen one may deduce a character formula for simple modules if p 2h 2 our results are a consequence of an extension to modular coefficients of a monoidal koszul duality equivalence established by bezrukavnikov and yun

detection of blood vessels in retinal images using two dimensional matched filters blood vessels usually have poor local contrast and the application of existing edge detection algorithms yield results which are not satisfactory an operator for feature extraction based on the optical and spatial properties of objects to be recognized is introduced the gray level profile of the cross section of a blood vessel is approximated by a gaussian shaped curve the concept of matched filter detection of signals is used to detect piecewise linear segments of blood vessels in these images twelve different templates that are used to search for vessel segments along all possible directions are constructed various issues related to the implementation of these matched filters are discussed the results are compared to those obtained with other methods

a new method for linear feature and junction enhancement in 2d images based on morphological operation oriented anisotropic gaussian function and hessian information abstract feature enhancement is an important preprocessing step in many image processing tasks it is the process of adjusting image intensities so that the enhanced results are more suitable for analysis good enhancement results for linear structures such as vessels or neurites can be used as inputs for segmentation and other operations in this paper a novel linear feature enhancement filter an adaptive multi scale morpho gaussian filter which can enhance and smooth linear features is proposed based on morphological operation anisotropic gaussian function and hessian information this filter can enhance and smooth along the local orientation of the linear structures and the hessian measurement is used to further enhance the linear features we utilize the hessian matrix to calculate the orientation information for our directional morphological operation and the oriented anisotropic gaussian smoothing we also propose a novel method for junction enhancement which can solve the problem of junction suppression we decompose the junctions and enhance along each linear structure within a junction region we present the test results of our algorithm on images of different types and compare our method with three existing methods the experimental results show that the proposed approach can achieve better results

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

kostant section universal centralizer and a modular derived satake equivalence we prove analogues of fundamental results of kostant on the universal centralizer of a connected reductive algebraic group for algebraically closed fields of positive characteristic with mild assumptions and for integral coefficients as an application we use these results to obtain a mixed modular analogue of the derived satake equivalence of bezrukavnikov finkelberg

joint volumetric extraction and enhancement of vasculature from low snr 3 d fluorescence microscopy images abstract to simultaneously overcome the challenges imposed by the nature of optical imaging characterized by a range of artifacts including space varying signal to noise ratio snr scattered light and non uniform illumination we developed a novel method that segments the 3 d vasculature directly from original fluorescence microscopy images eliminating the need for employing pre and post processing steps such as noise removal and segmentation refinement as used with the majority of segmentation techniques our method comprises two initialization and constrained recovery and enhancement stages the initialization approach is fully automated using features derived from bi scale statistical measures and produces seed points robust to non uniform illumination low snr and local structural variations this algorithm achieves the goal of segmentation via design of an iterative approach that extracts the structure through voting of feature vectors formed by distance local intensity gradient and median measures qualitative and quantitative analysis of the experimental results obtained from synthetic and real data prove the efficacy of this method in comparison to the state of the art enhancing segmenting methods the algorithmic simplicity freedom from having a priori probabilistic information about the noise and structural definition gives this algorithm a wide potential range of applications where i e structural complexity significantly complicates the segmentation problem

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

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

segmentation of vessel like patterns using mathematical morphology and curvature evaluation this paper presents an algorithm based on mathematical morphology and curvature evaluation for the detection of vessel like patterns in a noisy environment such patterns are very common in medical images vessel detection is interesting for the computation of parameters related to blood flow its tree like geometry makes it a usable feature for registration between images that can be of a different nature in order to define vessel like patterns segmentation is performed with respect to a precise model we define a vessel as a bright pattern piece wise connected and locally linear mathematical morphology is very well adapted to this description however other patterns fit such a morphological description in order to differentiate vessels from analogous background patterns a cross curvature evaluation is performed they are separated out as they have a specific gaussian like profile whose curvature varies smoothly along the vessel the detection algorithm that derives directly from this modeling is based on four steps 1 noise reduction 2 linear pattern with gaussian like profile improvement 3 cross curvature evaluation 4 linear filtering we present its theoretical background and illustrate it on real images of various natures then evaluate its robustness and its accuracy with respect to noise

equivariant sheaves on flag varieties we show that the borel equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of differential graded modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves this proves a conjecture of soergel and lunts in the case of flag varieties

general retinal vessel segmentation using regularization based multiconcavity modeling detecting blood vessels in retinal images with the presence of bright and dark lesions is a challenging unsolved problem in this paper a novel multiconcavity modeling approach is proposed to handle both healthy and unhealthy retinas simultaneously the differentiable concavity measure is proposed to handle bright lesions in a perceptive space the line shape concavity measure is proposed to remove dark lesions which have an intensity structure different from the line shaped vessels in a retina the locally normalized concavity measure is designed to deal with unevenly distributed noise due to the spherical intensity variation in a retinal image these concavity measures are combined together according to their statistical distributions to detect vessels in general retinal images very encouraging experimental results demonstrate that the proposed method consistently yields the best performance over existing state of the art methods on the abnormal retinas and its accuracy outperforms the human observer which has not been achieved by any of the state of the art benchmark methods most importantly unlike existing methods the proposed method shows very attractive performances not only on healthy retinas but also on a mixture of healthy and pathological retinas

retina verification system based on biometric graph matching this paper presents an automatic retina verification framework based on the biometric graph matching bgm algorithm the retinal vasculature is extracted using a family of matched filters in the frequency domain and morphological operators then retinal templates are defined as formal spatial graphs derived from the retinal vasculature the bgm algorithm a noisy graph matching algorithm robust to translation non linear distortion and small rotations is used to compare retinal templates the bgm algorithm uses graph topology to define three distance measures between a pair of graphs two of which are new a support vector machine svm classifier is used to distinguish between genuine and imposter comparisons using single as well as multiple graph measures the classifier achieves complete separation on a training set of images from the varia database 60 of the data equaling the state of the art for retina verification because the available data set is small kernel density estimation kde of the genuine and imposter score distributions of the training set are used to measure performance of the bgm algorithm in the one dimensional case the kde model is validated with the testing set a 0 eer on testing shows that the kde model is a good fit for the empirical distribution for the multiple graph measures a novel combination of the svm boundary and the kde model is used to obtain a fair comparison with the kde model for the single measure a clear benefit in using multiple graph measures over a single measure to distinguish genuine and imposter comparisons is demonstrated by a drop in theoretical error of between 60 and more than two orders of magnitude

solvability of some third order boundary value problems with asymmetric unbounded nonlinearities abstract in this paper we present existence and location results for the third order separated boundary value problems u t f t u t u t u t with the boundary conditions u a a u a b u b c or u a a c 1 u a c 2 u a b c 3 u b c 4 u b c with c 1 c 2 c 3 c 4 r and a b c r we assume f a b r 3 r is a continuous function satisfying one sided nagumo type condition which allows an asymmetric unbounded behaviour the arguments used concern leray schauder degree and lower and upper solution techniques

a novel vessel segmentation algorithm for pathological retina images based on the divergence of vector fields in this paper a method is proposed for detecting blood vessels in pathological retina images in the proposed method blood vessel like objects are extracted using the laplacian operator and noisy objects are pruned according to the centerlines which are detected using the normalized gradient vector field the method has been tested with all the pathological retina images in the publicly available stare database experiment results show that the method can avoid detecting false vessels in pathological regions and can produce reliable results for healthy regions

upper and lower solutions for second order boundary value problems with nonlinear boundary conditions on demontre des theoremes d existence en utilisant des solutions superieures et inferieures