monodromies at infinity of confluent a hypergeometric functions abstract we study the monodromies at infinity of confluent a hypergeometric functions introduced by adolphson 2 in particular we extend the result of 38 for non confluent a hypergeometric functions to the confluent case the integral representation by rapid decay homology cycles proved in 9 will play a central role in the proof

spectral isometries onto algebras having a separating family of finite dimensional irreducible representations we prove that if a is a complex unital semisimple banach algebra and b is a complex unital banach algebra having a separating family of finite dimensional irreducible representations then any unital linear operator from a onto b which preserves the spectral radius is a jordan morphism

periods for irregular singular connections on surfaces given an integrable connection on a smooth quasi projective algebraic surface u over a subfield k of the complex numbers we define rapid decay homology groups with respect to the associated analytic connection which pair with the algebraic de rham cohomology in terms of period integrals these homology groups generalize the analogous groups in the same situation over curves defined by bloch and esnault in dimension two however new features appear in this context which we explain in detail assuming a conjecture of sabbah on the formal classification of meromorphic connections on surfaces known to be true if the rank is lower than or equal to 5 we prove perfectness of the period pairing in dimension two

hierarchical shape segmentation and registration via topological features of laplace beltrami eigenfunctions this work introduces a method to hierarchically segment articulated shapes into meaningful parts and to register these parts across populations of near isometric shapes e g head arms legs and fingers of humans in different body postures the method exploits the isometry invariance of eigenfunctions of the laplace beltrami operator and uses topological features level sets at important saddles for the segmentation concepts from persistent homology are employed for a hierarchical representation for the elimination of topological noise and for the comparison of eigenfunctions the obtained parts can be registered via their spectral embedding across a population of near isometric shapes this work also presents the highly accurate computation of eigenfunctions and eigenvalues with cubic finite elements on triangle meshes and discusses the construction of persistence diagrams from the morse smale complex as well as the relation to size functions

geodesic matting a framework for fast interactive image and video segmentation and matting an interactive framework for soft segmentation and matting of natural images and videos is presented in this paper the proposed technique is based on the optimal linear time computation of weighted geodesic distances to user provided scribbles from which the whole data is automatically segmented the weights are based on spatial and or temporal gradients considering the statistics of the pixels scribbled by the user without explicit optical flow or any advanced and often computationally expensive feature detectors these could be naturally added to the proposed framework as well if desired in the form of weights in the geodesic distances an automatic localized refinement step follows this fast segmentation in order to further improve the results and accurately compute the corresponding matte function additional constraints into the distance definition permit to efficiently handle occlusions such as people or objects crossing each other in a video sequence the presentation of the framework is complemented with numerous and diverse examples including extraction of moving foreground from dynamic background in video natural and 3d medical images and comparisons with the recent literature

period determinant of an irregular connection over an elliptic curve abstract in this article we calculate the period of an irregular singular connection d d y on the legendre curve u y 2 x x 1 x we calculate its de rham cohomology and the cycles in the homology of the dual connection and describe the period matrix terasoma s work is introduced to approximate the direct image connection by a sequence of regular connections as an intermediate step where u a 1 spec k y x y y finally we will make the comparison of the period obtained by this approximation and that over u

direct mapping of hippocampal surfaces with intrinsic shape context we propose in this paper a new method for the mapping of hippocampal hc surfaces to establish correspondences between points on hc surfaces and enable localized hc shape analysis a novel geometric feature the intrinsic shape context is defined to capture the global characteristics of the hc shapes based on this intrinsic feature an automatic algorithm is developed to detect a set of landmark curves that are stable across population the direct map between a source and target hc surface is then solved as the minimizer of a harmonic energy function defined on the source surface with landmark constraints for numerical solutions we compute the map with the approach of solving partial differential equations on implicit surfaces the direct mapping method has the following properties 1 it has the advantage of being automatic 2 it is invariant to the pose of hc shapes in our experiments we apply the direct mapping method to study temporal changes of hc asymmetry in alzheimer s disease ad using hc surfaces from 12 ad patients and 14 normal controls our results show that the ad group has a different trend in temporal changes of hc asymmetry than the group of normal controls we also demonstrate the flexibility of the direct mapping method by applying it to construct spherical maps of hc surfaces spherical harmonics spharm analysis is then applied and it confirms our results on temporal changes of hc asymmetry in ad

robust surface reconstruction via laplace beltrami eigen projection and boundary deformation in medical shape analysis a critical problem is reconstructing a smooth surface of correct topology from a binary mask that typically has spurious features due to segmentation artifacts the challenge is the robust removal of these outliers without affecting the accuracy of other parts of the boundary in this paper we propose a novel approach for this problem based on the laplace beltrami lb eigen projection and properly designed boundary deformations using the metric distortion during the lb eigen projection our method automatically detects the location of outliers and feeds this information to a well composed and topology preserving deformation by iterating between these two steps of outlier detection and boundary deformation we can robustly filter out the outliers without moving the smooth part of the boundary the final surface is the eigen projection of the filtered mask boundary that has the correct topology desired accuracy and smoothness in our experiments we illustrate the robustness of our method on different input masks of the same structure and compare with the popular spharm tool and the topology preserving level set method to show that our method can reconstruct accurate surface representations without introducing artificial oscillations we also successfully validate our method on a large data set of more than 900 hippocampal masks and demonstrate that the reconstructed surfaces retain volume information accurately

linear maps preserving the spectral radius abstract let x be a complex banach space if b x b x is a surjective linear map such that a and a have the same spectral radius for every a b x then c where is either an algebra automorphism or an antiautomorphism of b x and c is a complex constant such that c 1

euler s constant euler s work and modern developments this paper has two parts the first part surveys euler s work on the constant 0 57721 bearing his name together with some of his related work on the gamma function values of the zeta function and divergent series the second part describes various mathematical developments involving euler s constant as well as another constant the euler gompertz constant these developments include connections with arithmetic functions and the riemann hypothesis and with sieve methods random permutations and random matrix products it also includes recent results on diophantine approximation and transcendence related to euler s constant

a riemann roch theorem for flat bundles with values in the algebraic chern simons theory our purpose in this paper is to continue the algebraic study of complex local systems on complex algebraic varieties we prove a riemann roch theorem for these objects using algebraic chern simons characteristic classes a complex local system e on a smooth projective complex variety x gives rise to a locally free analytic sheaf ean e c oan x which using gaga admits a canonical algebraic structure e the tautological analytic connection on e co x induces an integrable algebraic connection e e x combining gaga with the poincare lemma we see that the analytic cohomology of the local system can be identified with the hypercohomology of the algebraic de rham complex

confluent a hypergeometric functions and rapid decay homology cycles we study confluent a hypergeometric functions introduced by adolphson in particular we give their integral representations by using rapid decay homology cycles of hien the method of toric compactifications introduced in our previous works will be used to prove our main theorem moreover we apply it to obtain a formula for the asymptotic expansions at infinity of confluent a hypergeometric functions

hippocampal shape modeling based on a progressive template surface deformation and its verification accurately recovering the hippocampal shapes against rough and noisy segmentations is as challenging as achieving good anatomical correspondence between the individual shapes to address these issues we propose a mesh to volume registration approach characterized by a progressive model deformation our model implements flexible weighting scheme for model rigidity under a multi level neighborhood for vertex connectivity this method induces a large to small scale deformation of a template surface to build the pairwise correspondence by minimizing geometric distortion while robustly restoring the individuals shape characteristics we evaluated the proposed method s 1 accuracy and robustness in smooth surface reconstruction 2 sensitivity in detecting significant shape differences between healthy control and disease groups mild cognitive impairment and alzheimer s disease 3 robustness in constructing the anatomical correspondence between individual shape models and 4 applicability in identifying subtle shape changes in relation to cognitive abilities in a healthy population we compared the performance of the proposed method with other well known methods spharm pdm shapeworks and lddmm volume registration with template injection using various metrics of shape similarity surface roughness volume and shape deformity the experimental results showed that the proposed method generated smooth surfaces with less volume differences and better shape similarity to input volumes than others the statistical analyses with clinical variables also showed that it was sensitive in detecting subtle shape changes of hippocampus

periods of hodge structures and special values of the gamma function at the end of the 1970s gross and deligne conjectured that periods of geometric hodge structures with multiplication by an abelian number field are products of values of the gamma function at rational arguments with exponents determined by the hodge decomposition we prove an alternating variant of this conjecture for smooth projective varieties acted upon by an automorphism of finite order thus improving previous results of maillot and rossler the proof relies on a product formula for periods of regular singular connections due to saito and terasoma

surface interpolation from sparse cross sections using region correspondence the ability to estimate a surface from a set of cross sections allows calculation of the enclosed volume and the display of the surface in three dimensions this process has increasingly been used to derive useful information from medical data however extracting the cross sections segmenting can be very difficult and automatic segmentation methods are not sufficiently robust to handle all situations hence it is an advantage if the surface reconstruction algorithm can work effectively on a small number of cross sections in addition cross sections of medical data are often quite complex shape based interpolation is a simple and elegant solution to this problem although it has known limitations when handling complex shapes in this paper the shape based interpolation paradigm is extended to interpolate a surface through sparse complex cross sections providing a significant improvement over the authors previously published maximal disc guided interpolation the performance of this algorithm is demonstrated on various types of medical data x ray computed tomography magnetic resonance imaging and three dimensional ultrasound although the correspondence problem in general remains unsolved it is demonstrated that correct surfaces can be estimated from a limited amount of real data through the use of region rather than object correspondence

automated graph based analysis and correction of cortical volume topology the human cerebral cortex is topologically equivalent to a sheet and can be considered topologically spherical if it is closed at the brainstem low level segmentation of magnetic resonance mr imagery typically produces cerebral volumes whose tessellations are not topologically spherical the authors present a novel algorithm that analyzes and constrains the topology of a volumetric object graphs are formed that represent the connectivity of voxel segments in the foreground and background of the image these graphs are analyzed and minimal corrections to the volume are made prior to tessellation the authors apply the algorithm to a simple test object and to cerebral white matter masks generated by a low level tissue identification sequence the authors tessellate the resulting objects using the marching cubes algorithm and verify their topology by computing their euler characteristics a key benefit of the algorithm is that it localizes the change to a volume to the specific areas of its topological defects

periods for flat algebraic connections in previous work hien math ann 337 631 669 2007 we established a duality between the algebraic de rham cohomology of a flat algebraic connection on a smooth quasi projective surface over the complex numbers and the rapid decay homology of the dual connection relying on a conjecture by c sabbah in the present article we generalize this result to arbitrary dimensions unconditionally using recent results of t mochizuki

gersten s conjecture for thek theory with torsion coefficients of a discrete valuation ring between quillen k groups induced by the obvious inclusion functors are all zero in 6 quillen proved this conjecture if r is essentially of finite type over a field in the case of discrete valuation rings this conjecture reduces to conjecture let r be a discrete valuation ring with residue field k and field of fractions f then for all q z 0 the transfer map which is defined since every k vector space is of finite projective dimension as an r module k k k r is zero equivalently the map r ec f induced by the ring homomorphism r p f is injective a third equivalent formulation is that the boundary map

random walks for image segmentation a novel method is proposed for performing multilabel interactive image segmentation given a small number of pixels with user defined or predefined labels one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels by assigning each pixel to the label for which the greatest probability is calculated a high quality image segmentation may be obtained theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits this algorithm is formulated in discrete space i e on a graph using combinatorial analogues of standard operators and principles from continuous potential theory allowing it to be applied in arbitrary dimension on arbitrary graphs

gauss manin determinants for rank 1 irregular connections on curves let f u to rm spec k be a smooth open curve over a field k supset k where k is an algebraically closed field of characteristic 0 let nabla l to l otimes omega 1 u k be a possibly irregular absolutely integrable connection on a line bundle l a formula is given for the determinant of de rham cohomology with its gaus manin connection big det rf l otimes omega 1 u k det nabla gm big the formula is expressed as a norm from the curve of a cocycle with values in a complex defining algebraic differential characters 7 and this cocycle is shown to exist for connections of arbitrary rank

a real time algorithm for the approximation of level set based curve evolution in this paper we present a complete and practical algorithm for the approximation of level set based curve evolution suitable for real time implementation in particular we propose a two cycle algorithm to approximate level set based curve evolution without the need of solving partial differential equations pdes our algorithm is applicable to a broad class of evolution speeds that can be viewed as composed of a data dependent term and a curve smoothness regularization term we achieve curve evolution corresponding to such evolution speeds by separating the evolution process into two different cycles one cycle for the data dependent term and a second cycle for the smoothness regularization the smoothing term is derived from a gaussian filtering process in both cycles the evolution is realized through a simple element switching mechanism between two linked lists that implicitly represents the curve using an integer valued level set function by careful construction all the key evolution steps require only integer operations a consequence is that we obtain significant computation speedups compared to exact pde based approaches while obtaining excellent agreement with these methods for problems of practical engineering interest in particular the resulting algorithm is fast enough for use in real time video processing applications which we demonstrate through several image segmentation and video tracking experiments

simplex mesh diffusion snakes integrating 2d and 3d deformable models and statistical shape knowledge in a variational framework in volumetric medical imaging the boundaries of structures are frequently blurred due to insufficient resolution this artefact is particularly serious in structures such as articular joints where different cartilage surfaces appear to be linked at the contact regions traditional image segmentation techniques fail to separate such erroneously linked structures and a sensible approach has been the introduction of prior knowledge to the segmentation process although several 3d prior knowledge based techniques that could successfully segment these structures have been published most of them are pixel labelling schemes that generate pixellated images with serious geometric distortions the simplex mesh diffusion snakes segmentation technique presented here is an extension of the two dimensional diffusion snakes but without any restriction on the number of dimensions of the data set this technique integrates a simplex mesh a region based deformable model and statistical shape knowledge into a single energy functional so that it takes into account both the image information available directly from the data set and the shape statistics obtained from a training process the resulting segmentations converge correctly to well defined boundaries and provide a feasible location for those removed boundaries the algorithm has been evaluated using 2d and 3d data sets obtained with magnetic resonance imaging mri and has proved to be robust to most of the mri artefacts providing continuous and smooth curves or surfaces with sub pixel resolution additionally this novel technique opens a wide range of opportunities for segmentation and tracking time dependent 3d structures or data sets with more than three dimensions due to its non restrictive mathematical formulation

high order discontinuous galerkin methods for elliptic problems on surfaces we derive and analyze high order discontinuous galerkin methods for second order elliptic problems on implicitly defined surfaces in mathbb r 3 this is done by carefully adapting the unified discontinuous galerkin framework of d n arnold et al siam j numer anal 39 2002 pp 1749 1779 on a triangulated surface approximating the smooth surface we prove optimal error estimates in both a mesh dependent energy and l 2 norms numerical results validating our theoretical estimates are also presented

a topology preserving level set method for geometric deformable models active contour and surface models also known as deformable models are powerful image segmentation techniques geometric deformable models implemented using level set methods have advantages over parametric models due to their intrinsic behavior parameterization independence and ease of implementation however a long claimed advantage of geometric deformable models the ability to automatically handle topology changes turns out to be a liability in applications where the object to be segmented has a known topology that must be preserved we present a new class of geometric deformable models designed using a novel topology preserving level set method which achieves topology preservation by applying the simple point concept from digital topology these new models maintain the other advantages of standard geometric deformable models including subpixel accuracy and production of nonintersecting curves or surfaces moreover since the topology preserving constraint is enforced efficiently through local computations the resulting algorithm incurs only nominal computational overhead over standard geometric deformable models several experiments on simulated and real data are provided to demonstrate the performance of this new deformable model algorithm

shape based interpolation of multidimensional objects a shape based interpolation scheme for multidimensional images is presented this scheme consists of first segmenting the given image data into a binary image converting the binary image back into a gray image wherein the gray value of a point represents its shortest distance positive value for points of the object and negative for those outside from the cross sectional boundary and then interpolating the gray image the set of all points with nonnegative values associated with them in the interpolated image constitutes the interpolated object the method not only minimizes user involvement in interactive segmentation but also leads to more accurate representation and depiction of dynamic as well as static objects the general methodology and the implementation details of the method are presented and compared on a qualitative and quantitative basis to the existing methods the generality of the proposed scheme is illustrated with a number of medical imaging examples

efficient algorithms for globally optimal trajectories we present serial and parallel algorithms for solving a system of equations that arises from the discretization of the hamilton jacobi equation associated to a trajectory optimization problem of the following type a vehicle starts at a prespecified point x sub o and follows a unit speed trajectory x t inside a region in spl rscr sup m until an unspecified time t that the region is exited a trajectory minimizing a cost function of the form spl int sub 0 sup t r x t dt q x t is sought the discretized hamilton jacobi equation corresponding to this problem is usually solved using iterative methods nevertheless assuming that the function r is positive we are able to exploit the problem structure and develop one pass algorithms for the discretized problem the first algorithm resembles dijkstra s shortest path algorithm and runs in time o n log n where n is the number of grid points the second algorithm uses a somewhat different discretization and borrows some ideas from a variation of dial s shortest path algorithm 1969 that we develop here it runs in time o n which is the best possible under some fairly mild assumptions finally we show that the latter algorithm can be efficiently parallelized for two dimensional problems and with p processors its running time becomes o n p provided that p o spl radic n log n

using a deformable surface model to obtain a shape representation of the cortex this paper examines the problem of obtaining a mathematical representation of the outer cortex of the human brain which is a key problem in several applications including morphological analysis of the brain and spatial normalization and registration of brain images a parameterization of the outer cortex is first obtained using a deformable surface algorithm which motivated by the structure of the cortex is constructed to find the central layer of thick surfaces based on this parameterization a hierarchical representation of the outer cortical structure is proposed through its depth map and its curvature maps at various scales various experiments on magnetic resonance data are presented

the extremal mesh and the understanding of 3d surfaces this paper is about a new concept for the description of 3d smooth surfaces the extremal mesh in previous works we have shown how to extract the extremal lines from 3d images which are the lines where one of the two principal surface curvatures is locally extremal we have also shown how to extract the extremal points which are specific points where the two principal curvatures are both extremal the extremal mesh is the graph of the surface whose vertices are the extremal points and whose edges are the extremal lines it is invariant with respect to rigid transforms the good topological properties of this graph are ensured with a new local geometric invariant of 3d surfaces that we call the gaussian extremality and which allows to overcome orientation problems encountered with previous definitions of the extremal lines and points this paper presents also an algorithm to extract the extremal mesh from 3d images and experiments with synthetic and real 3d medical images show that this graph can be extremely precise and stable

metric optimization for surface analysis in the laplace beltrami embedding space in this paper we present a novel approach for the intrinsic mapping of anatomical surfaces and its application in brain mapping research using the laplace beltrami eigen system we represent each surface with an isometry invariant embedding in a high dimensional space the key idea in our system is that we realize surface deformation in the embedding space via the iterative optimization of a conformal metric without explicitly perturbing the surface or its embedding by minimizing a distance measure in the embedding space with metric optimization our method generates a conformal map directly between surfaces with highly uniform metric distortion and the ability of aligning salient geometric features besides pairwise surface maps we also extend the metric optimization approach for group wise atlas construction and multi atlas cortical label fusion in experimental results we demonstrate the robustness and generality of our method by applying it to map both cortical and hippocampal surfaces in population studies for cortical labeling our method achieves excellent performance in a cross validation experiment with 40 manually labeled surfaces and successfully models localized brain development in a pediatric study of 80 subjects for hippocampal mapping our method produces much more significant results than two popular tools on a multiple sclerosis study of 109 subjects

towards a holistic cortical thickness descriptor heat kernel based grey matter morphology signatures in this paper we propose a heat kernel based regional shape descriptor that may be capable of better exploiting volumetric morphological information than other available methods thereby improving statistical power on brain magnetic resonance imaging mri analysis the mechanism of our analysis is driven by the graph spectrum and the heat kernel theory to capture the volumetric geometry information in the constructed tetrahedral meshes in order to capture profound brain grey matter shape changes we first use the volumetric laplace beltrami operator to determine the point pair correspondence between white grey matter and csf grey matter boundary surfaces by computing the streamlines in a tetrahedral mesh secondly we propose multi scale grey matter morphology signatures to describe the transition probability by random walk between the point pairs which reflects the inherent geometric characteristics thirdly a point distribution model is applied to reduce the dimensionality of the grey matter morphology signatures and generate the internal structure features with the sparse linear discriminant analysis we select a concise morphology feature set with improved classification accuracies in our experiments the proposed work outperformed the cortical thickness features computed by freesurfer software in the classification of alzheimer s disease and its prodromal stage i e mild cognitive impairment on publicly available data from the alzheimer s disease neuroimaging initiative the multi scale and physics based volumetric structure feature may bring stronger statistical power than some traditional methods for mri based grey matter morphology analysis

character sums associated to finite coxeter groups we prove a character sum identity for coxeter arrangements which is a finite field analogue of macdonald s conjecture proved by opdam

smooth functional and structural maps on the neocortex via orthonormal bases of the laplace beltrami operator functional and structural maps such as a curvature cortical thickness and functional magnetic resonance imaging mri maps indexed over the local coordinates of the cortical manifold play an important role in neuropsychiatric studies due to the highly convoluted nature of the cerebral cortex and image quality these functions are generally uninterpretable without proper methods of association and smoothness onto the local coordinate system in this paper we generalized the spline smoothing problem wahba 1990 from a sphere to any arbitrary two dimensional 2 d manifold with boundaries we first seek a numerical solution to orthonormal basis functions of the laplace beltrami lb operator with neumann boundary conditions for a 2 d manifold m then solve the spline smoothing problem in a reproducing kernel hilbert space r k h s of real valued functions on manifold m with kernel constructed from the basis functions the explicit discrete lb representation is derived using the finite element method calculated directly on the manifold coordinates so that finding discrete lb orthonormal basis functions is equivalent to solving an algebraic eigenvalue problem and then smoothed functions in r k h s can be represented as a linear combination of the basis functions we demonstrate numerical solutions of spherical harmonics on a unit sphere and brain orthonormal basis functions on a planum temporale manifold then synthetic data is used to quantify the goodness of the smoothness compared with the ground truth and discuss how many basis functions should be incorporated in the smoothing we present applications of our approach to smoothing sulcal mean curvature cortical thickness and functional statistical maps on submanifolds of the neocortex

more elementary operators that are spectrally bounded we study spectrally bounded elementary operators of length two on a complex unital banach algebra a some related conditions are investigated as well in particular we show that if s is an elementary operator of length two such that s x is quasi nilpotent for every x a then s x 3 rada for every x a