conformal geometry and its applications on 3d shape matching recognition and stitching three dimensional shape matching is a fundamental issue in computer vision with many applications such as shape registration 3d object recognition and classification however shape matching with noise occlusion and clutter is a challenging problem in this paper we analyze a family of quasi conformal maps including harmonic maps conformal maps and least squares conformal maps with regards to 3d shape matching as a result we propose a novel and computationally efficient shape matching framework by using least squares conformal maps according to conformal geometry theory each 3d surface with disk topology can be mapped to a 2d domain through a global optimization and the resulting map is a diffeomorphism i e one to one and onto this allows us to simplify the 3d shape matching problem to a 2d image matching problem by comparing the resulting 2d parametric maps which are stable insensitive to resolution changes and robust to occlusion and noise therefore highly accurate and efficient 3d shape matching algorithms can be achieved by using the above three parametric maps finally the robustness of least squares conformal maps is evaluated and analyzed comprehensively in 3d shape matching with occlusion noise and resolution variation in order to further demonstrate the performance of our proposed method we also conduct a series of experiments on two computer vision applications i e 3d face recognition and 3d nonrigid surface alignment and stitching

homology and closure properties of autostackable groups abstract autostackability for finitely presented groups is a topological property of the cayley graph combined with formal language theoretic restrictions that implies solvability of the word problem the class of autostackable groups is known to include all asynchronously automatic groups with respect to a prefix closed normal form set and all groups admitting finite complete rewriting systems although groups in the latter two classes all satisfy the homological finiteness condition f p we show that the class of autostackable groups includes a group that is not of type f p 3 we also show that the class of autostackable groups is closed under graph products and extensions

structural image restoration through deformable templates abstract prior knowledge on the space of possible images is given in the form of a function or template in some domain the set of all possible true images is assumed to be formed by a composition of that function with continuous mappings of the domain into itself a prior gaussian distribution is given on the set of continuous mappings the observed image is assumed to be a degradation of the true image with additive noise given the observed image a posterior distribution is then obtained and has the form of a nonlinear perturbation of the gaussian measure on the space of mappings we present simulations of the posterior distribution that lead to structural reconstructions of the true image in the sense that it enables us to determine landmarks and other characteristic features of the image as well as to spot pathologies in it moreover we show that the reconstruction algorithm is relatively robust when the images are degraded by noise that is not necessarily additive

cohomologically generic 2 complexes and 3 dimensional poincare complexes this is part of a series of papers devoted to topological applications of deformation theory imp mp2 especially in low dimensional topology bp mp1 our lowdimensional program consists in fixing accessible homological invariants like the betti numbers or the cup product structure of the cohomology and then apply the machinery of deformation theory in order to both compute and distinguish more interesting invariants such as the graded lie algebra associated to the fundamental group gr l itself the entire homotopy type here we address the problem of computing gr ttl s a difficult task in general in terms of h s z the forthcoming bp and imp1 deal with the other above mentioned invariants leading in particular to invariants associated to links and 3 manifolds via deformation theory one may associate to any connected space s assuming h s z to be free of finite rank a graded lie algebra l defined as the quotient of the free lie algebra on has by the ideal generated by the image of the dual of the cup product h sah s h2s see sect 1 our main result reads

diffeomorphisms groups and pattern matching in image analysis in a previous paper it was proposed to see the deformations of a common pattern as the action of an infinite dimensional group we show in this paper that this approac h can be applied numerically for pattern matching in image analysis of digital images using lie group ideas we construct a distance between deformations defined through a metric given the cost of infinitesimal deformations then we propose a numerical scheme to solve a variational problem involving this distance and leading to a sub optimal gradient pattern matching its links with fluid models are established

constraints on deformable models recovering 3d shape and nongrid motion abstract inferring the 3d structures of nonrigidly moving objects from images is a difficult yet basic problem in computational vision our approach makes use of dynamic elastically deformable object models that offer the geometric flexibility to satisfy a diversity of real world visual constraints we specialize these models to include intrinsic forces inducing a preference for axisymmetry image based constraints are applied as extrinsic forces that mold the symmetry seeking model into shapes consistent with image data we describe an extrinsic force that applies constraints derived from profiles of monocularly viewed objects we generalize this constraint force to incorporate profile information from multiple views and use it to exploit binocular image data for time varying images the force becomes dynamic and the model is able to infer not only depth but nonrigid motion as well we demonstrate the recovery of 3d shape and nonrigid motion from natural imagery

non commutative graded algebras and their hilbert series abstract this paper begins to develop a theory of non commutative graded algebras and their hilbert series which parallels the extensive already existing theory on commutative graded algebras we begin with a discussion of order ideals of monomials which give us vector space bases for these algebras we develop next the concept of weak summand which is a special relationship a subalgebra may have to an algebra and the concept of strongly free sets strongly free sets have many properties which are analogous to the properties of regular sequences in commutative algebras we discuss combinatorially free sets which are useful for constructing examples of many of the preceding ideas we conclude with an application of our results to algebraic topology by showing that there is a finite cw complex with only nine positive dimensional cells whose loop space has an irrational poincare series

effective strong dimension in algorithmic information and computational complexity the two most important notions of fractal dimension are hausdorff dimension developed by hausdorff math ann 79 1919 pp 157 179 and packing dimension developed independently by tricot math proc cambridge philos soc 91 1982 pp 57 74 and sullivan acta math 153 1984 pp 259 277 both dimensions have the mathematical advantage of being defined from measures and both have yielded extensive applications in fractal geometry and dynamical systems lutz proceedings of the 15th ieee conference on computational complexity florence italy 2000 ieee computer society press piscataway nj 2000 pp 158 169 has recently proven a simple characterization of hausdorff dimension in terms of gales which are betting strategies that generalize martingales imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of hausdorff dimension including constructive computable polynomial space polynomial time and finite state dimensions work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science in this paper we show that packing dimension can also be characterized in terms of gales moreover even though the usual definition of packing dimension is considerably more complex than that of hausdorff dimension our gale characterization of packing dimension is an exact dual of and every bit as simple as the gale characterization of hausdorff dimension effectivizing our gale characterization of packing dimension produces a variety of effective strong dimensions which are exact duals of the effective dimensions mentioned above in general and in analogy with the classical fractal dimensions the effective strong dimension of a set or sequence is at least as great as its effective dimension with equality for sets or sequences that are sufficiently regular we develop the basic properties of effective strong dimensions and prove a number of results relating them to fundamental aspects of randomness kolmogorov complexity prediction boolean circuit size complexity polynomial time degrees and data compression aside from the above characterization of packing dimension our two main theorems are the following 1 if vec beta beta 0 beta 1 ldots is a computable sequence of biases that are bounded away from 0 and r is random with respect to vec beta then the dimension and strong dimension of r are the lower and upper average entropies respectively of vec beta 2 for each pair of delta 0 2 computable real numbers 0 alpha le beta le 1 there exists a in rm e such that the polynomial time many one degree of a has dimension alpha in e and strong dimension beta in e our proofs of these theorems use a new large deviation theorem for self information with respect to a bias sequence vec beta that need not be convergent

on shape of plane elastic curves we study shapes of planar arcs and closed contours modeled on elastic curves obtained by bending stretching or compressing line segments non uniformly along their extensions shapes are represented as elements of a quotient space of curves obtained by identifying those that differ by shape preserving transformations the elastic properties of the curves are encoded in riemannian metrics on these spaces geodesics in shape spaces are used to quantify shape divergence and to develop morphing techniques the shape spaces and metrics constructed are novel and offer an environment for the study of shape statistics elasticity leads to shape correspondences and deformations that are more natural and intuitive than those obtained in several existing models applications of shape geodesics to the definition and calculation of mean shapes and to the development of shape clustering techniques are also investigated

nasal similarity measure of 3d faces based on curve shape space abstract we propose a novel method for measuring the nasal similarity among 3d faces firstly we construct a representation for the nose shape which is composed of a set of geodesic curves each crosses the bridge of the nose next using these geodesic curves we formulate a similarity measure to compare among noses in the curve shape space under the riemannian framework the shape space is a quotient space for which the scaling translation and rotation are removed since the nose similarity measure is based on the shape comparison the proposed method has the following advantages 1 the similarity measure is robust to facial expressions since the nose is not affected by facial expressions 2 the geometric features of the nose shape match well with the human perception 3 the similarity measure is independent of the mesh grid because the chosen nose curves are not sensitive to the triangular mesh model we construct a nasal hierarchical structure for noses organization which is based on nose similarity measure results in our experiments we evaluate the performance of the proposed method and compare it with competing methods on three public face databases namely frgc2 0 texas3d and bosphorusdb the results show superiority of the proposed method in terms of both the speed and the accuracy when the nasal measurements are processed in the nasal hierarchical structure and the nasal samples with low sampling rate 5 25 of original point cloud

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

initial ideals veronese subrings and rates of algebras abstract let s be a polynomial ring over an infinite field and let i be a homogeneous ideal of s let t d be a polynomial ring whose variables correspond to the monomials of degree d in s we study the initial ideals of the ideals v d i t d that define the veronese subrings of s i in suitable orders they are easily deduced from the initial ideal of i we show that in v d i is generated in degree max reg i d 2 where reg i is the regularity of the ideal i in other words the d th veronese subrings of any commutative graded ring s i has a grobner basis of degree max i d 2 we also give bounds on the regularity of i in terms of the degrees of the generators of in i and some combinatorial data this implies a version of backelin s theorem that high veronese subrings of any ring are homogeneous koszul algebras in the sense of priddy trans amer math soc 152 1970 39 60 we also give a general obstruction for a homogeneous ideal i s to have an initial ideal in i that is generated by quadrics beyond the obvious requirement that i itself should be generated by quadrics and the stronger statement that s i is koszul we use the obstruction to show that in certain dimensions a generic complete intersection of quadrics cannot have an initial ideal that is generated by quadrics for the application to backelin s theorem we require a result of backelin whose proof has never appeared we give a simple proof of a sharpened version bounding the rate of growth of the degrees of generators for syzygies of any multihomogenous module over a polynomial ring modulo an ideal generated by monomials following a method of bruns and herzog

extracting kolmogorov complexity with applications to dimension zero one laws we apply results on extracting randomness from independent sources to extract kolmogorov complexity for any a e 0 given a string x with k x a x we show how to use a constant number of advice bits to efficiently compute another string y y w x with k y 1 e y this result holds for both unbounded and space bounded kolmogorov complexity we use the extraction procedure for space bounded complexity to establish zero one laws for the strong dimensions of complexity classes within espace the unbounded extraction procedure yields a zero one law for the constructive strong dimensions of turing degrees

on the laplace beltrami operator and brain surface flattening in this paper using certain conformal mappings from uniformization theory the authors give an explicit method for flattening the brain surface in a way which preserves angles from a triangulated surface representation of the cortex the authors indicate how the procedure may be implemented using finite elements further they show how the geometry of the brain surface may be studied using this approach

piecewise linear manifolds einstein metrics and ricci flows this article provides an attempt to extend concepts from the theory of riemannian manifolds to piecewise linear p l spaces in particular we propose an analogue of the ricci tensor which we give the name of an einstein vector field on a given set of p l spaces we define and discuss normalized einstein flows p l einstein metrics are defined and examples are provided criteria for flows to approach einstein metrics are formulated second variations of the total scalar curvature at a specific einstein space are calculated

teichmuller shape descriptor and its application to alzheimer s disease study we propose a novel method to apply teichmuller space theory to study the signature of a family of nonintersecting closed 3d curves on a general genus zero closed surface our algorithm provides an efficient method to encode both global surface and local contour shape information the signature teichmuller shape descriptor is computed by surface ricci flow method which is equivalent to solving an elliptic partial differential equation on surfaces and is numerically stable we propose to apply the new signature to analyze abnormalities in brain cortical morphometry experimental results with 3d mri data from alzheimer s disease neuroimaging initiative adni dataset 152 healthy control subjects versus 169 alzheimer s disease ad patients demonstrate the effectiveness of our method and illustrate its potential as a novel surface based cortical morphometry measurement in ad research

three dimensional face recognition an expression invariant 3d face recognition approach is presented our basic assumption is that facial expressions can be modelled as isometries of the facial surface this allows to construct expression invariant representations of faces using the bending invariant canonical forms approach the result is an efficient and accurate face recognition algorithm robust to facial expressions that can distinguish between identical twins the first two authors we demonstrate a prototype system based on the proposed algorithm and compare its performance to classical face recognition methods the numerical methods employed by our approach do not require the facial surface explicitly the surface gradients field or the surface metric are sufficient for constructing the expression invariant representation of any given face it allows us to perform the 3d face recognition task while avoiding the surface reconstruction stage

dimension 1 sequences are close to randoms abstract we show that a sequence has effective hausdorff dimension 1 if and only if it is coarsely similar to a martin lof random sequence more generally a sequence has effective dimension s if and only if it is coarsely similar to a weakly s random sequence further for any s t every sequence of effective dimension s can be changed on density at most h 1 t h 1 s of its bits to produce a sequence of effective dimension t and this bound is optimal

finite grobner shirshov bases for plactic algebras and biautomatic structures for plactic monoids this paper shows that every plactic algebra of finite rank admits a finite grobner shirshov basis the result is proved by using the combinatorial properties of young tableaux to construct a finite complete rewriting system for the corresponding plactic monoid which also yields the corollaries that plactic monoids of finite rank have finite derivation type and satisfy the homological finiteness properties left and right fp fp also answering a question of zelmanov we apply this rewriting system and other techniques to show that plactic monoids of finite rank are biautomatic

a combinatorial approach to the set theoretic solutions of the yang baxter equation a bijective map r x2 x2 where x x1 xn is a finite set is called a set theoretic solution of the yang baxter equation ybe if the braid relation r12r23r12 r23r12r23 holds in x3 a nondegenerate involutive solution x r satisfying r xx xx for all x x is called square free solution there exist close relations between the square free set theoretic solutions of ybe the semigroups of i type the semigroups of skew polynomial type and the bieberbach groups as it was first shown in a joint paper with michel van den bergh in this paper we continue the study of square free solutions x r and the associated yang baxter algebraic structures the semigroup s x r the group g x r and the k algebra a k x r over a field k generated by x and with quadratic defining relations naturally arising and uniquely determined by r we study the properties of the associated yang baxter structures and prove a conjecture of the present author that the three notions a square free solution of set theoretic ybe

cohen macaulayness of modules of invariants for sl2 abstract let u w be finite dimensional representations of g sl2 we give conditions under which u k w g is a cohen macaulay k w g module in particular we obtain an invariant theoretic proof of the fact that the trace ring of generic 2 2 matrices is cohen macaulay

brain surface conformal parameterization with the ricci flow in brain mapping research parameterized 3 d surface models are of great interest for statistical comparisons of anatomy surface based registration and signal processing here we introduce the theories of continuous and discrete surface ricci flow which can create riemannian metrics on surfaces with arbitrary topologies with user defined gaussian curvatures the resulting conformal parameterizations have no singularities and they are intrinsic and stable first we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves secondly we conformally parameterize each cortical surface to a parameter domain with a user designed gaussian curvature arrangement in the parameter domain a shape index based on conformal invariants is computed and inter subject cortical surface matching is performed by solving a constrained harmonic map we illustrate various target curvature arrangements and demonstrate the stability of the method using longitudinal data to map statistical differences in cortical morphometry we studied brain asymmetry in 14 healthy control subjects we used a manifold version of hotelling s t2 test applied to the jacobian matrices of the surface parameterizations a permutation test along with the cumulative distribution of p values were used to estimate the overall statistical significance of differences the results show our algorithm s power to detect subtle group differences in cortical surfaces

coordinate systems for conformal cerebellar flat maps methods quasi conformal flat maps of the human cerebellum 1 4 were created from a high resolution tl weighted mri volume 5 a topologically correct surface was produced from a cerebellar volume defined by a plane parallel to the posterior commisure obex line and orthogonal to a plane passing through the vermal midline our quasi conformal flattening procedure 2 3 was applied to this surface to produce flat maps the cortical surface was parcellated according to 6 and activations produced by a target interception task were imposed eleven readily identifiable anatomical landmarks e g the apex of the fourth ventricle and the rostral most tip of the lingula were defined for the hyperbolic coordinate system one landmark was used as the map center and a second to specify the map orientation we also used a polar coordinate system on the hyperbolic flat map with equiangular grid lines radiating from the map focus and equidistant circles surrounding the focus for the spherical coordinate system one landmark was used as the north pole one as the south pole and a third as a distinguished equatorial point we used the usual latitude and longitude coordinates similar to 7 results and conclusions a major benefit of quasi conformal flat maps is that they can be used to generate canonical surface based coordinate systems for a cortical surface since our fiat maps to the hyperbolic plane and the sphere are l l and onto the coordinate systems produced by these flat maps can be used to quantitatively localize structure and function furthermore these flat maps are mathematically unique different choices of landmarks and coordinate systems will increase the utility of flat maps for studying functional imaging dam we believe that the use of cerebellar flat maps and canonical coordinate systems based on reproducible anatomical landmarks will improve our ability to localize functional activity on the cerebellar cortex and to quantify anatomical and functional differences between individual subjects and groups of subjects

combinatorial ricci flows on surfaces we show that the analogue of hamilton s ricci flow in the combinatorial setting produces solutions which converge exponentially fast to thurston s circle packing on surfaces as a consequence a new proof of thurston s existence of circle packing theorem is obtained as another consequence ricci flow suggests a new algorithm to find circle packings

extracting information is hard a turing degree of non integral effective hausdorff dimension abstract we construct a 2 0 infinite binary sequence with effective hausdorff dimension 1 2 that does not compute a sequence of higher dimension introduced by lutz effective hausdorff dimension can be viewed as a measure of the information density of a sequence in particular the dimension of a 2 is the lim inf of the ratio between the information content and length of initial segments of a thus the main result demonstrates that it is not always possible to extract information from a partially random source to produce a sequence that has higher information density

quadratic algebras yang baxter equation and artin schelter regularity abstract we study two classes of quadratic algebras over a field k the class c n of all n generated pbw algebras with polynomial growth and finite global dimension and the class of quantum binomial algebras we show that a pbw algebra a is in c n iff its hilbert series is h a z 1 1 z n furthermore the class c n contains a unique up to isomorphism monomial algebra a k x 1 x n x j x i 1 i j n a surprising amount can be said when a is a quantum binomial algebra that is its defining relations are nondegenerate square free binomials x y c x y z t c x y k our main result shows that for an n generated quantum binomial algebra a the following conditions are equivalent i a is an artin schelter regular pbw algebra ii a is a yang baxter algebra that is the set of relations defines canonically a solution of the yang baxter equation iii a is a binomial skew polynomial ring with respect to an enumeration of x iv the koszul dual a is a quantum grassmann algebra

on monomial algebras of finite global dimension let g be an associative monomial k algebra if g is assumed to be finitely presented then either g contains a free subalgebra on two monomials or else g has polynomial growth if instead g is assumed to have finite global dimension then either g contains a free subalgebra or else g has a finite presentation and polynomial growth also a graded hopf algebra with generators in degree one and relations in degree two contains a free hopf subalgebra if the number of relations is small enough

optimal mass transport for shape matching and comparison surface based 3d shape analysis plays a fundamental role in computer vision and medical imaging this work proposes to use optimal mass transport map for shape matching and comparison focusing on two important applications including surface registration and shape space the computation of the optimal mass transport map is based on monge brenier theory in comparison to the conventional method based on monge kantorovich theory this method significantly improves the efficiency by reducing computational complexity from o n 2 to o n for surface registration problem one commonly used approach is to use conformal map to convert the shapes into some canonical space although conformal mappings have small angle distortions they may introduce large area distortions which are likely to cause numerical instability thus resulting failures of shape analysis this work proposes to compose the conformal map with the optimal mass transport map to get the unique area preserving map which is intrinsic to the riemannian metric unique and diffeomorphic for shape space study this work introduces a novel riemannian framework conformal wasserstein shape space by combing conformal geometry and optimal mass transport theory in our work all metric surfaces with the disk topology are mapped to the unit planar disk by a conformal mapping which pushes the area element on the surface to a probability measure on the disk the optimal mass transport provides a map from the shape space of all topological disks with metrics to the wasserstein space of the disk and the pullback wasserstein metric equips the shape space with a riemannian metric we validate our work by numerous experiments and comparisons with prior approaches and the experimental results demonstrate the efficiency and efficacy of our proposed approach

shape analysis of planar multiply connected objects using conformal welding shape analysis is a central problem in the field of computer vision in 2d shape analysis classification and recognition of objects from their observed silhouettes are extremely crucial but difficult it usually involves an efficient representation of 2d shape space with a metric so that its mathematical structure can be used for further analysis although the study of 2d simply connected shapes has been subject to a corpus of literatures the analysis of multiply connected shapes is comparatively less studied in this work we propose a representation for general 2d multiply connected domains with arbitrary topologies using conformal welding a metric can be defined on the proposed representation space which gives a metric to measure dissimilarities between objects the main idea is to map the exterior and interior of the domain conformally to unit disks and circle domains unit disk with several inner disks removed using holomorphic 1 forms a set of diffeomorphisms of the unit circle mathbb s 1 can be obtained which together with the conformal modules are used to define the shape signature a shape distance between shape signatures can be defined to measure dissimilarities between shapes we prove theoretically that the proposed shape signature uniquely determines the multiply connected objects under suitable normalization we also introduce a reconstruction algorithm to obtain shapes from their signatures this completes our framework and allows us to move back and forth between shapes and signatures with that a morphing algorithm between shapes can be developed through the interpolation of the beltrami coefficients associated with the signatures experiments have been carried out on shapes extracted from real images results demonstrate the efficacy of our proposed algorithm as a stable shape representation scheme

compression of data streams down to their information content according to the kolmogorov complexity every finite binary string is compressible to a shortest code its information content from which it is effectively recoverable we investigate the extent to which this holds for the infinite binary sequences streams we devise a new coding method that uniformly codes every stream x into an algorithmically random stream y in such a way that the first n bits of x are recoverable from the first i x upharpoonright n bits of y where i is any partial computable information content measure that is defined on all prefixes of x and where x upharpoonright n is the initial segment of x of length n as a consequence if g is any computable upper bound on the initial segment prefix free complexity of x then x is computable from an algorithmically random y with oracle use at most g alternatively making no use of such a computable bound g one can achieve an the oracle use bounded above by k x upharpoonright n log n this provides a strong analogue of shannon s source coding theorem for the algorithmic information theory

regularity of trace rings of generic matrices is called the ring of m generic n x n matrices g n the f subalgebra of m em generated by g and tr g is the ring of m generic n x n matrices and is denoted by t n these trace rings appear naturally in the study of finite dimensional representations of free algebras and in the invariant theory of n x n matrices 7 unlike rings of generic matrices t n shares some properties with commutative polynomial rings e g they are maximal orders and even unique factorization rings in the sense of chatters and jordan however their homological properties are far from being understood the main aim of this paper is to prove the following result

shape matching and object recognition using shape contexts we present a novel approach to measuring similarity between shapes and exploit it for object recognition in our framework the measurement of similarity is preceded by 1 solving for correspondences between points on the two shapes 2 using the correspondences to estimate an aligning transform in order to solve the correspondence problem we attach a descriptor the shape context to each point the shape context at a reference point captures the distribution of the remaining points relative to it thus offering a globally discriminative characterization corresponding points on two similar shapes will have similar shape contexts enabling us to solve for correspondences as an optimal assignment problem given the point correspondences we estimate the transformation that best aligns the two shapes regularized thin plate splines provide a flexible class of transformation maps for this purpose the dissimilarity between the two shapes is computed as a sum of matching errors between corresponding points together with a term measuring the magnitude of the aligning transform we treat recognition in a nearest neighbor classification framework as the problem of finding the stored prototype shape that is maximally similar to that in the image results are presented for silhouettes trademarks handwritten digits and the coil data set

the ricci flow on the 2 sphere the classical uniformization theorem interpreted differential geomet rically states that any riemannian metric on a 2 dimensional surface ispointwise conformal to a constant curvature metric thus one can con sider the question of whether there is a natural evolution equation whichconformally deforms any metric on a surface to a constant curvature met ric the primary interest in this question is not so much to give a newproof of the uniformization theorem but rather to understand nonlinearparabolic equations better especially those arising in differential geome try a sufficiently deep understanding of parabolic equations should yieldimportant new results in riemannian geometry the question in the preceding paragraph has been studied by richardhamilton 3 and brad osgood ralph phillips and peter sarnak 6 in 3 hamilton studied the following equation which we refer to as

optimizing equiareality of nurbs surfaces using composite mobius transformations the equiareality of nurbs surfaces greatly affects the results of visualization and tessellation applications especially when dealing with extruding and intruding shapes to obtain more equiareal parameterizations of nurbs surfaces than the mobius based method an optimization algorithm is presented in this paper based on the more flexible composite reparameterization functions for fixed knots the optimal composite reparameterization can be obtained using the levenberg marquardt method for a given tolerance a uniform subdivision scheme is interleaved with the optimization procedure and this process finishes until the change of the equiareal deviation energy is less than the given tolerance examples are given to show the performance of our algorithm for visualization and tessellation applications

generalized gradients priors on minimization flows this paper tackles an important aspect of the variational problem underlying active contours optimization by gradient flows classically the definition of a gradient depends directly on the choice of an inner product structure this consideration is largely absent from the active contours literature most authors explicitely or implicitely assume that the space of admissible deformations is ruled by the canonical l 2 inner product the classical gradient flows reported in the literature are relative to this particular choice here we investigate the relevance of using i other inner products yielding other gradient descents and ii other minimizing flows not deriving from any inner product in particular we show how to induce different degrees of spatial consistency into the minimizing flow in order to decrease the probability of getting trapped into irrelevant local minima we report numerical experiments indicating that the sensitivity of the active contours method to initial conditions which seriously limits its applicability and efficiency is alleviated by our application specific spatially coherent minimizing flows we show that the choice of the inner product can be seen as a prior on the deformation fields and we present an extension of the definition of the gradient toward more general priors

hyperbolic harmonic mapping for surface registration automatic computation of surface correspondence via harmonic map is an active research field in computer vision computer graphics and computational geometry it may help document and understand physical and biological phenomena and also has broad applications in biometrics medical imaging and motion capture industries although numerous studies have been devoted to harmonic map research limited progress has been made to compute a diffeomorphic harmonic map on general topology surfaces with landmark constraints this work conquers this problem by changing the riemannian metric on the target surface to a hyperbolic metric so that the harmonic mapping is guaranteed to be a diffeomorphism under landmark constraints the computational algorithms are based on ricci flow and nonlinear heat diffusion methods the approach is general and robust we employ our algorithm to study the constrained surface registration problem which applies to both computer vision and medical imaging applications experimental results demonstrate that by changing the riemannian metric the registrations are always diffeomorphic and achieve relatively high performance when evaluated with some popular surface registration evaluation standards

a theory of multiscale curvature based shape representation for planar curves a shape representation technique suitable for tasks that call for recognition of a noisy curve of arbitrary shape at an arbitrary scale or orientation is presented the method rests on the describing a curve at varying levels of detail using features that are invariant with respect to transformations that do not change the shape of the curve three different ways of computing the representation are described they result in three different representations the curvature scale space image the renormalized curvature scale space image and the resampled curvature scale space image the process of describing a curve at increasing levels of abstraction is referred to as the evolution or arc length evolution of that curve several evolution and arc length evolution properties of planar curves are discussed

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

trace rings of generic matrices are cohen macaulay in this paper we prove that trace rings of generic matrics are cohen macaulay theorem 7 3 6 this is done by relating this problem to a conjecture of stanley about modules of invariants under a reductive group we prove a slightly weakened version conjecture 3 4 of this conjecture in special cases theorem 6 1 8 in particular we obtain that conjecture 3 4 is true for sl2 remark 6 1 10 department of mathematics universiteitsplein 1 universitaire instelling antwer pen 2160 wilrijk belgium this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 44 27 utc all use subject to http about jstor org terms

non commutative resolutions of quotient singularities for reductive groups in this paper we generalize standard results about non commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups we show in particular that quotient singularities for reductive groups always have non commutative resolutions in an appropriate sense moreover we exhibit a large class of such singularities which have twisted non commutative crepant resolutions we discuss a number of examples both new and old that can be treated using our methods notably we prove that twisted non commutative crepant resolutions exist in previously unknown cases for determinantal varieties of symmetric and skew symmetric matrices in contrast to almost all prior results in this area our techniques are algebraic and do not depend on knowing a commutative resolution of the singularity

computing quasiconformal maps using an auxiliary metric and discrete curvature flow surface mapping plays an important role in geometric processing which induces both area and angular distortions if the angular distortion is bounded the mapping is called a quasiconformal mapping qc mapping many surface mappings in our physical world are quasiconformal the angular distortion of a qc mapping can be represented by the beltrami differentials according to qc teichmuller theory there is a one to one correspondence between the set of beltrami differentials and the set of qc surface mappings under normalization conditions therefore every qc surface mapping can be fully determined by the beltrami differential and reconstructed by solving the so called beltrami equation in this work we propose an effective method to solve the beltrami equation on general riemann surfaces the solution is a qc mapping associated with the prescribed beltrami differential the main strategy is to define an auxiliary metric am on the domain surface such that the original qc mapping becomes conformal under the auxiliary metric the desired qc mapping can then be obtained by using the conventional conformal mapping method in this paper we first formulate a discrete analogue of qc mappings on triangular meshes then we propose an algorithm to compute discrete qc mappings using the discrete yamabe flow method to the best of our knowledge it is the first work to compute the discrete qc mappings for general riemann surfaces especially with different topologies numerically the discrete qc mapping converges to the continuous solution as the mesh grid size approaches to 0 we tested our algorithm on surfaces scanned from real life with different topologies experimental results demonstrate the generality and accuracy of our auxiliary metric method

ricci flow embedding for rectifying non euclidean dissimilarity data pairwise dissimilarity representations are frequently used as an alternative to feature vectors in pattern recognition one of the problems encountered in the analysis of such data is that the dissimilarities are rarely euclidean while statistical learning algorithms often rely on euclidean dissimilarities such non euclidean dissimilarities are often corrected or a consistent euclidean geometry is imposed on them via embedding this paper commences by reviewing the available algorithms for analysing non euclidean dissimilarity data the novel contribution is to show how the ricci flow can be used to embed and rectify non euclidean dissimilarity data according to our representation the data is distributed over a manifold consisting of patches each patch has a locally uniform curvature and this curvature is iteratively modified by the ricci flow the raw dissimilarities are the geodesic distances on the manifold rectified euclidean dissimilarities are obtained using the ricci flow to flatten the curved manifold by modifying the individual patch curvatures we use two algorithmic components to implement this idea firstly we apply the ricci flow independently to a set of surface patches that cover the manifold second we use curvature regularisation to impose consistency on the curvatures of the arrangement of different surface patches we perform experiments on three real world datasets and use these to determine the importance of the different algorithmic components i e ricci flow and curvature regularisation we conclude that curvature regularisation is an essential step needed to control the stability of the piecewise arrangement of patches under the ricci flow

focusr feature oriented correspondence using spectral regularization a method for precise surface matching existing methods for surface matching are limited by the tradeoff between precision and computational efficiency here we present an improved algorithm for dense vertex to vertex correspondence that uses direct matching of features defined on a surface and improves it by using spectral correspondence as a regularization this algorithm has the speed of both feature matching and spectral matching while exhibiting greatly improved precision distance errors of 1 4 percent the method focusr incorporates implicitly such additional features to calculate the correspondence and relies on the smoothness of the lowest frequency harmonics of a graph laplacian to spatially regularize the features in its simplest form focusr is an improved spectral correspondence method that nonrigidly deforms spectral embeddings we provide here a full realization of spectral correspondence where virtually any feature can be used as an additional information using weights on graph edges but also on graph nodes and as extra embedded coordinates as an example the full power of focusr is demonstrated in a real case scenario with the challenging task of brain surface matching across several individuals our results show that combining features and regularizing them in a spectral embedding greatly improves the matching precision to a submillimeter level while performing at much greater speed than existing methods

independent minimum length programs to translate between given strings a string p is called a program to compute y given x if u p x y where u denotes universal programming language kolmogorov complexity k y x of y relative to x is defined as minimum length of a program to compute y given x let k x denote k x emptystring kolmogorov complexity of x and let i x y k x k y k x y the amount of mutual information in x y in the present paper we answer in the negative the following question posed in bennett et al ieee trans inform theory 44 4 1998 1407 1423 is it true that for any strings x y there are independent minimum length programs p q to translate between x y that is is it true that for any x y there are p q such that u p x y u q y x the length of p is k y x the length of q is k x y and i p q 0 where the last three equalities hold up to an additive o log k x y k y x term

on the homology of associative algebras we present a new free resolution for k as an g module where g is an associative augmented algebra over a field k the resolution reflects the combinatorial properties of g introduction let k be a field and let g be an associative augmented k algebra for many purposes one wishes to have a projective resolution of k as a g module the bar resolution is always easy to define but it is often too large to use in practice at the other extreme minimal resolutions may exist but they are often hard to write down in a way that is amenable to calculations the main theorem of this paper presents a compromise resolution though rarely minimal it is small enough to offer some bounds but explicit enough to facilitate calculations as it relies heavily upon combinatorial constructions it is best suited for analyzing otherwise tricky algebras given via generators and relations since several results we get as consequences of the main theorem have been obtained before through other means this paper may be viewed as generalizing and unifying several seemingly unrelated ideas in particular we are generalizing priddy s results on koszul algebras 12 extending homology computations by govorov 9 and backelin 3 and complementing bergman s methods regarding the diamond lemma 6 three results may be of interest the homology of the mod p steenrod algebra is given in terms of the homology of a new chain complex smaller than the a algebra in theorem 3 5 formula 16 offers an efficient algorithm for the determination of hilbert series and theorem 4 2 asserts the existence of new bounds on the torsion groups of commutative graded rings 1 definitions and the main theorem throughout this paper k denotes any field and g is an associative k algebra with unity the field k embeds in g via 77 k g and we suppose that g has an augmentation i e a k algebra map s g k for which 77 is a right inverse s denotes a set of generators for g as a k algebra and k s is the free associative k algebra with unity on s there is a canonical surjection f k s g once s is chosen and the augmentation e is determined once we know s x for each x e s in particular this means that k s may be augmented such that f becomes a map of augmented algebras to s we associate a function e s z called a grading in the absence of a more compelling choice we often take e to be grading by length i e e x 1 for received by the editors may 23 1983 and in revised form february 22 1984 this paper was the subject of an invited one hour address in boulder colorado during the week long ams summer program on combinatorics and algebra june 1983 1980 mathematics subject classification 1985 revision primary 16a62 secondary 13d03 55s10 1986 american mathematical society 0002 9947 86 1 00 25 per page

free products cyclic homology and the gauss manin connection abstract we use the techniques of cuntz and quillen to present a new approach to periodic cyclic homology our construction is based on a t d t i a noncommutative equivariant de rham complex of an associative algebra a here d is the karoubi de rham differential and i is an operation analogous to contraction with a vector field as a byproduct we give a simple explicit construction of the gauss manin connection introduced earlier by e getzler on the relative periodic cyclic homology of a flat family of associative algebras over a central base ring we introduce and study free product deformations of an associative algebra a new type of deformation over a not necessarily commutative base ring natural examples of free product deformations arise from preprojective algebras and group algebras for compact surface groups

ricci flow for 3d shape analysis ricci flow is a powerful curvature flow method which is invariant to rigid motion scaling isometric and conformal deformations we present the first application of surface ricci flow in computer vision previous methods based on conformal geometry which only handle 3d shapes with simple topology are subsumed by the ricci flow based method which handles surfaces with arbitrary topology we present a general framework for the computation of ricci flow which can design any riemannian metric by user defined curvature the solution to ricci flow is unique and robust to noise we provide implementation details for ricci flow on discrete surfaces of either euclidean or hyperbolic background geometry our ricci flow based method can convert all 3d problems into 2d domains and offers a general framework for 3d shape analysis we demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems such as 3d shape matching and registration and shape indexing surfaces with large nonrigid anisotropic deformations can be registered using ricci flow with constraints of feature points and curves we show how conformal equivalence can be used to index shapes in a 3d surface shape space with the use of teichmuller space coordinates experimental results are shown on 3d face data sets with large expression deformations and on dynamic heart data

morse theory from an algebraic viewpoint forman s discrete morse theory is studied from an algebraic viewpoint and we show how this theory can be extended to chain complexes of modules over arbitrary rings as applications we compute the homologies of a certain family of nilpotent lie algebras and show how the algebraic morse theory can be used to derive the classical anick resolution as well as a new two sided anick resolution

genus zero surface conformal mapping and its application to brain surface mapping we developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one forms for surfaces with or without boundaries gu and yau 2002 gu and yau 2003 for genus zero surfaces our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map in this paper we apply the algorithm to the cortical surface matching problem we use a mesh structure to represent the brain surface further constraints are added to ensure that the conformal map is unique empirical tests on magnetic resonance imaging mri data show that the mappings preserve angular relationships are stable in mris acquired at different times and are robust to differences in data triangulation and resolution compared with other brain surface conformal mapping algorithms our algorithm is more stable and has good extensibility