hard lefschetz theorem for valuations complex integral geometry and unitarily invariant valuations we obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets a version of the hard lefschetz theorem using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations it implies new integral geometric formulas for real submanifolds in hermitian spaces generalizing the classical kinematic formulas in euclidean

a linear programming approach for the weighted graph matching problem a linear programming lp approach is proposed for the weighted graph matching problem a linear program is obtained by formulating the graph matching problem in l sub 1 norm and then transforming the resulting quadratic optimization problem to a linear one the linear program is solved using a simplex based algorithm then approximate 0 1 integer solutions are obtained by applying the hungarian method on the real solutions of the linear program the complexity of the proposed algorithm is polynomial time and it is o n sup 6 l for matching graphs of size n the developed algorithm is compared to two other algorithms one is based on an eigendecomposition approach and the other on a symmetric polynomial transform experimental results showed that the lp approach is superior in matching graphs than both other methods

scale and rotation invariant matching using linearly augmented trees we propose a novel linearly augmented tree method for efficient scale and rotation invariant object matching the proposed method enforces pairwise matching consistency defined on trees and high order constraints on all the sites of a template the pairwise constraints admit arbitrary metrics while the high order constraints use l1 norms and therefore can be linearized such a linearly augmented tree formulation introduces hyperedges and loops into the basic tree structure but different from a general loopy graph its special structure allows us to relax and decompose the optimization into a sequence of tree matching problems that are efficiently solvable by dynamic programming the proposed method also works on continuous scale and rotation parameters we can match with a scale up to any large value with the same efficiency our experiments on ground truth data and a variety of real images and videos show that the proposed method is efficient accurate and reliable

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

sl m c equivariant and translation covariant continuous tensor valuations abstract the space of continuous sl m c equivariant m 2 and translation covariant valuations taking values in the space of real symmetric tensors on c m r 2 m of rank r 0 is completely described the classification involves the moment tensor valuation for r 1 and is analogous to the known classification of the corresponding tensor valuations that are sl 2 m r equivariant although the method of proof cannot be adapted

centro affine tensor valuations abstract all measurable sl n covariant symmetric tensor valuations on convex polytopes containing the origin in their interiors are completely classified it is shown that essentially the only examples of such valuations are the moment tensor and a tensor derived from l p surface area measures this generalizes and unifies earlier results for the scalar vector and matrix valued case

integral geometry of unitary area measures abstract the existence of kinematic formulas for area measures with respect to any connected closed subgroup of the orthogonal group acting transitively on the unit sphere is established in particular the kinematic operator for area measures is shown to have the structure of a co product in the case of the unitary group the algebra associated to this co product is described explicitly in terms of generators and relations as a consequence a simple algorithm that yields explicit kinematic formulas for unitary area measures is obtained

ellipsoids and matrix valued valuations a classification is obtained of borel measurable gl n covariant symmetric matrix valued valuations on the space of n dimensional convex polytopes the only ones turn out to be the moment matrix corresponding to the classical legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by lutwak yang and zhang a classical concept from mechanics is the legendre ellipsoid or ellipsoid of inertia 2k associated with a convex body k r it can be defined as the unique ellipsoid centered at the center of mass of k such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of k if we fix a scalar product x y for x y r 2k can be defined by the moment matrix m2 k of k this is the n n matrix with coefficients

an integer linear program for substitution tolerant subgraph isomorphism and its use for symbol spotting in technical drawings this paper tackles the problem of substitution tolerant subgraph isomorphism which is a specific class of error tolerant isomorphism this problem aims at finding a subgraph isomorphism of a pattern graph s in a target graph g this isomorphism only considers label substitutions and forbids vertex and edge insertion in g this kind of subgraph isomorphism is often needed in pattern recognition problems when graphs are attributed with real values and no exact matching can be found between attributes due to noise our proposal to solve the problem of substitution tolerant subgraph isomorphism relies on its formulation in the integer linear program ilp formalism using a general ilp solver the approach is able to find if one exists a mapping of a pattern graph into a target graph such that the topology of the searched graph is kept and the editing operations between the labels have a minimal cost this technique is evaluated on both a set of synthetic graphs and a problem of symbol detection in technical drawings in the second case document and symbol images are represented by vector attributed region adjacency graphs built from a segmentation process obtained results demonstrate the relevance of considering subgraph isomorphism as an optimization process

structure of the unitary valuation algebra s alesker has shown that if g is a compact subgroup of o n acting transitively on the unit sphere sn 1 then the vector space valg of continuous translation invariant g invariant convex valuations on rn has the structure of a finite dimensional graded algebra over r satisfying poincare duality we show that the kinematic formulas for g are determined by the product pairing using this result we then show that the algebra valu n is isomorphic to r s t fn 1 fn 2 where s t have degrees 2 and 1 respectively and the polynomial fi is the degree i term of the power series log 1 s t

difference bodies in complex vector spaces abstract a complete classification is obtained of continuous translation invariant minkowski valuations on an m dimensional complex vector space which are covariant under the complex special linear group

fisher information and matrix valued valuations abstract all continuous and affinely contravariant matrix valued valuations on the sobolev space w 1 2 r n are completely classified it is shown that there is a unique such valuation this valuation turns out to be the fisher information matrix

inexact bayesian point pattern matching for linear transformations we introduce a novel bayesian inexact point pattern matching model that assumes that a linear transformation relates the two sets of points the matching problem is inexact due to the lack of one to one correspondence between the point sets and the presence of noise the algorithm is itself inexact we use variational bayesian approximation to estimate the posterior distributions in the face of a problematic evidence term the method turns out to be similar in structure to the iterative closest point algorithm

an eigendecomposition approach to weighted graph matching problems an approximate solution to the weighted graph matching problem is discussed for both undirected and directed graphs the weighted graph matching problem is that of finding the optimum matching between two weighted graphs which are graphs with weights at each arc the proposed method uses an analytic instead of a combinatorial or iterative approach to the optimum matching problem using the eigendecompositions of the adjacency matrices in the case of the undirected graph matching problem or hermitian matrices derived from the adjacency matrices in the case of the directed graph matching problem a matching close to the optimum can be found efficiently when the graphs are sufficiently close to each other simulation results are given to evaluate the performance of the proposed method

symbol recognition by error tolerant subgraph matching between region adjacency graphs we propose an error tolerant subgraph isomorphism algorithm formulated in terms of region adjacency graphs rag a set of edit operations to transform one rag into another one are defined as regions are represented by polylines and string matching techniques are used to measure their similarity the algorithm follows a branch and bound approach driven by the rag edit operations this formulation allows matching computing under distorted inputs and also reaching a solution in a near polynomial time the algorithm has been used for recognizing symbols in hand drawn diagrams

boundary blow up solutions with a spike layer abstract we prove that for large 0 the boundary blow up problem u u u a 1 u in u has a solution with a spike located near the most centered part of our proof is based on the reduction method and uses a cut off technique to avoid the difficulties for the estimates near the boundary of

continuous rotation invariant valuations on convex sets the notion of valuation on convex sets can be considered as a generalization of the notion of measure which is deflned only on the class of convex compact sets it is well known that there are important and interesting examples of valuations on convex sets which are not measures in the usual sense as for example the mixed volumes basic deflnitions and some classical examples are discussed in section 2 of this paper for more detailed information we refer to the surveys mc sch and mc3 throughout this paper all the valuations are assumed to be continuous with respect to the hausdorfi metric note that the theory of valuations which are invariant or covariant with respect to translations belongs to the classical part of convex geometry there exists an explicit description of translation invariant continuous valuations on 1 and 2 due to hadwiger h1 the case of 2 is nontrivial continuous rigid motion invariant valuations on d are completely classifled by the remarkable hadwiger theorem as linear combinations of the quermassintegrals cf h2 or for a simpler proof k

factorized graph matching graph matching gm is a fundamental problem in computer science and it plays a central role to solve correspondence problems in computer vision gm problems that incorporate pairwise constraints can be formulated as a quadratic assignment problem qap although widely used solving the correspondence problem through gm has two main limitations 1 the qap is np hard and difficult to approximate 2 gm algorithms do not incorporate geometric constraints between nodes that are natural in computer vision problems to address aforementioned problems this paper proposes factorized graph matching fgm fgm factorizes the large pairwise affinity matrix into smaller matrices that encode the local structure of each graph and the pairwise affinity between edges four are the benefits that follow from this factorization 1 there is no need to compute the costly in space and time pairwise affinity matrix 2 the factorization allows the use of a path following optimization algorithm that leads to improved optimization strategies and matching performance 3 given the factorization it becomes straight forward to incorporate geometric transformations rigid and non rigid to the gm problem 4 using a matrix formulation for the gm problem and the factorization it is easy to reveal commonalities and differences between different gm methods the factorization also provides a clean connection with other matching algorithms such as iterative closest point experimental results on synthetic and real databases illustrate how fgm outperforms state of the art algorithms for gm the code is available at http humansensing cs cmu edu fgm

even valuations on convex bodies the notion of eveil valuation is introduced as a natural generalization of volume on cornpact convex subsets of euclidean space a recent characterization theorem for volume leads in turn to a connection between even valuations on compact convex sets and continuous functions on grassmannians this connection can be described in part using generating distributions for symmetric compact convex sets we also explore some consequences of these characterization results in convex and integral geometry recent interest in volume as a valuation on compact conlvex sets stems from hilbert s third problem which is actually an ancient problem recast in modern terms hilbert asked if two polytopes p and q can be each cut into a finite number of pieces p pn and q qm with each pi congruent to qi by a rigid euclidean motion provided that p and q have the same volume 18 this question was answered in the negative by max dehn 2 4 29 who found a functionlal on polytopes that is invariant under dissections over rigid motions while varying in value among polytopes of equal volume in other words the dehn invariant is a simple rigid motion invariant valuation on polytopes that is not equal to volume under any normalization dehn s solution left open the question of exactly what conditions oil p and q imply equidissectability over the group of rigid motions although this problem was solved by hadwiger in the case where only translations and no rotations nor reflections are permitted see 2 16 17 26 29 in the course of studying this and related problems hadwiger discovered a characterization of euclidean volume as a continuous rigid motion invariant simple valuation on compact convex sets that is a continuous rigid motion invariant valuation that vanishes on convex sets of less than full dimension this result led in turn to a complete characterization of all continuous rigid motion inivariant valuations on compact convex sets in 1r as consisting of a real n 4 dimensional vector space spanned by the intrinsic volumes or quermassintegrals 16 also 20 21 31 since many standard functionals and integral operators can be interpreted as invariant valuations such as intrinsic volumes mean projections crofton and kinematic formulas what came to be known as hadwiger s chtracterization theorem proved to be a valuable tool for generating quick and effortless proofs of many formulas and equations in integral geometry unfortunately hadwiger s original proof was long and difficult 161 while seeking a shorter proof of hadwiger s volume characterization the author discovered received by the editors june 24 1996 and in revised form september 29 1997 1991 mathematics subject classification primary 52a22 52a38 52a39 52b45 research supported in part by nsf grants dms 9022140 to msri and dms 9626688 to the author 1999 american mathematical society 71 this content downloaded from 207 46 13 52 on mon 24 oct 2016 04 15 57 utc all use subject to http about jstor org terms

projection bodies in complex vector spaces the space of minkowski valuations on an m dimensional complex vector space which are continuous translation invariant and contravariant under the complex special linear group is explicitly described each valuation with these properties is shown to satisfy geometric inequalities of the brunn minkowski aleksandrov fenchel and minkowski type

tensor valuations on lattice polytopes abstract the ehrhart polynomial and the reciprocity theorems by ehrhart macdonald are extended to tensor valuations on lattice polytopes a complete classification is established of tensor valuations of rank up to eight that are equivariant with respect to the special linear group over the integers and translation covariant every such valuation is a linear combination of the ehrhart tensors which is shown to no longer hold true for rank nine

projection bodies and valuations let be the projection operator which maps every polytope to its projection body it is well known that maps the set of polytopes pn in rn into pn that it is a valuation and that for every p pn p is affinely associated to p it is shown that these properties characterize the projection operator this proves a conjecture by lutwak

hermitian integral geometry we give in explicit form the principal kinematic formula for the action of the ane unitary group on c n together with a straightforward algebraic method for computing the full array of unitary kinematic formulas expressed in terms of certain convex valuations introduced essentially by h tasaki we introduce also several other canonical bases for the algebra of unitary invariant valuations explore their interrelations and characterize in these terms the cones of positive and monotone elements

moments and valuations all measurable and rm sl n covariant vector valued valuations on convex polytopes containing the origin in their interiors are completely classified the moment vector is shown to be essentially the only such valuation

classification of invariant valuations on the quaternionic plane abstract we describe the orbit space of the action of the group sp 2 sp 1 on the real grassmann manifolds gr k h 2 in terms of certain quaternionic matrices of moore rank not larger than 2 we then give a complete classification of valuations on the quaternionic plane h 2 which are invariant under the action of the group sp 2 sp 1