random convex hulls in a product of balls the convex hull of a set of points sampled independently and uniformly from the cartesian product of balls of various dimensions is investigated bounds on the asymptotic behavior of the expected combinatorial complexity volume and mean width are derived when the distribution is held fixed and the sample size approaches infinity the expected combinational complexity and volume are found to depend up to constant factors only on the greatest dimension of any factor ball and the number of balls of that dimension on the other hand the expected mean width depends only on the number of balls and the dimensions of the product

random polytopes and the efron stein jackknife inequality let k be a smooth convex body the convex hull of independent random points in k is a random polytope estimates for the variance of the volume and the variance of the number of vertices of a random polytope are obtained the essential step is the use of the efron stein jackknife inequality for the variance of symmetric statistics consequences are strong laws of large numbers for the volume and the number of vertices of the random polytope a conjecture of barany concerning random and best approximation of convex bodies is confirmed analogous results for random polytopes with vertices on the boundary of the convex body are given

a method for registration of 3 d shapes the authors describe a general purpose representation independent method for the accurate and computationally efficient registration of 3 d shapes including free form curves and surfaces the method handles the full six degrees of freedom and is based on the iterative closest point icp algorithm which requires only a procedure to find the closest point on a geometric entity to a given point the icp algorithm always converges monotonically to the nearest local minimum of a mean square distance metric and the rate of convergence is rapid during the first few iterations therefore given an adequate set of initial rotations and translations for a particular class of objects with a certain level of shape complexity one can globally minimize the mean square distance metric over all six degrees of freedom by testing each initial registration one important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model prior to shape inspection experimental results show the capabilities of the registration algorithm on point sets curves and surfaces

a law of large numbers for nearest neighbour statistics in practical data analysis methods based on proximity near neighbour relationships between sample points are important because these relations can be computed in time n log n as the number of points n associated with such methods are a class of random variables defined to be functions of a given point and its nearest neighbours in the sample if the sample points are independent and identically distributed the associated random variables will also be identically distributed but not independent despite this we show that random variables of this type satisfy a strong law of large numbers in the sense that their sample means converge to their expected values almost surely as the number of sample points n

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

random points and lattice points in convex bodies assume k subset of r d is a convex body and x is a large finite subset of k how many convex polytopes are there whose vertices belong to x is there a typical shape of such polytopes how well does the maximal such polytope which is actually the convex hull of x approximate k we are interested in these questions mainly in two cases the first is when x is a random sample of n uniform independent points from k in this case motivation comes from sylvester s famous four point problem and from the theory of random polytopes the second case is when x k boolean and z d where z d is the lattice of integer points in r d and the questions come from integer programming and geometry of numbers surprisingly or not so surprisingly the answers in the two cases are rather similar

automatic extraction of sulcal lines on cortical surfaces based on anisotropic geodesic distance analyzing cortical sulci is important for studying cortical morphology and brain functions although sulcal lines on cortical surfaces can be defined in various ways it is critical in a neuroimaging study to define a sulcal line along the valley of a cortical surface with a high curvature within a sulcus to extract the sulcal lines automatically we present a new geometric algorithm based on the computation of anisotropic skeletons of sulcal regions because anisotropic skeletons are highly adaptive to the anisotropic nature of the surface shape the resulting sulcal lines lie accurately on the valleys of the sulcal areas our sulcal lines remain unchanged under local shape variabilities in different human brains through experiments we show that the errors of the sulcal lines for both synthetic data and real cortical surfaces were nearly as constant as the function of random noise by measuring the changes in sulcal shape in alzheimer s disease ad patients we further investigated the effectiveness of the accuracy of our sulcal lines using a large sample of mri data this study involved 70 normal controls n men women 29 41 age mean sd 71 7 4 9 years and 100 ad subjects 37 63 72 3 5 5 we observe significantly lower absolute average mean curvature and shallower sulcal depth in ad subjects where the group difference becomes more significant if we measure the quantities along the sulcal lines rather than over the entire sulcal area the most remarkable difference in the ad patients was the average sulcal depth control 11 70 and ad 11 34

constrained convex bodies with extremal affine surface areas abstract given a convex body k r n and p r we introduce and study the extremal inner and outer affine surface areas i s p k sup k k as p k and o s p k inf k k as p k where as p k denotes the l p affine surface area of k and the supremum is taken over all convex subsets of k and the infimum over all convex compact subsets containing k the convex body that realizes i s 1 k in dimension 2 was determined in 3 where it was also shown that this body is the limit shape of lattice polytopes in k in higher dimensions no results are known about the extremal bodies we use a thin shell estimate of 23 and the lowner ellipsoid to give asymptotic estimates on the size of i s p k and o s p k surprisingly it turns out that both quantities are proportional to a power of volume

intrinsic volumes and f vectors of random polytopes on appelle polytope aleatoire k n l enveloppe convexe de n points choisis de facon aleatoire dans un corps convexe k on s interesse a l approximation stochastique de k par k n plus particulierement on obtient des resultats d approximation pour les esperances de v s k v s k n ou v s est le s ieme volume intrinseque du nombre de faces s dimensionnelles de k n et de la distance de hausdorff entre k et k n

central limit theorems for random polytopes let k be a smooth convex set the convex hull of independent random points in k is a random polytope central limit theorems for the volume and the number of i dimensional faces of random polytopes are proved as the number of random points tends to infinity one essential step is to determine the precise asymptotic order of the occurring variances

distributed channel synthesis two familiar notions of correlation are rediscovered as the extreme operating points for distributed synthesis of a discrete memoryless channel in which a stochastic channel output is generated based on a compressed description of the channel input wyner s common information is the minimum description rate needed however when common randomness independent of the input is available the necessary description rate reduces to shannon s mutual information this paper characterizes the optimal tradeoff between the amount of common randomness used and the required rate of description we also include a number of related derivations including the effect of limited local randomness rate requirements for secrecy applications to game theory and new insights into common information duality our proof makes use of a soft covering lemma known in the literature for its role in quantifying the resolvability of a channel the direct proof achievability constructs a feasible joint distribution over all parts of the system using a soft covering from which the behavior of the encoder and decoder is inferred with no explicit reference to joint typicality or binning of auxiliary interest this paper also generalizes and strengthens this soft covering tool

the combinatorial structure of random polytopes abstract choose n random points in r d let p n be their convex hull and denote by f i p n the number of i dimensional faces of p n a general method for computing the expectation of f i p n i 0 d 1 is presented this generalizes classical results of efron in the case i 0 and renyi and sulanke in the case i d 1 to arbitrary i for random points chosen in a smooth convex body a limit law for f i p n is proved as n for random points chosen in a polytope the expectation of f i p n is determined as n this implies an extremal property for random points chosen in a simplex

measuring brain variability by extrapolating sparse tensor fields measured on sulcal lines modeling and understanding the variability of brain structures is a fundamental problem in neurosciences improved mathematical representations of structural brain variation are needed to help detect and understand genetic or disease related sources of abnormality as well as to improve statistical power when integrating functional brain mapping data across subjects in this paper we develop a new mathematical model of normal brain variation based on a large set of cortical sulcal landmarks 72 per brain delineated in each of 98 healthy human subjects scanned with 3d mri age 51 8 6 2 years we propose an original method to compute an average representation of the sulcal curves which constitutes the mean anatomy after affine alignment of the individual data across subjects the second order moment distribution of the sulcal position is modeled as a sparse field of covariance tensors symmetric positive definite matrices to extrapolate this information to the full brain one has to overcome the limitations of the standard euclidean matrix calculus we propose an affine invariant riemannian framework to perform computations with tensors in particular we generalize radial basis function rbf interpolation and harmonic diffusion partial differential equations pdes to tensor fields as a result we obtain a dense 3d variability map that agrees well with prior results on smaller subject samples moreover leave one sulcus out tests show that our model is globally able to recover the missing information on brain variation when there is a consistent neighboring pattern of variability finally we propose an innovative method to analyze the asymmetry of brain variability as expected the greatest asymmetries are found in regions that includes the primary language areas interestingly any such asymmetries in anatomical variance if it remains after anatomical normalization could explain why there may be greater power to detect group activation in one hemisphere versus the other in fmri studies

hamilton jacobi skeleton on cortical surfaces in this paper we propose a new method to construct graphical representations of cortical folding patterns by computing skeletons on triangulated cortical surfaces in our approach a cortical surface is first partitioned into sulcal and gyral regions via the solution of a variational problem using graph cuts which can guarantee global optimality after that we extend the method of hamilton jacobi skeleton to subsets of triangulated surfaces together with a geometrically intuitive pruning process that can trade off between skeleton complexity and the completeness of representing folding patterns compared with previous work that uses skeletons of 3 d volumes to represent sulcal patterns the skeletons on cortical surfaces can be easily decomposed into branches and provide a simpler way to construct graphical representations of cortical morphometry in our experiments we demonstrate our method on two different cortical surface models its ability of capturing major sulcal patterns and its application to compute skeletons of gyral regions

automated 3 d extraction and evaluation of the inner and outer cortical surfaces using a laplacian map and partial volume effect classification accurate reconstruction of the inner and outer cortical surfaces of the human cerebrum is a critical objective for a wide variety of neuroimaging analysis purposes including visualization morphometry and brain mapping the anatomic segmentation using proximity asp algorithm previously developed by our group provides a topology preserving cortical surface deformation method that has been extensively used for the aforementioned purposes however constraints in the algorithm to ensure topology preservation occasionally produce incorrect thickness measurements due to a restriction in the range of allowable distances between the gray and white matter surfaces this problem is particularly prominent in pediatric brain images with tightly folded gyri this paper presents a novel method for improving the conventional asp algorithm by making use of partial volume information through probabilistic classification in order to allow for topology preservation across a less restricted range of cortical thickness values the new algorithm also corrects the classification of the insular cortex by masking out subcortical tissues for 70 pediatric brains validation experiments for the modified algorithm constrained laplacian asp clasp were performed by three methods i volume matching between surface masked gray matter gm and conventional tissue classified gm ii surface matching between simulated and clasp extracted surfaces and iii repeatability of the surface reconstruction among 16 mri scans of the same subject in the volume based evaluation the volume enclosed by the clasp wm and gm surfaces matched the classified gm volume 13 more accurately than using conventional asp in the surface based evaluation using synthesized thick cortex the average difference between simulated and extracted surfaces was 4 6 t 1 4 mm for conventional asp and 0 5 t 0 4 mm for clasp in a repeatability study clasp produced a 30 lower rms error for the gm surface and a 8 lower rms error for the

the likelihood encoder for lossy compression a likelihood encoder is studied in the context of lossy source compression the analysis of the likelihood encoder is based on the soft covering lemma it is demonstrated that the use of a likelihood encoder together with the soft covering lemma yields simple achievability proofs for classical source coding problems the cases of the point to point rate distortion function the rate distortion function with side information at the decoder i e the wyner ziv problem and the multi terminal source coding inner bound i e the berger tung problem are examined in this paper furthermore a non asymptotic analysis is used for the point to point case to examine the upper bound on the excess distortion provided by this method the likelihood encoder is also related to a recent alternative technique using the properties of random binning

automated 3 d extraction of inner and outer surfaces of cerebral cortex from mri automatic computer processing of large multidimensional images such as those produced by magnetic resonance imaging mri is greatly aided by deformable models which are used to extract identify and quantify specific neuroanatomic structures a general method of deforming polyhedra is presented here with two novel features first explicit prevention of self intersecting surface geometries is provided unlike conventional deformable models which use regularization constraints to discourage but not necessarily prevent such behavior second deformation of multiple surfaces with intersurface proximity constraints allows each surface to help guide other surfaces into place using model based constraints such as expected thickness of an anatomic surface these two features are used advantageously to identify automatically the total surface of the outer and inner boundaries of cerebral cortical gray matter from normal human mr images accurately locating the depths of the sulci even where noise and partial volume artifacts in the image obscure the visibility of sulci the extracted surfaces are enforced to be simple two dimensional manifolds having the topology of a sphere even though the data may have topological holes this automatic 3 d cortex segmentation technique has been applied to 150 normal subjects simultaneously extracting both the gray white and gray cerebrospinal fluid interface from each individual the collection of surfaces has been used to create a spatial map of the mean and standard deviation for the location and the thickness of cortical gray matter three alternative criteria for defining cortical thickness at each cortical location were developed and compared these results are shown to corroborate published postmortem and in vivo measurements of cortical thickness

multiple description coding with many channels an achievable region for the l channel multiple description coding problem is presented this region generalizes two channel results of el gamal and cover 1982 and of zhang and berger 1987 it further generalizes three channel results of gray and wyner 1974 and of zhang and berger a source that is successively refinable on chains is shown to be successively refinable on trees a new outer bound on the rate distortion rd region for memoryless gaussian sources with mean squared error distortion is also derived the achievable region meets this outer bound for certain symmetric cases

stochastische approximation konvexer polygone let k be a convex polygon with perimeter l and area f denote the perimeter of the convex hull of n uniformly and independently distributed points in k by ln and by fn its area as n tends to infinity l e l n c 1 k sqrt n o 1 n 1 varepsilon for any fixed 0 and f e f n c 2 k log n n o 1 n where the constants c1 k and c2 k are given explicitly this generalizes a result of renyi and sulanke

coverage processes on spheres and condition numbers for linear programming this paper has two agendas firstly we exhibit new results for coverage processes let p n m alpha be the probability that n spherical caps of angular radius alpha in s m do not cover the whole sphere s m we give an exact formula for p n m alpha in the case alpha in pi 2 pi and an upper bound for p n m alpha in the case alpha in 0 pi 2 which tends to p n m pi 2 when alpha to pi 2 in the case alpha in 0 pi 2 this yields upper bounds for the expected number of spherical caps of radius alpha that are needed to cover s m secondly we study the condition number mathscr c a of the linear programming feasibility problem exists x in mathbb r m 1 ax le0 x ne0 where a in mathbb r n times m 1 is randomly chosen according to the standard normal distribution we exactly determine the distribution of mathscr c a conditioned to a being feasible and provide an upper bound on the distribution function in the infeasible case using these results we show that mathbf e ln mathscr c a le2 ln m 1 3 31 for all n m the sharpest bound for this expectancy as of today both agendas are related through a result which translates between coverage and condition

stochastical approximation of convex bodies enveloppe convexe de points aleatoires sur la sphere enveloppe convexe de points aleatoires sur la frontiere d un corps convexe lisse

quantitative comparison and analysis of sulcal patterns using sulcal graph matching a twin study the global pattern of cortical sulci provides important information on brain development and functional compartmentalization sulcal patterns are routinely used to determine fetal brain health and detect cerebral malformations we present a quantitative method for automatically comparing and analyzing the sulcal pattern between individuals using a graph matching approach white matter surfaces were reconstructed from volumetric t1 mri data and sulcal pits the deepest points in local sulci were identified on this surface the sulcal pattern was then represented as a graph structure with sulcal pits as nodes the similarity between graphs was computed with a spectral based matching algorithm by using the geometric features of nodes 3d position depth and area and their relationship in particular we exploited the feature of graph topology the number of edges and the paths between nodes to highlight the interrelated arrangement and patterning of sulcal folds we applied this methodology to 48 monozygotic twins and showed that the similarity of the sulcal graphs in twin pairs was significantly higher than in unrelated pairs for all hemispheres and lobar regions consistent with a genetic influence on sulcal patterning this novel approach has the potential to provide a quantitative and reliable means to compare sulcal patterns

new results in binary multiple descriptions an encoder whose input is a binary equiprobable memoryless source produces one output of rate r 1 and another of rate r 2 let d 1 d 2 and d 0 respectively denote the average error frequencies with which the source data can be reproduced on the basis of the encoder output of rate r l only the encoder output of rate r 2 only and both encoder outputs the two descriptions problem is to determine the region r of all quintuples r 1 r 2 d 1 d 2 d 0 that are achievable in thc usual shannon sense let r d 1 d log 2 d 1 d log 2 1 d denote the error frequency rate distortion function of the source the no excess rate case prevails when r 1 r 2 r d 0 and the excess rate case when r 1 r 2 r d 0 denote the section of r at r 1 r 2 d 0 by d r 1 r 2 d 0 d 1 d 2 r 1 r 2 d 1 d 2 d 0 in r in the no excess rate case we show that a portion of the boundary of d r 1 r 2 d 0 coincides with the curve frac 1 2 d 1 2d 0 frac 1 2 d 2 2d 0 frac 1 2 1 2d 0 2 this curve is an extension of witsenhausen s hyperbola bound to the case d 0 0 it follows that the projection of r onto the d 1 d 2 plane at fixed d 0 consists of all d 1 geq d 0 and d 2 geq d 0 that lie on or above this hyperbola in the excess rate case we show by counterexample that the achievable region of el gamal and cover is not tight

variable length compression allowing errors this paper studies the fundamental limits of the minimum average length of lossless and lossy variable length compression allowing a nonzero error probability epsilon for lossless compression we give nonasymptotic bounds on the minimum average length in terms of erokhin s rate distortion function and we use those bounds to obtain a gaussian approximation on the speed of approach to the limit which is quite accurate for all but small blocklengths 1 epsilon k h mathsf s k v mathsf s 2 pi 1 2 mathop rm exp q 1 left epsilon right 2 2 where q 1 left cdot right is the functional inverse of the standard gaussian complementary cumulative distribution function and v mathsf s is the source dispersion a nonzero error probability thus not only reduces the asymptotically achievable rate by a factor of 1 epsilon but this asymptotic limit is approached from below i e larger source dispersions and shorter blocklengths are beneficial variable length lossy compression under an excess distortion constraint is shown to exhibit similar properties

one shot lossy quantum data compression we provide a framework for one shot quantum rate distortion coding in which the goal is to determine the minimum number of qubits required to compress quantum information as a function of the probability that the distortion incurred upon decompression exceeds some specified level we obtain a one shot characterization of the minimum qubit compression size for an entanglement assisted quantum rate distortion code in terms of the smooth max information a quantity previously employed in the one shot quantum reverse shannon theorem next we show how this characterization converges to the known expression for the entanglement assisted quantum rate distortion function for asymptotically many copies of a memoryless quantum information source finally we give a tight finite blocklength characterization for the entanglement assisted minimum qubit compression size of a memoryless isotropic qubit source subject to an average symbolwise distortion constraint

high resolution random mesh algorithms for creating a probabilistic 3d surface atlas of the human brain abstract striking variations exist across individuals in the internal and external geometry of the brain such normal variations in the size orientation topology and geometric complexity of cortical and subcortical structures have complicated the problem of quantifying deviations from normal anatomy and of developing standardized neuroanatomical atlases this paper describes the design implementation and results of a technique for creating a three dimensional 3d probabilistic surface atlas of the human brain we have developed implemented and tested a new 3d statistical method for assessing structural variations in a database of anatomic images the algorithm enables the internal surface anatomy of new subjects to be analyzed at an extremely local level the goal was to quantify subtle and distributed patterns of deviation from normal anatomy by automatically generating detailed probability maps of the anatomy of new subjects connected systems of parametric meshes were used to model the internal course of the following structures in both hemispheres the parieto occipital sulcus the anterior and posterior rami of the calcarine sulcus the cingulate and marginal sulci and the supracallosal sulcus these sulci penetrate sufficiently deeply into the brain to introduce an obvious topological decomposition of its volume architecture a family of surface maps was constructed encoding statistical properties of local anatomical variation within individual sulci a probability space of random transformations based on the theory of gaussian random fields was developed to reflect the observed variability in stereotaxic space of the connected system of anatomic surfaces a complete system of probability density functions was computed yielding confidence limits on surface variation the ultimate goal of brain mapping is to provide a framework for integrating functional and anatomical data across many subjects and modalities this task requires precise quantitative knowledge of the variations in geometry and location of intracerebral structures and critical functional interfaces the surface mapping and probabilistic techniques presented here provide a basis for the generation of anatomical templates and expert diagnostic systems which retain quantitative information on intersubject variations in brain architecture

determining the geometry of boundaries of objects from medial data we consider a region in r2 or r3 with generic smooth boundary b and blum medial axis m on which is defined a multivalued radial vector field u from points x on m to the points of tangency of the sphere at x with b we introduce a radial shape operator srad and an edge shape operator se which measure how u bends along m these are not traditional differential geometric shape operators nonetheless we derive all local differential geometric invariants of b from these operators this allows us to define from m u a geometric medial map on m which corresponds via a radial map from m to b to the differential geometric properties of b the geometric medial map also includes a description of the relative geometry of b this is defined using the relative critical set of the radius function r on m this set consists of a network of curves on m which describe where b is thickest and thinnest it is computed using the covariant derivative of the tangential component of the unit radial vector field we further determine how these invariants are related to the differential geometric invariants of m and how these invariants change under deforming diffeomorphisms of m

achievable rates for multiple descriptions consider a sequence of independent identically distributed i i d random variables x l x 2 cdots x n and a distortion measure d x i x i on the estimates x i of x i two descriptions i x in 1 2 cdots 2 nr 1 and j x in 1 2 cdots 2 nr 2 are given of the sequence x x 1 x 2 cdots x n from these two descriptions three estimates i x x2 j x and hat x o i x j x are formed with resulting expected distortions e frac 1 n sum n k 1 d x k hat x mk d m m 0 1 2 we find that the distortion constraints d 0 d 1 d 2 are achievable if there exists a probability mass distribution p x p hat x 1 hat x 2 hat x 0 x with ed x hat x m leq d m such that r 1 i x hat x 1 r 2 i x hat x 2 where i cdot denotes shannon mutual information these rates are shown to be optimal for deterministic distortion measures

a novel quantitative cross validation of different cortical surface reconstruction algorithms using mri phantom cortical surface reconstruction is important for functional brain mapping and morphometric analysis of the brain cortex several methods have been developed for the faithful reconstruction of surface models which represent the true cortical surface in both geometry and topology however there has been no explicit comparison study among those methods because each method has its own procedures file formats coordinate systems and use of the reconstructed surface there has also been no explicit evaluation method except visual inspection to validate the whole cortical surface models quantitatively in this study we presented a novel phantom based validation method of the cortical surface reconstruction algorithm and quantitatively cross validated the three most prominent cortical surface reconstruction algorithms which are used in freesurfer brainvisa and clasp respectively the validation included geometrical accuracy and mesh characteristics such as euler number fractal dimension fd total surface area and local density of points clasp showed the best geometric topologic accuracy and mesh characteristics such as fd and total surface area compared to freesurfer and brainvisa in the validation of local density of points freesurfer and brainvisa showed more even distribution of points on the cortical surface compared to clasp

variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points we show that the random point measures induced by vertices in the convex hull of a poisson sample on the unit ball when properly scaled and centered converge to those of a mean zero gaussian field we establish limiting variance and covariance asymptotics in terms of the density of the poisson sample similar results hold for the point measures induced by the maximal points in a poisson sample the approach involves introducing a generalized spatial birth growth process allowing for cell overlap

limit theorems for functionals of convex hulls in 4 a central limit theorem for the number of vertices of the convex hull of a uniform sample from the interior of a convex polygon is derived this is done by approximating the process of vertices of the convex hull by the process of extreme points of a poisson point process and by considering the latter process of extreme points as a markov process for a particular parametrization we show that this method can also be applied to derive limit theorems for the boundary length and for the area of the convex hull this extents results of renyi and sulanke 1963 and buchta 1984 and shows that the boundary length and the area have a strikingly different probabilistic behavior

trace a topological graph representation for automatic sulcal curve extraction a proper geometric representation of the cortical regions is a fundamental task for cortical shape analysis and landmark extraction however a significant challenge has arisen due to the highly variable convoluted cortical folding patterns in this paper we propose a novel topological graph representation for automatic sulcal curve extraction trace in practice the reconstructed surface suffers from noise influences introduced during image acquisition surface reconstruction in the presence of noise on the surface trace determines stable sulcal fundic regions by employing the line simplification method that prevents the sulcal folding pattern from being significantly smoothed out the sulcal curves are then traced over the connected graph in the determined regions by the dijkstra s shortest path algorithm for validation we used the state of the art surface reconstruction pipelines on a reproducibility data set the experimental results showed higher reproducibility and robustness to noise in trace than the existing method li et al 2010 with over 20 relative improvement in error for both surface reconstruction pipelines in addition the extracted sulcal curves by trace were well aligned with manually delineated primary sulcal curves we also provided a choice of parameters to control quality of the extracted sulcal curves and showed the influences of the parameter selection on the resulting curves

on exact and renyi common informations recently two extensions of wyner s common information exact and renyi common informations were introduced respectively by kumar li and el gamal kle and the present authors the class of common information problems involves determining the minimum rate of the common input to two independent processors needed to exactly or approximately generate a target joint distribution for the exact common information problem exact generation of the target distribution is required while for wyner s and alpha renyi common informations the relative entropy and renyi divergence with order alpha were respectively used to quantify the discrepancy between the synthesized and target distributions the exact common information is larger than or equal to wyner s common information however it was hitherto unknown whether the former is strictly larger than the latter for some joint distributions in this paper we first establish the equivalence between the exact and infty renyi common informations and then provide single letter upper and lower bounds for these two quantities for doubly symmetric binary sources we show that the upper and lower bounds coincide which implies that for such sources the exact and infty renyi common informations are completely characterized interestingly we observe that for such sources these two common informations are strictly larger than wyner s this answers an open problem posed by kle furthermore we extend wyner s infty renyi and exact common informations to sources with countably infinite or continuous alphabets including gaussian sources

spectral based automatic labeling and refining of human cortical sulcal curves using expert provided examples we present a spectral based method for automatically labeling and refining major sulcal curves of a human cerebral cortex given a set of input unlabeled sulcal curves automatically extracted from a cortical surface and a collection of expert provided examples labeled sulcal curves our objective is to identify the input major sulcal curves and assign their neuroanatomical labels and then refines these curves based on the expert provided example data without employing any atlas based registration scheme as preprocessing in order to construct the example data neuroanatomists manually labeled a set of 24 major sulcal curves 12 each for the left and right hemispheres for each individual subject according to a precise protocol we collected 30 sets of such curves from 30 subjects given the raw input sulcal curve set of a subject we choose the most similar example curve to each input curve in the set to label and refine the latter according to the former we adapt a spectral matching algorithm to choose the example curve by exploiting the sulcal curve features and their relationship the high dimensionality of sulcal curve data in spectral matching is addressed by using their multi resolution representations which greatly reduces time and space complexities our method provides consistent labeling and refining results even under high variability of cortical sulci across the subjects through experiments we show that the results are comparable in accuracy to those done manually most output curves exhibited accuracy values higher than 80 and the mean accuracy values of the curves in the left and the right hemispheres were 84 69 and 84 58 respectively

absolute continuity for curvature measures of convex sets ii this work is devoted to the investigation of the basic relationship between the geometric shape of a convex set and measure theoretic properties of the associated curvature and surface area measures we study geometric consequences of and conditions for absolute continuity of curvature and surface area measures with respect to d 1 dimensional hausdorff measure in euclidean space r our main results are two transfer principles which allow one to translate properties connected with the absolute continuity of the rth curvature measure of a convex body to dual properties related to the absolute continuity of the d 1 r th surface area measure of the polar body and conversely applications are also considered 2002

variance asymptotics and scaling limits for random polytopes abstract let k be a convex set in r d and let k be the convex hull of a homogeneous poisson point process p of intensity on k when k is a simple polytope we establish scaling limits as for the boundary of k in a vicinity of a vertex of k and we give variance asymptotics for the volume and k face functional of k k 0 1 d 1 resolving an open question posed in 17 the scaling limit of the boundary of k and the variance asymptotics are described in terms of a germ grain model consisting of cone like grains pinned to the extreme points of a poisson point process on r d 1 r having intensity d e d h d h d v

entanglement assisted capacity of a quantum channel and the reverse shannon theorem the entanglement assisted classical capacity of a noisy quantum channel c sub e is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel assuming that the sender and receiver have access to the resource of shared quantum entanglement which may be used up by the communication protocol we show that the capacity c sub e is given by an expression parallel to that for the capacity of a purely classical channel i e the maximum over channel inputs spl rho of the entropy of the channel input plus the entropy of the channel output minus their joint entropy the latter being defined as the entropy of an entangled purification of spl rho after half of it has passed through the channel we calculate entanglement assisted capacities for two interesting quantum channels the qubit amplitude damping channel and the bosonic channel with amplification attenuation and gaussian noise we discuss how many independent parameters are required to completely characterize the asymptotic behavior of a general quantum channel alone or in the presence of ancillary resources such as prior entanglement in the classical analog of entanglement assisted communication communication over a discrete memoryless channel dmc between parties who share prior random information we show that one parameter is sufficient i e that in the presence of prior shared random information all dmcs of equal capacity can simulate one another with unit asymptotic efficiency

limit theorems for the typical poisson voronoi cell and the crofton cell with a large inradius in this paper we are interested in the behavior of the typical poisson voronoi cell in the plane when the radius of the largest disk centered at the nucleus and contained in the cell goes to infinity we prove a law of large numbers for its number of vertices and the area of the cell outside the disk moreover for the latter we establish a central limit theorem as well as moderate deviation type results the proofs deeply rely on precise connections between poisson voronoi tessellations convex hulls of poisson samples and germ grain models in the unit ball besides we derive analogous facts for the crofton cell of a stationary poisson line process in the plane

supporting points processes and some of their applications we introduce a stochastic point process of s supporting points and prove that upon rescaling it converges to a gaussian field the notion of s supporting points specializes for adequately chosen s to pareto or more generally cone extremal points or to vertices of convex hulls or to centers of generalized voronoi tessellations in the models of large scale structure of the universe based on burgers equation the central limit theorems proven here imply i a the asymptotic normality for the number of convex hull vertices in large poisson sample from a simple polyhedra or for the number of pareto vector extremal points in poisson samples with independent coordinates

the convex hull of the integer points in a large ball where fk p denotes the number of k dimensional faces of the polytope p the limit as r of the average of r 2 3f0 pr on an interval r r h is determined by balog and deshoullier bd and turns out to be 3 453 h must be large our main result extends 1 1 to any d 2 and to any fk pr with k 0 d 1 theorem 1 for every d 2 there are constants c1 d and c2 d such that for all k 0 d 1

the communication complexity of correlation let x and y be finite nonempty sets and x y a pair of random variables taking values in x y we consider communication protocols between two parties alice and bob for generating x and y alice is provided an x x generated according to the distribution of x and is required to send a message to bob in order to enable him to generate y y whose distribution is the same as that of y x x both parties have access to a shared random string generated in advance let t x y be the minimum over all protocols of the expected number of bits alice needs to transmit to achieve this we show that i x y t x y i x y 2 log2 i x y o 1 we also consider the worst case communication required for this problem where we seek to minimize the average number of bits alice must transmit for the worst case x x we show that the communication required in this case is related to the capacity c e of the channel e derived from x y that maps x x to the distribution of y x x we also show that the required communication t e satisfies c e t e c e 2 log2 c e 1 o 1 using the first result we derive a direct sum theorem in communication complexity that substantially improves the previous such result shown by jain radhakrishnan and sen in proc 30th international colloquium of automata languages and programming icalp ser lecture notes in computer science vol 2719 2003 pp 300 315 these results are obtained by employing a rejection sampling procedure that relates the relative entropy between two distributions to the communication complexity of generating one distribution from the other

equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points a general formula is proved which relates the equiaffine inner parallel curves of a plane convex body and the probability that the convex hull of j independent random points is disjoint from the convex hull of k further independent random points this formula is applied to improve some well known results in geometric probability for example an estimate which was established for a special case by l c g rogers is obtained with the best possible bound and an asymptotic formula due to a renyi and r sulanke is extended to an asymptotic expansion

random points on the boundary of smooth convex bodies the convex hull of n independent random points chosen on the boundary of a convex body k c r d according to a given density function is a random polytope the expectation of its i th intrinsic volume for i 1 d is investigated in the case that the boundary of k is sufficiently smooth asymptotic expansions for these expected intrinsic volumes as n oo are derived

automated sulcal segmentation using watersheds on the cortical surface the human cortical surface is a highly complex folded structure sulci the spaces between the folds define location on the cortex and provide a parcellation into anatomically distinct areas a topic that has recently received increased attention is the segmentation of these sulci from magnetic resonance images with most work focusing on extracting either the sulcal spaces between the folds or curve representations of sulci unlike these methods we propose a technique that extracts actual regions of the cortical surface that surround sulci which we call sulcal regions the method is based on a watershed algorithm applied to a geodesic depth measure on the cortical surface a well known problem with the watershed algorithm is a tendency toward oversegmentation meaning that a single region is segmented as several pieces to address this problem we propose a postprocessing algorithm that merges appropriate segments from the watershed algorithm the sulcal regions are then manually labeled by simply selecting the appropriate regions with a mouse click and a preliminary study of sulcal depth is reported finally a scheme is presented for computing a complete parcellation of the cortical surface

central limit theorems for random polytopes in a smooth convex set let k be a smooth convex set with volume one in choose n random points in k independently according to the uniform distribution the convex hull of these points denoted by kn is called a random polytope we prove that several key functionals of kn satisfy the central limit theorem as n tends to infinity

strong functional representation lemma and applications to coding theorems this paper shows that for any random variables x and y it is possible to represent y as a function of x z such that z is independent of x and i x z y le log i x y 1 4 bits we use this strong functional representation lemma sfrl to establish a bound on the rate needed for one shot exact channel simulation for general discrete or continuous random variables strengthening the results by harsha et al and braverman and garg and to establish new and simple achievability results for one shot variable length lossy source coding multiple description coding and gray wyner system we also show that the sfrl can be used to reduce the channel with state noncausally known at the encoder to a point to point channel which provides a simple achievability proof of the gelfand pinsker theorem

on the variance of random polytopes abstract a random polytope is the convex hull of uniformly distributed random points in a convex body k a general lower bound on the variance of the volume and f vector of random polytopes is proved also an upper bound in the case when k is a polytope is given for polytopes as for smooth convex bodies the upper and lower bounds are of the same order of magnitude the results imply a law of large numbers for the volume and f vector of random polytopes when k is a polytope

exact channel synthesis we consider the exact channel synthesis problem this problem concerns the determination of the minimum amount of information required to create exact correlation remotely when there is a certain rate of randomness shared by two terminals this problem generalizes an existing approximate version in which the generated joint distribution is required to be close to a target distribution under the total variation tv distance measure instead being exactly equal to the target distribution we provide single letter inner and outer bounds on the admissible region of the shared randomness rate and the communication rate for the exact channel synthesis problem these two bounds coincide for doubly symmetric binary sources we observe that for such sources the admissible rate region for exact channel synthesis is strictly included in that for the tv approximate version we also extend the exact and tv approximate channel synthesis problems to sources with countably infinite alphabets and continuous sources the latter includes gaussian sources as by products lemmas concerning soft covering under renyi divergence measures are derived

random convex hulls and extreme value statistics in this paper we study the statistical properties of convex hulls of n random points in a plane chosen according to a given distribution the points may be chosen independently or they may be correlated after a non exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem we present a unifying approach based on the notion of support function of a closed curve and the associated cauchy s formulae that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics as an example of correlated points we study here in detail the case when the points represent the vertices of n independent random walks in the continuum time limit this reduces to n independent planar brownian trajectories for which we compute exactly for all n the mean perimeter and the mean area of their global convex hull our results have relevant applications in ecology in estimating the home range of a herd of animals some of these results were announced recently in a short communication phys rev lett 103 140602 2009

cones generated by random points on half spheres and convex hulls of poisson point processes let u 1 u 2 ldots be random points sampled uniformly and independently from the d dimensional upper half sphere we show that as n rightarrow infty the f vector of the d 1 dimensional convex cone c n generated by u 1 ldots u n weakly converges to a certain limiting random vector without any normalization we also show convergence of all moments of the f vector of c n and identify the limiting constants for the expectations we prove that the expected grassmann angles of c n can be expressed through the expected f vector this yields convergence of expected grassmann angles and conic intrinsic volumes and answers thereby a question of barany et al random struct algorithms 50 1 3 22 2017 https doi org 10 1002 rsa 20644 our approach is based on the observation that the random cone c n weakly converges after a suitable rescaling to a random cone whose intersection with the tangent hyperplane of the half sphere at its north pole is the convex hull of the poisson point process with power law intensity function proportional to vert x vert d gamma where gamma 1 we compute the expected number of facets the expected intrinsic volumes and the expected t functional of this random convex hull for arbitrary gamma 0

on the role of the refinement layer in multiple description coding and scalable coding we clarify the relationship among several existing achievable multiple description rate distortion regions by investigating the role of refinement layer in multiple description coding specifically we show that the refinement layer in the el gamal cover egc scheme and the venkataramani kramer goyal vkg scheme can be removed as a consequence the egc region is equivalent to the egc region an antecedent version of the egc region while the vkg region when specialized to the 2 description case is equivalent to the zhang berger zb region moreover we prove that for multiple description coding with individual and hierarchical distortion constraints the number of layers in the vkg scheme can be significantly reduced when only certain weighted sum rates are concerned the role of refinement layer in scalable coding a special case of multiple description coding is also studied

the redundancy of source coding with a fidelity criterion 1 known statistics the problem of redundancy of source coding with respect to a fidelity criterion is considered for any fixed rate r 0 and any memoryless source with finite source and reproduction alphabets and a common distribution p the nth order distortion redundancy d sub n r of fixed rate coding is defined as the minimum of the difference between the expected distortion per symbol of any block code with length n and rate r and the distortion rate function d p r of the source p it is demonstrated that for sufficiently large n d sub n r is equal to spl part spl part r d p r ln n 2n o ln n n where spl part spl part r d p r is the partial derivative of d p r evaluated at r and assumed to exist for any fixed distortion level d 0 and any memoryless source p the nth order rate redundancy r sub n d of coding at fixed distortion level d or by using d semifaithful codes is defined as the minimum of the difference between the expected rate per symbol of any d semifaithful code of length n and the rate distortion function r p d of p evaluated at d it is proved that for sufficiently large n r sub n d is upper bounded by ln n n o ln n n and lower bounded by in n 2n o in n n as a by product the lower bound of r sub n d derived in this paper gives a positive answer to a conjecture proposed by yu and speed 1993

random polytopes in a convex polytope independence of shape and concentration of vertices write d for the set of all convex bodies convex compact sets with nonempty interior in d define o g l d as the set of those k e 5 b d with vol k 1 fix k e g i d and choose points x l x e k randomly independently and according to the uniform distribution on k then k c o n v x l xn is a random polytope in k write e k n for the expectation of the random variable v o l k k n e k n shows how well k approximates k in volume on the average groemer grl proved that among all convex bodies k e o g l d the ellipsoids are approximated worst i e

joint sulcal detection on cortical surfaces with graphical models and boosted priors in this paper we propose an automated approach for the joint detection of major sulci on cortical surfaces by representing sulci as nodes in a graphical model we incorporate markovian relations between sulci and formulate their detection as a maximum a posteriori map estimation problem over the joint space of major sulci to make the inference tractable a sample space with a finite number of candidate curves is automatically generated at each node based on the hamilton jacobi skeleton of sulcal regions using the adaboost algorithm we learn both individual and pairwise shape priors of sulcal curves from training data which are then used to define potential functions in the graphical model based on the connection between adaboost and logistic regression finally belief propagation is used to perform the map inference and select the joint detection results from the sample spaces of candidate curves in our experiments we quantitatively validate our algorithm with manually traced curves and demonstrate the automatically detected curves can capture the main body of sulci very accurately a comparison with independently detected results is also conducted to illustrate the advantage of the joint detection approach

on the mean width of random polytopes on the boundary of a d dimensional convex body a probability distribution with a positive continuous density function g is given the convex hull of n points chosen independently according to g is a random polytope the asymptotic behaviour n of the expected value of the mean width of the random polytope is determined

automatic fiber bundle segmentation in massive tractography datasets using a multi subject bundle atlas this paper presents a method for automatic segmentation of white matter fiber bundles from massive dmri tractography datasets the method is based on a multi subject bundle atlas derived from a two level intra subject and inter subject clustering strategy this atlas is a model of the brain white matter organization computed for a group of subjects made up of a set of generic fiber bundles that can be detected in most of the population each atlas bundle corresponds to several inter subject clusters manually labeled to account for subdivisions of the underlying pathways often presenting large variability across subjects an atlas bundle is represented by the multi subject list of the centroids of all intra subject clusters in order to get a good sampling of the shape and localization variability the atlas composed of 36 known deep white matter bundles and 47 superficial white matter bundles in each hemisphere was inferred from a first database of 12 brains it was successfully used to segment the deep white matter bundles in a second database of 20 brains and most of the superficial white matter bundles in 10 subjects of the same database

limit theorems for convex hulls it is shown that the process of vertices of the convex hull of a uniform sample from the interior of a convex polygon converges locally after rescaling to a strongly mixing markov process as the sample size tends to infinity the structure of the limiting markov process is determined explicitly and from this a central limit theorem for the number of vertices of the convex hull is derived similar results are given for uniform samples from the unit disk