combinatorial entropy power inequalities a preliminary study of the stam region we initiate the study of the stam region defined as the subset of the positive orthant in r 2 n 1 that arises from considering the entropy powers of subset sums of n independent random vectors in a euclidean space of finite dimension we show that the class of fractionally superadditive set functions provides an outer bound to the stam region resolving a conjecture of barron and madiman on the other hand the entropy power of a sum of independent random vectors is not supermodular in any dimension we also develop some qualitative properties of the stam region showing for instance that its closure is a logarithmically convex cone

beyond the entropy power inequality via rearrangements a lower bound on the renyi differential entropy of a sum of independent random vectors is demonstrated in terms of rearrangements for the special case of boltzmann shannon entropy this lower bound is better than that given by the entropy power inequality several applications are discussed including a new proof of the classical entropy power inequality and an entropy inequality involving symmetrization of levy processes

a robust real time ellipse detector abstract in this paper we present an improved ellipse detector that may be used in real time face detection in this algorithm we first extract edges from the image using a robust edge detector which is then followed by a rule based method finally we decompose the parameter space of the hough transform to achieve computational efficiency our extensive experimental results show indeed that the proposed detector is capable of detecting ellipse features with an excellent accuracy under various image conditions at a high speed 10 frames per second on a pentium iii 500 mhz pc

a strengthened entropy power inequality for log concave densities we show that shannon s entropy power inequality admits a strengthened version in the case in which the densities are log concave in such a case in fact one can extend the blachman stam argument to obtain a sharp inequality for the second derivative of shannon s entropy functional with respect to the heat semigroup

entropy and asymptotic geometry of non symmetric convex bodies abstract we extend to the general not necessarily centrally symmetric setting a number of basic results of local theory which were known before for centrally symmetric bodies and were using very essentially the symmetry in their proofs some of these extensions look surprising the main additional tool is a study of volume behavior around the centroid of the body

intrinsic entropies of log concave distributions the entropy of a random variable is well known to equal the exponential growth rate of the volumes of its typical sets in this paper we show that for any log concave random variable x the sequence of the lfloor n theta rfloor text th intrinsic volumes of the typical sets of x in dimensions n geq 1 grows exponentially with a well defined rate we denote this rate by h x theta and call it the theta text th intrinsic entropy of x we show that h x theta is a continuous function of theta over the range 0 1 thereby providing a smooth interpolation between the values 0 and h x at the endpoints 0 and 1 respectively

low complexity finite element algorithms for the de rham complex on simplices we combine recently developed finite element algorithms based on bernstein polynomials m ainsworth g andriamaro and o davydov siam j sci comput 33 2011 pp 3087 3109 r c kirby numer math 117 2011 pp 631 652 with the explicit basis construction of the finite element exterior calculus d n arnold r s falk and r winther comput methods appl mech engrg 198 2009 pp 1660 1672 to give a family of algorithms for the de rham complex on simplices that achieves stiffness matrix construction and matrix free action in optimal complexity these algorithms are based on realizing the exterior calculus bases as short combinations of bernstein polynomials numerical results confirm optimal scaling of the algorithms and favorable comparison with fenics at high polynomial order as well additional empirical studies show that very high accuracy is achieved in the mixed discretization of the poisson equation and the maxwell eigenvalue problem as the polynomial degree increases

edcircles a real time circle detector with a false detection control we propose a real time parameter free circle detection algorithm that has high detection rates produces accurate results and controls the number of false circle detections the algorithm makes use of the contiguous connected set of edge segments produced by our parameter free edge segment detector the edge drawing parameter free edpf algorithm hence the name edcircles the proposed algorithm first computes the edge segments in a given image using edpf which are then converted into line segments the detected line segments are converted into circular arcs which are joined together using two heuristic algorithms to detect candidate circles and near circular ellipses the candidates are finally validated by an a contrario validation step due to the helmholtz principle which eliminates false detections leaving only valid circles and near circular ellipses we show through experimentation that edcircles works real time 10 20ms for 640x480 images has high detection rates produces accurate results and is very suitable for the next generation real time vision applications including automatic inspection of manufactured products eye pupil detection circular traffic sign detection etc

a short note on a bernstein bezier basis for the pyramid we introduce a bernstein bezier basis for the pyramid whose restriction to the face reduces to the bernstein bezier basis on the triangle or quadrilateral the basis satisfies the standard positivity and partition of unity properties common to bernstein polynomials and spans the same space as nonpolynomial pyramid bases in the literature procedures for differentiation and integration of these basis functions are also discussed

bernstein bezier bases for tetrahedral finite elements abstract we present a new set of basis functions for h curl conforming h div conforming and l 2 conforming finite elements of arbitrary order on a tetrahedron the basis functions are expressed in terms of bernstein polynomials and augment the natural h 1 conforming bernstein basis the basis functions respect the differential operators namely the gradients of the high order h 1 conforming bernstein bezier basis functions form part of the h curl conforming basis and the curl of the high order non gradients h curl conforming basis functions form part of the h div conforming basis and the divergence of the high order non curl h div conforming basis functions form part of the l 2 conforming basis procedures are given for the efficient computation of the mass and stiffness matrices with these basis functions without using quadrature rules for piecewise constant coefficients on affine tetrahedra numerical results are presented to illustrate the use of the basis to approximate representative problems

entropy bounds on abelian groups and the ruzsa divergence over the past few years a family of interesting new inequalities for the entropies of sums and differences of random variables has been developed by ruzsa tao and others motivated by analogous results in additive combinatorics this paper extends these earlier results to the case of random variables taking values in mathbb r n or more generally in arbitrary locally compact and polish abelian groups we isolate and study a key quantity the ruzsa divergence between two probability distributions and we show that its properties can be used to extend the earlier inequalities to the present general setting the new results established include several variations on the theme that the entropies of the sum and the difference of two independent random variables severely constrain each other although the setting is quite general the results are already of interest and new for random vectors in mathbb r n in that special case we discuss quantitative bounds for the stability of the equality conditions in the entropy power inequality a reverse entropy power inequality for log concave random vectors an information theoretic analog of the rogers shephard inequality for convex bodies and consequences of some of our results to determinant inequalities for sums of positive definite matrices moreover by considering various multiplicative subgroups of the complex plane one obtains new inequalities for the differential entropies of products and ratios of nonzero complex valued random variables

edge curvature and convexity based ellipse detection method in this paper we propose a novel ellipse detection method for real images the proposed method uses the information of edge curvature and their convexity in relation to other edge contours as clues for identifying edge contours that can be grouped together a search region is computed for every edge contour that contains other edge contours eligible for grouping with the current edge contour a two dimensional hough transform is performed in an intermediate step in which we use a new relationship score for ranking the edge contours in a group instead of the conventional histogram count the score is found to be more selective and thus more efficient in addition we use three novel saliency criteria that are non heuristic and consider various aspects for quantifying the goodness of the detected elliptic hypotheses and finally selecting good elliptic hypotheses the thresholds for selection of elliptic hypotheses are determined by the detected hypotheses themselves such that the selection is free from human intervention the method requires a few seconds in most cases so it is suitable for practical applications the performance of the proposed ellipse detection method has been tested on a dataset containing 1200 synthetic images and the caltech 256 dataset containing real images in both cases the results show that the proposed ellipse detection method performs far better than existing methods and is close to the ideal results with precision recall and f measure all very close to 1 further the method is robust to the increase in the complexity of the images such as overlapping ellipses occluded ellipses while the performance of the contemporary methods deteriorates significantly

fast inversion of the simplicial bernstein mass matrix we consider the mass matrix arising from bernstein polynomials as finite element shape functions in particular we give an explicit formula for its eigenvalues and exact characterization of the eigenspaces in terms of the bernstein representation of orthogonal polynomials we then derive a fast algorithm for solving linear systems involving the element mass matrix after establishing these results we describe the application of bernstein techniques to the discontinuous galerkin finite element method for hyperbolic conservation laws obtaining optimal complexity algorithms finally we present numerical results investigating the accuracy of the mass inversion algorithms and the scaling of total run time for the function evaluation needed in dg time stepping

a new entropy power inequality a strengthened version of shannon s entropy power inequality for the case where one of the random vectors involved is gaussian is proved in particular it is shown that if independent gaussian noise is added to an arbitrary multivariate random variable the entropy power of the resulting random variable is a concave function of the variance power of the added noise the strengthened inequality is shown to hold for the class of stable distributions

reverse brunn minkowski and reverse entropy power inequalities for convex measures we develop a reverse entropy power inequality for convex measures which may be seen as an anegeometric inverse of the entropy power inequality of shannon and stam the specialization of this inequality to log concave measures may be seen as a version of milman s reverse brunn minkowski inequality the proof relies on a demonstration of new relationships between the entropy of high dimensional random vectors and the volume of convex bodies and on a study of eective supports of convex measures both of which are of independent interest as well as on milman s deep technology of m ellipsoids and on certain information theoretic inequalities as a by product we also give a

concentration of the information in data with log concave distributions on which the distribution of xis supported in this case eh x is essentially the number of bits neededto represent x by a coding scheme that minimizes average code length sha48 in the continuous case with reference measure dx one may stillcall eh x the information content even though the coding interpretation nolonger holds in statistics one may think of the information content as thelog likelihood function the average value of the information content of xis known more com monly as the entropy indeed the entropy of xis de ned byh x zf x logf x dx elogf x

randomized circle detection with isophotes curvature analysis circle detection is a critical issue in image analysis and object detection although hough transform based solvers are largely used randomized approaches based on the iterative sampling of the edge pixels are object of research in order to provide solutions less computationally expensive this work presents a randomized iterative work flow which exploits geometrical properties of isophotes in the image to select the most meaningful edge pixels and to classify them in subsets of equal isophote curvature the analysis of candidate circles is then performed with a kernel density estimation based voting strategy followed by a refinement algorithm based on linear error compensation the method has been applied to a set of real images on which it has also been compared with two leading state of the art approaches and hough transform based solutions the achieved results show how discarding up to 57 of unnecessary edge pixels it is able to accurately detect circles within a limited number of iterations maintaining a sub pixel accuracy even in the presence of high level of noise highlightsan iterative randomized circle detection algorithm is proposed curvature of isophotes is used to reduce the number of necessary iterations edges on circumference s are selected by a kernel density estimation strategy sub pixel accuracy was maintained on real images even in presence of high noise levels isophotes analysis reduces the influence of the used edge map on the results

on the similarity of the entropy power inequality and the brunn minkowski inequality corresp the entropy power inequality states that the effective variance entropy power of the sum of two independent random variables is greater than the sum of their effective variances the brunn minkowski inequality states that the effective radius of the set sum of two sets is greater than the sum of their effective radii both these inequalities are recast in a form that enhances their similarity in spite of this similarity there is as yet no common proof of the inequalities nevertheless their intriguing similarity suggests that new results relating to entropies from known results in geometry and vice versa may be found two applications of this reasoning are presented first an isoperimetric inequality for entropy is proved that shows that the spherical normal distribution minimizes the trace of the fisher information matrix given an entropy constraint just as a sphere minimizes the surface area given a volume constraint second a theorem involving the effective radii of growing convex sets is proved

generalized entropy power inequalities and monotonicity properties of information new families of fisher information and entropy power inequalities for sums of independent random variables are presented these inequalities relate the information in the sum of n independent random variables to the information contained in sums over subsets of the random variables for an arbitrary collection of subsets as a consequence a simple proof of the monotonicity of information in central limit theorems is obtained both in the setting of independent and identically distributed i i d summands as well as in the more general setting of independent summands with variance standardized sums

otlines a novel line detection algorithm without the interference of smooth curves abstract we propose a novel line detection algorithm that avoids interpreting a smooth curve in an image as many straight segments the algorithm starts from a contiguous connected set of edge chains extracted by the edge drawing parameter free edpf algorithm such that a problem of line detection in the image space is transformed to a problem of line segments and circular arcs identified for each edge chain the core of our approach is the proposed orientation transformation ot algorithm based on principal component analysis hence our algorithm is named otlines with the ot algorithm each edge chain is not only decomposed into a series of straight line segments and smooth curve candidates at the estimated corners but is also mapped to a series of oblique lines and horizontal lines in a new space called the orientation space based on the results of the transformation line segments and circular arcs can be detected simultaneously in both the orientation and image spaces using principal component analysis the performance of our proposed algorithm is validated using a set of typical images we also compare our algorithm with existing algorithms our novel algorithm is very effective at segmenting the lines in synthetic and natural images no parameters in the proposed algorithm must be tuned during its application

a fast and robust circle detection method using isosceles triangles sampling circle detection using randomized sampling has been developed in recent years to reduce computational intensity however randomized sampling is sensitive to noise that can lead to reduced accuracy and false positive candidates to improve on the robustness of randomized circle detection under noisy conditions this paper presents a new methodology for circle detection based upon randomized isosceles triangles sampling it is shown that the geometrical property of isosceles triangles provides a robust criterion to find relevant edge pixels which in turn offers an efficient means to estimate the centers and radii of circles for best efficiency the estimated results given by the sampling from individual connected components of the edge map were analyzed using a simple clustering approach to further improve on the accuracy we applied a two step refinement process using chords and linear error compensation with gradient information of the edge pixels extensive experiments using both synthetic and real images have been performed the results are compared to leading state of the art algorithms and it is shown that the proposed methodology has a number of advantages it is efficient in finding circles with a low number of iterations it has high rejection rate of false positive circle candidates and it has high robustness against noise all this makes it adaptive and useful in many vision applications a new circle detection algorithm based on randomized sampling of isosceles triangles its is presented sampling of isosceles triangles provides distinctive probability distribution for circular shapes a low amount of iterations are needed by using a simple clustering approch the proposed algorithm has short execution time high accuracy and high robustness against noise

a strong entropy power inequality when one of the random summands is gaussian we sharpen the entropy power inequality epi in terms of the strong data processing function for gaussian channels among other consequences this strong epi generalizes the vector extension of costa s epi to non gaussian channels in a precise sense this leads to a new reverse epi and as a corollary sharpens stam s uncertainty principle relating entropy power and fisher information or equivalently gross logarithmic sobolev inequality applications to network information theory are also given including a short self contained proof of the rate region for the two encoder quadratic gaussian source coding problem and a new outer bound for the one sided gaussian interference channel

efficient sampling strategy and refinement strategy for randomized circle detection circle detection is fundamental in pattern recognition and computer vision the randomized approach has received much attention for its computational benefit when compared with the hough transform in this paper a multiple evidence based sampling strategy is proposed to speed up the randomized approach next an efficient refinement strategy is proposed to improve the accuracy based on different kinds of ten test images experimental results demonstrate the computation saving and accuracy effects when plugging the proposed strategies into three existing circle detection methods

solution of shannon s problem on the monotonicity of entropy it is shown that if x1 x2 are independent and identically distributed square integrable random variables then the entropy of the normalized sum ent x1 xn over n is an increasing function of n this resolves an old problem which goes back to 6 7 5 the result also has a version for non identically distributed random variables or random vectors

convex bodies and norms associated to convex measures isotropy like properties are considered for finite measures with heavy tails as a basic tool we extend k ball s relationship between convex bodies and finite logarithmically concave measures to a larger class of distributions satisfying convexity conditions of the brunn minkowski type

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

a central limit theorem for convex sets we show that there exists a sequence varepsilon n searrow0 for which the following holds let k n be a compact convex set with a non empty interior let x be a random vector that is distributed uniformly in k then there exist a unit vector in n t0 and 0 such that sup a subset mathbb r left textit prob langle x theta rangle in a frac 1 sqrt 2 pi sigma int ae frac t t 0 2 2 sigma 2 dt right leq varepsilon n qquad ast where the supremum runs over all measurable sets a and where denotes the usual scalar product in n furthermore under the additional assumptions that the expectation of x is zero and that the covariance matrix of x is the identity matrix we may assert that most unit vectors satisfy with t0 0 and 1 corresponding principles also hold for multi dimensional marginal distributions of convex sets

simple proof of the concavity of the entropy power with respect to added gaussian noise a very simple proof of m h costa s result see ibid vol it 31 p 751 60 1985 that the entropy power of x sub t x n o ti is concave in t is derived as an immediate consequence of an inequality concerning fisher information this relationship between fisher information and entropy is found to be useful for proving the central limit theorem thus one who seeks new entropy inequalities should try first to find new equalities about fisher information or at least to exploit the existing ones in new ways

on the entropy power inequality for the renyi entropy of order 0 1 using a sharp version of the reverse young inequality and a renyi entropy comparison result due to fradelizi madiman and wang 2016 the authors derive renyi entropy power inequalities for log concave random vectors when renyi parameters belong to 0 1 furthermore the estimates are shown to be sharp up to absolute constants

lsd a fast line segment detector with a false detection control we propose a linear time line segment detector that gives accurate results a controlled number of false detections and requires no parameter tuning this algorithm is tested and compared to state of the art algorithms on a wide set of natural images

a method for solving the multiple ellipses detection problem in this paper the multiple ellipses detection problem on the basis of a data points set coming from a number of ellipses in the plane not known in advance is considered an ellipse is considered as a mahalanobis circle with some positive definite matrix a very efficient method for solving this problem is proposed this method very successfully combines the well known direct least squares method and the ransac algorithm with a realistic statistical model of multiple ellipses in the plane the method is illustrated and tested on numerous synthetic and real world applications the method was also compared with other similar methods in the case when a data points set comes from a number of ellipses with clear edges the proposed method gives results similar to other known methods however when a data points set comes from a number of ellipses with noisy edges the proposed method performs significantly better than the other methods we should emphasize the advantage and utility of the proposed methods in a variety of applications such as medical image analysis and ultrasound image segmentation highlightsransac based method for solving the multiple ellipses detection problem is proposed the proposed method shows high efficiency the method has the potential to solve real time applications data is considered as a realization of a realistic statistical model pseudocodes of all proposed algorithms are provided

subcell flux limiting for high order bernstein finite element discretizations of scalar hyperbolic conservation laws abstract this work extends the concepts of algebraic flux correction and convex limiting to continuous high order bernstein finite element discretizations of scalar hyperbolic problems using an array of adjustable diffusive fluxes the standard galerkin approximation is transformed into a nonlinear high resolution scheme which has the compact sparsity pattern of the piecewise linear or multilinear subcell discretization the representation of this scheme in terms of invariant domain preserving states makes it possible to prove the validity of local discrete maximum principles under cfl like conditions in contrast to predictor corrector approaches based on the flux corrected transport methodology the proposed flux limiting strategy is monolithic i e limited antidiffusive terms are incorporated into the well defined residual of a nonlinear semi discrete problem a stabilized high order galerkin discretization is recovered if no limiting is performed in the limited version the compact stencil property prevents direct mass exchange between nodes that are not nearest neighbors a formal proof of sparsity is provided for simplicial and box elements the involved element contributions can be calculated efficiently making use of matrix free algorithms and precomputed element matrices of the reference element numerical studies for q 2 discretizations of linear and nonlinear two dimensional test problems illustrate the virtues of monolithic convex limiting based on subcell flux decompositions

multiple ellipses detection in noisy environments a hierarchical approach detection of multiple ellipses in noisy environments is a basic yet challenging task in many vision related problems the key area of difficulty is on distinguishing the pixels pertaining to each target in the presence of noise to tackle with the issue we propose a hierarchical approach which is motivated by the fact that any segment of an ellipse can identify itself in ellipse reconstruction first we find all the neat edges without any branches followed by an ellipse fitting on each of them second some target candidates are estimated based on the neat edges by a proposed grouping strategy finally the targets are detected based on the candidates by a proposed selective competitive algorithm to distinguish the true pixels of each target a real application of the proposed method is illustrated in addition to some other demonstrative experiments

volumes of restricted minkowski sums and the free analogue of the entropy power inequality in noncommutative probability theory independence can be based on free products instead of tensor products this yields a highly noncommutative theory free probability theory for an introduction see 9 the analogue of entropy in the free context was introduced by the second named author in 8 here we show that shannon s entropy power inequality 6 1 has an analogue for the free entropy x theorem 2 1

affine moments of a random vector an affine invariant pth moment measure is defined for a random vector and used to prove sharp moment entropy inequalities that are more general and stronger than standard moment entropy inequalities

power histogram for circle detection on images the detection of circles from digital images is very important to shape recognition in this paper we propose and analyze a novel method based on power histograms to detect circles in an image the key idea is to transform an image into several power histograms that are computed from the pairwise distance products referring to the rays emanated from some reference points on the image according to the circle power theorem the edge pixels on a circle share the same circle powers with respect to an arbitrary point on the image the peaks in the power histograms provide strong evidence for the existence of real circles a multi reference cross validation among various peaks can systematically remove randomness and eliminate false detections experimental results over several synthetic as well as natural images with varying ranges of complexity validate the efficacy of the proposed method in terms of its accuracy simplicity and robustness highlightsa new and simple circle detector based on the power theorem of circles is proposed transforms the edge pixel pairs into 1d power space and detects peaks in power histogram for circle locating incorporates the inherent geometric properties of circle and generates only a few number of candidate circles for validation a robust peak detection and cross validation algorithm for enhancing the robustness of the method experiments on synthetic hand sketched and natural images are carried out

the entropy per coordinate of a random vector is highly constrained under convexity conditions the entropy per coordinate in a log concave random vector of any dimension with given density at the mode is shown to have a range of just 1 uniform distributions on convex bodies are at the lower end of this range the distribution with i i d exponentially distributed coordinates is at the upper end and the normal is exactly in the middle thus in terms of the amount of randomness as measured by entropy per coordinate any log concave random vector of any dimension contains randomness that differs from that in the normal random variable with the same maximal density value by at most 1 2 as applications we obtain an information theoretic formulation of the famous hyperplane conjecture in convex geometry entropy bounds for certain infinitely divisible distributions and quantitative estimates for the behavior of the density at the mode on convolution more generally one may consider so called convex or hyperbolic probability measures on euclidean spaces we give new constraints on entropy per coordinate for this class of measures which generalize our results under the log concavity assumption expose the extremal role of multivariate pareto type distributions and give some applications