rigidity and energy suppose one holds two sticks in the form of a cross in one hand and places a rubber band in tension around the four ends when it is released it comes to rest in the shape of a convex quadrilateral in a plane it always returns to the same shape no matter how it is distorted as long as the ends do not slip and the rubber band does not break this is a very simple example of a rigid framework of the type we discuss here energy explains why a framework such as the one above is rigid the rubber band deforms in such a way as to minimize the total energy in the framework only in the final deformed shape will the framework have a minimum energy this idea of introducing energy functions is very useful it can be used to prove a key lemma that was used by cauchy 5 in 1813 to show that convex polyhedral surfaces are rigid it can explain why some and perhaps all of r buckminster fuller s tensegrity structures 10 stay up when applied to spider webs it shows why they can only take on certain geometric shapes it also can be used to prove conjecture 6 of branko gri inbaum and g c shephard in their lectures on lost mathematics 12 this conjecture says that if a framework in the shape of a convex polygon with rods sticks on the boundary and cables inside is rigid in the plane so is the framework obtained by reversing the roles of rods and cables it is interesting to compare some of these results with the opening a rm theorem of axel schur 18 see chern 6 also which is a very close smooth analogue to cauchy s lemma schur s theorem says that if a convex planar smooth arc the arm is opened that is it is moved to another position with the same length but with corresponding points having smaller curvature then the two ends are moved apart another amusing application is to show that a regular pentagon in 4 space a pentagon with all 5 sides equal and all 5 angles between the sides equal has its angles bounded between 36 and 108 a comment of o bottema in 4

algebraic tensegrity form finding abstract this paper concerns the form finding problem for general and symmetric tensegrity structures with shape constraints a number of different geometries are treated and several fundamental properties of tensegrity structures are identified that simplify the form finding problem the concept of a tensegrity invariance similarity transformation is defined and it is shown that tensegrity equilibrium is preserved under affine node position transformations this result provides the basis for a new tensegrity form finding tool the generality of the problem formulation makes it suitable for the automated generation of the equations and their derivatives state of the art numerical algorithms are applied to solve several example problems examples are given for tensegrity plates shell class symmetric tensegrity structures and structures generated by applying similarity transformation

a fast phase space method for computing creeping rays creeping rays can give an important contribution to the solution of medium to high frequency scattering problems they are generated at the shadow lines of the illuminated scatterer by grazing incident rays and propagate along geodesics on the scatterer surface continuously shedding diffracted rays in their tangential direction in this paper we show how the ray propagation problem can be formulated as a partial differential equation pde in a three dimensional phase space to solve the pde we use a fast marching method the pde solution contains information about all possible creeping rays this information includes the phase and amplitude of the field which are extracted by a fast post processing computationally the cost of solving the pde is less than tracing all rays individually by solving a system of ordinary differential equations we consider an application to mono static radar cross section problems where creeping rays from all illumination angles must be computed the numerical results of the fast phase space method and a comparison with the results of ray tracing are presented

matrix analysis of statically and kinematically indeterminate frameworks abstract the paper is concerned with the structural mechanics of assemblies of bars and pinjoints particularly where they are simultaneously statically and kinematically indeterminate the physical significance of the four linear algebraic vector subspaces of the equilibrium matrix is examined and an algorithm is set up which determines the rank of the matrix and the bases for the four subspaces in particular this algorithm gives full details of any states of self stress and modes of inextensional deformation which an assembly may possess a scheme is devised for the segregation of inextensional modes into rigid body modes up to six of these may be allowed by the foundation constraints and internal mechanisms in some circumstances a state of self stress may impart first order stiffness to an inextensional mode a matrix method for detecting this effect is devised and it is shown that if there is no state of self stress which imparts first order stiffness to a given mode then that mode can undergo rather large distortion which involves either zero change in length of the bars or possibly changes in length of third or higher order in the displacements the significance of negative stiffness as indicated by the matrix method is discussed the paper contains simple examples which illustrate all of the main points of the work

generalised correlation for multi feature correspondence computing correspondences between pairs of images is fundamental to all structures from motion algorithms correlation is a popular method to estimate similarity between patches of images in the standard formulation the correlation function uses only one feature such as the gray level values of a small neighbourhood research has shown that different features such as colour edge strength corners texture measures work better under di0erent conditions we propose a framework of generalized correlation that can compute a real valued similarity measure using a feature vector whose components can be dissimilar the framework can combine the e0ects of di0erent image features such as multi spectral features edges corners texture measures etc into a single similarity measure in a 3exible manner additionally it can combine results of di0erent window sizes used for correlation with proper weighting for each relative importances of the features can be estimated from the image itself for accurate correspondence in this paper we present the framework of generalised correlation provide a few examples demonstrating its power as well as discuss the implementation issues 2002 published by elsevier science ltd on behalf of pattern recognition society

fast computation of weighted distance functions and geodesics on implicit hyper surfaces 730 an algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hyper surfaces is introduced in this paper the basic idea is to approximate the intrinsic weighted distance by the euclidean weighted distance computed in a band surrounding the implicit hyper surface in the embedding space thereby performing all the computations in a cartesian grid with classical and efficient numerics based on work on geodesics on riemannian manifolds with boundaries we bound the error between the two distance functions we show that this error is of the same order as the theoretical numerical error in computationally optimal hamilton jacobi based algorithms for computing distance functions in cartesian grids therefore we can use these algorithms modified to deal with spaces with boundaries and obtain also for the case of intrinsic distance functions on implicit hyper surfaces a computationally efficient technique the approach can be extended to solve a more general class of hamilton jacobi equations defined on the implicit surface following the same idea of approximating their solutions by the solutions in the embedding euclidean space the framework here introduced thereby allows for the computations to be performed on a cartesian grid with computationally optimal algorithms in spite of the fact that the distance and hamilton jacobi equations are intrinsic to the implicit hyper surface c 2001 academic press

using selective correlation coefficient for robust image registration a new method is proposed for robust image registration named selective correlation coefficient in order to search images under ill conditioned illumination or partial occlusion a correlation mask image is generated for selecting pixels of an image before matching the mask image can be derived from a binary coded increment sign image defined from any object image and the template the mask rate of occluded regions is theoretically expected to be 0 5 while unoccluded regions have much lower rate than 0 5 robustness for ill conditioned environment can be realized since inconsistent brightness of occluded regions can be omitted from the mask operation furthermore the mask enhancement procedure is proposed to get more stable robustness the effectiveness of masking increases by the procedure resulting in the rate around 0 7 for masking of occluded regions this paper includes a theoretical modeling and analysis of the proposed method and some experimental results with real images

ordinal measures for image correspondence we present ordinal measures of association for image correspondence in the context of stereo linear correspondence measures like correlation and the sum of squared difference between intensity distributions are known to be fragile ordinal measures which are based on relative ordering of intensity values in windows rank permutations have demonstrable robustness by using distance metrics between two rank permutations ordinal measures are defined these measures are independent of absolute intensity scale and invariant to monotone transformations of intensity values like gamma variation between images we have developed simple algorithms for their efficient implementation experiments suggest the superiority of ordinal measures over existing techniques under nonideal conditions these measures serve as a general tool for image matching that are applicable to other vision problems such as motion estimation and texture based image retrieval

an o 1 disparity refinement method for stereo matching disparity refinement is the final step but also the timing bottleneck of stereo matching due to its high computational complexity weighted media filter refinement method and non local refinement method are two typical refinement methods with o n computational complexity for each pixel where n indicates the maximum disparity this paper presents an o 1 disparity refinement method based on belief aggregation and belief propagation the aggregated belief which means the possibility of correct disparity value is efficiently computed on a minimum spanning tree first and then the belief aggregation is fast performed on another minimum spanning tree in two sequential passes first from leaf nodes to root then from root to leaf nodes only 2 additions and 4 multiplications are required for each pixel at all disparity levels so the computational complexity is o 1 performance evaluation on middlebury data sets shows that the proposed method has good performances both in accuracy and speed we proposed an o 1 disparity refinement method for each pixel speed evaluation shows constant result due to o 1 computational complexity accuracy evaluation shows better performance than 2 typical refinement methods

a new approach to the analytical and numerical form finding of tensegrity structures we develop a new formulation for the form finding of tensegrity structures in which the primary variables are the cartesian components of element lengths both an analytical and a numerical implementation of the formulation are described each require a description of the connectivity of the tensegrity with the iterative numerical method also requiring a random starting vector of member force densities the analytical and numerical form finding of tensegrity structures is demonstrated through six examples and the results obtained are compared and contrasted with those available in the literature to verify the accuracy and viability of the suggested methods

advanced form finding for cable strut structures a numerical method is presented for form finding of cable strut structures the topology and the types of members are the only information that is required in this form finding process dummy members are used to transform the cable strut structure with supports into self stressed system without supports the requirement on rank deficiencies of the force density and equilibrium matrices for the purpose of obtaining a non degenerate d dimensional self stressed structure has been explicitly discussed the spectral decomposition of the force density matrix and the singular value decomposition of the equilibrium matrix are performed iteratively to find the feasible sets of nodal coordinates and force densities which satisfy the minimum required rank deficiencies of the force density and equilibrium matrices respectively based on numerical examples it is found that the proposed method is very efficient robust and versatile in searching self equilibrium configurations of cable strut structures

robust image registration by increment sign correlation abstract a novel and robust statistic as a similarity measure for robust image registration is proposed the statistic is named as increment sign correlation because it is based on the average evaluation of incremental tendency of brightness in adjacent pixels it is formalized to be a binary distribution or a gaussian distribution for a large image size through statistical analysis and modeling by utilizing the proposed statistical model for example we can theoretically determine a reasonable value of threshold for verification of matching this sign correlation can also be proved to expectedly have the constant value 0 5 for any uncorrelated images to a template image and then the property of the constancy can be utilized to analyze the high robustness for occlusion the good performance for the case of saturation or highlight can also be proved through theoretical analysis and fundamental experiments a basic algorithm for image scanning search and registration over a large scene is represented with a technique for a fast version by the branch and bound approach many experimental evidences with real images are provided and discussed

similarity measures for image matching despite occlusions in stereo vision in the context of computer vision matching can be done with similarity measures this paper presents the classification of these measures into five families in addition 18 measures based on robust statistics previously proposed in order to deal with the problem of occlusions are studied and compared to the state of the art a new evaluation protocol and new analyses are proposed and the results highlight the most efficient measures first near occlusions the smooth median powered deviation and second near discontinuities a non parametric transform based measure census

a direct approach to design of geometry and forces of tensegrity systems in the process of designing a tensegrity system some constraints are usually introduced for geometry and or forces to ensure uniqueness of the solution because the tensegrity systems are underdetermined in most cases in this paper a new approach is presented to enable designers to specify independent sets of axial forces and nodal coordinates consecutively under the equilibrium conditions and the given constraints to satisfy the distinctly different requirements of architects and structural engineers the proposed method can be used very efficiently for practical applications because only linear algebraic equations are to be solved and no equation of kinematics or material property is needed some numerical examples are given to show not only efficiency of the proposed method but also its ability of searching new configurations

fast phase space computation of multiple arrivals we present a fast general computational technique for computing the phase space solution of static hamilton jacobi equations starting with the liouville formulation of the characteristic equations we derive escape equations which are static time independent eulerian pdes they represent all arrivals to the given boundary from all possible starting configurations the solution is numerically constructed through a one pass formulation building on ideas from semi lagrangian methods dijkstra like methods for the eikonal equation and ordered upwind methods to compute all possible trajectories corresponding to all possible boundary conditions the technique is of computational order o n log n where n is the total number of points in the computational phase space domain any particular set of boundary conditions then is extracted through rapid post processing suggestions are made for speeding up the algorithm in the case when the particular distribution of sources is provided in advance as an application we apply the technique to the problem of computing first multiple and most energetic arrivals to the eikonal equation

cross trees edge and superpixel priors based cost aggregation for stereo matching in this paper we propose a novel cross trees structure to perform the non local cost aggregation strategy and the cross trees structure consists of a horizontal tree and a vertical tree compared to other spanning trees the significant superiorities of the cross trees are that the trees constructions are efficient and the trees are exactly unique since the constructions are independent on any local or global property of the image itself additionally two different priors edge prior and superpixel prior are proposed to tackle the false cost aggregations which cross the depth boundaries hence our method contains two different algorithms in terms of cross trees prior by traversing the two crossed trees successively a fast non local cost aggregation algorithm is performed twice to compute the aggregated cost volume performance evaluation on the 27 middlebury data sets shows that both our algorithms outperform the other two tree based non local methods namely minimum spanning tree mst and segment tree st graphical abstractdisplay omitted highlightswe propose a novel cross trees structure to perform the non local cost aggregation the trees constructions are unique and independent on the image itself we propose and incorporate two priors into the non local framework new edge weights function is designed according to the trees and the priors optimal support regions can be chose by cutting the cost aggregation flow on paths

adaptive force density method for form finding problem of tensegrity structures a numerical method is presented for form finding of tensegrity structures eigenvalue analysis and spectral decomposition are carried out iteratively to find the feasible set of force densities that satisfies the requirement on rank deficiency of the equilibrium matrix with respect to the nodal coordinates the equilibrium matrix is shown to correspond to the geometrical stiffness matrix in the conventional finite element formulation a unique and non degenerate configuration of the structure can then be obtained by specifying an independent set of nodal coordinates a simple explanation is given for the required rank deficiency of the equilibrium matrix that leads to a non degenerate structure several numerical examples are presented to illustrate the robustness as well as the strong ability of searching new configurations of the proposed method

static and dynamic analyses of tensegrity structures part ii quasi static analysis linearized lagrangian equations developed in the first part of the paper were employed for static analyses of cyclic cylindrical tensegrity modules linearized equilibrium equations at natural configurations were used to investigate initial shape static and kinematic indeterminancy pre stress and infinitesimal mechanism modes and the sensitivity analysis of initial geometry linearized equilibrium equations at pre stressed initial configurations were utilized to investigate pre stress stiffening and to distinguish first order mechanisms from higher order mechanisms to estimate critical loads for bar buckling and cable slacking nonlinear equilibrium equations were employed to compute element forces further the equivalence between the twist angle theorem obtained from a geometrical consideration and the equilibrium analysis was established for cyclic cylindrical tensegrity modules it is concluded that infinitesimal mechanism modes and pre stresses characterize the static and dynamic response of tensegrity structures

short note fast geodesics computation with the phase flow method this paper introduces a novel approach for rapidly computing a very large number of geodesics on a smooth surface the idea is to apply the recently developed phase flow method l ying e j candes the phase flow method j comput phys to appear an efficient and accurate technique for constructing phase maps for nonlinear ordinary differential equations on invariant manifolds which are here the unit tangent bundles of the surfaces under study we show how to rapidly construct the whole geodesic flow map which then allows computing any geodesic by straightforward local interpolation an operation with constant complexity a few numerical experiments complement our study and demonstrate the effectiveness of our approach

discriminant analysis and similarity measure the cosine similarity measure is often applied after discriminant analysis in pattern recognition this paper first analyzes why the cosine similarity is preferred by establishing the connection between the cosine similarity based decision rule in the discriminant analysis framework and the bayes decision rule for minimum error the paper then investigates the challenges inherent of the cosine similarity and presents a new similarity that overcomes these challenges the contributions of the paper are thus three fold first the application of the cosine similarity after discriminant analysis is discovered to have its theoretical roots in the bayes decision rule second some inherent problems of the cosine similarity such as its inadequacy in addressing distance and angular measures are discussed finally a new similarity measure which overcomes the problems by integrating the absolute value of the angular measure and the lp norm the distance measure is presented to enhance pattern recognition performance the effectiveness of the proposed new similarity measure in the discriminant analysis framework is evaluated using a large scale grand challenge problem namely the face recognition grand challenge frgc problem experimental results using 36 818 frgc images on the most challenging frgc experiment the frgc experiment 4 show that the new similarity measure improves face recognition performance upon other popular similarity measures such as the cosine similarity measure the normalized correlation and the euclidean distance measure highlightsthe connection between the cosine similarity and the bayes decision rule discuss some inherent challenges of the cosine similarity a new similarity measure that integrates both angular and distance measure experiments on the frgc version 2 database 36 818 images

a multiple baseline stereo a stereo matching method that uses multiple stereo pairs with various baselines generated by a lateral displacement of a camera to obtain precise distance estimates without suffering from ambiguity is presented matching is performed simply by computing the sum of squared difference ssd values the ssd functions for individual stereo pairs are represented with respect to the inverse distance and are then added to produce the sum of ssds this resulting function is called the sssd in inverse distance it is shown that the sssd in inverse distance function exhibits a unique and clear minimum at the correct matching position even when the underlying intensity patterns of the scene include ambiguities or repetitive patterns the authors first define a stereo algorithm based on the sssd in inverse distance and present a mathematical analysis to show how the algorithm can remove ambiguity and increase precision experimental results with real stereo images are presented to demonstrate the effectiveness of the algorithm

structural computations with the singular value decomposition of the equilibrium matrix abstract the singular value decomposition of the equilibrium matrix makes it possible to answer any question of a static kinematic or static kinematic nature for any structural assembly within a unified computational framework the paper introduces this decomposition relates it to the physical properties of a given assembly and derives formulae for stresses displacements etc a simple example illustrates the calculations