view variation of point set and line segment features the variation with respect to view of 2d features defined for projections of 3d point sets and line segments is studied it is established that general case view invariants do not exist for any number of points given true perspective weak perspective or orthographic projection models feature variation under the weak perspective approximation is then addressed though there are no general case weak perspective invariants there are special case invariants of practical importance those cited in the literature are derived from linear dependence relations and the invariance of this type of relation to linear transformation the variation with respect to view is studied for an important set of 2d line segment features the relative orientation size and position of one line segment with respect to another the analysis includes an evaluation criterion for feature utility in terms of view variation this relationship is a function of both the feature and the particular configuration of 3d line segments the use of this information in objection recognition is demonstrated for difficult discrimination tasks

on the liouville theorem for the stationary navier stokes equations in a critical space abstract in this paper we prove the liouville type theorem for the stationary navier stokes equations on r 3 more specifically if a solution u h 1 r 3 to the stationary navier stokes system satisfies additional conditions characterized by the decays near infinity and by the oscillation then we show that u 0

recovering surface shape and orientation from texture texture provides an important source of information about the three dimensional structure of visible surfaces particularly for stationary monocular views to recover 3d structure the distorting effects of projection must be distinguished from properties of the texture on which the distortion acts this requires that assumptions must be made about the texture yet the unpredictability of natural textures precludes the use of highly restrictive assumptions the recovery method reported in this paper exploits the minimal assumption that textures do not mimic projective effects this assumption determines the strategy of attributing as much as possible of the variation observed in the image to projection equivalently the interpretation is chosen for which the texture prior to projection is made as uniform as possible this strategy was implemented using statistical methods first for the restricted case of planar surfaces and then by extension for curved surfaces the technique was applied successfully to natural images

encoding 2d range maximum queries we consider the two dimensional range maximum query 2d rmq problem given an array containing elements from an ordered set encode the array so that the position of the maximum element in any specified range of rows and range of columns can be found efficiently we focus on determining the effective entropy of 2d rmq i e how many bits are needed to encode an array so that 2d rmq queries can be answered without accessing the array we give tight upper and lower bounds on the expected effective entropy for the case when a contains independent identically distributed random values and give new upper and lower bounds for the case when the array contains few rows the latter results improve upon the upper and lower bounds by brodal et al 4 we also give some efficient data structures for 2d rmq whose space usage is close to the effective entropy

a graduated assignment algorithm for graph matching a graduated assignment algorithm for graph matching is presented which is fast and accurate even in the presence of high noise by combining graduated nonconvexity two way assignment constraints and sparsity large improvements in accuracy and speed are achieved its low order computational complexity o lm where l and m are the number of links in the two graphs and robustness in the presence of noise offer advantages over traditional combinatorial approaches the algorithm not restricted to any special class of graph is applied to subgraph isomorphism weighted graph matching and attributed relational graph matching to illustrate the performance of the algorithm attributed relational graphs derived from objects are matched then results from twenty five thousand experiments conducted on 100 mode random graphs of varying types graphs with only zero one links weighted graphs and graphs with node attributes and multiple link types are reported no comparable results have been reported by any other graph matching algorithm before in the research literature twenty five hundred control experiments are conducted using a relaxation labeling algorithm and large improvements in accuracy are demonstrated

a simple optimal representation for balanced parentheses we consider succinct or highly space efficient representations of a static string consisting of n pairs of balanced parentheses which support natural operations such as finding the matching parenthesis for a given parenthesis or finding the pair of parentheses that most tightly enclose a given pair this problem was considered by jacobson space efficient static trees and graphs in proc of the 30th focs 1989 pp 549 554 and munro and raman succinct representation of balanced parentheses and static trees siam j comput 31 2001 762 776 who gave o n bit and 2n o n bit representations respectively that supported the above operations in o 1 time on the ram model of computation this data structure is a fundamental tool in succinct representations and has applications in representing suffix trees ordinal trees planar graphs and permutations we consider the practical performance of parenthesis representations first we give a new 2n o n bit representation that supports all the above operations in o 1 time this representation is conceptually simpler its space bound has a smaller o n term and it also has a simple and uniform o n time and space construction algorithm we implement our data structure and a variant of jacobson s and evaluate their practical performance speed and memory usage when used in a succinct representation of trees derived from xml documents as a baseline we compare our representations against a widely used implementation of the standard dom document object model representation of xml documents both succinct representations use orders of magnitude less space than dom and tree traversal operations are usually only slightly slower than in dom

computing correspondences in a sequence of non rigid shapes abstract image correspondences are computed using a new relaxation technique applied to edge features each image is represented by a network of nodes connected by branches the nodes represent the locations of corners or other conspicuous points on the contours of the image the branches may be real image contours or imaginary lines drawn to link the nodes nodes in one image are matched to nodes in the next image using a measure of the length and angular separation between connecting branches an iterative relaxation scheme searches for consistency in the structure and relative motion of neighboring branches and nodes and rewards the matching probability of each node accordingly around 15 20 iterations are needed for convergence the matching process favors structural rigidity on a local scale while allowing global image to image deformations to occur as in the case of fluid motion examples are given as evidence that the algorithm can match patterns undergoing distortion rotation and scaling

compressed suffix arrays and suffix trees with applications to text indexing and string matching the proliferation of online text such as found on the world wide web and in online databases motivates the need for space efficient text indexing methods that support fast string searching we model this scenario as follows consider a text t consisting of n symbols drawn from a fixed alphabet sigma the text t can be represented in n lg sigma bits by encoding each symbol with lg sigma bits the goal is to support fast online queries for searching any string pattern p of m symbols with t being fully scanned only once namely when the index is created at preprocessing time the text indexing schemes published in the literature are greedy in terms of space usage they require omega n lg n additional bits of space in the worst case for example in the standard unit cost ram suffix trees and suffix arrays need omega n memory words each of omega lg n bits these indexes are larger than the text itself by a multiplicative factor of omega smash lg sigma n which is significant when sigma is of constant size such as in textsc ascii or textsc unicode on the other hand these indexes support fast searching either in o m lg sigma time or in o m lg n time plus an output sensitive cost o mathit occ for listing the mathit occ pattern occurrences we present a new text index that is based upon compressed representations of suffix arrays and suffix trees it achieves a fast smash o m lg sigma n lg sigma epsilon n search time in the worst case for any constant 0 epsilon leq 1 using at most smash bigl epsilon 1 o 1 bigr n lg sigma bits of storage our result thus presents for the first time an efficient index whose size is provably linear in the size of the text in the worst case and for many scenarios the space is actually sublinear in practice as a concrete example the compressed suffix array for a typical 100 mb textsc ascii file can require 30 40 mb or less while the raw suffix array requires 500 mb our theoretical bounds improve emph both time and space of previous indexing schemes listing the pattern occurrences introduces a sublogarithmic slowdown factor in the output sensitive cost giving o mathit occ smash lg sigma epsilon n time as a result when the patterns are sufficiently long we can use auxiliary data structures in o n lg sigma bits to obtain a total search bound of o m lg sigma n mathit occ time which is optimal

relaxation labeling with learning automata relaxation labeling processes are a class of mechanisms that solve the problem of assigning labels to objects in a manner that is consistent with respect to some domain specific constraints we reformulate this using the model of a team of learning automata interacting with an environment or a high level critic that gives noisy responses as to the consistency of a tentative labeling selected by the automata this results in an iterative linear algorithm that is itself probabilistic using an explicit definition of consistency we give a complete analysis of this probabilistic relaxation process using weak convergence results for stochastic algorithms our model can accommodate a range of uncertainties in the compatibility functions we prove a local convergence result and show that the point of convergence depends both on the initial labeling and the constraints the algorithm is implementable in a highly parallel fashion

on a consistency measure for object labeling problems abstract the problem of defining a consistency measure for object labeling problems is addressed the optimal probabilistic measures are derived for both discrete and fuzzy solutions alternative fuzzy consistency measures are suggested based on the error each object has in estimating its label using the labels of its neighbors in general the best estimation is impractical from computational complexity aspects as well as information considerations however if the information available about the problem is probabilities of pairs of events the best estimation is a linear form

the robust sequential estimator a general approach and its application to surface organization in range data presents an autonomous statistically robust sequential function approximation approach to simultaneous parameterization and organization of possibly partially occluded surfaces in noisy outlier ridden not gaussian functional range data at the core of this approach is the robust sequential estimator a robust extension to the method of sequential least squares unlike most existing surface characterization techniques the authors method generates complete surface hypotheses in parameter space given a noisy depth map of an unknown 3 d scene the algorithm first selects appropriate seed points representing possible surfaces for each nonredundant seed it chooses the best approximating model from a given set of competing models using a modified akaike information criterion with this best model each surface is expanded from its seed over the entire image and this step is repeated for all seeds those points which appear to be outliers with respect to the model in growth are not included in the possibly disconnected surface point regions are deleted from each newly grown surface in the prune stage noise outliers or coincidental surface alignment may cause some points to appear to belong to more than one surface these ambiguities are resolved by a weighted voting scheme within a 5 spl times 5 decision window centered around the ambiguous point the isolated point regions left after the resolve stage are removed and any missing points in the data are filled by the surface having a majority consensus in an 8 neighborhood

shape matching of two dimensional objects in this paper we present results in the areas of shape matching of nonoccluded and occluded two dimensional objects shape matching is viewed as a segment matching problem unlike the previous work the technique is based on a stochastic labeling procedure which explicitly maximizes a criterion function based on the ambiguity and inconsistency of classification to reduce the computation time the technique is hierarchical and uses results obtained at low levels to speed up and improve the accuracy of results at higher levels this basic technique has been extended to the situation where various objects partially occlude each other to form an apparent object and our interest is to find all the objects participating in the occlusion in such a case several hierarchical processes are executed in parallel for every object participating in the occlusion and are coordinated in such a way that the same segment of the apparent object is not matched to the segments of different actual objects these techniques have been applied to two dimensional simple closed curves represented by polygons and the power of the techniques is demonstrated by the examples taken from synthetic aerial industrial and biological images where the matching is done after using the actual segmentation methods

probabilistic 3d object recognition a probabilistic 3d object recognition algorithm is presented in order to guide the recognition process the probability that match hypotheses between image features and model features are correct is computed a model is developed which uses the probabilistic peaking effect of measured angles and ratios of lengths by tracing iso angle and iso ratio curves on the viewing sphere the model also accounts for various types of uncertainty in the input such as incomplete and inexact edge detection for each match hypothesis the pose of the object and the pose uncertainty which is due to the uncertainty in vertex position are recovered this is used to find sets of hypotheses which reinforce each other by matching features of the same object with compatible uncertainty regions a probabilistic expression is used to rank these hypothesis sets the hypothesis sets with the highest rank are output the algorithm has been fully implemented and tested on real images

space efficient preprocessing schemes for range minimum queries on static arrays given a static array of n totally ordered objects the range minimum query problem is to build a data structure that allows us to answer efficiently subsequent on line queries of the form what is the position of a minimum element in the subarray ranging from i to j we focus on two settings where 1 the input array is available at query time and 2 the input array is available only at construction time in setting 1 we show new data structures a of size frac 2n c n theta bigl frac n lg lg n c n lg n bigr bits and query time o c n for any positive integer function c n in o bigl n varepsilon bigr for an arbitrary constant 0 varepsilon 1 or b with o nh k o n bits and o 1 query time where h k denotes the empirical entropy of k th order of the input array in setting 2 we give a data structure of size 2n o n bits and query time o 1 all data structures can be constructed in linear time and almost in place

liouville type theorems for the 3d stationary navier stokes mhd and hall mhd equations abstract in this paper we consider the liouville type problem of the three dimensional stationary incompressible navier stokes equations and establish an improved liouville type result only under the condition that the l p norm of a velocity u is finite moreover these results are extended to the mhd and hall mhd equations

relaxation matching techniques a comparison many different relaxation schemes have been proposed for image analysis tasks we have developed a general matching procedure for comparing semantic network descriptions of images and we have implemented a variety of relaxation techniques an automatic segmentation and description system is used to produce the image representations so that the matching procedures must cope with variations in feature values missing objects and possible multiple matches this environment is used to test different relaxation matching schemes under a variety of conditions the best performance of those we compared in terms of the number of iterations and the number of errors is for the gradient based optimization approach of faugeras and price the related optimization approach of hummel and zucker performed almost as well with differences primarily in difficult matches i e where much of the evidence is against the match for instance poor segmentations the product combination rule proposed by peleg was extremely fast indeed too fast to work when global context is needed the classical rosenfeld hummel and zucker method is included for historical comparisons and performed only adequately producing fewer correct matches and taking more iterations

matching invariant to translations rotations and scale changes abstract an optimization approach to invariant matching is proposed in this approach an object or a pattern is invariantly represented by an object centred description called an attributed relational structure ars embedding invariant properties and relations between the primitives of the pattern such as line segments and points noise effect is taken into account such that a scene can consist of noisy sub parts of a model the matching is then to find the optimal mapping between the arss of the scene and the model a gain functional is formulated to measure the goodness of fit and is to be maximized by using the relaxation labelling method experiments are shown to illustrate the matching algorithm and to demonstrate that the approach is truly invariant to arbitrary translations rotations and scale changes under noise

liouville type theorems for the forced euler equations and the navier stokes equations in this paper we study the liouville type properties for solutions to the steady incompressible euler equations with forces in mathbb r n if we assume single signedness condition on the force then we can show that a c 1 mathbb r n solution v p with v 2 p in l frac q 2 mathbb r n q in frac 3n n 1 infty is trivial v 0 for the solution of the steady navier stokes equations satisfying v x to 0 as x to infty the condition int mathbb r 3 delta v frac 6 5 dx infty which is stronger than the important d condition int mathbb r 3 nabla v 2 dx infty but both having the same scaling property implies that v 0 in the appendix we reprove theorem 1 1 chae commun math phys 273 203 215 2007 using the self similar euler equations directly

genetic algorithms for ambiguous labelling problems abstract consistent labelling problems frequently have more than one solution most work in the field has aimed at disambiguating early in the interpretation process using only local evidence this paper starts with a review of the literature on labelling problems and ambiguity based on this review we propose a strategy for simultaneously extracting multiple related solutions to the consistent labelling problem in a preliminary experimental study we show that an appropriately modified genetic algorithm is a robust tool for finding multiple solutions to the consistent labelling problem these solutions are related by common labellings of the most strongly constrained junctions we have proposed three run time measures of algorithm performance the maximum fitness of the genetic algorithm s population its shannon entropy and the total hamming distance between its distinct members the results to date indicate that when the shannon entropy falls below a certain threshold new solutions are unlikely to emerge and that most of the diversity in the population disappears within the first few generations

on the foundations of relaxation labeling processes a large class of problems can be formulated in terms of the assignment of labels to objects frequently processes are needed which reduce ambiguity and noise and select the best label among several possible choices relaxation labeling processes are just such a class of algorithms they are based on the parallel use of local constraints between labels this paper develops a theory to characterize the goal of relaxation labeling the theory is founded on a definition of con sistency in labelings extending the notion of constraint satisfaction in certain restricted circumstances an explicit functional exists that can be maximized to guide the search for consistent labelings this functional is used to derive a new relaxation labeling operator when the restrictions are not satisfied the theory relies on variational cal culus it is shown that the problem of finding consistent labelings is equivalent to solving a variational inequality a procedure nearly identical to the relaxation operator derived under restricted circum stances serves in the more general setting further a local convergence result is established for this operator the standard relaxation labeling formulas are shown to approximate our new operator which leads us to conjecture that successful applications of the standard methods are explainable by the theory developed here observations about con vergence and generalizations to higher order compatibility relations are described

deformable shape detection and description via model based region grouping a method for deformable shape detection and recognition is described deformable shape templates are used to partition the image into a globally consistent interpretation determined in part by the minimum description length principle statistical shape models enforce the prior probabilities on global parametric deformations for each object class once trained the system autonomously segments deformed shapes from the background while not merging them with adjacent objects or shadows the formulation can be used to group image regions obtained via any region segmentation algorithm e g texture color or motion the recovered shape models can be used directly in object recognition experiments with color imagery are reported

a computational model of view degeneracy we quantify the observation by kender and freudenstein 1987 that degenerate views occupy a significant fraction of the viewing sphere surrounding an object for a perspective camera geometry we introduce a computational model that can be used to estimate the probability that a view degeneracy will occur in a random view of a polyhedral object for a typical recognition system parameterization view degeneracies typically occur with probabilities of 20 percent and depending on the parameterization as high as 50 percent we discuss the impact of view degeneracy on the problem of object recognition and for a particular recognition framework relate the cost of object disambiguation to the probability of view degeneracy to reduce this cost we incorporate our model of view degeneracy in an active focal length control paradigm that balances the probability of view degeneracy with the camera field of view in order to validate both our view degeneracy model as well as our active focal length control model a set of experiments are reported using a real recognition system operating on real images

on view likelihood and stability we define two measures on views view likelihood and view stability view likelihood measures the probability that a certain view of a given 3d object is observed it may be used to identify typical or characteristic views view stability measures how little the image changes as the viewpoint is slightly perturbed it may be used to identify generic views both definitions are shown to be identical up to the prior probability of camera orientations and determined by the 2d metric used to compare images we analytically derive the stability and likelihood measures for two feature based 2d metrics where the most stable and most likely view is shown to be the flattest view of the 3d shape incorporating view likelihood or stability in 3d object recognition and 3d reconstruction increases the chance of robust performance in particular we propose to use these measures to enhance 3d object recognition and 3d reconstruction algorithms by adding a second step where the most likely solution is selected among all feasible solutions these applications are demonstrated using simulated and real images

dpa a deterministic approach to the map problem deterministic pseudo annealing dpa is a new deterministic optimization method for finding the maximum a posteriori map labeling in a markov random field in which the probability of a tentative labeling is extended to a merit function on continuous labelings this function is made convex by changing its definition domain this unambiguous maximization problem is solved and the solution is followed down to the original domain yielding a good if suboptimal solution to the original labeling assignment problem the performance of dpa is analyzed on randomly weighted graphs

simple parallel hierarchical and relaxation algorithms for segmenting noncausal markovian random fields the modeling and segmentation of images by mrf s markov random fields is treated these are two dimensional noncausal markovian stochastic processes two conceptually new algorithms are presented for segmenting textured images into regions in each of which the data are modeled as one of c mrf s the algorithms are designed to operate in real time when implemented on new parallel computer architectures that can be built with present technology a doubly stochastic representation is used in image modeling here a gaussian mrf is used to model textures in visible light and infrared images and an autobinary or autoternary etc mrf to model a priori information about the local geometry of textured image regions for image segmentation the true texture class regions are treated either as a priori completely unknown or as a realization of a binary or ternary etc mrf in the former case image segmentation is realized as true maximum likelihood estimation in the latter case it is realized as true maximum a posteriori likelihood segmentation in addition to providing a mathematically correct means for introducing geometric structure the autobinary or ternary etc mrf can be used in a generative mode to generate image geometries and artificial images and such simulations constitute a very powerful tool for studying the effects of these models and the appropriate choice of model parameters the first segmentation algorithm is hierarchical and uses a pyramid like structure in new ways that exploit the mutual dependencies among disjoint pieces of a textured region

probabilistic relaxation and the hough transform abstract we discuss the recent developments in probabilistic relaxation which is used as a tool for contextual sensory data interpretation the relationship of this technique with the hough transform is then established focusing on the generalised hough transform ght we show that the label probability updating formula of the probabilistic relaxation process exploiting binary relations between object primitives under the assumption that the primitives convey weak context exhibits very close similarity to the voting function employed by a computationally efficient ght we argue that the relationship could be exploited by importing the positive features of the respective techniques to benefit one another specific suggestions for enhancing the respective techniques are mentioned they include the adoption of the representational efficiency of the hough transform to reduce the computational complexity of probabilistic relaxation vice versa in the case of the generalised hough transform it is pointed out that the effect of an unknown object transformation could be dealt with by means of a parallel cooperative interpretation rather than by means of an exhaustive search through the parameter space of a transformation group this has implications in terms of both the storage and computational requirements of the ght it also opens the possibility of using the ght for detecting objects subject to more complex transformations such as affine or perspective the relationship also suggests the possibility of using alternative voting functions which may speed up the object detection process

the probabilistic peaking effect of viewed angles and distances with application to 3 d object recognition two novel probabilistic models for viewed angles and distances are derived using an observability sphere method the method which is based on the assumption that the prior probability density is isotropic for all viewing orientations can be used for the computation of observation probabilities for object s aspects features and probability densities of their quantitative attributes using the sphere it is discovered that the probability densities of viewed angles distances and even projected curvature have sharp peaks at their original values from this peaking effect it is concluded that in most cases the values of angles and distances are being altered only slightly by the imaging process and they can still serve as a strong cue for model based recognition the probabilistic models for 3 d object recognition from monocular images are used to form the angular elements that are needed the objects are represented by their linear features and specific points primitives using the joint density model of angles and distances the probabilities of initial matching hypotheses and mutual information coefficients are estimated these results are then used for object recognition by optimal matching search and stochastic labeling schemes various synthetic and real objects are recognized by this approach

three dimensional descriptions based on the analysis of the invariant and quasi invariant properties of some curved axis generalized cylinders we address the recovery of object level 3d descriptions of some classes of curved axis generalized cylinders for this the first part of the paper analyzes the projective properties of two common generic shapes planar right constant generalized cylinders prcgcs and circular planar right generalized cylinders circular prgcs the properties we analyze include new geometric invariant and quasi invariant properties of the orthographic projection of the above shapes and a useful classification of their structural properties as functions of their pose the second part of the paper describes an implemented system which detects and recovers prcgcs and circular prgcs from an intensity image in the presence of noise surface markings shadows and partial occlusion the methods exploit the projective properties to hypothesize and verify relevant curved axis objects thus explicitly using the three dimensionality of the objects and of the desired descriptions this work extends past work on the recovery of volumetric shapes from an intensity image by addressing new primitives deriving new properties and by developing a system that recovers them from an intensity image we demonstrate our method on several real intensity images

structural matching in computer vision using probabilistic relaxation in this paper we develop the theory of probabilistic relaxation for matching features extracted from 2d images derive as limiting cases the various heuristic formulae used by researchers in matching problems and state the conditions under which they apply we successfully apply our theory to the problem of matching and recognizing aerial road network images based on road network models and to the problem of edge matching in a stereo pair for this purpose each line network is represented by an attributed relational graph where each node is a straight line segment characterized by certain attributes and related with every other node via a set of binary relations

image registration and object recognition using affine invariants and convex hulls this paper is concerned with the problem of feature point registration and scene recognition from images under weak perspective transformations which are well approximated by affine transformations and under possible occlusion and or appearance of new objects it presents a set of local absolute affine invariants derived from the convex hull of scattered feature points e g fiducial or marking points corner points inflection points etc extracted from the image the affine invariants are constructed from the areas of the triangles formed by connecting three vertices among a set of four consecutive vertices quadruplets of the convex hull and hence do make direct use of the area invariance property associated with the affine transformation because they are locally constructed they are very well suited to handle the occlusion and or appearance of new objects these invariants are used to establish the correspondences between the convex hull vertices of a test image with a reference image in order to undo the affine transformation between them a point matching approach for recognition follows this the time complexity for registering l feature points on the test image with n feature points of the reference image is of order o n spl times l the method has been tested on real indoor and outdoor images and performs well

improving consistency and reducing ambiguity in stochastic labeling an optimization approach we approach the problem of labeling a set of objects from a quantitative standpoint we define a world model in terms of transition probabilities and propose a definition of a class of global criteria that combine both ambiguity and consistency a projected gradient algorithm is developed to minimize the criterion we show that the minimization procedure can be implemented in a highly parallel manner results are shown on several examples and comparisons are made with relaxation labeling techniques

relaxation labeling algorithm for information integration and its convergence relaxation labeling is an iterative and parallel processing scheme which has been used to implement information integration in the framework of relaxation labeling two sources of given information is encoded in a form of constraints and that of certainty measures of the labels respectively with this formulation the information integration is implemented by the relaxation operator by forcing neighborhood constraint satisfaction various versions of relaxation labeling algorithm have been proposed by other researchers and their applications to image processing scene analysis object identification have produced satisfactory and partially satisfactory results in the past as an iterative algorithm the convergence properties of relaxation labeling have been investigated by some researchers a new relaxation labeling algorithm is developed and presented in this paper the motivation as well as the performance of the new algorithm is discussed its local and global convergence properties are established analytically and numerically it is also demonstrated that the convergence analysis of the new algorithm can be easily generalized

encoding nearest larger values in nearest larger value nlv problems we are given an array a 1 n of numbers and need to preprocess a to answer queries of the following form given any index i in 1 n return a nearest index j such that a j a i we consider the variant where the values in a are distinct and we wish to return an index j such that a j a i and j i is minimized the nondirectional nlv nnlv problem we consider nnlv in the encoding model where the array a is deleted after preprocessing and note that nnlv encoding problem has an unexpectedly rich structure the effective entropy optimal space usage of the problem depends crucially on details in the definition of the problem using a new path compressed representation of binary trees that may have other applications we encode nnlv in 1 9n o n bits and answer queries in o 1 time

segmentation of images having unimodal distributions a gradient relaxation method based on maximizing a criterion function is studied and compared to the nonlinear probabilistic relaxation method for the purpose of segmentation of images having unimodal distributions although both methods provide comparable segmentation results the gradient method has the additional advantage of providing control over the relaxation process by choosing three parameters which can be tuned to obtain the desired segmentation results at a faster rate examples are given on two different types of scenes

on liouville type theorems for the steady navier stokes equations in r3 abstract in this paper we prove three different liouville type theorems for the steady navier stokes equations in r 3 in the first theorem we improve logarithmically the well known l 9 2 r 3 result in the second theorem we present a sufficient condition for the trivially of the solution v 0 in terms of the head pressure q 1 2 v 2 p the imposed integrability condition here has the same scaling property as the dirichlet integral in the last theorem we present fubini type condition which guarantee v 0

liouville theorems for the navier stokes equations and applications we study bounded ancient solutions of the navier stokes equations these are solutions with bounded velocity defined in rn 1 0 in two space dimensions we prove that such solutions are either constant or of the form u x t b t depending on the exact definition of admissible solutions the general 3 dimensional problem seems to be out of reach of existing techniques but partial results can be obtained in the case of axisymmetric solutions we apply these results to some scenarios of potential singularity formation for axi symmetric solutions and obtain extensions of results in a recent paper by chen strain tsai and yau 4

recovery of the three dimensional shape of an object from a single view given a single picture which is a projection of a three dimensional scene onto the two dimensional picture plane we usually have definite ideas about the 3 d shapes of objects to do this we need to use assumptions about the world and the image formation process since there exist a large number of shapes which can produce the same picture the purpose of this paper is to identify some of these assumptions mostly geometrical ones by demonstrating how the theory and techniques which exploit such assumptions can provide a systematic shape recovery method the method consists of two parts the first is the application of the origami theory which models the world as a collection of plane surfaces and recovers the possible shapes qualitatively the second is the technique of mapping image regularities into shape constraints for recovering the probable shapes quantitatively actual shape recovery from a single view is demonstrated for the scenes of an object such as a box and a chair given a single image the method recovers the 3 d shapes of an object in it and generates images of the same object as we would see it from other directions