a mathematical formulation for reactive transport that eliminates mineral concentrations we present a concise and general mathematical formulation for reactive transport in groundwater for general applications by means of linear algebraic manipulations of the stoichiometric coefficients of the chemical reactions we are able to reduce the number of unknowns of the equations to be solved to the number of degrees of freedom according to thermodynamic rules we present six formulations that differ from each other by number and type of unknowns and discuss their advantages and disadvantages with respect to the two most important numerical solution methods the sequential iteration approach sia and the direct substitution approach dsa our conclusion is that the proposed reduction of the number of variables is of special interest for the dsa we have applied one of these formulations to an example of the flushing of saline water by fresh water

computing the visual hull of solids of revolution abstract the visual hull vh s v of an object s relative to a viewing region v is a geometric entity useful for silhouette based image understanding for instance two objects can be distinguished from their silhouettes only if their visual hulls are different an object can be exactly reconstructed from its silhouettes only if equal to its visual hull the visual hull idea also allows to define a measure of the reconstruction accuracy of an object from its silhouettes there are two main cases of visual hull the internal visual hull if v is only bounded by s and the external visual hull if v is bounded by the convex hull of s this paper addresses the problem of computing both visual hulls of solids of revolution two cases are considered smooth polynomial and piecewise linear generators the algorithm has been implemented for the latter case the surface of the visual hulls when it is not coincident with the surface of s turns out to consist of segments of cones annular rings and hyperboloids of one sheet

a global strategy for solving reactive transport equations reactive transport models are complex nonlinear partial differential algebraic equations pdae coupling the transport engine with the geochemical operator we propose an efficient and robust global numerical method based on a method of lines and differential algebraic equations dae solvers combined with a newton method using a powerful sparse linear solver numerical experiments show the performances of the method we also propose a unified framework to describe classical methods such as sequential non iterative approach sequential iterative approach direct substitution approach and compare them with our global method

surface shape from the deformation of apparent contours the spatiotemporal analysis of deforming silhouettes apparent contours is here extended using the mathematics of perspective projections and tools from differential geometry analysis of the image motion of a silhouette or apparent contour enables computation of local surface curvature along the corresponding contour generator on the surface assuming viewer motion is known to perform the analysis a spatiotemporal parameterization of image curve motion is needed but is underconstrained a manifestation of the well known aperture problem it is shown that an epipolar parameterization is most naturally matched to the recovery of surface curvature

on computing structural changes in evolving surfaces and their appearance as a surface undergoes a one parameter family of deformations its shape and its appearance change smoothly except at certain critical parameter values where abrupt structural changes occur this paper considers the case of surfaces defined as the zero set of smooth density functions undergoing a gaussian diffusion process and addresses the problem of computing the critical parameter values corresponding to structural changes in the parabolic curves of a surface and in its aspect graph an algorithm based on homotopy continuation and curve tracing is proposed in the case of polynomial density functions whose zero set is an algebraic surface it has been implemented and examples are presented

a similarity based aspect graph approach to 3d object recognition this paper describes a view based method for recognizing 3d objects from 2d images we employ an aspect graph structure where the aspects are not based on the singularities of visual mapping but are instead formed using a notion of shape similarity between views specifically the viewing sphere is endowed with a metric of dis similarity for each pair of views and the problem of aspect generation is viewed as a segmentation of the viewing sphere into homogeneous regions the viewing sphere is sampled at regular 5 degree intervals and the similarity metric is used in an iterative procedure to combine views into aspects with a prototype representing each aspect this is done in a region growing regime which stands in contrast to the usual edge detection styles to computing the aspect graph the aspect growth is constrained such that two aspects of an object remain distinct under the given similarity metric once the database of 3d objects is organized as a set of aspects and prototypes for these aspects for each object unknown views of database objects are compared with the prototypes and the results are ordered by similarity we use two similarity metrics for shape one based on curve matching and the other based on matching shock graphs which for a database of 64 objects and unknown views of objects from the database give a recall rate of 90 3 74 2 59 7 and 95 2 69 0 57 5 respectively for the top three matchess cumulative recall rate based on the top three matches is 98 and 100 respectively the result of indexing unknown views of objects not in the database also produce intuitive matches we also develop a hierarchical indexing scheme to prune unlikely objects at an early stage to improve the efficiency of indexing resulting in savings of 35 at the top level and of 55 at the next level cumulatively

line drawing interpretation bilateral symmetry results previously derived by the author are used to investigate the implication of bilateral symmetry in line drawings it is shown that the line drawing of an orthographically protected surface of revolution exhibits bilateral symmetry about the projection of its axis of revolution irrespective of the viewing direction barring one exception a bilaterally symmetric line drawing is necessarily the orthographic projection of a local surface of revolution whenever its symmetry axis continues to be the projection of a fixed line in space under perturbation of viewpoint the axis of revolution is the invariant preimage of the symmetry axis various line drawing causes are detailed which facilitate the deduction of invariant preimages of symmetry axes and consequently of local surfaces of revolution

generalised epipolar constraints in this paper we will discuss structure and motion problems for curved surfaces these will be studied using the silhouettes or apparent contours in the images the problem of determining camera motion from the apparent contours of curved three dimensional surfaces is studied it will be shown how special points called epipolar tangency points or frontier points can be used to solve this problem a generalised epipolar constraint is introduced which applies to points curves as well as to apparent contours of surfaces the theory is developed for both continuous and discrete motion known and unknown orientation calibrated and uncalibrated perspective weak perspective and orthographic cameras results of an iterative scheme to recover the epipolar line structure from real image sequences using only the outlines of curved surfaces is presented a statistical evaluation is performed to estimate the stability of the solution it is also shown how the motion of the camera from a sequence of images can be obtained from the relative motion between image pairs

epipolar geometry from profiles under circular motion addresses the problem of motion estimation from profiles apparent contours of an object rotating on a turntable in front of a single camera a practical and accurate technique for solving this problem from profiles alone is developed it is precise enough to reconstruct the shape of the object no correspondences between points or lines are necessary symmetry of the surface of revolution swept out by the rotating object is exploited to obtain the image of the rotation axis and the homography relating epipolar lines in two views robustly and elegantly these together with geometric constraints for images of rotating objects are used to obtain first the image of the horizon which is the projection of the plane that contains the camera centers and then the epipoles thus fully determining the epipolar geometry of the image sequence the estimation of this geometry by this sequential approach avoids many of the problems found in other algorithms the search for the epipoles by far the most critical step is carried out as a simple 1d optimization parameter initialization is trivial and completely automatic at all stages after the estimation of the epipolar geometry the euclidean motion is recovered using the fixed intrinsic parameters of the camera obtained either from a calibration grid or from self calibration techniques finally the spinning object is reconstructed from its profiles using the motion estimated in the previous stage results from real data are presented demonstrating the efficiency and usefulness of the proposed methods

what s next an interactive next best view approach shape from silhouettes algorithms use either a surface or a volumetric approach a problem with the volumetric approach is that in general we know neither the accuracy of the reconstruction nor where to locate new viewpoints for improving the accuracy in this paper we present a general approach to interactive object specific volumetric algorithms based on a necessary condition for the best possible reconstruction to have been performed the outlined approach can be applied to any class of objects as an example of this approach an interactive algorithm has been implemented for convex polyhedra

how far 3d shapes can be understood from 2d silhouettes each 2d silhouette of a 3d unknown object o constrains o inside the volume obtained by back projecting the silhouette from the corresponding viewpoint a set of silhouettes specifies a boundary volume r obtained by intersecting the volumes due to each silhouette r more or less closely approximates o depending on the viewpoints and the object itself this approach to the reconstruction of 3d objects is usually referred to as volume intersection this correspondence addresses the problem of inferring the shape of the unknown object o from the reconstructed object r for doing this the author divides the points of the surface of r into hard points which belong to the surface of any possible object originating r and soft points which may or may not belong to o the author considers two cases in the first case r is the closest approximation of o which can be obtained from its silhouettes i e its visual hull in the second case r is a generic reconstructed object in both cases the author supplies necessary and sufficient conditions for a point to be hard and gives rules for computing the hard surfaces

3d surface reconstruction using occluding contours this paper addresses the problem of 3d surface reconstruction using image sequences it has been shown that shape recovery from three or more occluding contours of the surface is possible given a known camera motion several algorithms which have been recently proposed allow such a reconstruction under the assumption of a linear camera motion a new approach is presented which deals with the reconstruction problem directly from a discrete point of view first a theoretical study of the epipolar correspondence between occluding contours is achieved a correct depth formulation is then derived from a local approximation of the surface up to order two this allows the local shape to be estimated given three consecutive contours without any constraints on the camera motion experimental results are presented for both synthetic and real data

projective reconstruction and invariants from multiple images this correspondence investigates projective reconstruction of geometric configurations seen in two or more perspective views and the computation of projective invariants of these configurations from their images a basic tool in this investigation is the fundamental matrix that describes the epipolar correspondence between image pairs it is proven that once the epipolar geometry is known the configurations of many geometric structures for instance sets of points or lines are determined up to a collineation of projective 3 space spl pscrsup 3 by their projection in two independent images this theorem is the key to a method for the computation of invariants of the geometry invariants of six points in spl pscrsup 3 and of four lines in spl pscrsup 3 are defined and discussed an example with real images shows that they are effective in distinguishing different geometrical configurations since the fundamental matrix is a basic tool in the computation of these invariants new methods of computing the fundamental matrix from seven point correspondences in two images or six point correspondences in three images are given

computing the aspect graph for line drawings of polyhedral objects an algorithm for computing the aspect graph for polyhedral objects is described the aspects graph is a representation of three dimensional objects by a set of two dimensional views the set of viewpoints on the gaussian sphere is partitioned into regions such that in each region the qualitative structure of the line drawing remains the same at the boundaries between adjacent regions are the accidental viewpoints where the structure for the line drawing changes it is shown that for polyhedral objects there are two fundamental visual events 1 the projections of an edge and a vertex coincide and 2 the projections of three nonadjacent edges intersect at a point the geometry of the object is reflected in the locus of the accidental viewpoints the algorithm computes the partition together with a representative view for each region of the partition

global bifurcation sets and stable projections of nonsingular algebraic surfaces the view graph of a surface is a planar graph whose nodes are the stable views projections of the surface and whose edges represent transitional views of codimension one the space of all directions of orthogonal projection can be identified with the projective plane the set of bad projection directions associated with the degenerate views of positive codimension forms a graph in the projective plane the view bifurcation set this graph is dual to the view graph and divides the projective plane into a certain number of connected regions whose representatives are the nodes of the view graph we assume that the projected surface is nonsingular and parameterized by polynomials of degree d we present an estimate for the number of nodes in the view graph in terms of d and describe symbolic algorithms for computing the bifurcation set and the view graph of a surface from a parametrization

acoustic analogy and alternative theories for jet noise prediction several methods for the prediction of jet noise are described all but one of the noise prediction schemes are based on lighthill s or lilley s acoustic analogy whereas the other is the jet noise generation model recently proposed by tam and auriault in all of the approaches some assumptions must be made concerning the statistical properties of the turbulent sources in each case the characteristic scales of the turbulence are obtained from a solution of the reynolds averaged navier stokes equation using a kappa sigma turbulence model it is shown that for the same level of empiricism tam and auriault s model yields better agreement with experimental noise measurements than the acoustic analogy it is then shown that this result is not because of some fundamental flaw in the acoustic analogy approach but instead is associated with the assumptions made in the approximation of the turbulent source statistics if consistent assumptions are made both the acoustic analogy and tam and auriault s model yield identical noise predictions in conclusion a proposal is presented for an acoustic analogy that provides a clearer identification of the equivalent source mechanisms as is a discussion of noise prediction issues that remain to be resolved

the visual hull of smooth curved objects the visual hull is a geometric entity that relates the shape of an object to its silhouettes or shadows this paper develops the theory of the visual hull of generic smooth objects we show that the visual hull can be constructed using surfaces which partition the viewpoint space of the aspect graph of the object the surfaces are those generated by the visual events tangent crossing and triple point an analysis based on the shape of the object at the tangency points of these surfaces allows pruning away many surfaces and patches not relevant to the construction an algorithm for computing the visual hull is outlined

computing exact aspect graphs of curved objects algebraic surfaces this article presents an algorithm for computing the exact aspect graph of an opaque solid bounded by a smooth algebraic surface orthographic projection is assumed the algorithm is based on a catalog of visual events available from singularity theory it uses curve tracing cell decomposition homotopy continuation and ray tracing to construct the regions of the view sphere delineated by visual event curves the algorithm has been fully implemented and examples are presented

operator splitting for chemical reaction systems with fast chemistry abstract splitting methods are frequently used for large scale chemical reaction systems the main reason is the reduced computational cost for solving the subproblems in comparison to the time integration of the full problem however a splitting error is introduced the most popular splitting schemes are the first order lie trotter splitting and the second order strang splitting however in case of stiff differential equations the strang splitting suffers from order reduction and both schemes have order one for stiff differential equations therefore the step size restrictions due to the low order can result in a prohibitive small step size hence splitting schemes with order larger than one are necessary for stiff differential equations the richardson extrapolation of the lie trotter splitting is a second order scheme in this paper we examine differential equations with a fast chemical source term and a slow transport term thus all stiffness of the system is related to the chemical source term we show that the richardson extrapolation does not suffer from order reduction in this case furthermore we perform a stability analysis of the extrapolated splitting scheme for a linear test problem thereby the operators of the test problem do not commute

robust automatic selection of optimal views in multi view free form object recognition this paper addresses the issue of automatic selection of the best and the optimum number of representative views for each object in a database that can enable the accurate recognition of that object from any single arbitrary view of the object the object boundary in each view is represented by its curvature scale space css image the css representation has been selected for mpeg 7 standardisation as a contour shape descriptor the paper also presents a novel method for fusion of results from combined shape descriptors the utilisation of this method for multi view three dimensional 3 d object recognition has been explored the object boundary of each view is represented effectively using the css technique moment invariants and fourier descriptors it has been shown that the results obtained from the fusion method are superior to the results obtained from any single technique the method has been tested on a collection of free form 3 d objects each object has been modelled using an optimal number of silhouette contours obtained from different viewpoints this number varies depending on the complexity of the object and the measure of expected accuracy a comprehensive analysis of the performance of the system has been given

affine reconstruction of curved surfaces from uncalibrated views of apparent contours in this paper we consider uncalibrated reconstruction of curved surfaces from apparent contours since apparent contours are not fixed features viewpoint independent we cannot directly apply the recent results of the uncalibrated reconstruction from fixed features we show that nonetheless curved surfaces can be reconstructed up to an affine ambiguity from their apparent contours viewed from uncalibrated cameras with unknown linear translations furthermore we show that even if the reconstruction is nonmetric non euclidean we can still extract useful information for many computer vision applications just from the apparent contours we first show that if the camera motion is linear translation but arbitrary direction and magnitude the epipolar geometry can be recovered from the apparent contours without using any optimization process the extracted epipolar geometry is next used for reconstructing curved surfaces from the deformations of the apparent contours viewed from uncalibrated cameras the result is applied to distinguishing curved surfaces from fixed features in images it is also shown that the time to contact to the curved surfaces can be computed from simple measurements of the apparent contours

real time object recognition in sparse range images using error surface embedding a novel object recognition algorithm is introduced to identify objects and recover their pose from sparse range data the method is based upon comparing the 7 d error surfaces of objects in various poses which result from the registration error function between two convolved surfaces the objects and their pose values are encoded by a small set of feature vectors extracted from the minima of the error surfaces the problem of object recognition is thus reduced to comparing these feature vectors to find the corresponding error surfaces between the runtime data and a preprocessed database the algorithm called potential well space embedding pwse has been implemented and tested on both simulated and real data the experimental results show the technique to be both effective and efficient executing at 122 frames per second on standard hardware and with recognition rates exceeding 97 for a database of 60 objects the performance of pwse on the large size database was also evaluated on the princeton shape benchmark containing 1 814 objects in addition it functions well with very sparse data possibly comprising only hundreds of points per image and is shown to be robust to measurement error and outliers with some small modifications we applied pwse to the problem of object class recognition in experiments with the princeton shape benchmark pwse is able to provides better classification rates than the previous methods in terms of nearest neighbour classification

the scale space aspect graph currently the aspect graph is computed from the theoretical standpoint of perfect resolution in object shape the viewpoint and the projected image this means that the aspect graph may include details that an observer could never see in practice introducing the notion of scale into the aspect graph framework provides a mechanism for selecting a level of detail that is large enough to merit explicit representation this effectively allows control over the number of nodes retained in the aspect graph this paper introduces the concept of the scale space aspect graph defines three different interpretations of the scale dimension and presents a detailed example for a simple class of objects with scale defined in terms of the spatial extent of features in the image

on recognizing and positioning curved 3 d objects from image contours an approach for explicitly relating the shape of image contours to models of curved three dimensional objects is presented this relationship is used for object recognition and positioning object models consist of collections of parametric surface patches and their intersection curves this includes nearly all representations used in computer aided geometric design and computer vision the image contours considered are the projections of surface discontinuities and occluding contours elimination theory provides a method for constructing the implicit equation of these contours for an object observed under orthographic or perspective projection this equation is parameterized by the object s position and orientation with respect to the observer determining these parameters is reduced to a fitting problem between the theoretical contour and the observed data points the proposed approach readily extends to parameterized models it has been implemented for a simple world composed of various surfaces of revolution and tested on several real images

new two dimensional slope limiters for discontinuous galerkin methods on arbitrary meshes summary in this paper we introduce an extension of van leer s slope limiter for two dimensional discontinuous galerkin dg methods on arbitrary unstructured quadrangular or triangular grids the aim is to construct a non oscillatory shock capturing dg method for the approximation of hyperbolic conservative laws without adding excessive numerical dispersion unlike some splitting techniques that are limited to linear approximations on rectangular grids in this work the solution is approximated by means of piecewise quadratic functions the main idea of this new reconstructing and limiting technique follows a well known approach where local maximum principle regions are defined by enforcing some constraints on the reconstruction of the solution numerical comparisons with some existing slope limiters on structured as well as on unstructured meshes show a superior accuracy of our proposed slope limiters copyright 2004 john wiley sons ltd

visibility occlusion and the aspect graph this chapter studies the ways in which the topology of the image of a polyhedron changes with changing viewpoint we catalog the ways that the topological appearance or aspect can change this enables us to find maximal regions of viewpoints of the same aspect we use these techniques to construct the viewpoint space partition vsp a partition of viewpoint space into maximal regions of constant aspect and its dual the aspect graph here we present tight bounds on the maximum size of the vsp and the aspect graph and give algorithms for their construction first in the convex case and then in the general case in particular we give bounds on the maximum size of n 2 and n 6 under an orthographic projection viewing model and of n 3 and n 9 under a perspective viewing model the algorithms make use of a new representation of the appearance of polyhedra from all viewpoints called the aspect representation or asp we believe that this representation is one of the significant contributions of this paper

cylindrical algebraic decomposition i the basic algorithm given a set of r variate integral polynomials a cylindrical algebraic decomposition cad of euclidean r space e r partitions e r into connected subsets compatible with the zeros of the polynomials by compatible with the zeros of the polynomials we mean that on each subset of e r each of the polynomials either vanishes everywhere or nowhere for example consider the bivariate polynomial y 4 2 y 3 y 2 3 x 2 y 2 x 4

using extremal boundaries for 3 d object modeling the extremal boundaries of 3 d curved objects are the images of special curves drawn on the object and are called rims they are viewpoint dependent and characterized by the fact that the optical rays of their points are tangential to the surface of the object the mathematics of the relationship between the extremal boundaries and the surface of the object is studied this study makes it possible to design an algorithm for detecting those boundaries in the images that are likely to be extremal once this has been done one can reconstruct the rims and compute the differential properties of the surface of the object along them up to the second order if a qualitative description is sufficient the sign of the gaussian curvature of the surface along the rim can be computed in a much simpler way experimental results are presented on synthetic and real images the work provides a better understanding of the relationship between the apparent and real shape of a 3 d object as well as algorithms for reconstructing the local shape of such an object along the rims

the fundamental matrix theory algorithms and stability analysis in this paper we analyze in some detail the geometry of a pair of cameras i e a stereo rig contrarily to what has been done in the past and is still done currently for example in stereo or motion analysis we do not assume that the intrinsic parameters of the cameras are known coordinates of the principal points pixels aspect ratio and focal lengths this is important for two reasons first it is more realistic in applications where these parameters may vary according to the task active vision second the general case considered here captures all the relevant information that is necessary for establishing correspondences between two pairs of images this information is fundamentally projective and is hidden in a confusing manner in the commonlyused formalism of the essential matrix introduced by longuet higgins 1981 this paper clarifies the projective nature of the correspondence problem in stereo and shows that the epipolar geometry can be summarized in one 3 x 3 matrix of rank 2 which we propose to call the fundamental matrix after this theoretical analysis we embark on the task of estimating the fundamental matrix from point corre spondences a task which is of practical importance we analyze theoretically and compare experimentally using synthetic and real data several methods of estimation the problem of the stability of the estimation is studied from two complementary viewpoints first we show that there is an interesting relationship between the fundamental matrix and three dimensional planes which induce homographies between the images and create unstabilities in the estimation procedures second we point to a deep relation between the unstability of the estimation procedure and the presence in the scene of so called critical surfaces which have been studied in the context of motion analysis finally we conclude by stressing the fact that we believe that the fundamental matrix will play a crucial role in future applications of three dimensional computer vision by greatly increasing its versatility robustness and hence applicability to real difficult problems

a structural approach to representation of curved objects abstract representation of complex curved objects is still a challenging and difficult problem in this paper we present a scheme for multi level structural qualitative representation of object surfaces the analysis begins with local face based descriptions of object surfaces surface primitive graphs and global high level structural and qualitative features are derived from these local descriptions the final descriptions global feature graphs are based on the quasi convexity concavity structures incorporating quantized directional features of closed boundaries of cross sections along several directions the global feature graphs have the following properties 1 they provide a characterization of global structural qualitative properties of the object 2 the graphs are composed of a few components nodes and edges with rich features and give compact descriptions of the objects 3 the global feature graph representations are robust and qualitative in that they are not affected by geometrical quantitative properties of the shape such as curvatures length and position 4 the representation has clear and rigorous properties furthermore we present some theoretical basis for object recognition using the global features graphs

strategies for solving index one dae with non negative constraints liquid liquid extraction modeling leads to solve an index one dae system for the sake of robustness it is desirable to account for non negative constraints based on the dassl architecture a classical index one dae solver we propose and compare three different strategies to implement these bound constraints each of these strategies corresponds to a different newton modification clipping damping or interior point method the comparisons are made on two test cases the robertson ode problem and an example from liquid liquid extraction modeling

noise from fine scale turbulence of nonaxisymmetric jets the noise from the fine scale turbulence of high speed nonaxisymmetric jets is considered a prediction method is developed by extending the work of tam and auriault a set of improved numerical boundary conditions for use in nonaxisymmetricjet mean flow and turbulence calculation is developed these new boundary conditions allow the computation to be carried out in a smaller computation domain it is known that nonaxisymmetric mean flow has a significant impact on the radiated noise spectrum and directivity through refraction in the tam and auriault theory this effect is accounted for by means of the adjoint green s function here the adjoint green s function method is extended to nonaxisymmetric mean flows the adjoint green s function is first recast into the solution of a sound scattering problem the sound scattering problem is then solved computationally by computational aeroacoustics methods extensive comparisons between calculated and experimentally measured jet noise spectra are presented they include both rectangular and elliptic jets at supersonic and subsonic mach numbers

localizing overlapping parts by searching the interpretation tree this paper discusses how local measurements of positions and surface normals may be used to identify and locate overlapping objects the objects are modeled as polyhedra or polygons having up to six degrees of positional freedom relative to the sensors the approach operates by examining all hypotheses about pairings between sensed data and object surfaces and efficiently discarding inconsistent ones by using local constraints on distances between faces angles between face normals and angles relative to the surface normals of vectors between sensed points the method described here is an extension of a method for recognition and localization of nonoverlapping parts previously described in 18 and 15

eigenpaxels and a neural network approach to image classification a expansion encoding approach to image classification is presented localized principal components or eigenpaxels are used as a set of basis functions to represent images that is principal component analysis is applied locally rather than on the entire image the eigenpaxels are statistically determined using a database of the images of interest classification based on visual similarity is achieved through the use of a single layer error correcting neural network expansion encoding and the technique of subsampling are key elements in the processing stages of the eigenpaxel algorithm tested using a database of frontal face images consisting of 40 individuals the algorithm exhibits equivalent performance to other comparable but more cumbersome methods in addition the technique is shown to be robust to various types of image noise

projective visual hulls this article presents a novel method for computing the visual hull of a solid bounded by a smooth surface and observed by a finite set of cameras the visual hull is the intersection of the visual cones formed by back projecting the silhouettes found in the corresponding images we characterize its surface as a generalized polyhedron whose faces are visual cone patches edges are intersection curves between two viewing cones and vertices are frontier points where the intersection of two cones is singular or intersection points where triples of cones meet we use the mathematical framework of oriented projective differential geometry to develop an image based algorithm for computing the visual hull this algorithm works in a weakly calibrated setting that is it only requires projective camera matrices or equivalently fundamental matrices for each pair of cameras the promise of the proposed algorithm is demonstrated with experiments on several challenging data sets and a comparison to another state of the art method

a reduction scheme for coupled multicomponent transport reaction problems in porous media generalization to problems with heterogeneous equilibrium reactions 1 in this article a systematic approach for the efficient computation of the transport and reaction of a multispecies multireaction system is proposed the objective of this approach is to reformulate the given system of differential or differential algebraic equations in such a way that the couplings and the nonlinearities are concentrated in a reduced number of equations if compared to the original formulation while some linear equations decouple from the system the resulting system is handled in the spirit of a global implicit approach one step method avoiding operator splitting techniques the reduction of the problem size proposed in this article helps to limit the large computational costs of numerical simulations for such problems the reduction mechanism is a generalization of the method proposed in a previous paper now problems with mixed mobile immobile species homogeneous heterogeneous kinetic equilibrium reactions are considered while the previous publication was restricted to problems without heterogeneous equilibrium reactions such as equilibrium sorption an application of the reduction mechanism to an example problem is given in order to investigate the reduction of the number of coupled nonlinear equations and to compare it to other methods

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

computing the generalized aspect graph for objects with moving parts algorithms for computing the aspect graph representation are generalized to include a larger more realistic domain of objects known as articulated assemblies those objects composed of rigid parts with articulated connections allowed between parts the generalization suggests two slightly different representations one that directly summarizes the possible general views of the object and another hierarchical form summarizing the possible general configurations and their respective views algorithms are outlined for computing both representations the generalized aspect graphs of assemblies formed using translational connections are examined

global surface reconstruction by purposive control of observer motion abstract what viewpoint control strategies are important for performing global visual exploration tasks such as searching for specific surface markings building a global model of an arbitrary object or recognizing an object in this paper we consider the task of purposefully controlling the motion of an active monocular observer in order to recover a global description of a smooth arbitrarily shaped object we formulate global surface reconstruction as the task of controlling the motion of the observer so that the visible rim slides over the maximal connected reconstructible surface regions intersecting the visible rim at the initial viewpoint we show that these regions are bounded by a subset of the visual event curves defined on the surface by studying the epipolar parameterization we develop two basic strategies that allow reconstruction of a surface region around any point in a reconstructible surface region these strategies control viewpoint to achieve and maintain a well defined geometric relationship with the object s surface rely only on information extracted directly from images e g tangents to the occluding contour and are simple enough to be performed in real time we then show how global surface reconstruction can be provably achieved by 1 appropriately integrating these strategies to iteratively grow the reconstructed regions and 2 obeying four simple rules

shape and motion from image streams under orthography a factorization method inferring scene geometry and camera motion from a stream of images is possible in principle but is an ill conditioned problem when the objects are distant with respect to their size we have developed a factorization method that can overcome this difficulty by recovering shape and motion under orthography without computing depth as an intermediate step an image stream can be represented by the 2fxp measurement matrix of the image coordinates of p points tracked through f frames we show that under orthographic projection this matrix is of rank 3 based on this observation the factorization method uses the singular value decomposition technique to factor the measurement matrix into two matrices which represent object shape and camera rotation respectively two of the three translation components are computed in a preprocessing stage the method can also handle and obtain a full solution from a partially filled in measurement matrix that may result from occlusions or tracking failures the method gives accurate results and does not introduce smoothing in either shape or motion we demonstrate this with a series of experiments on laboratory and outdoor image streams with and without occlusions

acquiring 3 d models from sequences of contours this paper explores shape from contour for acquiring 3 d graphics models in this method a continuous sequence of images is taken as an object rotates a smooth convex shape can be estimated instantaneously from its contour and by the first derivative of contour movement trace of contour or contour distribution with time we also analyze shapes that do not satisfy the conditions of smoothness and visibility which are indispensable for modeling an object a region that does not expose as contour yields a nonsmoothness in the tracked contour movement we can thus detect such a region by contour distribution filtering and extract its accurate location by computing the left and right derivatives of the distribution this has not been studied previously these unknown regions are obtained for further investigation using other visual cues a general approach for building a geometrical object model using contours is then described the entire process from silhouettes to a 3 d model is based local computation this is promising for producing shapes in real time our direct goal is to establish 3 d graphics models of human faces for the growing needs of visual communications we have obtained some good results

interpreting line drawings of curved objects in this thesis we study the problem of interpreting line drawings of scenes composed or opaque regular solid objects bounded by piecewise smooth surfaces with no markings or texture on them it is assumed that the line drawing has been formed by orthographic projection of such a scene under general viewpoint that the line drawing is error free and that there are no lines due to shadows or specularities our definition implicitly excludes laminae wires and the apices of cones a major component of the interpretation of line drawings is line labelling by line labelling we mean a classification of each image curve as corresponding to either a depth or orientation discontinuity in the scene and b further subclassification of each kind of discontinuity for a depth discontinuity we determine whether it is a limb a locus of points on the surface where the line of sight is tangent to the surface or an occluding edge a tangent plane discontinuity of the surface for an orientation discontinuity we determine whether it corresponds to a convex or concave edge this thesis presents the first mathematically rigorous scheme for labelling line drawings of the class of scenes described by analysing the projection of the neighborhoods of different kinds of points on a piecewise smooth surface we are able to catalog all local labelling possibilities for the different types of junctions in a line drawing an algorithm is developed which utilizes this catalog to determine all legal labellings of the line drawing a local minimum complexity rule at each vertex select those labellings which correspond to the minimum number of faces meeting at the vertex is used in order to prune highly counter intuitive interpretations the labelling scheme was implemented and tested on a number of line drawings the labellings obtained are few and by and large in accordance with human interpretations

efficiently computing and representing aspect graphs of polyhedral objects an efficient algorithm and a data structure for computing and representing the aspect graph of polyhedral objects under orthographic projection are presented the aspect graph is an approach to representing 3 d objects by a set of 2 d views for the purpose of object recognition in this approach the viewpoint space is partitioned into regions such that in each region the qualitative structure of the line drawing does not change the viewing data of an object is the partition of the viewpoint space together with a representative view in each region the algorithm computes the viewing data for line drawings of polyhedral objects under orthographic projection

the geometry of view space of opaque objects bounded by smooth surfaces abstract a view of a smooth opaque object is a line drawing consisting of contour fragments which form closed loops terminate or form t junctions given a compact object we have a global decomposition of the space of camera positions into cells from which topologically equivalent views are obtained to apply certain image based object recognition techniques based on characteristic views of an object one needs to know this cellular decomposition of view space the boundaries of such cells of stable views are formed by the view bifurcation set from which transitional degenerate views are seen we derive a finite list of geometrical models of the view bifurcation set the view bifurcation set of any particular object is a combination of such model surfaces from this list to match a characteristic view with an image one has to find the visible fragments of the apparent contour in the image and we briefly describe an observation which might lead to an algorithm for finding such contour fragments

the enumerative geometry of projective algebraic surfaces and the complexity of aspect graphs the aspect graph is a popular viewer centered representation that enumerates all the topologically distinct views of an object building the aspect graph requires partitioning viewpoint space in view equivalent cells by a certain number of visual event surfaces if the object is piecewise smooth algebraic then all visual event surfaces are either made of lines having specified contacts with the object or made of lines supporting the points of contacts of planes having specified contacts with the object in this paper we present a general framework for studying the enumerative properties of line and plane systems the context is that of enumerative geometry and intersection theory in particular we give exact results for the degrees of all visual event surfaces coming up in the construction of aspect graphs of piecewise smooth algebraic bodies we conclude by giving a bound on the number of topologically distinet views of such objects

determining the epipolar geometry and its uncertainty a review two images of a single scene object are related by the epipolar geometry which can be described by a 3 3 singular matrix called the essential matrix if images internal parameters are known or the fundamental matrix otherwise it captures all geometric information contained in two images and its determination is very important in many applications such as scene modeling and vehicle navigation this paper gives an introduction to the epipolar geometry and provides a complete review of the current techniques for estimating the fundamental matrix and its uncertainty a well founded measure is proposed to compare these techniques projective reconstruction is also reviewed the software which we have developed for this review is available on the internet

topological recognition of polyhedral objects from multiple views abstract the aspect graph ag has been proposed as a viewer centered tool for object recognition although the aspects could be used for recognition from one image the full information stored in the ag aspects and visual events suggests a multiple views possibly active approach the topological nature of the ag also suggests a topological match of images of the unknown object and stored aspects in this paper we present a theoretical investigation on the use of the ag for topological recognition from multiple views of polyhedral objects first we discuss the topological matching process and give a suitable topological definition of aspect moreover since topological identification is approximate we tackle the problem of understanding the ability of the ag to topologically discriminate different objects more precisely we address the question how similar are two polyhedra with the same ag the isomorphism of polyhedra is chosen as the reference similarity condition the cases of general and convex polyhedra are discussed under perspective and parallel projections in practice it could be difficult to identify the visual events from the images of the unknown object thus we also compare isomorphism and the similarity induced by i a topologically reduced aspect graph ii the set of topologically different aspects neglecting visual events

computing exact aspect graphs of curved objects solid of revolution this paper introduces a new approach to computing the exact orthographic aspect graph of curved objects curves corresponding to various visual events partition the view sphere into regions where the image structure is stable a catalog of these events for piecewise smooth objects is available from singularity theory for a solid of revolution whose generator is an algebraic curve each visual event is characterized by a system of polynomial equations whose roots can be computed by continuation methods within each region the stable image structure is characterized by a variation of cylindrical algebraic decomposition and ray tracing this approach has been implemented and several examples are presented

the visual hull concept for silhouette based image understanding many algorithms for both identifying and reconstructing a 3 d object are based on the 2 d silhouettes of the object in general identifying a nonconvex object using a silhouette based approach implies neglecting some features of its surface as identification clues the same features cannot be reconstructed by volume intersection techniques using multiple silhouettes of the object this paper addresses the problem of finding which parts of a nonconvex object are relevant for silhouette based image understanding for this purpose the geometric concept of visual hull of a 3 d object is introduced this is the closest approximation of object s that can be obtained with the volume intersection approach it is the maximal object silhouette equivalent to s i e which can be substituted for s without affecting any silhouette only the parts of the surface of s that also lie on the surface of the visual hull can be reconstructed or identified using silhouette based algorithms the visual hull depends not only on the object but also on the region allowed to the viewpoint two main viewing regions result in the external and internal visual hull in the former case the viewing region is related to the convex hull of s in the latter it is bounded by s the internal visual hull also admits an interpretation not related to silhouettes algorithms for computing visual hulls are presented and their complexity analyzed in general the visual hull of a 3 d planar face object turns out to be bounded by planar and curved patches

spectral embedding of graphs in this paper we explore how to embed symbolic relational graphs with unweighted edges in a pattern space we adopt a graph spectral approach we use the leading eigenvectors of the graph adjacency matrix to define eigenmodes of the adjacency matrix for each eigenmode we compute vectors of spectral properties these include the eigenmode perimeter eigenmode volume cheeger number inter mode adjacency matrices and intermode edge distance we embed these vectors in a pattern space using two contrasting approaches the first of these involves performing principal or independent components analysis on the covariance matrix for the spectral pattern vectors the second approach involves performing multidimensional scaling on the l2 norm for pairs of pattern vectors we illustrate the utility of the embedding methods on neighbourhood graphs representing the arrangement of corner features in 2d images of 3d polyhedral objects two problems are investigated the first of these is the clustering of graphs representing distinct objects viewed from different directions the second is the identification of characteristic views of single objects these two studies reveal that both embedding methods result in well structured view spaces for graph data extracted from 2d views of 3d objects

modeling of chemically reactive groundwater transport a complete reactive groundwater transport model must account for both chemical and transport processes for the chemical processes one has to decide whether to formulate them kinetically or to assume a local chemical equilibrium state this decision in the chemical model part determines the mathematical structure of the overall model in a kinetic formulation the linear partial differential equations of the transport have to be coupled with a nonlinear system of ordinary differential equations describing the kinetic development whereas in an equilibrium formulation the equations of the transport are coupled to a nonlinear system of algebraic equations describing the equilibrium state basically two kinds of methods for solving reactive transport systems may be distinguished namely one step methods which simultaneously solve the transport and the chemical model parts and two step methods which solve these model parts separately we here present a sequential two step method for kinetic transport models and an iterative two step method for equilibrium transport models we conduct a timescale analysis to check whether the error of the sequential two step method is tolerable and whether a given kinetic transport system can be reduced to an equilibrium one the numerical methods and the timescale analysis are applied to two test cases zysset et al 1994 present a further application of the kinetic transport model to laboratory column experiments governed by biodegradation

local image features resulting from 3 dimensional geometric features illumination and movement i we study images of smooth or piecewise smooth objects illuminated by a single light source with only background illumination from other sources the objects may have geometric features f namely surface markings boundary edges creases and corners and shade features s namely shade curves and cast shadow curves we determine the local stable interactions between these and apparent contours c for the various configurations of f s c and we concisely summarize them using an alphabet of local curve configurations we further determine the generic transitions for the configurations resulting from viewer movement these classifications are obtained using the methods of singularity theory which allows us to ensure that our lists are complete in some cases correcting earlier attempts at similar classifications

reply by the authors to c k w tam the prediction of noise generation and radiation by turbulence has been the subject of continuous research for over fifty years the essential problem is how to model the noise sources when one s knowledge of the detailed space time properties of the turbulence is limited we attempted to provide a comparison of models based on acoustic analogies and recent alternative models our goal was to demonstrate that the predictive capabilities of any model are based on the choice of the turbulence property that is modeled as a source of noise our general definition of an acoustic analogy is a rearrangement of the equations of motion into the form l u q where l is a linear operator that reduces to an acoustic propagation operator outside a region upsilon u is a variable that reduces to acoustic pressure or a related linear acoustic variable outside upsilon and q is a source term that can be meaningfully estimated without knowing u and tends to zero outside upsilon