Shape Modeling

Knowledge about the biological variability of anatomical objects is essential for statistical shape analysis and discrimination between healthy and pathological structures. This paper describes a novel approach that incorporates the variability of an object population into the generation of a characteristic 3D shape model. The proposed shape representation is a coarse-scale sampled medial description derived from a fine-scale spherical harmonics (SPHARM) boundary description. This medial description is composed of a net of medial samples (m-rep) with fixed graph properties. The medial model is computed automatically from a predefined shape space using pruned 3D Voronoi skeletons. A new method determines the stable medial branching topology from the shape space. An intrinsic coordinate system and an implicit correspondence between shapes is defined on the medial manifold. Several studies of biological structures clearly demonstrate that the novel representation has the promise to describe shape changes in a natural and intuitive way. A new medial shape similarity study of group differences between monozygotic and dizygotic twins in lateral ventricle shape demonstrates the meaningful and powerful representation of local and global form.

This article surveys deformable models, a promising and vigorously researched computer-assisted medical image analysis technique. Among model-based techniques, deformable models offer a unique and powerful approach to image analysis that combines geometry, physics, and approximation theory. They have proven to be effective in segmenting, matching, and tracking anatomic structures by exploiting (bottom-up) constraints derived from the image data together with (top-down) a priori knowledge about ...

8 typically slower than ϳ1 km s −1 ) might differ significantly from what is assumed by current modelling efforts 27 . The expected equation-of-state differences among small bodies (ice versus rock, for instance) presents another dimension of study; having recently adapted our code for massively parallel architectures (K. M. Olson and E.A, manuscript in preparation), we are now ready to perform a more comprehensive analysis.

With the advent of Concurrent Engineering (CE), the engineering product design process has c h g e d substantially. Different factors of the product life cycle, such as the manufacturing process, part assembly and quality control, can now all be simultaneously involved in the design phase. To achieve CE traditional Computer-Aided Design systems are inadequate. Cooperative product design needs more powerful modelling techniques and geometric representation schemes than those offered by actual commercial CAD systems. I n this paper a multiple-view CAD representation, called Intermediate Model, is presented. The Intermediate Model has been defined with the aim to upgrade a traditional CAD system to a feature-based modelling system which allows to represent different context dependent views of the product.

This article presents a 3D shape matching method for 3D mesh models applied to content-based search in database of 3D objects. The approach is based on the multiresolution Reeb graph (MRG) proposed by Hilaga et al 1 . MRG provides a rich representation of shapes able in particular to embed the object topology. In our framework, we consider 3D mesh models of various geometrical complexity, of different resolution, and when available with color texture map. The original approach, mainly based on the 3D object topology, is not accurate enough to obtain satisfying matching. Therefore we propose to reinforce the topological consistency conditions of the matching and to merge within the graph geometrical and visual information to improve matching and calculation of shape similarity between models. Besides, all these new attributes can be freely weighted to fit the user requirements for object retrieval. We obtain a flexible multiresolutional and multicriteria representation that we called augmented multiresolution Reeb graph (aMRG). The approach has been tested and compared with other methods. It reveals very performant for the retrieval and the classification of similar 3D shapes.

Communicated by (xxxxxxxxxx) Recent works in geometric modeling show the advantage of local differential coordinates in various surface processing applications. In this paper we advocate surface representation via differential coordinates as a basis to interactive mesh editing. One of the main challenges in editing a mesh is to retain the visual appearance of the surface after applying various modifications. The differential coordinates capture the local geometric details and therefore are a natural surface representation for editing applications. The coordinates are obtained by applying a linear operator to the mesh geometry. Given suitable deformation constraints, the mesh geometry is reconstructed from the differential representation by solving a sparse linear system. The differential coordinates are not rotation-invariant and thus their rotation must be explicitly handled in order to retain the correct orientation of the surface details. We suggest two methods for computing the local rotations: the first estimates them heuristically using a deformation which only preserves the underlying smooth surface, and the second estimates the rotations implicitly through a variational representation of the problem. We show that the linear reconstruction system can be solved fast enough to guarantee interactive response time thanks to a precomputed factorization of the coefficient matrix. We demonstrate that our approach enables to edit complex meshes while retaining the shape of the details in their natural orientation.

This paper presents work in progress and continues a project devoted to developing shape modeling system for surface generation and enhancement. A local mesh enhancement based on statistical characteristics of an initial triangle mesh is proposed for mesh simplification. Experimental results are included to demonstrate the functionality of our simplification algorithm.

A rational boundary Gregory patch is characterized by the facts that any n-sided loop can be smoothly interpolated and that it can be smoothly connected to an adjacent patch. Thus, it is well-suited to interpolate complicated wire frames in shape modeling. Although a rational boundary Gregory patch can be exactly converted to a rational B6zier patch to enable the exchange of data, problems of high degree and singularity tend to arise as a result of conversion. This paper presents an algorithm that can approximately convert a rational boundary Gregory patch to a bicubic nonuniform B-spline surface. The approximating surface has C x continuity between its inner patches.

Recognizing 3D objects in the presence of noise, varying mesh resolution, occlusion and clutter is a very challenging task. This paper presents a novel method named Rotational Projection Statistics (RoPS). It has three major modules: local reference frame (LRF) definition, RoPS feature description and 3D object recognition. We propose a novel technique to define the LRF by calculating the scatter matrix of all points lying on the local surface. RoPS feature descriptors are obtained by rotationally projecting the neighboring points of a feature point onto 2D planes and calculating a set of statistics (including low-order central moments and entropy) of the distribution of these projected points. Using the proposed LRF and RoPS descriptor, we present a hierarchical 3D object recognition algorithm. The performance of the proposed LRF, RoPS descriptor and object recognition algorithm was rigorously tested on a number of popular and publicly available datasets. Our proposed techniques exhibited superior performance compared to existing techniques. We also showed that our method is robust with respect to noise and varying mesh resolution. Our RoPS based algorithm achieved recognition rates of 100, 98.9, 95.4 and 96.0 % respectively when tested on the Bologna, UWA, Queen's and Ca' Foscari Venezia Datasets.

Symbolic computation of NURBS plays an important role in many areas of NURBS-based geometric computation and design. However, any nontrivial symbolic computation, especially when rational B-splines are involved, would typically result in B-splines with high degrees. In this paper we develop degree reduction strategies for NURBS symbolic computation on curves. The specific topics we consider include zero curvatures and critical curvatures of plane curves, various ruled surfaces related to space curves, and point/curve bisectors and curve/curve bisectors.

Experimental and theoretical investigations indicate particle shape and size influence biomass particle dynamics, including drying, heating rate, and reaction rate. Experimental samples include disc/flake-like, cylindrical/cylinder-like, and equant (nearly spherical) shapes of wood particles with similar particle masses and volumes but different surface areas. Small samples (320 lm) passed through a laboratory entrained-flow reactor in a nitrogen atmosphere and a maximum reactor wall temperature of 1600 K. Large samples were suspended in the center of a single-particle reactor. Experimental data indicate that equant particles react more slowly than the other shapes, with the difference becoming more significant as particle mass or aspect ratio increases and reaching a factor of two or more for particles with sizes over 10 mm. A one-dimensional, time-dependent particle model simulates the rapid pyrolysis process of particles with different shapes. The model characterizes particles in three basic shapes (sphere, cylinder, and flat plate). With the particle geometric information (particle aspect ratio, volume, and surface area) included, this model simulates the devolatilization process of biomass particles of any shape. Model simulations of the three shapes show satisfactory agreement with the experimental data. Model predictions show that both particle shape and size affect the product yield distribution. Near-spherical particles exhibit lower volatile and higher tar yields relative to aspherical particles with the same mass under similar conditions. Volatile yields decrease with increasing particle size for particles of all shapes. Assuming spherical or isothermal conditions for biomass particles leads to large errors at most biomass particle sizes of practical interest.

We propose two mechanisms that can be used to generate complex geometry: generalized subdivision and mesh-based parametrized L-Systems. Instead of using standard subdivision, which uses the same subdivision rule at each level of the subdivision process, in order to converge to a limit surface, we employ a generalized approach, that allows different subdivision rules at each level. By limiting the variations at each level, it is possible to ensure convergence. Mesh-based parametrized L-Systems represent an extension to L-systems which associates symbols to the faces in a mesh. Thereby complex geometry can be introduced into an existing mesh by using procedural substitution rules. Combining both these mechanisms, a wide variety of complex models can be easily generated from very compact representations.

We have analyzed photometric lightcurves of 30 asteroids, and present here the obtained shapes, rotational periods and pole directions. We also present new photometric observations of five asteroids. The shape models indicate the existence of many features of varying degrees of irregularity. Even large main-belt asteroids display such features, so the resulting poles and periods are more consistent than those obtained by simple ellipsoid-like models. In some cases the new rotational parameters are rather different from those obtained previously, and in a few cases there were no proper previous estimates at all. and 694 Ekard. New observations of 21 Lutetia, 37 Fides, 85 Io, 129 Antigone, and 135 Hertha are presented in Section 4. In Section 5 we sum up the paper and discuss some future prospects of the analysis of photometric and complementary data.

8 typically slower than ϳ1 km s −1 ) might differ significantly from what is assumed by current modelling efforts 27 . The expected equation-of-state differences among small bodies (ice versus rock, for instance) presents another dimension of study; having recently adapted our code for massively parallel architectures (K. M. Olson and E.A, manuscript in preparation), we are now ready to perform a more comprehensive analysis.

Mr. Mahmoodul Islam, Lecturer, Faculty of Engineering, for constant motivation and support during the course of our work in the last few months. We truly appreciate and value his esteemed guidance and encouragement from the beginning to the end of this project.

The creation of building and facility models is a tedious and complicated task. Existing CAD models are typically not well suited since they contain too much or not enough detail; the manual modeling approach does not scale; different views on the same model are needed, and different levels of detail and abstraction; and finally, conventional modeling tools are inappropriate for models with many internal parameter dependencies. As a solution to this problem we propose a combination of a procedural approach with shape grammars. The model is created in a top-down manner; high-level changeability and re-usability are much less of a problem; and it can be interactively evaluated to provide different views at runtime. We present some insights on the relation between imperative and declarative grammar descriptions, and show a detailed case study with facility surveillance as a practical application.

Computer-Aided Geometric Design modelers are now based on powerful mathematical curve and surface models, but there is still a considerable need for efficient tools to handle, analyze and modify these objects. Designing product shapes using geometric operations on free-form curves and surfaces is still a tedious task. Moreover, designers would prefer to use meaningful tools to concentrate on design objectives expressed in terms of functionalities and constraints related to engineering topics and technical matters.

Advances in digital design and fabrication technologies are leading toward single fabrication systems capable of producing almost any complete functional object. We are proposing a new paradigm for manufacturing, which we call Universal Desktop Fabrication (UDF), and a framework for its development. UDF will be a coherent system of volumetric digital design software able to handle infinite complexity at any spatial resolution and compact, automated, multi-material digital fabrication hardware. This system aims to be inexpensive, simple, safe and intuitive to operate, open to user modification and experimentation, and capable of rapidly manufacturing almost any arbitrary, complete, high-quality, functional object. Through the broad accessibility and generality of digital technology, UDF will enable vastly more individuals to become innovators of technology, and will catalyze a shift from specialized mass production and global transportation of products to personal customization and point-of-use manufacturing. Likewise, the inherent accuracy and speed of digital computation will allow processes that significantly surpass the practical complexity of the current design and manufacturing systems. This transformation of manufacturing will allow for entirely new classes of human-made, peer-produced, micro-engineered objects, resulting in more dynamic and natural interactions with the world. We describe and illustrate our current results in UDF hardware and software, and describe future development directions.

Very large graphics models are common in a number of applications, and many di erent simplication methods have been recently developed. Some of them support the construction of multiresolution representations of the input meshes. On the basis of these innovative techniques, we foresee a modeling framework based on three separate stages shape modeling, multiresolution encoding and resolution modeling, and propose a new approach to the last stage, resolution modeling, which i s highly general, user-driven and not strictly tied to a particular simpli cation method. The approach proposed is based on a multiresolution representation scheme for triangulated, 2manifold meshes, the Hypertriangulation Model HyT. This scheme allows to selectively walk" along the multiresolution surface, moving between adjacent faces e ciently. A prototypal resolution modeling system, Zeta, has been implemented to allow i n teractive modeling of surface details and evaluated on several practical models. It supports: e cient extraction of xed resolution representations; uni ed management of selective re nement and selective simpli cation; easy composition of the selective re nement simpli cation actions, with no cracks in the variable resolution mesh produced; multiresolution editing; interactive response times.

Industrial designers make sketches and physical models to start and develop ideas and concept designs. Such representations have advantages that they support fast, intuitive, rich, sensory exploration of solutions. Although existing tools and techniques provide adequate support where the shape of the product is concerned, the exploration of surface qualities such as material and printed graphics is supported to a much lesser extent. Moreover, there are no tools that have the fluency of sketching that allow combined exploration of shape, material, and their interactions. This paper evaluates Skin, an augmented reality tool designed to solve these two shortcomings. By projecting computer-generated images onto the shape model Skin allows for a "sketchy" tangible interaction where designers can explore surface qualities on a three-dimensional physical shape model. The tool was evaluated in three design situations in the domain of ceramics design. In each case, we found that the joint exploration of shape and surface provided creative benefits in the form of new solutions; in addition, a gain in efficiency was found in at least one case. The results show that joint exploration of shape and surface can be effectively supported with tangible augmented reality techniques and suggest that this can be put to practical use in industry today.

We present a new and efficient algorithm to accurately polygonize an implicit surface generated by multiple Boolean operations with globally deformed primitives. Our algorithm is special in the sense that it can be applied to objects with both an implicit and a parametric representation, such as superquadrics, supershapes, and Dupin cyclides. The input is a Constructive Solid Geometry tree (CSG tree) that contains the Boolean operations, the parameters of the primitives, and the global deformations. At each node of the CSG tree, the implicit formulations of the subtrees are used to quickly determine the parts to be transmitted to the parent node, while the primitives' parametric definition are used to refine an intermediary mesh around the intersection curves. The output is both an implicit equation and a mesh representing its solution. For the resulting object, an implicit equation with guaranteed differential properties is obtained by simple combinations of the primitives' implicit equations using R-functions. Depending on the chosen R-function, this equation is continuous and can be differentiable everywhere. The primitives' parametric representations are used to directly polygonize the resulting surface by generating vertices that belong exactly to the zero-set of the resulting implicit equation. The proposed approach has many potential applications, ranging from mechanical engineering to shape recognition and data compression. Examples of complex objects are presented and commented on to show the potential of our approach for shape modeling.

Gothic architecture, and especially window tracery, exhibits quite complex geometric shape configurations. But this complexity is achieved by combining only a few basic geometric patterns. We present some principles of this long-standing domain, together with some delicate details, and show how the constructions of some prototypic Gothic windows can be formalized using our Generative Modeling Language (GML). The emphasis of our procedural approach is on modularization, so that complex configurations can be obtained from combining elementary constructions. Different combinations of specific parametric features can be grouped together, which leads to the concept of styles. They permit to differentiate between the basic shape and its appearance, i.e., in a particular ornamental decoration. This leads to an extremely compact representation for a whole class of shapes, which can nevertheless be quickly evaluated to obtain a connected manifold mesh of a particular window instance. The resulting mesh may also contain free-form surface parts, represented as subdivision surfaces.

Two design solutions are proposed for the development of an adaptive compliant aircraft wing frame using reconfigurable honeycomb core based on a network of joints with hidden topological surfaces. The first design solution utilizes the lever actuators integrated with the honeycomb structure of the wing frame, the second one uses actuators in the form of mechatronic nodes. In addition, this research outlines the development potential of the ideas on the using of smart materials and artificial muscles. Applied use of the dynamic spatial forms can be extended to many industrial and utility products and, in general, is aimed at filling the object-spatial environment with objects of dynamic functionality and aesthetics. The objective of future studies in this topic is to implement the topological optimization of the reconfigurable honeycomb core and selecting various cell form combinations optimal for each instance.

AIntract--This paper proposes a novel shape representation scheme based on CAD database specifications, and an optimization matching algorithm using the least.squares criterion to determine the pose and scale of objects, and perform out-of-profile inspection. The shapes of interest are two-dimensional (2D) profiles generated by projecting 3D objects onto a 2D inspection plane, the boundaries of which are composed of straight-line segments and circular arcs. This procedure can be applied to automated inspection of machined parts using machine vision systems. The shape model is characterized by four global parameters, based on which all the geometric primitives of the shape are expressed in analytic form. The matching procedure is to find the least-squares error between the true profile of the object and the sensed data, and to satisfy all the geometric constraints among primitives simultaneously. The advantage of the proposed model is that all the primitives of the shape are represented by the same (four) parameters in implicit equations, so that the spatial relationships among them are completely specified by the equations. Based on this representation scheme, the subsequent matching problem is formulated as an unconstrained optimization model, and solved by a gradient-based descent algorithm. In the experimental study, the results generated from the proposed technique are compared with those from two of the most commonly used machine vision approaches: Fourier Descriptors (FDs) and moments. This comparison shows that the method is superior to FDs and moments in error minimization. In addition, a complete procedure for automated design, manufacturing and machine vision inspection of production parts is described.

The ability to adapt is certainly one of the best qualities possessed by life forms on earth, an essential component for the survival of species. Talking about adaptation in architecture means conceiving the building no longer as a closed system, but as an organism that transforms itself to adapt to changing external and internal conditions. The research presented here shows the experiments of scholars who throughout history have supported and fuelled the concepts of transformability, transportability and adaptability. Among the 'pioneers' of the adaptive dimension in architecture, we find the famous architects Richard Buckminster Fuller and Konrad Wachsmann. The first linked its name to the geodesic domes: light, economic and transportable roofs, which can be used in various circumstances. Instead, Wachsmann's thought was oriented towards the innovation of construction procedures through prefabrication and standardization. Taking these two inventors as a starting point, it is possible to outline one of the most flourishing chapters in the history of architecture, where imagination meets the need to create extraordinary works.

This paper discusses some of the properties of irregular particles that are of interest to engineers, including volume, density and surface area. Numerical and statistical information on these properties is essential (a) for a better understanding of particulates, (b) to suggest more efficient ways to utilize particulate materials and (c) to permit the creation of mathematical models that can reduce the need for lengthy real-world testing. While the motivation, examples and applications are from the construction materials industry, the results should be of interest to others. Measurement techniques used included X-ray computed tomography (CT) and multiple projected images, augmented by traditional laboratory techniques. To compare the results of these techniques, a set of 12 rocks were studied of which six were between 19 mm and 12.7 mm (0.75 in. to 0.5 in.) in size, and six were between 12.7 mm and 6.3 mm (0.5 in. to 0.25 in.) in size. Microfine versions of these rocks (< 80 Am equivalent spherical diameter) were also studied and compared. The shapes of the rocks were studied by relating three dimensions to their volume and surface area. These three physical dimensions were defined in two different ways: direct measurement of three unique orthogonal dimensions on the rock surface, and dimensions obtained from the use of absolute first moments of volume and principal second moments of volume. These measurements and calculated moments allowed the development of three-parameter equivalent shape models based on rectangular parallelepipeds and tri-axial ellipsoids. All types of three-parameter equivalent shape models considered provided acceptable accuracy in predicting both volume and surface area, with the box models being generally more physical and realistic than the ellipsoid-based equivalent shape models for the type of rocks considered. D

Hair designing is one of crucial components of the hair simulation tasks. The efficiency of hair modeling is very much determined by the interactivity and the ease-to-use the designing tools within an application. This paper presents a unified framework that uses the various key techniques developed for specific tasks in hair simulation and to realize the ultimate goal of 'virtual hair-dressing room' that is simple to use but quite effective for generating fast hairstyles. Successful attempts have been made to handle the different challenging issues involved in simulation of hair at interactive rates. Effort has been put in developing methodologies for hair shape modeling, hair dynamics and hair rendering. A user friendly interface controlled by a haptic device facilitates designer's interactivity with the hairstyles. Furthermore, designer's visualization is enhanced by using real time animation and interactive rendering. Animation is done using a modified Free Form Deformation (FFD) technique that has been effectively adapted to various hairstyles. Hair Rendering is performed using an efficient scattering based technique, displaying hair with its various optical effects.

This paper takes a closer look at the existing multi-step diamond shaped models for creative problem solving (CPS). A case history of a real-life technical problem in which CPS techniques were used is our source of inspiration for some new ideas about approaching CPS. We propose three concurrent processes: Content finding, Acceptance finding and Information finding. In concrete in-company projects, these three processes need to be managed simultaneously, which leads to a fourth overarching process: project management. Content finding is concerned with the process the creative session members are going through based on people's own active knowledge and ideas and on sharing their mental models to get new ideas. Acceptance finding is concerned with the co-creation of new and additional mental systems that are needed for bringing new ideas into good currency within the existing organization and goes beyond agreement on implementation plans. Information finding is concerned with gathering additional knowledge on the ideas that are not readily available during the session. Finally, Project management is concerned with organizing and leading the creative session and in the embedding of the project into the larger organization.

Style sketches are an essential tool for the expression and the definition of the shape of industrial products. We present a novel wavelet approach for sketch segmentation and editing expressly addressed to the designer needs. Starting from a 2D curve drawn by a user with a graphics tablet, our algorithm converts it in a B-spline representation, detects the style features, allows interactive smoothing and continuous local and global editing. The novelty compared to the existing literature is the combination of: (i) unified wavelet approach to segmentation, smoothing and editing, (ii) support for endpoint C 0 and C 1 continuity constraints and (iii) implementation with an interactive pen-tablet interface. Our approach requires low memory footprints with time complexity linear in the number of control points. We validated the algorithm with synthetic and practical tests.

Shape complexity has recently received attention from different fields, such as computer vision and psychology. In this paper, integral geometry and information theory tools are applied to quantify the shape complexity from two different perspectives: from the inside of the object, we evaluate its degree of structure or correlation between its surfaces (inner complexity), and from the outside, we compute its degree of interaction with the circumscribing sphere (outer complexity). Our shape complexity measures are based on the following two facts: uniformly distributed global lines crossing an object define a continuous information channel and the continuous mutual information of this channel is independent of the object discretisation and invariant to translations, rotations, and changes of scale. The measures introduced in this paper can be potentially used as shape descriptors for object recognition, image retrieval, object localisation, tumour analysis, and protein docking, among others.

Among the different experimental strategies available in capillary electrophoresis (CE) to determine binding parameters, affinity capillary electrophoresis (ACE) has been the most widely embraced due to its easiness of implementation and of data handling. Ligand-substrate binding constants are thus directly derived from the substrate migration time shifts resulting from the variation of ligand concentration introduced in a background electrolyte. Classically, the substrate migration time is measured on top of the electrophoretic peak, assuming symmetrical peak shape. Depending on both substrate and ligand concentrations that may be required to meet detection sensitivity or complexation conditions, zonal migrations in ACE may, however, produce triangular peak shape, most often due to pronounced electromigration dispersion (EMD), and this may result in positively or negatively erroneous migration time assessments. In this work, EMD distorted triangular peak shapes obtained in the course of host-guest complexation studies were fitted with the Haarhoff-Van der Linde function, allowing better estimation of migration time. The model systems studied were those of b-cyclodextrin and naproxen, 2-naphthalenesulfonate, or 1-adamantanecarboxylate. The impact of this correction on binding isotherms and binding constant evaluation was exemplified. Furthermore, in situations where the substrate concentration injected by far overtakes that of the ligand in the electrolyte, the interest in this peak shape correction was discussed in connection with the question of whether the free ligand concentration can be still considered equal to the ligand concentration introduced, a question that still remains under debate nowadays. 2005,[3094][3095][3096][3097][3098][3099][3100][3101][3102][3103][3104] Peak shape modeling by Haarhoff-Van der Linde function

In this work we introduce a new method for testing spectral line shape models and optimizing the parameters in any parametric model. Given some general parametric line shape and a piece of a spectrum, the method finds the optimal values for the parameters and gives a number which tells how well the spectrum under consideration is explained by that model. The number of spectral lines under analysis may be more than one, and their exact locations need not be known. This characteristic follows from the property that the method does not need the information about line positions and amplitudes at all. Thus, in the absence of a singlet line, a set of overlapping lines can also be analyzed. The analysis is carried out in the signal domain by utilizing linear prediction. Application examples of the method to a molecular spectrum measured in gas phase are given. The results suggest that the Voigt line shape, despite its common use, is not a correct model in molecular spectroscopy. Its limitations become evident when one is trying to enhance the resolution by linear prediction, which requires detailed knowledge of the line shape. Instead a stochastic model, which is also tested, turns out to be rather promising.

This paper addresses the problem of recovering 3D nonrigid shape models from image sequences. For example, given a video recording of a talking person, we would like to estimate a 3D model of the lips and the full face and its internal modes of variation. Many solutions that recover 3D shape from 2D image sequences have been proposed; these so-called structure-from-motion techniques usually assume that the 3D object is rigid. For example, Tomasi and Kanades' factorization technique is based on a rigid shape matrix, which produces a tracking matrix of rank 3 under orthographic projection. We propose a novel technique based on a non-rigid model, where the 3D shape in each frame is a linear combination of a set of basis shapes. Under this model, the tracking matrix is of higher rank, and can be factored in a three-step process to yield pose, configuration and shape. To the best of our knowledge, this is the first model free approach that can recover from single-view video sequences nonrigid shape models. We demonstrate this new algorithm on several video sequences. We were able to recover 3D non-rigid human face and animal models with high accuracy.

This paper introduces the project ADIAC (Automatic Diatom Identi cation and Classi cation), which started in May 1998 and which is nanced by the European MAST (Marine Science and Technology) programme. The main goal is to develop algorithms for an automatic identi cation of diatoms using image information: both valve shape (contour) and ornamentation. The paper presents the goals of the project as well as rst results on shape modeling and contour extraction. Public data are available in order to create student projects beyond the ADIAC partnership. For further information see http://www.ualg.pt/adiac

Revised (revised date) Communicated by (Name of Editor) Originally developed as an exhibition for the Providence Art Clubs' Dodge Gallery in March of 1996, "Surface Beyond the Third Dimension" continues its existence today as a virtual experience on the internet. Here, the authors take you on a tour of the exhibit and describe both the artwork and the mathematics underlying the images, whose subjects range from projections of spheres and tori in four-space, to images of complex functions, to views of the Klein bottle. Along the way, the authors introduce some of the issues that arise from this dual presentation approach, and point out some of the enhancements available in the virtual gallery that are not possible in the physical one.

Gothic architecture, and especially window tracery, exhibits quite complex geometric shape configurations. But this complexity is achieved by combining only a few basic geometric patterns. We present some principles of this long-standing domain, together with some delicate details, and show how the constructions of some prototypic Gothic windows can be formalized using our Generative Modeling Language (GML). The emphasis of our procedural approach is on modularization, so that complex configurations can be obtained from combining elementary constructions. Different combinations of specific parametric features can be grouped together, which leads to the concept of styles. They permit to differentiate between the basic shape and its appearance, i.e., in a particular ornamental decoration. This leads to an extremely compact representation for a whole class of shapes, which can nevertheless be quickly evaluated to obtain a connected manifold mesh of a particular window instance. The resulting mesh may also contain free-form surface parts, represented as subdivision surfaces.

producing compressed yet precise data structures, thus providing inter-operability between current and future computer platforms crucial to archiving. Examples of CSG reconstruction of historical temples and FRep modeling of traditional lacquer ware are given. We examine the application of fitting of a parameterized FRep model to a cloud of data points as a step towards automation of the modeling process. Virtual venues for public access to cultural heritage objects including real time interactive simulation of cultural heritage sites over the Web are discussed and illustrated.