3D Well-composed Polyhedral Complexes

A binary three-dimensional (3D) image $I$ is well-composed if the boundary surface of its continuous analog is a 2D manifold. Since 3D images are not often well-composed, there are several voxel-based methods ("repairing" algorithms) for turning them into well-composed ones but these methods either do not guarantee the topological equivalence between the original image and its corresponding well-composed one or involve sub-sampling the whole image. In this paper, we present a method to locally "repair" the cubical complex $Q(I)$ (embedded in $\mathbb{R}^3$) associated to $I$ to obtain a polyhedral complex $P(I)$ homotopy equivalent to $Q(I)$ such that the boundary of every connected component of $P(I)$ is a 2D manifold. The reparation is performed via a new codification system for $P(I)$ under the form of a 3D grayscale image that allows an efficient access to cells and their faces

Embedded Implicit Stand-ins for Animated Meshes: a Case of Hybrid Modelling

In this paper we address shape modelling problems, encountered in computer animation and computer games development that are difficult to solve just using polygonal meshes. Our approach is based on a hybrid modelling concept that combines polygonal meshes with implicit surfaces. A hybrid model consists of an animated polygonal mesh and an approximation of this mesh by a convolution surface stand-in that is embedded within it or is attached to it. The motions of both objects are synchronised using a rigging skeleton. This approach is used to model the interaction between an animated mesh object and a viscoelastic substance, normally modelled in implicit form. The adhesive behaviour of the viscous object is modelled using geometric blending operations on the corresponding implicit surfaces. Another application of this approach is the creation of metamorphosing implicit surface parts that are attached to an animated mesh. A prototype implementation of the proposed approach and several examples of modelling and animation with near real-time preview times are presented

Cell AT-models for digital volumes

In [4], given a binary 26-adjacency voxel-based digital volume V, the homological information (that related to n-dimensional holes: connected components, ”tunnels” and cavities) is extracted from a linear map (called homology gradient vector field) acting on a polyhedral cell complex P(V) homologically equivalent to V. We develop here an alternative way for constructing P(V) based on homological algebra arguments as well as a new more efficient algorithm for computing a homology gradient vector field based on the contractibility of the maximal cells of P(V)

Automatic tailoring and cloth modelling for animation characters.

The construction of realistic characters has become increasingly important to the production of blockbuster films, TV series and computer games. The outfit of character plays an important role in the application of virtual characters. It is one of the key elements reflects the personality of character. Virtual clothing refers to the process that constructs outfits for virtual characters, and currently, it is widely used in mainly two areas, fashion industry and computer animation. In fashion industry, virtual clothing technology is an effective tool which creates, edits and pre-visualises cloth design patterns efficiently. However, using this method requires lots of tailoring expertises. In computer animation, geometric modelling methods are widely used for cloth modelling due to their simplicity and intuitiveness. However, because of the shortage of tailoring knowledge among animation artists, current existing cloth design patterns can not be used directly by animation artists, and the appearance of cloth depends heavily on the skill of artists. Moreover, geometric modelling methods requires lots of manual operations. This tediousness is worsen by modelling same style cloth for different characters with different body shapes and proportions. This thesis addresses this problem and presents a new virtual clothing method which includes automatic character measuring, automatic cloth pattern adjustment, and cloth patterns assembling. There are two main contributions in this research. Firstly, a geodesic curvature flow based geodesic computation scheme is presented for acquiring length measurements from character. Due to the fast growing demand on usage of high resolution character model in animation production, the increasing number of characters need to be handled simultaneously as well as improving the reusability of 3D model in film production, the efficiency of modelling cloth for multiple high resolution character is very important. In order to improve the efficiency of measuring character for cloth fitting, a fast geodesic algorithm that has linear time complexity with a small bounded error is also presented. Secondly, a cloth pattern adjusting genetic algorithm is developed for automatic cloth fitting and retargeting. For the reason that that body shapes and proportions vary largely in character design, fitting and transferring cloth to a different character is a challenging task. This thesis considers the cloth fitting process as an optimization procedure. It optimizes both the shape and size of each cloth pattern automatically, the integrity, design and size of each cloth pattern are evaluated in order to create 3D cloth for any character with different body shapes and proportions while preserve the original cloth design. By automating the cloth modelling process, it empowers the creativity of animation artists and improves their productivity by allowing them to use a large amount of existing cloth design patterns in fashion industry to create various clothes and to transfer same design cloth to characters with different body shapes and proportions with ease

Brief introduction to tropical geometry

The paper consists of lecture notes for a mini-course given by the authors at the G\"okova Geometry \& Topology conference in May 2014. We start the exposition with tropical curves in the plane and their applications to problems in classical enumerative geometry, and continue with a look at more general tropical varieties and their homology theories.Comment: 75 pages, 37 figures, many examples and exercise