Computation with Curved Shapes: Towards Freeform Shape Generation in Design

Shape computations are a formal representation that specify particular aspects of the design process with reference to form. They are defined according to shape grammars, where manipulations of pictorial representations of designs are formalised by shapes and rules applied to those shapes. They have frequently been applied in architecture in order to formalise the stylistic properties of a given corpus of designs, and also to generate new designs within those styles. However, applications in more general design fields have been limited. This is largely due to the initial definitions of the shape grammar formalism which are restricted to rectilinear shapes composed of lines, planes or solids. In architecture such shapes are common but in many design fields, for example industrial design, shapes of a more freeform nature are prevalent. Accordingly, the research described in this thesis is concerned with extending the applicability of the shape grammar formalism such that it enables computation with freeform shapes. Shape computations utilise rules in order to manipulate subshapes of a design within formal algebras. These algebras are specified according to embedding properties and have previously been defined for rectilinear shapes. In this thesis the embedding properties of freeform shapes are explored and the algebras are extended in order to formalise computations with such shapes. Based on these algebras, shape operations are specified and algorithms are introduced that enable the application of rules to shapes composed of freeform B´ezier curves. Implementation of the algorithms enables the application of shape grammars to shapes of a more freeform nature than was previously possible. Within this thesis shape grammar implementations are introduced in order to explore both theoretical issues that arise when considering computation with freeform shapes and practical issues concerning the application of shape computation as a model for design and as a mode for generating freeform shapes

Discrete Optimization in Early Vision - Model Tractability Versus Fidelity

Early vision is the process occurring before any semantic interpretation of an image takes place. Motion estimation, object segmentation and detection are all parts of early vision, but recognition is not. Some models in early vision are easy to perform inference with---they are tractable. Others describe the reality well---they have high fidelity. This thesis improves the tractability-fidelity trade-off of the current state of the art by introducing new discrete methods for image segmentation and other problems of early vision. The first part studies pseudo-boolean optimization, both from a theoretical perspective as well as a practical one by introducing new algorithms. The main result is the generalization of the roof duality concept to polynomials of higher degree than two. Another focus is parallelization; discrete optimization methods for multi-core processors, computer clusters, and graphical processing units are presented. Remaining in an image segmentation context, the second part studies parametric problems where a set of model parameters and a segmentation are estimated simultaneously. For a small number of parameters these problems can still be optimally solved. One application is an optimal method for solving the two-phase Mumford-Shah functional. The third part shifts the focus to curvature regularization---where the commonly used length and area penalization is replaced by curvature in two and three dimensions. These problems can be discretized over a mesh and special attention is given to the mesh geometry. Specifically, hexagonal meshes in the plane are compared to square ones and a method for generating adaptive meshes is introduced and evaluated. The framework is then extended to curvature regularization of surfaces. Finally, the thesis is concluded by three applications to early vision problems: cardiac MRI segmentation, image registration, and cell classification

Invariant-based approach to symmetry class detection

In this paper, the problem of the identification of the symmetry class of a given tensor is asked. Contrary to classical approaches which are based on the spectral properties of the linear operator describing the elasticity, our setting is based on the invariants of the irreducible tensors appearing in the harmonic decomposition of the elasticity tensor [Forte-Vianello, 1996]. To that aim we first introduce a geometrical description of the space of elasticity tensors. This framework is used to derive invariant-based conditions that characterize symmetry classes. For low order symmetry classes, such conditions are given on a triplet of quadratic forms extracted from the harmonic decomposition of the elasticity tensor $C$, meanwhile for higher-order classes conditions are provided in terms of elements of $H^{4}$, the higher irreducible space in the decomposition of $C$. Proceeding in such a way some well known conditions appearing in the Mehrabadi-Cowin theorem for the existence of a symmetry plane are retrieved, and a set of algebraic relations on polynomial invariants characterizing the orthotropic, trigonal, tetragonal, transverse isotropic and cubic symmetry classes are provided. Using a genericity assumption on the elasticity tensor under study, an algorithm to identify the symmetry class of a large set of tensors is finally provided.Comment: 32 page