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Über dieses Buch

Based on the second Women in Shape (WiSH) workshop held in Sirince, Turkey in June 2016, these proceedings offer the latest research on shape modeling and analysis and their applications. The 10 peer-reviewed articles in this volume cover a broad range of topics, including shape representation, shape complexity, and characterization in solving image-processing problems. While the first six chapters establish understanding in the theoretical topics, the remaining chapters discuss important applications such as image segmentation, registration, image deblurring, and shape patterns in digital fabrication. The authors in this volume are members of the WiSH network and their colleagues, and most were involved in the research groups formed at the workshop. This volume sheds light on a variety of shape analysis methods and their applications, and researchers and graduate students will find it to be an invaluable resource for further research in the area.



Chapter 1. Medial Fragments for Segmentation of Articulating Objects in Images

The Blum medial axis is known to provide a useful representation of pre-segmented shapes. Very little work to date, however, has examined its usefulness for extracting objects from natural images. We propose a method for combining fragments of the medial axis, generated from the Voronoi diagram of an edge map of a natural image, into a coherent whole. Using techniques from persistent homology and graph theory, we combine image cues with geometric cues from the medial fragments to aggregate parts of the same object into a larger whole. We demonstrate our method on images containing articulating objects, with an eye to future work applying articulation-invariant measures on the medial axis for shape matching between images.
Erin Chambers, Ellen Gasparovic, Kathryn Leonard

Chapter 2. Scaffolding a Skeleton

The goal of this paper is to construct a quadrilateral mesh around a one-dimensional skeleton that is as coarse as possible, the “scaffold.” A skeleton allows one to quickly describe a shape, in particular a complex shape of high genus. The constructed scaffold is then a potential support for the surface representation: it provides a topology for the mesh, a domain for parametric representation (a quad-mesh is ideal for tensor product splines), or, together with the skeleton, a grid support on which to project an implicit surface that is naturally defined by the skeleton through convolution. We provide a constructive algorithm to derive a quad-mesh scaffold with topologically regular cross-sections (which are also quads) and no T-junctions. We show that this construction is optimal in the sense that no coarser quad-mesh with topologically regular cross-sections may be constructed. Finally, we apply an existing rotation minimization algorithm along the skeleton branches, which produces a mesh with a natural edge flow along the shape.
Athina Panotopoulou, Elissa Ross, Kathrin Welker, Evelyne Hubert, Géraldine Morin

Chapter 3. Convolution Surfaces with Varying Radius: Formulae for Skeletons Made of Arcs of Circles and Line Segments

In skeleton-based geometric modeling, convolution is an established technique: smooth surfaces around a skeleton made of curves are given as the level set of a convolution field. Varying the radius or making the surface scale sensitive along the skeleton are desirable features. This article provides the related necessary closed-form formulae of the convolution fields when the skeleton is made of arcs of circle and line segments. For the family of power inverse kernels, closed-form formulae are exhibited in terms of recurrence relationships. These are obtained by creative telescoping. This novel technique is described from a practitioner point of view so as to be applied to other families of kernels or skeleton primitives. The newly obtained formulae are applied to obtain convolution surfaces around G1 skeleton curves, some of them closed curves. Having arcs of circles in addition to line segments allows to demonstrably improve the visual quality of the generated surface with a lower number of skeleton primitives.
Alvaro Javier Fuentes Suárez, Evelyne Hubert

Chapter 4. Exploring 2D Shape Complexity

In this paper, we explore different notions of shape complexity, drawing from established work in mathematics, computer science, and computer vision. Our measures divide naturally into three main categories: skeleton-based, symmetry-based, and those based on boundary sampling. We apply these to an established library of shapes, using k-medoids clustering to understand what aspects of shape complexity are captured by each notion. Our contributions include a new measure of complexity based on the Blum medial axis and the notion of persistent complexity as captured by histograms at multiple scales rather than a single numerical value.
Erin Chambers, Tegan Emerson, Cindy Grimm, Kathryn Leonard

Chapter 5. Phase Field Topology Constraints

This paper presents a morphological approach to extract topologically critical regions in phase field models. There are a few studies regarding topological properties of phase fields. One line of work related to our problem addresses constrained phase field evolution. This approach is based on modifying the optimization problem to limit connectedness of the interface. However, this approach results in a complex optimization problem, and it provides nonlocal control. We adapted a non-simple point concept from digital topology to local regions using structuring masks. These regions can be used to constrain the evolution locally. Besides this approach is flexible as it allows the design of structuring elements. Such a study to define topological structures specific to phase field dynamics has not been done to our knowledge.
Rüyam Acar, Necati Sağırlı

Chapter 6. Adaptive Deflation Stopped by Barrier Structure for Equating Shape Topologies Under Topological Noise

Using level sets of a pair of transformations, we adaptively bring two shapes to be matched to a comparable topology prior to a correspondence search. One of the transformations readily provides a central structure for each shape. We utilize the central structure as a reference volume for scale normalization. By adaptively dilating the central structure with the help of the second transformation, we construct what we refer to as the barrier structure. The barrier structure is used to automatically stop topology equating adaptive deflations. Illustrative experiments using different datasets demonstrate that our approach provides robust solutions for the topological noise caused by localized touches or spurious links that connect different shape parts.
Asli Genctav, Sibel Tari

Chapter 7. Joint Segmentation and Nonlinear Registration Using Fast Fourier Transform and Total Variation

Image segmentation and registration play active roles in machine vision and medical image analysis of historical data. We explore the joint problem of segmenting and registering a template (current) image given a reference (past) image. We formulate the joint problem as a minimization of a functional that integrates two well-studied approaches in segmentation and registration: geodesic active contours and nonlinear elastic registration. The template image is modeled as a hyperelastic material (St. Venant-Kirchhoff model) which undergoes deformations under applied forces. To segment the deforming template, a two-phase level set-based energy is introduced together with a weighted total variation term that depends on gradient features of the deforming template. This particular choice allows for fast solution using the dual formulation of the total variation. This allows the segmenting front to accurately track spontaneous changes in the shape of objects embedded in the template image as it deforms. To solve the underlying registration problem, we use gradient descent and adopt an implicit-explicit method and use the fast Fourier transform.
Thomas Atta-Fosu, Weihong Guo

Chapter 8. Multi-parameter Mumford-Shah Segmentation

Mumford-Shah functional has two parameters that define a two-dimensional scale space of solutions. Instead of using the solution obtained at a predetermined fine-tuned parameter setting, we consider solutions at multiple parameter settings simultaneously. Using multiple solutions, we construct pixel-based features and employ them to extract shapes in images. We experiment with both synthetic and real images.
Murat Genctav, Sibel Tari

Chapter 9. L 1-Regularized Inverse Problems for Image Deblurring via Bound- and Equality-Constrained Optimization

Image deblurring is typically modeled as an ill-posed, linear inverse problem. By adding an L 1-penalty to the negative-log likelihood function, the resulting minimization problem becomes well-posed. Moreover, the penalty enforces sparsity. The difficulty with L 1-penalties, however, is that they are non-differentiable. Here we replace the L 1-penalty by a linear penalty together with bound and equality constraints. We consider two statistical models for measurement error: Gaussian and Poisson. In either case, we obtain a bound- and equality-constrained minimization problem, which we solve using an iterative augmented Lagrangian (AL) method. Each iteration of the AL method requires the solution of a bound-constrained minimization problem, which is convex-quadratic in the Gaussian case and convex in the Poisson case. We recommend two highly efficient methods for the solution of these subproblems that allows us to apply the AL method to large-scale imaging examples. Results are shown on synthetic data in one and two dimensions, as well as on a radiograph used to calibrate the transmission curve of a pulsed-power X-ray source at a US Department of Energy radiography facility.
Johnathan M. Bardsley, Marylesa Howard

Chapter 10. Shape Patterns in Digital Fabrication: A Survey on Negative Poisson’s Ratio Metamaterials

Poisson’s ratio for solid materials is defined as the ratio of the lateral length shrinkage to the longitudinal part extension on a simple tension test. While Poisson’s ratio for almost every material in nature is a positive number, materials having negative Poisson’s ratio may be engineered. We survey computational works toward design and fabrication of negative Poisson’s ratio materials focusing on shape patterns from macro to micro scale. Specifically, we cover folding, knitting, and repeatedly ordering geometric structures, i.e., symmetry. Both pattern design and the numerical aspects of the problem yield various future research possibilities.
Bengisu Yılmaz, Venera Adanova, Rüyam Acar, Sibel Tari
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