Subset Warping: Rubber Sheeting with Cuts

Image warping, often referred to as "rubber sheeting" represents the deformation of a domain image space into a range image space. In this paper, a technique is described which extends the definition of a rubber-sheet transformation to allow a polygonal region to be warped into one or more subsets of itself, where the subsets may be multiply connected. To do this, it constructs a set of "slits" in the domain image, which correspond to discontinuities in the range image, using a technique based on generalized Voronoi diagrams. The concept of medial axis is extended to describe inner and outer medial contours of a polygon. Polygonal regions are decomposed into annular subregions, and path homotopies are introduced to describe the annular subregions. These constructions motivate the definition of a ladder, which guides the construction of grid point pairs necessary to effect the warp itself

Image morphological processing

Mathematical Morphology with applications in image processing and analysis has been becoming increasingly important in today\u27s technology. Mathematical Morphological operations, which are based on set theory, can extract object features by suitably shaped structuring elements. Mathematical Morphological filters are combinations of morphological operations that transform an image into a quantitative description of its geometrical structure based on structuring elements. Important applications of morphological operations are shape description, shape recognition, nonlinear filtering, industrial parts inspection, and medical image processing. In this dissertation, basic morphological operations, properties and fuzzy morphology are reviewed. Existing techniques for solving corner and edge detection are presented. A new approach to solve corner detection using regulated mathematical morphology is presented and is shown that it is more efficient in binary images than the existing mathematical morphology based asymmetric closing for corner detection. A new class of morphological operations called sweep mathematical morphological operations is developed. The theoretical framework for representation, computation and analysis of sweep morphology is presented. The basic sweep morphological operations, sweep dilation and sweep erosion, are defined and their properties are studied. It is shown that considering only the boundaries and performing operations on the boundaries can substantially reduce the computation. Various applications of this new class of morphological operations are discussed, including the blending of swept surfaces with deformations, image enhancement, edge linking and shortest path planning for rotating objects. Sweep mathematical morphology is an efficient tool for geometric modeling and representation. The sweep dilation/erosion provides a natural representation of sweep motion in the manufacturing processes. A set of grammatical rules that govern the generation of objects belonging to the same group are defined. Earley\u27s parser serves in the screening process to determine whether a pattern is a part of the language. Finally, summary and future research of this dissertation are provided

Parametric freeform-based construction site layout optimization

Traditional approaches to the construction site layout problem have been focused mainly on rectilinear facilities where the importance proximity measures are mainly based on Cartesian distances between the centroids of the facilities. This is a fair abstraction of the problem; however it ignores the fact that many facilities on construction sites assume non-rectilinear shapes that allow for better compaction within tight sites. The main focus of this research is to develop a new approach of modeling site facilities to surpass limitations and inefficiencies of previous models and to ensure a more realistic approach to construction site layout problems. A construction site layout optimization model was developed that can suit both static and dynamic site layouts. The developed model is capable of modeling any rectilinear and non-rectilinear site shapes, especially splines, since it utilizes a parametric modeling software. The model also has the ability to mimic the “dynamic” behavior of the objects’ shapes through the introduction and development of three different algorithms for dynamic shapes; where the geometrical shapes representing site facilities automatically modify their geometrical forms to fit in strict areas on site. Moreover, the model provides different proximity measures and distance measurement techniques rather than the normal centroidal Cartesian distances used in most models. The new proximity measures take into consideration actual movement between the facilities including any passageways or access roads on site. Furthermore, the concept of selective zoning was introduced and a corresponding algorithm was provided; where the concept significantly enhances optimization efficiency by minimizing the number of solutions through selection of pre-determined movement zones on site. Soft constraints for buffer zones around the site facilities were developed as well. The site layout modeling was formulated on commercial parametric modeling tools (Rhino® and Grasshopper®) and the optimization was performed through genetic algorithms. After each of the algorithms was verified and validated, a case study of a real dynamic site layout planning problem was made to validate the comprehensive model combining all of the modules together. Different proximity measures and distance measurement techniques were considered, along with different static and dynamic geometrical shapes for the temporary facilities. The model produced valid near-optimum solutions, a comparison was then made between the layout that is produced with the model and the layout that would have been produced by other models to demonstrate the capabilities and advantages of the produced model

Lattice Point Asymptotics and Volume Growth on Teichmuller space

We apply some of the ideas of the Ph.D. Thesis of G. A. Margulis to Teichmuller space. Let x be a point in Teichmuller space, and let B_R(x) be the ball of radius R centered at x (with distances measured in the Teichmuller metric). We obtain asymptotic formulas as R tends to infinity for the volume of B_R(x), and also for for the cardinality of the intersection of B_R(x) with an orbit of the mapping class group.Comment: 45 Pages, 2 figures. Minor correction