Bézier Method For Image Processing

This project concerns about Bézier method in Computer Aided GeometricDesign (CAGD) involving Bézier Curve and Bézier Surface which widely related to the other theorem and method. The aim of this project is to introduce the basic of Bézier method and then generate the Bézier curves, Bézier surfaces, theory and properties and develop Bézier method in image processing application specifically image compression by using MATLAB

Shape analysis of the corpus callosum of autistic and normal subjects in neuroimaging.

Early detection of human disease in today’s society can have an enormous impact on the severity of the disease that is manifested. Disease such as Autism and Dyslexia, which have no current cure or proven mechanism as to how they develop, can often have an adverse physical and physiological impact on the lifestyle of a human being. Although these disease are not fully curable, the severity handicaps that accompany them can be significantly reduced with the proper therapy, and thus the earlier that the disease is detected the faster therapy can be administered. The research in this thesis is an attempt at studying discriminatory shape measures of some brain structures that are known to carry changes from autistics to normal individuals. The focus will be on the corpus callosum. There has been considerable research done on the brain scans (MRI, CT) of autistic individuals vs. control (normal) individuals to observe any noticeable discrepancies through statistical analysis. The most common and powerful tool to analyze structures of the brain, once a specific region has been segmented, is using Registration to match like structures and record their error. The ICP algorithm (Iterative Closest Point) is commonly used to accomplish this task. Many techniques such as level sets and statistical methods can be used for segmentation. The Corpus Callosum (CC) and the cortical surface of the brain are currently where most Autism analysis is performed. It has been observed that the gyrification of the cortical surface is different in the two groups, and size as well as shape of the CC. An analysis approach for autism MRI is quite extensive and involves many steps. This thesis is limited to examination of shape measures of the CC that lend discrimination ability to distinguish between normal and autistic individuals from T1-weigheted MRI scans. We will examine two approaches for shape analysis, based on the traditional Fourier Descriptors (FD) method and shape registration (SR) using the procrustes technique. MRI scans of 22 autistic and 16 normal individuals are used to test the approaches developed in this thesis. We show that both FD and SR may be used to extract features to discriminate between the two populations with accuracy levels over 80% up to 100% depending on the technique

A novel parallel algorithm for surface editing and its FPGA implementation

A thesis submitted to the University of Bedfordshire in partial fulfilment of the requirements for the degree of Doctor of PhilosophySurface modelling and editing is one of important subjects in computer graphics. Decades of research in computer graphics has been carried out on both low-level, hardware-related algorithms and high-level, abstract software. Success of computer graphics has been seen in many application areas, such as multimedia, visualisation, virtual reality and the Internet. However, the hardware realisation of OpenGL architecture based on FPGA (field programmable gate array) is beyond the scope of most of computer graphics researches. It is an uncultivated research area where the OpenGL pipeline, from hardware through the whole embedded system (ES) up to applications, is implemented in an FPGA chip. This research proposes a hybrid approach to investigating both software and hardware methods. It aims at bridging the gap between methods of software and hardware, and enhancing the overall performance for computer graphics. It consists of four parts, the construction of an FPGA-based ES, Mesa-OpenGL implementation for FPGA-based ESs, parallel processing, and a novel algorithm for surface modelling and editing. The FPGA-based ES is built up. In addition to the Nios II soft processor and DDR SDRAM memory, it consists of the LCD display device, frame buffers, video pipeline, and algorithm-specified module to support the graphics processing. Since there is no implementation of OpenGL ES available for FPGA-based ESs, a specific OpenGL implementation based on Mesa is carried out. Because of the limited FPGA resources, the implementation adopts the fixed-point arithmetic, which can offer faster computing and lower storage than the floating point arithmetic, and the accuracy satisfying the needs of 3D rendering. Moreover, the implementation includes Bézier-spline curve and surface algorithms to support surface modelling and editing. The pipelined parallelism and co-processors are used to accelerate graphics processing in this research. These two parallelism methods extend the traditional computation parallelism in fine-grained parallel tasks in the FPGA-base ESs. The novel algorithm for surface modelling and editing, called Progressive and Mixing Algorithm (PAMA), is proposed and implemented on FPGA-based ES’s. Compared with two main surface editing methods, subdivision and deformation, the PAMA can eliminate the large storage requirement and computing cost of intermediated processes. With four independent shape parameters, the PAMA can be used to model and edit freely the shape of an open or closed surface that keeps globally the zero-order geometric continuity. The PAMA can be applied independently not only FPGA-based ESs but also other platforms. With the parallel processing, small size, and low costs of computing, storage and power, the FPGA-based ES provides an effective hybrid solution to surface modelling and editing

Manifolds.jl: An Extensible Julia Framework for Data Analysis on Manifolds

For data given on a nonlinear space, like angles, symmetric positive matrices, the sphere, or the hyperbolic space, there is often enough structure to form a Riemannian manifold. We present the Julia package Manifolds.jl, providing a fast and easy to use library of Riemannian manifolds and Lie groups. We introduce a common interface, available in ManifoldsBase.jl, with which new manifolds, applications, and algorithms can be implemented. We demonstrate the utility of Manifolds.jl using B\'ezier splines, an optimization task on manifolds, and a principal component analysis on nonlinear data. In a benchmark, Manifolds.jl outperforms existing packages in Matlab or Python by several orders of magnitude and is about twice as fast as a comparable package implemented in C++

Non-Uniform Rational B-Splines and Rational Bezier Triangles for Isogeometric Analysis of Structural Applications

Isogeometric Analysis (IGA) is a major advancement in computational analysis that bridges the gap between a computer-aided design (CAD) model, which is typically constructed using Non-Uniform Rational B-splines (NURBS), and a computational model that traditionally uses Lagrange polynomials to represent the geometry and solution variables. In IGA, the same shape functions that are used in CAD are employed for analysis. The direct manipulation of CAD data eliminates approximation errors that emanate from the process of converting the geometry from CAD to Finite Element Analysis (FEA). As a result, IGA allows the exact geometry to be represented at the coarsest level and maintained throughout the analysis process. While IGA was initially introduced to streamline the design and analysis process, this dissertation shows that IGA can also provide improved computational results for complex and highly nonlinear problems in structural mechanics. This dissertation addresses various problems in structural mechanics in the context of IGA, with the use of NURBS and rational Bézier triangles for the description of the parametric and physical spaces. The approaches considered here show that a number of important properties (e.g., high-order smoothness, geometric exactness, reduced number of degrees of freedom, and increased flexibility in discretization) can be achieved, leading to improved numerical solutions. Specifically, using B-splines and a layer-based discretization, a distributed plasticity isogeometric frame model is formulated to capture the spread of plasticity in large-deformation frames. The modeling approach includes an adaptive analysis where the structure of interest is initially modeled with coarse mesh and knots are inserted based on the yielding information at the quadrature points. It is demonstrated that improvement on efficiency and convergence rates is attained. With NURBS, an isogeometric rotation-free multi-layered plate formulation is developed based on a layerwise deformation theory. The derivation assumes a separate displacement field expansion within each layer, and considers transverse displacement component as C0-continuous at dissimilar material interfaces, which is enforced via knot repetition. The separate integration of the in-plane and through-thickness directions allows to capture the complete 3D stresses in a 2D setting. The proposed method is used to predict the behavior of advanced materials such as laminated composites, and the results show advantages in efficiency and accuracy. To increase the flexibility in discretizing complex geometries, rational Bézier triangles for domain triangulation is studied. They are further coupled with a Delaunay-based feature-preserving discretization algorithm for static bending and free vibration analysis of Kirchhoff plates. Lagrange multipliers are employed to explicitly impose high-order continuity constraints and the augmented system is solved iteratively without increasing the matrix size. The resulting discretization is geometrically exact, admits small geometric features, and constitutes C1-continuity. The feature-preserving rational Bézier triangles are further applied to smeared damage modeling of quasi-brittle materials. Due to the ability of Lagrange multipliers to raise global continuity to any desired order, the implicit fourth- and sixth-order gradient damage models are analyzed. The inclusion of higher-order terms in the nonlocal Taylor expansion improves solution accuracy. A local refinement algorithm that resolves marked regions with high resolution while keeping the resulting mesh conforming and well-conditioned is also utilized to improve efficiency. The outcome is a unified modeling framework where the feature-preserving discretization is able to capture the damage initiation and early-stage propagation, and the local refinement technique can then be applied to adaptively refine the mesh in the direction of damage propagation.PHDCivil EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147668/1/ningliu_1.pd

Iterative geometric design for architecture

This work investigates on computer aided integrated architectural design and production. The aim is to provide integral solutions for the design and the production of geometrically complex free-form architecture. Investigations on computer aided geometric design and integrated manufacturing are carried out with equal importance. This research is considering an integral and interdisciplinary approach, including computer science, mathematics and architecture. Inspired by fractal geometry, the IFS formalism is studied with regards to discrete architectural geometric design. The geometric design method studied provides new shape control possibilities unifying two separate design paradigms of rough and smooth objects. Capable to design fractal geometric figures, the method also covers the generation of classical objects such as conics and NURBS-curves. Close attention has been paid to the design of iterative free-form surfaces, which are composed entirely out of planar elements. A surface method based on projected vector sums is proposed. The resulting geometric figures are expressed in a discrete form and can be easily translated into a coherent set of constructional elements. The studies for translation of the geometrical elements into constructional elements consider integrated manufacturing. Addressing and numbering of the elements by iterative geometric design are investigated and compared to lexicographically ordered addressing systems, in order to provide an adequate data structure for the design, production and assembly of the constructional elements. For the generation of the data describing constructional elements, problems related to thickening and offset meshes are discussed. Once the global geometry of the constructional part has been computed, parameters are defined for generic automated detailing. Hereby the entire description of the constructional elements is completed. These elements are mapped and packed with regards to the coordinate system of a CNC-machine and the properties and the dimensions of the raw material, providing the complete set of workshop plans needed for integrated manufacturing. For automated generation of machine instructions (G-code), machining strategies – depending on the type of machine used, tool and material properties – are elaborated. Finally, the integrated digital design methods studied within the scope of this thesis are tested and verified by the realization of different reduced scale prototypes. The studied applications range from bearing vault structures to fractal and smooth timber panel shell structures. The developed methods have shown to be efficient for the design and the realization of geometrically complex architectural objects. The required planning effort to handle and manipulate the design and the production data has been greatly reduced. Some of the proposed methods have proved to be robust and general enough to be applied on real world applications. Iterative geometric design provides high degree of design possibilities offering an efficient tool for the creation of smooth and rough free form objects. The possibility to incorporate successive folds in free-form objects allows structural applications

Semi-sharp creases on subdivision curves and surfaces

We explore a method for generalising Pixar semi-sharp creases from the univariate cubic case to arbitrary degree subdivision curves. Our approach is based on solving simple matrix equations. The resulting schemes allow for greater flexibility over existing methods, via control vectors. We demonstrate our results on several high-degree univariate examples and explore analogous methods for subdivision surfacesThis work was supported by the Engineering and Physical Sciences Research Council [EP/H030115/1].This is the author accepted manuscript and will be under embargo until the 23rd of August 2015. The final version has been published in Computer Graphics Forum here: http://onlinelibrary.wiley.com/doi/10.1111/cgf.12447/abstract

Robust multigrid methods for Isogeometric discretizations applied to poroelasticity problems

El análisis isogeométrico (IGA) elimina la barrera existente entre elementos finitos (FEA) y el diseño geométrico asistido por ordenador (CAD). Debido a esto, IGA es un método novedoso que está recibiendo una creciente atención en la literatura y recientemente se ha convertido en tendencia. Muchos esfuerzos están siendo puestos en el diseño de solvers eficientes y robustos para este tipo de discretizaciones. Dada la optimalidad de los métodos multimalla para elementos finitos, la aplicación de estosmétodos a discretizaciones isogeométricas no ha pasado desapercibida. Nosotros pensamos firmemente que los métodos multimalla son unos candidatos muy prometedores a ser solvers eficientes y robustos para IGA y por lo tanto en esta tesis apostamos por su aplicación. Para contar con un análisis teórico para el diseño de nuestros métodos multimalla, el análisis local de Fourier es propuesto como principal análisis cuantitativo. En esta tesis, a parte de considerar varios problemas escalares, prestamos especial atención al problema de poroelasticidad, concretamente al modelo cuasiestático de Biot para el proceso de consolidación del suelo. Actualmente, el diseño de métodos multimalla robustos para problemas poroelásticos respecto a parámetros físicos o el tamaño de la malla es un gran reto. Por ello, la principal contribución de esta tesis es la propuesta de métodos multimalla robustos para discretizaciones isogeométricas aplicadas al problema de poroelasticidad.La primera parte de esta tesis se centra en la construcción paramétrica de curvas y superficies dado que estas técnicas son la base de IGA. Así, la definición de los polinomios de Bernstein y curvas de Bézier se presenta como punto de partida. Después, introducimos los llamados B-splines y B-splines racionales no uniformes (NURBS) puesto que éstas serán las funciones base consideradas en nuestro estudio.La segunda parte trata sobre el análisis isogeométrico propiamente dicho. En esta parte, el método isoparamétrico es explicado al lector y se presenta el análisis isogeométrico de algunos problemas. Además, introducimos la formulación fuerte y débil de los problemas anteriores mediante el método de Galerkin y los espacios de aproximación isogeométricos. El siguiente punto de esta tesis se centra en los métodos multimalla. Se tratan las bases de los métodos multimalla y, además de introducir algunos métodos iterativos clásicos como suavizadores, también se introducen suavizadores por bloques como los métodos de Schwarz multiplicativos y aditivos. Llegados a esta parte, nos centramos en el LFA para el diseño de métodos multimalla robustos y eficientes. Además, se explican en detalle el análisis estándar y el análisis basado en ventanas junto al análisis de suavizadores por bloques y el análisis para sistemas de ecuaciones en derivadas parciales.Tras introducir las discretizaciones isogeométricas, los métodos multimalla y el LFA como análisis teórico, nuestro propósito es diseñar métodos multimalla eficientes y robustos respecto al grado polinomial de los splines para discretizaciones isogeométricas de algunos problemas escalares. Así, mostramos que el uso de métodos multimalla basados en suavizadores de tipo Schwarz multiplicativo o aditivo produce buenos resultados y factores de convergencia asintóticos robustos. La última parte de esta tesis está dedicada al análisis isogeométrico del problema de poroelasticidad. Para esta tarea, se introducen el modelo de Biot y su discretización isogeométrica. Además, presentamos una novedosa estabilización de masa para la formulación de dos campos de las ecuaciones de Biot que elimina todas las oscilaciones no físicas en la aproximación numérica de la presión. Después, nos centramos en dos tipos de solvers para estas ecuaciones poroelásticas: Solvers desacoplados y solvers monolíticos. En el primer grupo, le dedicamos una especial atención al método fixed-stress y a un método iterativo propuesto por nosotros que puede ser aplicado de forma automática a partir de la estabilización de masa ya mencionada.Por otro lado, realizamos un análisis de von Neumann para este método iterativo aplicado al problema de Terzaghi y demostramos su estabilidad y convergencia para los pares de elementos Q1 Q1, Q2 Q1 y Q3 Q2 (con suavidad global C1). Respecto al grupo de solvers monolíticos, nosotros proponemos métodos multimalla basados en suavizadores acoplados y desacoplados. En esta parte, métodosIsogeometric analysis (IGA) eliminates the gap between finite element analysis (FEA) and computer aided design (CAD). Due to this, IGA is an innovative approach that is receiving an increasing attention in the literature and it has recently become a trending topic. Many research efforts are being devoted to the design of efficient and robust solvers for this type of discretization. Given the optimality of multigrid methods for FEA, the application of these methods to IGA discretizations has not been unnoticed. We firmly think that they are a very promising approach as efficient and robust solvers for IGA and therefore in this thesis we are concerned about their application. In order to give a theoretical support to the design of multigrid solvers, local Fourier analysis (LFA) is proposed as the main quantitative analysis. Although different scalar problems are also considered along this thesis, we make a special focus on poroelasticity problems. More concretely, we focus on the quasi-static Biot's equations for the soil consolidation process. Nowadays, it is a very challenging task to achieve robust multigrid solvers for poroelasticity problems with respect physical parameters and/or the mesh size. Thus, the main contribution of this thesis is to propose robust multigrid methods for isogeometric discretizations applied to poroelasticity problems. The first part of this thesis is devoted to the introduction of the parametric construction of curves and surfaces since these techniques are the basis of IGA. Hence, with the definition of Bernstein polynomials and B\'ezier curves as a starting point, we introduce B-splines and non-uniform rational B-splines (NURBS) since these will be the basis functions considered for our numerical experiments. The second part deals with the isogeometric analysis. In this part, the isoparametric approach is explained to the reader and the isogeometric analysis of some scalar problems is presented. Hence, the strong and weak formulations by means of Galerkin's method are introduced and the isogeometric approximation spaces as well. The next point of this thesis consists of multigrid methods. The basics of multigrid methods are explained and, besides the presentation of some classical iterative methods as smoothers, block-wise smoothers such as multiplicative and additive Schwarz methods are also introduced. At this point, we introduce LFA for the design of efficient and robust multigrid methods. Furthermore, both standard and infinite subgrids local Fourier analysis are explained in detail together with the analysis for block-wise smoothers and the analysis for systems of partial differential equations. After the introduction of isogeometric discretizations, multigrid methods as our choice of solvers and LFA as theoretical analysis, our goal is to design efficient and robust multigrid methods with respect to the spline degree for IGA discretizations of some scalar problems. Hence, we show that the use of multigrid methods based on multiplicative or additive Schwarz methods provide a good performance and robust asymptotic convergence rates. The last part of this thesis is devoted to the isogeometric analysis of poroelasticity. For this task, Biot's model and its isogeometric discretization are introduced. Moreover, we present an innovative mass stabilization of the two-field formulation of Biot's equations that eliminates all the spurious oscillations in the numerical approximation of the pressure. Then, we deal with two types of solvers for these poroelastic equations: Decoupled and monolithic solvers. In the first group we devote special attention to the fixed-stress split method and a mass stabilized iterative scheme proposed by us that can be automatically applied from the mass stabilization formulation mentioned before. In addition, we perform a von Neumann analysis for this iterative decoupled solver applied to Terzaghi's problem and demonstrate that it is stable and convergent for pairs Q1-Q1, Q2-Q1 and Q3-Q2 (with global smoothness C1). Regarding the group of monolithic solvers, we propose multigrid methods based on coupled and decoupled smoothers. Coupled additive Schwarz methods are proposed as coupled smoothers for isogeometric Taylor-Hood elements. More concretely, we propose a 51-point additive Schwarz method for the pair Q2-Q1. In the last part, we also propose to use an inexact version of the fixed-stress split algorithm as decoupled smoother by applying iterations of different additive Schwarz methods for each variable. For the latter approach, we consider the pairs of elements Q2-Q1 and Q3-Q2 (with global smoothness C1). Finally, thanks to LFA we manage to design efficient and robust multigrid solvers for the Biot's equations and some numerical results are shown.<br /