Semi-Sparsity for Smoothing Filters

In this paper, we propose an interesting semi-sparsity smoothing algorithm based on a novel sparsity-inducing optimization framework. This method is derived from the multiple observations, that is, semi-sparsity prior knowledge is more universally applicable, especially in areas where sparsity is not fully admitted, such as polynomial-smoothing surfaces. We illustrate that this semi-sparsity can be identified into a generalized $L_0$-norm minimization in higher-order gradient domains, thereby giving rise to a new "feature-aware" filtering method with a powerful simultaneous-fitting ability in both sparse features (singularities and sharpening edges) and non-sparse regions (polynomial-smoothing surfaces). Notice that a direct solver is always unavailable due to the non-convexity and combinatorial nature of $L_0$-norm minimization. Instead, we solve the model based on an efficient half-quadratic splitting minimization with fast Fourier transforms (FFTs) for acceleration. We finally demonstrate its versatility and many benefits to a series of signal/image processing and computer vision applications

Analysis of Bas-Relief Generation Techniques

Simplifying the process of generating relief sculptures has been an interesting topic of research in the past decade. A relief is a type of sculpture that does not entirely extend into three-dimensional space. Instead, it has details that are carved into a flat surface, like wood or stone, such that there are slight elevations from the flat plane that define the subject of the sculpture. When viewed orthogonally straight on, a relief can look like a full sculpture or statue in the respect that a full sense of depth from the subject can be perceived. Creating such a model manually is a tedious and difficult process, akin to the challenges a painter may face when designing a convincing painting. Like with painting, certain digital tools (3D modeling programs most commonly) can make the process a little easier, but can still take a lot of time to obtain sufficient details. To further simplify the process of relief generation, a sizable amount of research has gone into developing semi-automated processes of creating reliefs based on different types of models. These methods can vary in many ways, including the type of input used, the computational time required, and the quality of the resulting model. The performance typically depends on the type of operations applied to the input model, and usually user-specified parameters to modify its appearance. In this thesis, we try to accomplish a few related topics. First, we analyze previous work in the field and briefly summarize the procedures to emphasize a variety of ways to solve the problem. We then look at specific algorithms for generating reliefs from 2D and 3D models. After explaining two of each type, a “basic” approach, and a more sophisticated one, we compare the algorithms based on their difficulty to implement, the quality of the results, and the time to process. The final section will include some more sample results of the previous algorithms, and will suggest possible ideas to enhance their results, which could be applied in continuing research on the topic

Development and applications of the Finite Point Method to compressible aerodynamics problems

This work deals with the development and application of the Finite Point Method (FPM) to compressible aerodynamics problems. The research focuses mainly on investigating the capabilities of the meshless technique to address practical problems, one of the most outstanding issues in meshless methods. The FPM spatial approximation is studied firstly, with emphasis on aspects of the methodology that can be improved to increase its robustness and accuracy. Suitable ranges for setting the relevant approximation parameters and the performance likely to be attained in practice are determined. An automatic procedure to adjust the approximation parameters is also proposed to simplify the application of the method, reducing problem- and user-dependence without affecting the flexibility of the meshless technique. The discretization of the flow equations is carried out following wellestablished approaches, but drawing on the meshless character of the methodology. In order to meet the requirements of practical applications, the procedures are designed and implemented placing emphasis on robustness and efficiency (a simplification of the basic FPM technique is proposed to this end). The flow solver is based on an upwind spatial discretization of the convective fluxes (using the approximate Riemann solver of Roe) and an explicit time integration scheme. Two additional artificial diffusion schemes are also proposed to suit those cases of study in which computational cost is a major concern. The performance of the flow solver is evaluated in order to determine the potential of the meshless approach. The accuracy, computational cost and parallel scalability of the method are studied in comparison with a conventional FEM-based technique. Finally, practical applications and extensions of the flow solution scheme are presented. The examples provided are intended not only to show the capabilities of the FPM, but also to exploit meshless advantages. Automatic hadaptive procedures, moving domain and fluid-structure interaction problems, as well as a preliminary approach to solve high-Reynolds viscous flows, are a sample of the topics explored. All in all, the results obtained are satisfactorily accurate and competitive in terms of computational cost (if compared with a similar mesh-based implementation). This indicates that meshless advantages can be exploited with efficiency and constitutes a good starting point towards more challenging applications.En este trabajo se aborda el desarrollo del Método de Puntos Finitos (MPF) y su aplicación a problemas de aerodinámica de flujos compresibles. El objetivo principal es investigar el potencial de la técnica sin malla para la solución de problemas prácticos, lo cual constituye una de las limitaciones más importantes de los métodos sin malla. En primer lugar se estudia la aproximación espacial en el MPF, haciendo hincapié en aquéllos aspectos que pueden ser mejorados para incrementar la robustez y exactitud de la metodología. Se determinan rangos adecuados para el ajuste de los parámetros de la aproximación y su comportamiento en situaciones prácticas. Se propone además un procedimiento de ajuste automático de estos parámetros a fin de simplificar la aplicación del método y reducir la dependencia de factores como el tipo de problema y la intervención del usuario, sin afectar la flexibilidad de la técnica sin malla. A continuación se aborda el esquema de solución de las ecuaciones del flujo. La discretización de las mismas se lleva a cabo siguiendo métodos estándar, pero aprovechando las características de la técnica sin malla. Con el objetivo de abordar problemas prácticos, se pone énfasis en la robustez y eficiencia de la implementación numérica (se propone además una simplificación del procedimiento de solución). El comportamiento del esquema se estudia en detalle para evaluar su potencial y se analiza su exactitud, coste computacional y escalabilidad, todo ello en comparación con un método convencional basado en Elementos Finitos. Finalmente se presentan distintas aplicaciones y extensiones de la metodología desarrollada. Los ejemplos numéricos pretenden demostrar las capacidades del método y también aprovechar las ventajas de la metodología sin malla en áreas en que la misma puede ser de especial interés. Los problemas tratados incluyen, entre otras características, el refinamiento automático de la discretización, la presencia de fronteras móviles e interacción fluido-estructura, como así también una aplicación preliminar a flujos compresibles de alto número de Reynolds. Los resultados obtenidos muestran una exactitud satisfactoria. Además, en comparación con una técnica similar basada en Elementos Finitos, demuestran ser competitivos en términos del coste computacional. Esto indica que las ventajas de la metodología sin malla pueden ser explotadas con eficiencia, lo cual constituye un buen punto de partida para el desarrollo de ulteriores aplicaciones.Postprint (published version

Invariant Reconstruction of Curves and Surfaces with Discontinuities with Applications in Computer Vision

The reconstruction of curves and surfaces from sparse data is an important task in many applications. In computer vision problems the reconstructed curves and surfaces generally represent some physical property of a real object in a scene. For instance, the sparse data that is collected may represent locations along the boundary between an object and a background. It may be desirable to reconstruct the complete boundary from this sparse data. Since the curves and surfaces represent physical properties, the characteristics of the reconstruction process differs from straight forward fitting of smooth curves and surfaces to a set of data in two important areas. First, since the collected data is represented in an arbitrarily chosen coordinate system, the reconstruction process should be invariant to the choice of the coordinate system (except for the transformation between the two coordinate systems). Secondly, in many reconstruction applications the curve or surface that is being represented may be discontinuous. For example in the object recognition problem if the object is a box there is a discontinuity in the boundary curve at the comer of the box. The reconstruction problem will be cast as an ill-posed inverse problem which must be stabilized using a priori information relative to the constraint formation. Tikhonov regularization is used to form a well posed mathematical problem statement and conditions for an invariant reconstruction are given. In the case where coordinate system invariance is incorporated into the problem, the resulting functional minimization problems are shown to be nonconvex. To form a valid convex approximation to the invariant functional minimization problem a two step algorithm is proposed. The first step forms an approximation to the curve (surface) which is piecewise linear (planar). This approximation is used to estimate curve (surface) characteristics which are then used to form an approximation of the nonconvex functional with a convex functional. Several example applications in computer vision for which the invariant property is important are presented to demonstrate the effectiveness of the algorithms. To incorporate the fact that the curves and surfaces may have discontinuities the minimizing functional is modified. An important property of the resulting functional minimization problems is that convexity is maintained. Therefore, the computational complexity of the resulting algorithms are not significantly increased. Examples are provided to demonstrate the characteristics of the algorithm

Development and applications of the finite point method to compressible aerodynamics problems

This work deals with the development and application of the Finite Point Method (FPM) to compressible aerodynamics problems. The research focuses mainly on investigating the capabilities of the meshless technique to address practical problems, one of the most outstanding issues in meshless methods. The FPM spatial approximation is studied firstly, with emphasis on aspects of the methodology that can be improved to increase its robustness and accuracy. Suitable ranges for setting the relevant approximation parameters and the performance likely to be attained in practice are determined. An automatic procedure to adjust the approximation parameters is also proposed to simplify the application of the method, reducing problem- and user-dependence without affecting the flexibility of the meshless technique. The discretization of the flow equations is carried out following wellestablished approaches, but drawing on the meshless character of the methodology. In order to meet the requirements of practical applications, the procedures are designed and implemented placing emphasis on robustness and efficiency (a simplification of the basic FPM technique is proposed to this end). The flow solver is based on an upwind spatial discretization of the convective fluxes (using the approximate Riemann solver of Roe) and an explicit time integration scheme. Two additional artificial diffusion schemes are also proposed to suit those cases of study in which computational cost is a major concern. The performance of the flow solver is evaluated in order to determine the potential of the meshless approach. The accuracy, computational cost and parallel scalability of the method are studied in comparison with a conventional FEM-based technique. Finally, practical applications and extensions of the flow solution scheme are presented. The examples provided are intended not only to show the capabilities of the FPM, but also to exploit meshless advantages. Automatic hadaptive procedures, moving domain and fluid-structure interaction problems, as well as a preliminary approach to solve high-Reynolds viscous flows, are a sample of the topics explored. All in all, the results obtained are satisfactorily accurate and competitive in terms of computational cost (if compared with a similar mesh-based implementation). This indicates that meshless advantages can be exploited with efficiency and constitutes a good starting point towards more challenging applications

A Smooth Lattice construction of the Oppenheimer-Snyder spacetime

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime. The results are in excellent agreement with theory and numerical results from other authors.Comment: 60 pages, 28 figure

Efficient Algorithms for Mumford-Shah and Potts Problems

In this work, we consider Mumford-Shah and Potts models and their higher order generalizations. Mumford-Shah and Potts models are among the most well-known variational approaches to edge-preserving smoothing and partitioning of images. Though their formulations are intuitive, their application is not straightforward as it corresponds to solving challenging, particularly non-convex, minimization problems. The main focus of this thesis is the development of new algorithmic approaches to Mumford-Shah and Potts models, which is to this day an active field of research. We start by considering the situation for univariate data. We find that switching to higher order models can overcome known shortcomings of the classical first order models when applied to data with steep slopes. Though the existing approaches to the first order models could be applied in principle, they are slow or become numerically unstable for higher orders. Therefore, we develop a new algorithm for univariate Mumford-Shah and Potts models of any order and show that it solves the models in a stable way in O(n^2). Furthermore, we develop algorithms for the inverse Potts model. The inverse Potts model can be seen as an approach to jointly reconstructing and partitioning images that are only available indirectly on the basis of measured data. Further, we give a convergence analysis for the proposed algorithms. In particular, we prove the convergence to a local minimum of the underlying NP-hard minimization problem. We apply the proposed algorithms to numerical data to illustrate their benefits. Next, we apply the multi-channel Potts prior to the reconstruction problem in multi-spectral computed tomography (CT). To this end, we propose a new superiorization approach, which perturbs the iterates of the conjugate gradient method towards better results with respect to the Potts prior. In numerical experiments, we illustrate the benefits of the proposed approach by comparing it to the existing Potts model approach from the literature as well as to the existing total variation type methods. Hereafter, we consider the second order Mumford-Shah model for edge-preserving smoothing of images which –similarly to the univariate case– improves upon the classical Mumford-Shah model for images with linear color gradients. Based on reformulations in terms of Taylor jets, i.e. specific fields of polynomials, we derive discrete second order Mumford-Shah models for which we develop an efficient algorithm using an ADMM scheme. We illustrate the potential of the proposed method by comparing it with existing methods for the second order Mumford-Shah model. Further, we illustrate its benefits in connection with edge detection. Finally, we consider the affine-linear Potts model for the image partitioning problem. As many images possess linear trends within homogeneous regions, the classical Potts model frequently leads to oversegmentation. The affine-linear Potts model accounts for that problem by allowing for linear trends within segments. We lift the corresponding minimization problem to the jet space and develop again an ADMM approach. In numerical experiments, we show that the proposed algorithm achieves lower energy values as well as faster runtimes than the method of comparison, which is based on the iterative application of the graph cut algorithm (with α-expansion moves)