Optimal polygonal L1 linearization and fast interpolation of nonlinear systems

The analysis of complex nonlinear systems is often carried out using simpler piecewise linear representations of them. A principled and practical technique is proposed to linearize and evaluate arbitrary continuous nonlinear functions using polygonal (continuous piecewise linear) models under the L1 norm. A thorough error analysis is developed to guide an optimal design of two kinds of polygonal approximations in the asymptotic case of a large budget of evaluation subintervals N. The method allows the user to obtain the level of linearization (N) for a target approximation error and vice versa. It is suitable for, but not limited to, an efficient implementation in modern Graphics Processing Units (GPUs), allowing real-time performance of computationally demanding applications. The quality and efficiency of the technique has been measured in detail on two nonlinear functions that are widely used in many areas of scientific computing and are expensive to evaluate.

Optimal Piecewise Linear Function Approximation for GPU-based Applications

Many computer vision and human-computer interaction applications developed in recent years need evaluating complex and continuous mathematical functions as an essential step toward proper operation. However, rigorous evaluation of this kind of functions often implies a very high computational cost, unacceptable in real-time applications. To alleviate this problem, functions are commonly approximated by simpler piecewise-polynomial representations. Following this idea, we propose a novel, efficient, and practical technique to evaluate complex and continuous functions using a nearly optimal design of two types of piecewise linear approximations in the case of a large budget of evaluation subintervals. To this end, we develop a thorough error analysis that yields asymptotically tight bounds to accurately quantify the approximation performance of both representations. It provides an improvement upon previous error estimates and allows the user to control the trade-off between the approximation error and the number of evaluation subintervals. To guarantee real-time operation, the method is suitable for, but not limited to, an efficient implementation in modern Graphics Processing Units (GPUs), where it outperforms previous alternative approaches by exploiting the fixed-function interpolation routines present in their texture units. The proposed technique is a perfect match for any application requiring the evaluation of continuous functions, we have measured in detail its quality and efficiency on several functions, and, in particular, the Gaussian function because it is extensively used in many areas of computer vision and cybernetics, and it is expensive to evaluate.

The Zero-Removing Property and Lagrange-Type Interpolation Series

The classical Kramer sampling theorem, which provides a method for obtaining orthogonal sampling formulas, can be formulated in a more general nonorthogonal setting. In this setting, a challenging problem is to characterize the situations when the obtained nonorthogonal sampling formulas can be expressed as Lagrange-type interpolation series. In this article a necessary and sufficient condition is given in terms of the zero removing property. Roughly speaking, this property concerns the stability of the sampled functions on removing a finite number of their zeros.

Analytic Kramer kernels, Lagrange-type interpolation series and de Branges spaces

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. In particular, when the involved kernel is analytic in the sampling parameter it can be stated in an abstract setting of reproducing kernel Hilbert spaces of entire functions which includes as a particular case the classical Shannon sampling theory. This abstract setting allows us to obtain a sort of converse result and to characterize when the sampling formula associated with an analytic Kramer kernel can be expressed as a Lagrange-type interpolation series. On the other hand, the de Branges spaces of entire functions satisfy orthogonal sampling formulas which can be written as Lagrange-type interpolation series. In this work some links between all these ideas are established.

An interpolation tool for aerodynamic mesh deformation problems based on octree decomposition

Desarrollo de algoritmo de interpolación basado en descomposición octree y funciones radiales de soporte compacto para movimiento de mallas en problemas aerolásticos

A Case Study of Geometric Modelling via 3-D Point Interpolation for the Bathymetry of the Rabasa Lakes (Alicante, Spain)

The interpolation of points by means of Information Technology programs appears as a technical tool of some relevancy in the hydrogeology in general and in the study of the humid zones in particular. Our approach has been the determination of the 3-D geometry of the humid zones of major depth of the Rabasa Lakes. To estimate the topography of the lake bed, we proceed to acquire information in the field by means of sonar and GPS equipment. A total of 335 points were measured both on the perimeter and in the lake bed. In a second stage, this information was used in a kriging program to obtain the bathymetry of the wetland. This methodology is demonstrated as one of the most reliable and cost-efficient for the 3-D analysis of this type of water masses. The bathymetric study of the zone allows us to characterize the mid- and long-term hydrological evolution of the lakes by means of depth-area-volume curves.

Los elementos mixtos en la teoría de elementos de contorno

En este trabajo se expone la formulación del B.I.E.M. (en problemas de potencial)con elementos mixtos, que representan una interpolación lineal en la función de campo y constante en su derivada. El objetivo primordial de dicha formulación, es el soslayar los problemas que se presentan con los planteamientos anteriores, cuando existen puntos singulares. Los ejemplos que se incluyen, resueltos mediante un programa desarrollado en un miniordenador, confirman que el método propuesto consigue mejores resultados, sin ninguna complicación adicional sobre la formulación general, y con un ahorro significativo en cuanto al número de incógnitas y ecuaciones que se han de resolver = In this paper the mixed B.I.E.M. for bidimensional potential problems is presented. The interpolation of the field variable is done by piecewise constant one. The main idea is to swep out the solution the parasitic disturbances introduced near singular points by a finite value although an important byproduct is the possibility of conexion with domain methods, as F.E.M., in which the same kind of interpolation is worked. Results are very good, as shown by the exemples, and also interesting is the reduction in computer time comparatively to the classical B.l.E.M approach.

Meromorphic continuation of functions and arbitrary distribution of interpolation points

We characterize the region of meromorphic continuation of an analytic function ff in terms of the geometric rate of convergence on a compact set of sequences of multi-point rational interpolants of ff. The rational approximants have a bounded number of poles and the distribution of interpolation points is arbitrary.

Sampling associated with resolvent-type kernels and lagrange-type interpolation series

In this paper a new class of Kramer kernels is introduced, motivated by the resolvent of a symmetric operator with compact resolvent. The article gives a necessary and sufficient condition to ensure that the associ- ated sampling formula can be expressed as a Lagrange-type interpolation series. Finally, an illustrative example, taken from the Hamburger moment problem theory, is included.

Advanced characterization of a coffee fermenting tank by multi-distributed wireless sensors: spatial interpolation and phase diagrams.

In coffee processing the fermentation stage is considered one of the critical operations by its impact on the final quality of the product. However, the level of control of the fermentation process on each farm is often not adequate; the use of sensorics for controlling coffee fermentation is not common. The objective of this work is to characterize the fermentation temperature in a fermentation tank by applying spatial interpolation and a new methodology of data analysis based on phase space diagrams of temperature data, collected by means of multi-distributed, low cost and autonomous wireless sensors. A real coffee fermentation was supervised in the Cauca region (Colombia) with a network of 24 semi-passive TurboTag RFID temperature loggers with vacuum plastic cover, submerged directly in the fermenting mass. Temporal evolution and spatial distribution of temperature is described in terms of the phase diagram areas which characterizes the cyclic behaviour of temperature and highlights the significant heterogeneity of thermal conditions at different locations in the tank where the average temperature of the fermentation was 21.2 °C, although there were temperature ranges of 4.6°C, and average spatial standard deviation of ±1.21ºC. In the upper part of the tank we found high heterogeneity of temperatures, the higher temperatures and therefore the higher fermentation rates. While at the bottom, it has been computed an area in the phase diagram practically half of the area occupied by the sensors of the upper tank, therefore this location showed higher temperature homogeneity

Advanced Characterisation of a Coffee Fermenting Tank by Multi-distributed Wireless Sensors: Spatial Interpolation and Phase Space Graphs

The fermentation stage is considered to be one of the critical steps in coffee processing due to its impact on the final quality of the product. The objective of this work is to characterise the temperature gradients in a fermentation tank by multi-distributed, low-cost and autonomous wireless sensors (23 semi-passive TurboTag® radio-frequency identifier (RFID) temperature loggers). Spatial interpolation in polar coordinates and an innovative methodology based on phase space diagrams are used. A real coffee fermentation process was supervised in the Cauca region (Colombia) with sensors submerged directly in the fermenting mass, leading to a 4.6 °C temperature range within the fermentation process. Spatial interpolation shows a maximum instant radial temperature gradient of 0.1 °C/cm from the centre to the perimeter of the tank and a vertical temperature gradient of 0.25 °C/cm for sensors with equal polar coordinates. The combination of spatial interpolation and phase space graphs consistently enables the identification of five local behaviours during fermentation (hot and cold spots).

Volume Haptics and Dendritic Spine Editing

Esta tesis se ha desarrollado en el contexto del proyecto Cajal Blue Brain, una iniciativa europea dedicada al estudio del cerebro. Uno de los objetivos de esta iniciativa es desarrollar nuevos métodos y nuevas tecnologías que simplifiquen el análisis de datos en el campo neurocientífico. El presente trabajo se ha centrado en diseñar herramientas que combinen información proveniente de distintos canales sensoriales con el fin de acelerar la interacción y análisis de imágenes neurocientíficas. En concreto se estudiará la posibilidad de combinar información visual con información háptica. Las espinas dendríticas son pequeñas protuberancias que recubren la superficie dendrítica de muchas neuronas del cerebro. A día de hoy, se cree que tienen un papel clave en la transmisión de señales neuronales. Motivo por el cual, el interés por parte de la comunidad científica por estas estructuras ha ido en aumento a medida que las técnicas de adquisición de imágenes mejoraban hasta alcanzar una calidad suficiente para analizar dichas estructuras. A menudo, los neurocientíficos utilizan técnicas de microscopía con luz para obtener los datos que les permitan analizar estructuras neuronales tales como neuronas, dendritas y espinas dendríticas. A pesar de que estas técnicas ofrezcan ciertas ventajas frente a su equivalente electrónico, las técnicas basadas en luz permiten una menor resolución. En particular, estructuras pequeñas como las espinas dendríticas pueden capturarse de forma incorrecta en las imágenes obtenidas, impidiendo su análisis. En este trabajo, se presenta una nueva técnica, que permite editar imágenes volumétricas, mediante un dispositivo háptico, con el fin de reconstruir de los cuellos de las espinas dendríticas. Con este objetivo, en un primer momento se desarrolló un algoritmo que proporciona retroalimentación háptica en datos volumétricos, completando la información que provine del canal visual. Dicho algoritmo de renderizado háptico permite a los usuarios tocar y percibir una isosuperficie en el volumen de datos. El algoritmo asegura un renderizado robusto y eficiente. Se utiliza un método basado en las técnicas de “marching tetrahedra” para la extracción local de una isosuperficie continua, lineal y definida por intervalos. La robustez deriva tanto de una etapa de detección de colisiones continua de la isosuperficie extraída, como del uso de técnicas eficientes de renderizado basadas en un proxy puntual. El método de “marching tetrahedra” propuesto garantiza que la topología de la isosuperficie extraída coincida con la topología de una isosuperficie equivalente determinada utilizando una interpolación trilineal. Además, con el objetivo de mejorar la coherencia entre la información háptica y la información visual, el algoritmo de renderizado háptico calcula un segundo proxy en la isosuperficie pintada en la pantalla. En este trabajo se demuestra experimentalmente las mejoras en, primero, la etapa de extracción de isosuperficie, segundo, la robustez a la hora de mantener el proxy en la isosuperficie deseada y finalmente la eficiencia del algoritmo. En segundo lugar, a partir del algoritmo de renderizado háptico propuesto, se desarrolló un procedimiento, en cuatro etapas, para la reconstrucción de espinas dendríticas. Este procedimiento, se puede integrar en los cauces de segmentación automática y semiautomática existentes como una etapa de pre-proceso previa. El procedimiento está diseñando para que tanto la navegación como el proceso de edición en sí mismo estén controlados utilizando un dispositivo háptico. Se han diseñado dos experimentos para evaluar esta técnica. El primero evalúa la aportación de la retroalimentación háptica y el segundo se centra en evaluar la idoneidad del uso de un háptico como dispositivo de entrada. En ambos casos, los resultados demuestran que nuestro procedimiento mejora la precisión de la reconstrucción. En este trabajo se describen también dos casos de uso de nuestro procedimiento en el ámbito de la neurociencia: el primero aplicado a neuronas situadas en la corteza cerebral humana y el segundo aplicado a espinas dendríticas situadas a lo largo de neuronas piramidales de la corteza del cerebro de una rata. Por último, presentamos el programa, Neuro Haptic Editor, desarrollado a lo largo de esta tesis junto con los diferentes algoritmos ya mencionados. ABSTRACT This thesis took place within the Cajal Blue Brain project, a European initiative dedicated to the study of the brain. One of the main goals of this project is the development of new methods and technologies simplifying data analysis in neuroscience. This thesis focused on the development of tools combining information originating from distinct sensory channels with the aim of accelerating both the interaction with neuroscience images and their analysis. In concrete terms, the objective is to study the possibility of combining visual information with haptic information. Dendritic spines are thin protrusions that cover the dendritic surface of numerous neurons in the brain and whose function seems to play a key role in neural circuits. The interest of the neuroscience community toward those structures kept increasing as and when acquisition methods improved, eventually to the point that the produced datasets enabled their analysis. Quite often, neuroscientists use light microscopy techniques to produce the dataset that will allow them to analyse neuronal structures such as neurons, dendrites and dendritic spines. While offering some advantages compared to their electronic counterpart, light microscopy techniques achieve lower resolutions. Particularly, small structures such as dendritic spines might suffer from a very low level of fluorescence in the final dataset, preventing further analysis. This thesis introduces a new technique enabling the edition of volumetric datasets in order to recreate dendritic spine necks using a haptic device. In order to fulfil this objective, we first presented an algorithm to provide haptic feedback directly from volumetric datasets, as an aid to regular visualization. The haptic rendering algorithm lets users perceive isosurfaces in volumetric datasets, and it relies on several design features that ensure a robust and efficient rendering. A marching tetrahedra approach enables the dynamic extraction of a piecewise linear continuous isosurface. Robustness is derived using a Continuous Collision Detection step coupled with acknowledged proxy-based rendering methods over the extracted isosurface. The introduced marching tetrahedra approach guarantees that the extracted isosurface will match the topology of an equivalent isosurface computed using trilinear interpolation. The proposed haptic rendering algorithm improves the coherence between haptic and visual cues computing a second proxy on the isosurface displayed on screen. Three experiments demonstrate the improvements on the isosurface extraction stage as well as the robustness and the efficiency of the complete algorithm. We then introduce our four-steps procedure for the complete reconstruction of dendritic spines. Based on our haptic rendering algorithm, this procedure is intended to work as an image processing stage before the automatic segmentation step giving the final representation of the dendritic spines. The procedure is designed to allow both the navigation and the volume image editing to be carried out using a haptic device. We evaluated our procedure through two experiments. The first experiment concerns the benefits of the force feedback and the second checks the suitability of the use of a haptic device as input. In both cases, the results shows that the procedure improves the editing accuracy. We also report two concrete cases where our procedure was employed in the neuroscience field, the first one concerning dendritic spines in the human cortex, the second one referring to an ongoing experiment studying dendritic spines along dendrites of mouse cortical pyramidal neurons. Finally, we present the software program, Neuro Haptic Editor, that was built along the development of the different algorithms implemented during this thesis, and used by neuroscientists to use our procedure.

Aplicación de Métodos Meshless al Análisis de Problemas de Autovalores

La presente Tesis Doctoral aborda la aplicación de métodos meshless, o métodos sin malla, a problemas de autovalores, fundamentalmente vibraciones libres y pandeo. En particular, el estudio se centra en aspectos tales como los procedimientos para la resolución numérica del problema de autovalores con estos métodos, el coste computacional y la viabilidad de la utilización de matrices de masa o matrices de rigidez geométrica no consistentes. Además, se acomete en detalle el análisis del error, con el objetivo de determinar sus principales fuentes y obtener claves que permitan la aceleración de la convergencia. Aunque en la actualidad existe una amplia variedad de métodos meshless en apariencia independientes entre sí, se han analizado las diferentes relaciones entre ellos, deduciéndose que el método Element-Free Galerkin Method [Método Galerkin Sin Elementos] (EFGM) es representativo de un amplio grupo de los mismos. Por ello se ha empleado como referencia en este análisis. Muchas de las fuentes de error de un método sin malla provienen de su algoritmo de interpolación o aproximación. En el caso del EFGM ese algoritmo es conocido como Moving Least Squares [Mínimos Cuadrados Móviles] (MLS), caso particular del Generalized Moving Least Squares [Mínimos Cuadrados Móviles Generalizados] (GMLS). La formulación de estos algoritmos indica que la precisión de los mismos se basa en los siguientes factores: orden de la base polinómica p(x), características de la función de peso w(x) y forma y tamaño del soporte de definición de esa función. Se ha analizado la contribución individual de cada factor mediante su reducción a un único parámetro cuantificable, así como las interacciones entre ellos tanto en distribuciones regulares de nodos como en irregulares. El estudio se extiende a una serie de problemas estructurales uni y bidimensionales de referencia, y tiene en cuenta el error no sólo en el cálculo de autovalores (frecuencias propias o carga de pandeo, según el caso), sino también en términos de autovectores. This Doctoral Thesis deals with the application of meshless methods to eigenvalue problems, particularly free vibrations and buckling. The analysis is focused on aspects such as the numerical solving of the problem, computational cost and the feasibility of the use of non-consistent mass or geometric stiffness matrices. Furthermore, the analysis of the error is also considered, with the aim of identifying its main sources and obtaining the key factors that enable a faster convergence of a given problem. Although currently a wide variety of apparently independent meshless methods can be found in the literature, the relationships among them have been analyzed. The outcome of this assessment is that all those methods can be grouped in only a limited amount of categories, and that the Element-Free Galerkin Method (EFGM) is representative of the most important one. Therefore, the EFGM has been selected as a reference for the numerical analyses. Many of the error sources of a meshless method are contributed by its interpolation/approximation algorithm. In the EFGM, such algorithm is known as Moving Least Squares (MLS), a particular case of the Generalized Moving Least Squares (GMLS). The accuracy of the MLS is based on the following factors: order of the polynomial basis p(x), features of the weight function w(x), and shape and size of the support domain of this weight function. The individual contribution of each of these factors, along with the interactions among them, has been studied in both regular and irregular arrangement of nodes, by means of a reduction of each contribution to a one single quantifiable parameter. This assessment is applied to a range of both one- and two-dimensional benchmarking cases, and includes not only the error in terms of eigenvalues (natural frequencies or buckling load), but also of eigenvectors

Analysis of frame-compatible subsampling structures for efficient 3DTV broadcast

The evolution of the television market is led by 3DTV technology, and this tendency can accelerate during the next years according to expert forecasts. However, 3DTV delivery by broadcast networks is not currently developed enough, and acts as a bottleneck for the complete deployment of the technology. Thus, increasing interest is dedicated to ste-reo 3DTV formats compatible with current HDTV video equipment and infrastructure, as they may greatly encourage 3D acceptance. In this paper, different subsampling schemes for HDTV compatible transmission of both progressive and interlaced stereo 3DTV are studied and compared. The frequency characteristics and preserved frequency content of each scheme are analyzed, and a simple interpolation filter is specially designed. Finally, the advantages and disadvantages of the different schemes and filters are evaluated through quality testing on several progressive and interlaced video sequences.

Kernel bandwidth estimation for moving object detection in non-stabilized cameras

The evolution of the television market is led by 3DTV technology, and this tendency can accelerate during the next years according to expert forecasts. However, 3DTV delivery by broadcast networks is not currently developed enough, and acts as a bottleneck for the complete deployment of the technology. Thus, increasing interest is dedicated to ste-reo 3DTV formats compatible with current HDTV video equipment and infrastructure, as they may greatly encourage 3D acceptance. In this paper, different subsampling schemes for HDTV compatible transmission of both progressive and interlaced stereo 3DTV are studied and compared. The frequency characteristics and preserved frequency content of each scheme are analyzed, and a simple interpolation filter is specially designed. Finally, the advantages and disadvantages of the different schemes and filters are evaluated through quality testing on several progressive and interlaced video sequences.