About the maximal rank of 3-tensors over the real and the complex number field

High dimensional array data, tensor data, is becoming important in recent days. Then maximal rank of tensors is important in theory and applications. In this paper we consider the maximal rank of 3 tensors. It can be attacked from various viewpoints, however, we trace the method of Atkinson-Stephens(1979) and Atkinson-Lloyd(1980). They treated the problem in the complex field, and we will present various bounds over the real field by proving several lemmas and propositions, which is real counterparts of their results.Comment: 13 pages, no figure v2: correction and improvemen

Rank of 3-tensors with 2 slices and Kronecker canonical forms

Tensor type data are becoming important recently in various application fields. We determine a rank of a tensor T so that A+T is diagonalizable for a given 3-tensor A with 2 slices over the complex and real number field.Comment: 12 pages, no figur

Spectral-Spatial Method for Hyperspectral Image classification in Noisy Environment

International audienc

Convenient Tail Bounds for Sums of Random Tensors

This work prepares new probability bounds for sums of random, inde-pendent, Hermitian tensors. These probability bounds characterize large-deviation behavior of the extreme eigenvalue of the sums of random tensors. We extend Laplace transform method and Lieb’s concavity theorem from matrices to tensors, and apply these tools to generalize the classical bounds associated with the names Chernoff, Ben-nett, and Bernstein from the scalar to the tensor setting. Tail bounds for the norm of a sum of random rectangular tensors are also derived from corollaries of random Hermitian tensors cases. The proof mechanism can also be applied to tensor-valued martingales and tensor-based Azuma, Hoeffding and McDiarmid inequalities are es-tablished

Robust perceptual organization techniques for analysis of color images

Esta tesis aborda el desarrollo de nuevas técnicas de análisis robusto de imágenes estrechamente relacionadas con el comportamiento del sistema visual humano. Uno de los pilares de la tesis es la votación tensorial, una técnica robusta que propaga y agrega información codificada en tensores mediante un proceso similar a la convolución. Su robustez y adaptabilidad han sido claves para su uso en esta tesis. Ambas propiedades han sido verificadas en tres nuevas aplicaciones de la votación tensorial: estimación de estructura, detección de bordes y segmentación de imágenes adquiridas mediante estereovisión.El mayor problema de la votación tensorial es su elevado coste computacional. En esta línea, esta tesis propone dos nuevas implementaciones eficientes de la votación tensorial derivadas de un análisis en profundidad de esta técnica.A pesar de su capacidad de adaptación, esta tesis muestra que la formulación original de la votación tensorial (a partir de aquí, votación tensorial clásica) no es adecuada para algunas aplicaciones, dado que las hipótesis en las que se basa no se ajustan a todas ellas. Esto ocurre particularmente en el filtrado de imágenes en color. Así, esta tesis muestra que, más que un método, la votación tensorial es una metodología en la que la codificación y el proceso de votación pueden ser adaptados específicamente para cada aplicación, manteniendo el espíritu de la votación tensorial.En esta línea, esta tesis propone un marco unificado en el que se realiza a la vez el filtrado de imágenes y la detección robusta de bordes. Este marco de trabajo es una extensión de la votación tensorial clásica en la que el color y la probabilidad de encontrar un borde en cada píxel se codifican mediante tensores, y en el que el proceso de votación se basa en un conjunto de criterios perceptuales relacionados con el modo en que el sistema visual humano procesa información. Los avances recientes en la percepción del color han sido esenciales en el diseño de dicho proceso de votación.Este nuevo enfoque ha sido efectivo, obteniendo excelentes resultados en ambas aplicaciones. En concreto, el nuevo método aplicado al filtrado de imágenes tiene un mejor rendimiento que los métodos del estado del arte para ruido real. Esto lo hace más adecuado para aplicaciones reales, donde los algoritmos de filtrado son imprescindibles. Además, el método aplicado a detección de bordes produce resultados más robustos que las técnicas del estado del arte y tiene un rendimiento competitivo con relación a la completitud, discriminabilidad, precisión y rechazo de falsas alarmas.Además, esta tesis demuestra que este nuevo marco de trabajo puede combinarse con otras técnicas para resolver el problema de segmentación robusta de imágenes. Los tensores obtenidos mediante el nuevo método se utilizan para clasificar píxeles como probablemente homogéneos o no homogéneos. Ambos tipos de píxeles se segmentan a continuación por medio de una variante de un algoritmo eficiente de segmentación de imágenes basada en grafos. Los experimentos muestran que el algoritmo propuesto obtiene mejores resultados en tres de las cinco métricas de evaluación aplicadas en comparación con las técnicas del estado del arte, con un coste computacional competitivo.La tesis también propone nuevas técnicas de evaluación en el ámbito del procesamiento de imágenes. En concreto, se proponen dos métricas de filtrado de imágenes con el fin de medir el grado en que un método es capaz de preservar los bordes y evitar la introducción de defectos. Asimismo, se propone una nueva metodología para la evaluación de detectores de bordes que evita posibles sesgos introducidos por el post-procesado. Esta metodología se basa en cinco métricas para estimar completitud, discriminabilidad, precisión, rechazo de falsas alarmas y robustez. Por último, se proponen dos nuevas métricas no paramétricas para estimar el grado de sobre e infrasegmentación producido por los algoritmos de segmentación de imágenes.This thesis focuses on the development of new robust image analysis techniques more closely related to the way the human visual system behaves. One of the pillars of the thesis is the so called tensor voting technique. This is a robust perceptual organization technique that propagates and aggregates information encoded by means of tensors through a convolution like process. Its robustness and adaptability have been one of the key points for using tensor voting in this thesis. These two properties are verified in the thesis by applying tensor voting to three applications where it had not been applied so far: image structure estimation, edge detection and image segmentation of images acquired through stereo vision.The most important drawback of tensor voting is that its usual implementations are highly time consuming. In this line, this thesis proposes two new efficient implementations of tensor voting, both derived from an in depth analysis of this technique.Despite its adaptability, this thesis shows that the original formulation of tensor voting (hereafter, classical tensor voting) is not adequate for some applications, since the hypotheses from which it is based are not suitable for all applications. This is particularly certain for color image denoising. Thus, this thesis shows that, more than a method, tensor voting can be thought of as a methodology in which the encoding and voting process can be tailored for every specific application, while maintaining the tensor voting spirit.By following this reasoning, this thesis proposes a unified framework for both image denoising and robust edge detection.This framework is an extension of the classical tensor voting in which both color and edginess the likelihood of finding an edge at every pixel of the image are encoded through tensors, and where the voting process takes into account a set of plausible perceptual criteria related to the way the human visual system processes visual information. Recent advances in the perception of color have been essential for designing such a voting process.This new approach has been found effective, since it yields excellent results for both applications. In particular, the new method applied to image denoising has a better performance than other state of the art methods for real noise. This makes it more adequate for real applications, in which an image denoiser is indeed required. In addition, the method applied to edge detection yields more robust results than the state of the art techniques and has a competitive performance in recall, discriminability, precision, and false alarm rejection.Moreover, this thesis shows how the results of this new framework can be combined with other techniques to tackle the problem of robust color image segmentation. The tensors obtained by applying the new framework are utilized to classify pixels into likely homogeneous and likely inhomogeneous. Those pixels are then sequentially segmented through a variation of an efficient graph based image segmentation algorithm. Experiments show that the proposed segmentation algorithm yields better scores in three of the five applied evaluation metrics when compared to the state of the art techniques with a competitive computational cost.This thesis also proposes new evaluation techniques in the scope of image processing. First, two new metrics are proposed in the field of image denoising: one to measure how an algorithm is able to preserve edges, and the second to measure how a method is able not to introduce undesirable artifacts. Second, a new methodology for assessing edge detectors that avoids possible bias introduced by post processing is proposed. It consists of five new metrics for assessing recall, discriminability, precision, false alarm rejection and robustness. Finally, two new non parametric metrics are proposed for estimating the degree of over and undersegmentation yielded by image segmentation algorithms

Investigation of Multi-dimensional Tensor Multi-task Learning for Modeling Alzheimer's Disease Progression

Machine learning (ML) techniques for predicting Alzheimer's disease (AD) progression can significantly assist clinicians and researchers in constructing effective AD prevention and treatment strategies. The main constraints on the performance of current ML approaches are prediction accuracy and stability problems in medical small dataset scenarios, monotonic data formats (loss of multi-dimensional knowledge of the data and loss of correlation knowledge between biomarkers) and biomarker interpretability limitations. This thesis investigates how multi-dimensional information and knowledge from biomarker data integrated with multi-task learning approaches to predict AD progression. Firstly, a novel similarity-based quantification approach is proposed with two components: multi-dimensional knowledge vector construction and amalgamated magnitude-direction quantification of brain structural variation, which considers both the magnitude and directional correlations of structural variation between brain biomarkers and encodes the quantified data as a third-order tensor to address the problem of monotonic data form. Secondly, multi-task learning regression algorithms with the ability to integrate multi-dimensional tensor data and mine MRI data for spatio-temporal structural variation information and knowledge were designed and constructed to improve the accuracy, stability and interpretability of AD progression prediction in medical small dataset scenarios. The algorithm consists of three components: supervised symmetric tensor decomposition for extracting biomarker latent factors, tensor multi-task learning regression and algorithmic regularisation terms. The proposed algorithm aims to extract a set of first-order latent factors from the raw data, each represented by its first biomarker, second biomarker and patient sample dimensions, to elucidate potential factors affecting the variability of the data in an interpretable manner and can be utilised as predictor variables for training the prediction model that regards the prediction of each patient as a task, with each task sharing a set of biomarker latent factors obtained from tensor decomposition. Knowledge sharing between tasks improves the generalisation ability of the model and addresses the problem of sparse medical data. The experimental results demonstrate that the proposed approach achieves superior accuracy and stability in predicting various cognitive scores of AD progression compared to single-task learning, benchmarks and state-of-the-art multi-task regression methods. The proposed approach identifies brain structural variations in patients and the important brain biomarker correlations revealed by the experiments can be utilised as potential indicators for AD early identification