A Novel Representation for Two-dimensional Image Structures

This paper presents a novel approach towards two-dimensional (2D) image structures modeling. To obtain more degrees of freedom, a 2D image signal is embedded into a certain geometric algebra. Coupling methods of differential geometry, tensor algebra, monogenic signal and quadrature filter, we can design a general model for 2D structures as the monogenic extension of a curvature tensor. Based on it, a local representation for the intrinsically two-dimensional (i2D) structure is derived as the monogenic curvature signal. From it, independent features of local amplitude, phase and orientation are simultaneously extracted. Besides, a monogenic curvature scale-space can be built by applying a Poisson kernel to the monogenic curvature signal. Compared with the other related work, the remarkable advantage of our approach lies in the rotationally invariant phase evaluation of 2D structures in a multi-scale framework, which delivers access to phase-based processing in many computer vision tasks

The hidden geometric character of relativistic quantum mechanics

The presentation makes use of geometric algebra, also known as Clifford algebra, in 5-dimensional spacetime. The choice of this space is given the character of first principle, justified solely by the consequences that can be derived from such choice and their consistency with experimental results. Given a metric space of any dimension, one can define monogenic functions, the natural extension of analytic functions to higher dimensions; such functions have null vector derivative and have previously been shown by other authors to play a decisive role in lower dimensional spaces. All monogenic functions have null Laplacian by consequence; in an hyperbolic space this fact leads inevitably to a wave equation with plane-like solutions. This is also true for 5-dimensional spacetime and we will explore those solutions, establishing a parallel with the solutions of the Dirac equation. For this purpose we will invoke the isomorphism between the complex algebra of 4x4 matrices, also known as Dirac's matrices. There is one problem with this isomorphism, because the solutions to Dirac's equation are usually known as spinors (column matrices) that don't belong to the 4x4 matrix algebra and as such are excluded from the isomorphism. We will show that a solution in terms of Dirac spinors is equivalent to a plane wave solution. Just as one finds in the standard formulation, monogenic functions can be naturally split into positive/negative energy together with left/right ones. This split is provided by geometric projectors and we will show that there is a second set of projectors providing an alternate 4-fold split. The possible implications of this alternate split are not yet fully understood and are presently the subject of profound research.Comment: 29 pages. Small changes in V3 suggested by refere

Signal Modeling for Two-Dimensional Image Structures and Scale-Space Based Image Analysis

Model based image representation plays an important role in many computer vision tasks. Consequently, it is of high significance to model image structures with more powerful representation capabilities. In the literature, there exist bulk of researches for intensity based modeling. However, most of them suffer from the illumination variation. On the other hand, phase information, which carries most essential structural information of the original signal, has the advantage of being invariant to the brightness change. Therefore, phase based image analysis is advantageous when compared to purely intensity based approaches. This thesis aims to propose novel image representations for 2D image structures, from which useful local features can be extracted, which are useful for phase based image analysis. The first approach presents a 2D rotationally invariant quadrature filter. This model is able to handle superimposed intrinsically two-dimensional (i2D) patterns with flexible angles of intersection. Hence, it can be regarded as an extension of the structure multivector. The second approach is the monogenic curvature tensor. Coupling methods of differential geometry, tensor algebra, monogenic signal and quadrature filter, we can design a general model for 2D structures as the monogenic extension of a curvature tensor. Based on it, local representations for the intrinsically one-dimensional (i1D) and i2D structures are derived as the monogenic signal and the generalized monogenic curvature signal, respectively. From them, independent features of local amplitude, phase and orientation are simultaneously extracted. Besides, a generalized monogenic curvature scale-space can be built by applying a Poisson kernel to the monogenic curvature tensor. Compared with other related work, the remarkable advantage of our approach lies in the rotationally invariant phase evaluation of 2D structures in a multi-scale framework, which delivers access to phase-based processing in many computer vision tasks. To demonstrate the efficiency and power of the theoretic framework, some computer vision applications are presented, which include the phase based image reconstruction, detecting i2D image structures using local phase and monogenic curvature tensor for optical flow estimation

Eigenvalues of conformally invariant operators on spheres

Spectrum of a certain class of first order conformally invariant operators on the sphere is explicitly computed. The class contains the (elliptic verions of) Rarita-Schwinger operator and its higher spin analogues.Comment: 14 page

Fluids mobilization in Arabia Terra, Mars: depth of pressurized reservoir from mounds self-similar clustering

Arabia Terra is a region of Mars where signs of past-water occurrence are recorded in several landforms. Broad and local scale geomorphological, compositional and hydrological analyses point towards pervasive fluid circulation through time. In this work we focus on mound fields located in the interior of three casters larger than 40 km (Firsoff, Kotido and unnamed crater 20 km to the east) and showing strong morphological and textural resemblance to terrestrial mud volcanoes and spring-related features. We infer that these landforms likely testify the presence of a pressurized fluid reservoir at depth and past fluid upwelling. We have performed morphometric analyses to characterize the mound morphologies and consequently retrieve an accurate automated mapping of the mounds within the craters for spatial distribution and fractal clustering analysis. The outcome of the fractal clustering yields information about the possible extent of the percolating fracture network at depth below the craters. We have been able to constrain the depth of the pressurized fluid reservoir between ~2.5 and 3.2 km of depth and hence, we propose that mounds and mounds alignments are most likely associated to the presence of fissure ridges and fluid outflow. Their process of formation is genetically linked to the formation of large intra-crater bulges previously interpreted as large scale spring deposits. The overburden removal caused by the impact crater formation is the inferred triggering mechanism for fluid pressurization and upwelling, that through time led to the formation of the intra-crater bulges and, after compaction and sealing, to the widespread mound fields in their surroundings