2,348 research outputs found
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
Programming matrix optics into Mathematica
The various non-linear transformations incurred by the rays in an optical
system can be modelled by matrix products up to any desired order of
approximation. Mathematica software has been used to find the appropriate
matrix coefficients for the straight path transformation and for the
transformations induced by conical surfaces, both direction change and position
offset. The same software package was programmed to model optical systems in
seventh-order. A Petzval lens was used to exemplify the modelling power of the
program.Comment: 15 page
A General Method for the Determination of Matrix Coefficients for High Order Optical System Modelling
The non-linear transformations incurred by the rays in an optical system can
be suitably described by matrices to any desired order of approximation. In
systems composed of uniform refractive index elements, each individual ray
refraction or translation has an associated matrix and a succession of
transformations correspond to the product of the respective matrices. This
paper describes a general method to find the matrix coefficients for
translation and surface refraction irrespective of the surface shape or the
order of approximation. The choice of coordinates is unusual as the orientation
of the ray is characterised by the direction cosines, rather than slopes; this
is shown to greatly simplify and generalise coefficient calculation. Two
examples are shown in order to demonstrate the power of the method: The first
is the determination of seventh order coefficients for spherical surfaces and
the second is the determination of third order coefficients for a toroidal
surface.Comment: 12 page
Wavefront and ray-density plots using seventh-order matrices
The optimization of an optical system benefits greatly from a study of its
aberrations and an identification of each of its elements' contribution to the
overall aberration figures. The matrix formalism developed by one of the
authors was the object of a previous paper and allows the expression of
image-space coordinates as high-order polynomials of object-space coordinates.
In this paper we approach the question of aberrations, both through the
evaluation of the wavefront evolution along the system and its departure from
the ideal spherical shape and the use of ray density plots. Using seventh-order
matrix modeling, we can calculate the optical path between any two points of a
ray as it travels along the optical system and we define the wavefront as the
locus of the points with any given optical path; the results are presented on
the form of traces of the wavefront on the tangential plane, although the
formalism would also permit sagital plane plots. Ray density plots are obtained
by actual derivation of the seventh-order polynomials.Comment: 15 pages, 5 figure
A Framework for Fast Image Deconvolution with Incomplete Observations
In image deconvolution problems, the diagonalization of the underlying
operators by means of the FFT usually yields very large speedups. When there
are incomplete observations (e.g., in the case of unknown boundaries), standard
deconvolution techniques normally involve non-diagonalizable operators,
resulting in rather slow methods, or, otherwise, use inexact convolution
models, resulting in the occurrence of artifacts in the enhanced images. In
this paper, we propose a new deconvolution framework for images with incomplete
observations that allows us to work with diagonalized convolution operators,
and therefore is very fast. We iteratively alternate the estimation of the
unknown pixels and of the deconvolved image, using, e.g., an FFT-based
deconvolution method. This framework is an efficient, high-quality alternative
to existing methods of dealing with the image boundaries, such as edge
tapering. It can be used with any fast deconvolution method. We give an example
in which a state-of-the-art method that assumes periodic boundary conditions is
extended, through the use of this framework, to unknown boundary conditions.
Furthermore, we propose a specific implementation of this framework, based on
the alternating direction method of multipliers (ADMM). We provide a proof of
convergence for the resulting algorithm, which can be seen as a "partial" ADMM,
in which not all variables are dualized. We report experimental comparisons
with other primal-dual methods, where the proposed one performed at the level
of the state of the art. Four different kinds of applications were tested in
the experiments: deconvolution, deconvolution with inpainting, superresolution,
and demosaicing, all with unknown boundaries.Comment: IEEE Trans. Image Process., to be published. 15 pages, 11 figures.
MATLAB code available at
https://github.com/alfaiate/DeconvolutionIncompleteOb
A convex formulation for hyperspectral image superresolution via subspace-based regularization
Hyperspectral remote sensing images (HSIs) usually have high spectral
resolution and low spatial resolution. Conversely, multispectral images (MSIs)
usually have low spectral and high spatial resolutions. The problem of
inferring images which combine the high spectral and high spatial resolutions
of HSIs and MSIs, respectively, is a data fusion problem that has been the
focus of recent active research due to the increasing availability of HSIs and
MSIs retrieved from the same geographical area.
We formulate this problem as the minimization of a convex objective function
containing two quadratic data-fitting terms and an edge-preserving regularizer.
The data-fitting terms account for blur, different resolutions, and additive
noise. The regularizer, a form of vector Total Variation, promotes
piecewise-smooth solutions with discontinuities aligned across the
hyperspectral bands.
The downsampling operator accounting for the different spatial resolutions,
the non-quadratic and non-smooth nature of the regularizer, and the very large
size of the HSI to be estimated lead to a hard optimization problem. We deal
with these difficulties by exploiting the fact that HSIs generally "live" in a
low-dimensional subspace and by tailoring the Split Augmented Lagrangian
Shrinkage Algorithm (SALSA), which is an instance of the Alternating Direction
Method of Multipliers (ADMM), to this optimization problem, by means of a
convenient variable splitting. The spatial blur and the spectral linear
operators linked, respectively, with the HSI and MSI acquisition processes are
also estimated, and we obtain an effective algorithm that outperforms the
state-of-the-art, as illustrated in a series of experiments with simulated and
real-life data.Comment: IEEE Trans. Geosci. Remote Sens., to be publishe
Wigner distribution transformations in high-order systems
By combining the definition of the Wigner distribution function (WDF) and the
matrix method of optical system modeling, we can evaluate the transformation of
the former in centered systems with great complexity. The effect of stops and
lens diameter are also considered and are shown to be responsible for
non-linear clipping of the resulting WDF in the case of coherent illumination
and non-linear modulation of the WDF when the illumination is incoherent. As an
example, the study of a single lens imaging systems illustrates the
applicability of the method.Comment: 16 pages, 7 figures. To appear in J. of Comp. and Appl. Mat
Crystalline lens imaging with a slit-scanning system
[Introduction] The human crystalline lens plays a fundamental role on the retinal image quality. The ability of the lens to change its shape and, consequently, the total refractive power of the eye, during accommodation, enables one to focus objects at a wide range of distances. Knowledge regarding the crystalline lens’ geometric properties provides insights into the focusing mechanism of the visual system. Namely, the study of changes in surface shape and curvatures of the lens during accommodation is essential for a better understanding of the accommodation mechanism itself [1,2]; the age-related changes in these geometric properties of the lens are necessary to explain the etiology and progression of presbyopia [3,4]. Furthermore, crystalline lens’ optical properties, such as focal length and spherical aberrations, are closely related to its geometric properties. The aim of this work is to present a slit-scanning tomography system that’s capable of imaging the anterior chamber of the eye. We overview the applicability of our device to the crystalline lens.(undefined
- …