438 research outputs found
Super-Resolution in Phase Space
This work considers the problem of super-resolution. The goal is to resolve a
Dirac distribution from knowledge of its discrete, low-pass, Fourier
measurements. Classically, such problems have been dealt with parameter
estimation methods. Recently, it has been shown that convex-optimization based
formulations facilitate a continuous time solution to the super-resolution
problem. Here we treat super-resolution from low-pass measurements in Phase
Space. The Phase Space transformation parametrically generalizes a number of
well known unitary mappings such as the Fractional Fourier, Fresnel, Laplace
and Fourier transforms. Consequently, our work provides a general super-
resolution strategy which is backward compatible with the usual Fourier domain
result. We consider low-pass measurements of Dirac distributions in Phase Space
and show that the super-resolution problem can be cast as Total Variation
minimization. Remarkably, even though are setting is quite general, the bounds
on the minimum separation distance of Dirac distributions is comparable to
existing methods.Comment: 10 Pages, short paper in part accepted to ICASSP 201
On Unlimited Sampling
Shannon's sampling theorem provides a link between the continuous and the
discrete realms stating that bandlimited signals are uniquely determined by its
values on a discrete set. This theorem is realized in practice using so called
analog--to--digital converters (ADCs). Unlike Shannon's sampling theorem, the
ADCs are limited in dynamic range. Whenever a signal exceeds some preset
threshold, the ADC saturates, resulting in aliasing due to clipping. The goal
of this paper is to analyze an alternative approach that does not suffer from
these problems. Our work is based on recent developments in ADC design, which
allow for ADCs that reset rather than to saturate, thus producing modulo
samples. An open problem that remains is: Given such modulo samples of a
bandlimited function as well as the dynamic range of the ADC, how can the
original signal be recovered and what are the sufficient conditions that
guarantee perfect recovery? In this paper, we prove such sufficiency conditions
and complement them with a stable recovery algorithm. Our results are not
limited to certain amplitude ranges, in fact even the same circuit architecture
allows for the recovery of arbitrary large amplitudes as long as some estimate
of the signal norm is available when recovering. Numerical experiments that
corroborate our theory indeed show that it is possible to perfectly recover
function that takes values that are orders of magnitude higher than the ADC's
threshold.Comment: 11 pages, 4 figures, copy of initial version to appear in Proceedings
of 12th International Conference on Sampling Theory and Applications (SampTA
Implementation of Motion Without Movement on Real 3D Objects
Researchers have developed a technique such that when the colors of a static image are toggled on a screen, the illusion of continuous movement in a certain direction is produced. This phenomenon is known as motion without movement. In our research we aim to extend the applications of this technique and apply it to real three-dimensional objects. In order to achieve the projection of images onto three-dimensional shapes, we use projectors called shader lamps, which apply an algorithm to a two-dimensional image so that it appears undistorted when projected onto a three-dimensional object. The resulting effect is that a static object appears to be moving continuously in a desired direction. In addition to applying the motion without movement technique to entire objects, we also examine its use on parts of an image as small as a pixel. Using a technique called optical flow, we determine the exact movement oparts of an image by taking a second image similar to the original, where image shapes have moved and determine in which direction. Finally, we extend the motion without movement technique beyond its previous applications to use on color rather than solely grayscale images, thus producing even more realistic results
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