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The two-point source longitudinal resolution of three-dimensional integral imaging depends on several factors including the number of sensors, sensor pixel size, pitch between sensors, and the lens point spread function. We assume the two-point sources to be resolved if their point spread functions can be resolved in any one of the sensors. Previous studies of integral imaging longitudinal resolution either rely on geometrical optics formulation or assume the point spread function to be of sub-pixel size, thus neglecting the effect of the lens. These studies also assume both point sources to be in focus in captured elemental images. More importantly, the previous analysis does not consider the effect of noise. In this manuscript, we use the Gaussian process-based two-point source resolution criterion to overcome these limitations. We compute the circle of confusion to model the out-of-focus blurring effect. The Gaussian process-based two-point source resolution criterion allows us to study the effect of noise on the longitudinal resolution. In the absence of noise, we also present a simple analytical expression for longitudinal resolution which approximately matches the Gaussian process-based formulation. Also, we investigate the dependence of the longitudinal resolution on the parallax of the integral imaging system. We present optical experiments to validate our results. The experiments demonstrate agreement with our Gaussian process-based two-point source resolution criteria.
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Stable super-resolution of images: theoretical study
Abstract We study the ubiquitous super-resolution problem, in which one aims at localizing positive point sources in an image, blurred by the point spread function of the imaging device. To recover the point sources, we propose to solve a convex feasibility program, which simply finds a non-negative Borel measure that agrees with the observations collected by the imaging device. In the absence of imaging noise, we show that solving this convex program uniquely retrieves the point sources, provided that the imaging device collects enough observations. This result holds true if the point spread function of the imaging device can be decomposed into horizontal and vertical components and if the translations of these components form a Chebyshev system, i.e., a system of continuous functions that loosely behave like algebraic polynomials. Building upon the recent results for one-dimensional signals, we prove that this super-resolution algorithm is stable, in the generalized Wasserstein metric, to model mismatch (i.e., when the image is not sparse) and to additive imaging noise. In particular, the recovery error depends on the noise level and how well the image can be approximated with well-separated point sources. As an example, we verify these claims for the important case of a Gaussian point spread function. The proofs rely on the construction of novel interpolating polynomials—which are the main technical contribution of this paper—and partially resolve the question raised in Schiebinger et al. (2017, Inf. Inference, 7, 1–30) about the extension of the standard machinery to higher dimensions.
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- Award ID(s):
- 1704204
- PAR ID:
- 10253370
- Date Published:
- Journal Name:
- Information and Inference: A Journal of the IMA
- Volume:
- 10
- Issue:
- 1
- ISSN:
- 2049-8772
- Page Range / eLocation ID:
- 161 to 193
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imaiai/iaaa029
