Towards a Mathematical Theory of Super‐resolution

By E.J. Candes, Carlos Fernandez-Granda. 2012

1,273 citations recorded by the source corpus.

This paper develops a mathematical theory of super‐resolution. Broadly speaking, super‐resolution is the problem of recovering the fine details of an object—the high end of its spectrum—from coarse scale information only—from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in [0,1] and with unknown complex‐valued amplitudes. We only observe Fourier samples of this object up to a frequency cutoff fc. We show that one can super‐resolve these point sources with infinite precision—i.e., recover the exact locations and amplitudes—by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least 2/fc. This result extends to higher dimensions and other models. In one dimension, for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super‐resolved signal is expected to degrade when both the noise level and the super‐resolution factor vary. © 2014 Wiley Periodicals, Inc.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

1. Lihua Xie
2. Zai Yang
3. Jian Li
4. P. Stoica
5. Gongguo Tang
6. Badri Narayan Bhaskar
7. B. Recht
8. Yonina C. Eldar

In Computer Vision and Pattern Recognition

1. Yonina C. Eldar
2. Zhao Tan
3. Lihua Xie
4. G. Peyr'e
5. Vincent Duval
6. Arye Nehorai
7. P. Stoica
8. Zai Yang

Citations in the corpus

• On Gridless Sparse Methods for Line Spectral Estimation From Complete and Incomplete Data (2014, positive): “In particular, the noiseless complete data case is studied in [28], where it is shown that the frequencies can be exactly recovered provided that they are appropriately separated.”
• Direction of Arrival Estimation Using Co-Prime Arrays: A Super Resolution Viewpoint (2013, methodology): “The super resolution theory developed in [14] can be implemented for coprime arrays to demonstrate that we can detect up to O(MN) sources as long as the distance between any two sources is on the order of 1 MN .”
• Sparse Methods for Direction-of-Arrival Estimation (2016, methodology): “Additionally, with the development of recent gridless sparse methods it was shown that the local coherence between nearly located atoms actually does not matter for the convex relaxation approach if the true DOAs are appropriately separated [97] (details will be provided in Section 6).”
• Atomic Norm Denoising With Applications to Line Spectral Estimation (2012, background): “Indeed, when there is no noise, Candés and FernandezGranda showed the dual solution recovers these frequencies exactly under mild technical conditions [26].”
• Direct Localization for Massive MIMO (2016, neutral): “, achieve super-resolution) [29], [35], [36].”
• Exact Support Recovery for Sparse Spikes Deconvolution (2013, positive): “For deconvolution from ideal low-pass measurements, the ground-breaking paper [8] shows that it is indeed possible to construct a dual certificate by solving a linear system when the input Diracs are well-separated.”