A mathematical theory of super-resolution and two-point resolution
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866910704599040000 |
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| author | Liu, Ping Ammari, Habib |
| author_facet | Liu, Ping Ammari, Habib |
| contents | This paper focuses on the fundamental aspects of super-resolution, particularly addressing the stability of super-resolution and the estimation of two-point resolution. Our first major contribution is the introduction of two location-amplitude identities that characterize the relationships between locations and amplitudes of true and recovered sources in the one-dimensional super-resolution problem. These identities facilitate direct derivations of the super-resolution capabilities for recovering the number, location, and amplitude of sources, significantly advancing existing estimations to levels of practical relevance. As a natural extension, we establish the stability of a specific $l_0$ minimization algorithm in the super-resolution problem.
The second crucial contribution of this paper is the theoretical proof of a two-point resolution limit in multi-dimensional spaces. The resolution limit is expressed as:
\[
R = \frac{4\arcsin \left(\left(\fracσ{m_{\min}}\right)^{\frac{1}{2}} \right)}Ω
\]
for $\fracσ{m_{\min}}\leq\frac{1}{2}$, where $\fracσ{m_{\min}}$ represents the inverse of the signal-to-noise ratio ($\mathrm{SNR}$) and $Ω$ is the cutoff frequency. It also demonstrates that for resolving two point sources, the resolution can exceed the Rayleigh limit $\fracπΩ$ when the signal-to-noise ratio (SNR) exceeds $2$. Moreover, we find a tractable algorithm that achieves the resolution $R$ when distinguishing two sources. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_15208 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A mathematical theory of super-resolution and two-point resolution Liu, Ping Ammari, Habib Image and Video Processing 94A08, 94A12, 42A10, 15A09 This paper focuses on the fundamental aspects of super-resolution, particularly addressing the stability of super-resolution and the estimation of two-point resolution. Our first major contribution is the introduction of two location-amplitude identities that characterize the relationships between locations and amplitudes of true and recovered sources in the one-dimensional super-resolution problem. These identities facilitate direct derivations of the super-resolution capabilities for recovering the number, location, and amplitude of sources, significantly advancing existing estimations to levels of practical relevance. As a natural extension, we establish the stability of a specific $l_0$ minimization algorithm in the super-resolution problem. The second crucial contribution of this paper is the theoretical proof of a two-point resolution limit in multi-dimensional spaces. The resolution limit is expressed as: \[ R = \frac{4\arcsin \left(\left(\fracσ{m_{\min}}\right)^{\frac{1}{2}} \right)}Ω \] for $\fracσ{m_{\min}}\leq\frac{1}{2}$, where $\fracσ{m_{\min}}$ represents the inverse of the signal-to-noise ratio ($\mathrm{SNR}$) and $Ω$ is the cutoff frequency. It also demonstrates that for resolving two point sources, the resolution can exceed the Rayleigh limit $\fracπΩ$ when the signal-to-noise ratio (SNR) exceeds $2$. Moreover, we find a tractable algorithm that achieves the resolution $R$ when distinguishing two sources. |
| title | A mathematical theory of super-resolution and two-point resolution |
| topic | Image and Video Processing 94A08, 94A12, 42A10, 15A09 |
| url | https://arxiv.org/abs/2211.15208 |