<commit_before>---
layout: post
title: "[SIAM NEWS] Super-resolution and Compressed Sensing"
author: "李军"
categories: journal
tags: [siam, compressed sensing]
image:
  feature:
  teaser:
  credit: 
  creditlink: ""
---

By <u><NAME></u>

October 01, 2013

[<NAME> is a Ph.D. student in the Department of Electrical Engineering at Stanford University.]

It has long been known that a lens, however perfect, has limited resolving power.

人们早就知道，不管镜头多么完美，分辨率都很有限。 

As <NAME> pointed out in his seminal 1891 paper “On Pin-hole Photography,” diffraction imposes an inflexible limit on the resolution of any optical system. As a result, in such fields as microscopy, astronomy, and medical imaging, it is often challenging to discern cellular structures, far-away galaxies, or incipient tumours from the available data. Super-resolution aims to meet precisely that challenge—to uncover fine-scale structure from coarse-scale measurements. This problem also arises in electronic imaging, where photon shot noise constrains the minimum possible pixel size, and in many other applications, including signal processing, spectroscopy, radar, and seismology.

正如1891年Lord Rayleigh在其开创性论文“On Pin-hole Photography”中指出，衍射对任何光学系统的分辨率都有一个严格的限制。因此，在显微镜、天文学和医学成像等领域中，从现有数据中分辨出细胞结构、遥远星系或早期肿瘤常常是极具挑战性的。超分辨率正是为了应对这一挑战，从粗尺度测量中发现细尺度结构。这个问题也出现在电子成像（这里光子散粒噪声限制了最小可能的像素大小）和许多其他应用（包括信号处理、光谱学、雷达和地震学等）中。

In its most general formulation, the problem of super-resolution is ill posed. This becomes apparent in the frequency domain. Low-resolution measurements essentially erase the high end of the spectrum; consequently, we cannot hope to retrieve the missing fine-scale information without prior knowledge of our object of interest. Under one popular assumption, we can model the signal as a superposition of point sources or spikes, which might represent celestial bodies in astronomy, molecules in fluorescence microscopy, or line spectra in signal processing. Figure 1 illustrates the resulting inverse problem: The aim is to estimate the train of spikes at the upper left from the blurry measurements at the upper right or, equivalently, to extrapolate the high frequencies at the lower left from the truncated spectrum at the lower right. Under what conditions can we hope to achieve this? The answer must surely involve exploiting the sparsity of the signal, which suggests connections with compressed sensing.

在最一般的形式中，超分辨率问题是病态的。这在频域中非常明显。低分辨率测量本质上消除了光谱的高端，因此，我们不可能在没有我们感兴趣对象的先验知识的情况下恢复丢失的细尺度信息。在一个流行的假设下，我们可以将信号建模成点源或尖峰的叠加，这或许可以表示天文学中的天体、荧光显微镜中的分子或信号处理中的线谱等。图1说明了相应的反问题：其目的是估计右上角模糊测量中左上角的一系列尖峰，或等价地推断右下角截断频谱中左下角的高频。我们希望在什么条件下做到这一点呢？答案肯定包括运用信号的稀疏性，这表明与压缩感知有关。

![image](https://github.com/brucejunlee/brucejunlee.github.io/raw/master/assets/img/siam-cs-superresolution-1.jpg)
 
Figure 1. Schematic illustration of the super-resolution problem. Left, a highly resolved signal and its spectrum. Right, low-pass measurements of the signal and the corresponding truncated spectrum.

In a nutshell, compressed sensing allows the recovery of a low-dimensional structure embedded in a high-dimensional space. Surprisingly, this is achieved by taking non-adaptive randomized measurements and solving a tractable convex program. When applied to our model of interest, compressed-sensing theory guarantees that we can reconstruct any sequence of k spikes from approximately Ck random Fourier coefficients by solving the ℓ<sub>1</sub>-norm minimization problem

简单来讲，压缩感知可以从高维空间中恢复低维结构。令人惊讶的是，这是通过先进行非自适应随机测量然后求解一个易处理的凸规划来实现的。当运用到我们的有趣模型时，压缩感知理论保证了我们可以通过求解ℓ<sub>1</sub>范数最小化问题

$$min \limits_x ||x||_{ℓ_1}$$
<commit_msg>add all<commit_after>---
layout: post
title: "[SIAM NEWS] Super-resolution and Compressed Sensing"
author: "李军"
categories: journal
tags: [siam, compressed sensing]
image:
  feature:
  teaser:
  credit: 
  creditlink: ""
---

By <u><NAME></u>

October 01, 2013

[<NAME> is a Ph.D. student in the Department of Electrical Engineering at Stanford University.]

It has long been known that a lens, however perfect, has limited resolving power.

人们早就知道，不管镜头多么完美，分辨率都很有限。 

As <NAME> pointed out in his seminal 1891 paper “On Pin-hole Photography,” diffraction imposes an inflexible limit on the resolution of any optical system. As a result, in such fields as microscopy, astronomy, and medical imaging, it is often challenging to discern cellular structures, far-away galaxies, or incipient tumours from the available data. Super-resolution aims to meet precisely that challenge—to uncover fine-scale structure from coarse-scale measurements. This problem also arises in electronic imaging, where photon shot noise constrains the minimum possible pixel size, and in many other applications, including signal processing, spectroscopy, radar, and seismology.

正如1891年<NAME>在其开创性论文“On Pin-hole Photography”中指出，衍射对任何光学系统的分辨率都有一个严格的限制。因此，在显微镜、天文学和医学成像等领域中，从现有数据中分辨出细胞结构、遥远星系或早期肿瘤常常是极具挑战性的。超分辨率正是为了应对这一挑战，从粗尺度测量中发现细尺度结构。这个问题也出现在电子成像（这里光子散粒噪声限制了最小可能的像素大小）和许多其他应用（包括信号处理、光谱学、雷达和地震学等）中。

In its most general formulation, the problem of super-resolution is ill posed. This becomes apparent in the frequency domain. Low-resolution measurements essentially erase the high end of the spectrum; consequently, we cannot hope to retrieve the missing fine-scale information without prior knowledge of our object of interest. Under one popular assumption, we can model the signal as a superposition of point sources or spikes, which might represent celestial bodies in astronomy, molecules in fluorescence microscopy, or line spectra in signal processing. Figure 1 illustrates the resulting inverse problem: The aim is to estimate the train of spikes at the upper left from the blurry measurements at the upper right or, equivalently, to extrapolate the high frequencies at the lower left from the truncated spectrum at the lower right. Under what conditions can we hope to achieve this? The answer must surely involve exploiting the sparsity of the signal, which suggests connections with compressed sensing.

在最一般的形式中，超分辨率问题是病态的。这在频域中非常明显。低分辨率测量本质上消除了光谱的高端，因此，我们不可能在没有我们感兴趣对象的先验知识的情况下恢复丢失的细尺度信息。在一个流行的假设下，我们可以将信号建模成点源或尖峰的叠加，这或许可以表示天文学中的天体、荧光显微镜中的分子或信号处理中的线谱等。图1说明了相应的反问题：其目的是估计右上角模糊测量中左上角的一系列尖峰，或等价地推断右下角截断频谱中左下角的高频。我们希望在什么条件下做到这一点呢？答案肯定包括运用信号的稀疏性，这表明与压缩感知有关。

![image](https://github.com/brucejunlee/brucejunlee.github.io/raw/master/assets/img/siam-cs-superresolution-1.jpg)
 
Figure 1. Schematic illustration of the super-resolution problem. Left, a highly resolved signal and its spectrum. Right, low-pass measurements of the signal and the corresponding truncated spectrum.

In a nutshell, compressed sensing allows the recovery of a low-dimensional structure embedded in a high-dimensional space. Surprisingly, this is achieved by taking non-adaptive randomized measurements and solving a tractable convex program. When applied to our model of interest, compressed-sensing theory guarantees that we can reconstruct any sequence of k spikes from approximately Ck random Fourier coefficients by solving the ℓ<sub>1</sub>-norm minimization problem

简单来讲，压缩感知可以从高维空间中恢复低维结构。令人惊讶的是，这是通过先进行非自适应随机测量然后求解一个易处理的凸规划来实现的。当运用到我们的有趣模型时，压缩感知理论保证了我们可以通过求解ℓ<sub>1</sub>范数最小化问题

$$min\limits_x ||x||_{ℓ_1}$$

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