陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「Compressed sensing and single-pixel cameras」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

I’ve had a number of people ask me (especially in light of some recent publicity) exactly what “ compressed sensing ” means, and how a “ single pixel camera ” could possibly work (and how it might be advantageous over traditional cameras in certain circumstances). There is a large literature on the subject , but as the field is relatively recent, there does not yet appear to be a good non-technical introduction to the subject. So here’s my stab at the topic, which should hope

For sake of concreteness I’ll primarily discuss the camera application, although compressed sensing is a more general measurement paradigm which is applicable to other contexts than imaging (e.g. astronomy, MRI, statistical selection, etc.), as I’ll briefly remark upon at the end of this post.

已知结果和反例

The purpose of a camera is, of course, to record images. To simplify the discussion, let us think of an image as a rectangular array, e.g. a 1024 x 2048 array of pixels (thus there are 2 megapixels in all). To ignore the (minor) issue of colour, let us assume that we are just taking a black-and-white picture, so that each pixel is measured in grayscale as an integer (e.g. an 8-bit integer from 0 to 255, or a 16-bit integer from 0 to 65535) which signifies the intensity of eac

Now, to oversimplify things quite a bit, a traditional digital camera would take one measurement of intensity for each of its pixels (so, about 2 million measurements in the above example), resulting in a relatively large image file (2MB if one uses 8-bit grayscale, or 4MB if one uses 16-bit grayscale). Mathematically, this file can be represented by a very high-dimensional vector of numbers (in this example, the dimension is about 2 million).

证明或构造的主线

Before I get to the new story of “compressed sensing”, I have to first quickly review the somewhat older story of plain old “compression”. (Those who already know how image compression works can skip forward a few paragraphs.)

The images described above can take up a lot of disk space on the camera (or on some computer where the images are later uploaded), and also take a non-trivial amount of time (and energy) to transfer from one medium to another. So, it is common practice to get the camera to compress the image, from an initial large size (e.g. 2MB) to a much smaller size (e.g. 200KB, which is 10% of the size). The thing is that while the space of all images has 2MB worth of “degrees of freedom

阅读时建议盯住的点

How can one compress an image? There are many ways, some of which are rather technical, but let me try to give a non-technical (and slightly inaccurate) sketch of how it is done. It is quite typical for an image to have a large featureless component – for instance, in a landscape, up to half of the picture might be taken up by a monochromatic sky background. Suppose for instance that we locate a large square, say pixels, which are all exactly the same colour – e.g. all white.

Now, the above algorithm is not all that effective in practice, as it does not cope well with sharp transitions from one colour to another. It turns out to be better to work not with average colours in squares, but rather with average colour imbalances in squares – the extent to which the intensity on (say) the right half of the square is higher on average than the intensity on the left. One can formalise this by using the (two-dimensional) Haar wavelet system . It then turns

值得单独记下的条目

• Matching pursuit : locate a wavelet whose signature seems to correlate with the data collected; remove all traces of that signature from the data; and repeat until we have totally “explained” the data collected in terms of wavelet signature

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源