科学与数学nautil.us
傅里叶变换：光学、MP3 与 JPEG 背后的数学工具
Fourier transform – A math tool used in optics, MP3s, JPEGs and more (2013)
2013 — 2017，它一次次回到首页
黑点是上首页（≥ 50 分）的那次投稿，空心点是没冲上去的投稿，点越大分数越高。
导读
这篇 Nautilus 的科普文章介绍了傅里叶变换这一数学工具，解释它如何把信号分解为不同频率的正弦波，并说明这一原理在 MP3 音频压缩、JPEG 图像压缩和光学等领域的应用。文章配有直观的图示，帮助读者建立对频域分析的直觉。
为什么经久不衰：傅里叶变换是数字时代几乎所有信号与图像处理的基础，这篇通俗导读配图友好，多年后仍是入门的好起点。
HN 评论精选
认为维基百科讲得更清楚，文章应明确说明时域/频域转换并解释为何选正弦波作基
这篇文章有几张不错的图，但我觉得维基百科在描述傅里叶变换方面做得更好。文章暗示了但没有明确说出的一个要点是：傅里叶变换能在波的时间表示和频率表示之间转换。另一个值得至少简单提及的问题是：正弦波（或复指数）到底好在哪？为什么用它们而不是阶跃函数来分解许多物理现象？哪怕举几个简单例子也会有帮助。以上听起来可能苛刻，并非本意，只是觉得稍加展开能让人更好地理解这个算法及其应用。
ptero查看英文原文
This article has several nice plots, but IMO the wikipedia does a better job describing Fourier transforms than this article. A few things that article alludes to, but not clearly states is FT ability to convert between time and frequency representation of a wave. Another question worth at least cursory mention: what is so great about sine waves (or complex exponents)? Why are they better than, say, step functions for decomposition of many physical phenomena? Even a couple of quick examples would be helpful (drop a stone into a lake; what shape are the waves? extra credit -- why?). Above may sound harsh, which was not my intent; I just think expanding a little can give a much better view of the algorithm and its applications.
指出文章混淆了模拟与离散分析：数字信号天然带宽受限，MP3 的有损来自心理声学压缩而非傅里叶变换本身
这篇帖子很好，但在谈 MP3 和有损压缩时有点误导，把模拟傅里叶分析和离散分析混为一谈。对数字信号来说，决定可表示最高频率的是采样率，不是 MP3 在“扔掉高音”——任何数字信号都是如此。离散傅里叶变换其实是无损的，只是带宽受限。发烧友偏好 FLAC 而非 MP3 的原因，是 MP3 基于心理声学原理丢掉了更多其他信息，而不只是扔掉高音。
aliston查看英文原文
This is a great post, but it's a little bit misleading when talking about MP3s and lossy compression and conflates analog fourier analysis with discrete analysis. When you're talking about a digital signal, it is the sample rate that determines the maximum frequency you can represent. It's not MP3s that "throw out the really high notes" -- it's any digital signal. A discrete fourier transform actually is lossless, but it is bandwidth limited. The reason audiophiles prefer Flac to MP3s, for instance, is because MP3s do more than just "throw out the high notes." Both are bandwidth limited, but MP3s also throw out other information based on psychoacoustic principles.
纠正文章对二维变换的描述：实际是逐行逐列的可分离变换，JPEG 用 DCT 分块降低分量精度
这篇帖子对二维傅里叶变换作用于图像的方式有些误述。二维 DFT（更准确说是 DCT）并不涉及视频中那种在平面上描画形状的复杂操作，而是把图像的每一行当作信号/波形，像素强度作为振幅。傅里叶类变换是可分离的，二维情形等价于先对每行做变换再对结果列做变换。JPEG 这类变换编码器做的是降低这些分量的精度，使许多分量变为零，这对熵编码有利，且我们几乎察觉不到。当然 JPEG 只对 8x8 块操作而非整幅图像。
0x09查看英文原文
There's a bit of a mischaracterization of the way 2 dimensional Fourier transforms operate on images in this post. The 2D DFT (or DCT rather) doesn't deal with anything as complex as tracing shapes on a 2D plane like seen in the video. What it does is treat each 1 dimensional line of the image as a signal/waveform, with the pixel intensity as amplitude. Fourier-family transforms are separable, so the 2D (and ND) case is equivalent to the transform of each line followed by a transform along the resulting columns. So the actual representation that arises from this looks like so, with each image representing the corresponding cosine component: http://0x09.net/img/dct32.png (here grey is 0) Visually most of the sinusoidal components here are zero or nearly so. However if we scale them logarithmically, we'll see that it's actually not so: http://0x09.net/img/dct32log.png What transform coders like JPEG do is reduce the precision of these components, causing many of them to become zero. Which is good for the entropy coder, and mostly imperceptible to us. Of course JPEG operates on 8x8 blocks only * rather than a whole image like here. It's hard to imagine this as an image, so here's a progressive sum starting from the second term, which essentially demonstrates an inverse DCT: http://0x09.net/img/idct32.png mind that 0 is adjusted to grey in this rendering, and the brightness of the result is not an artifact of the transform. It's easier to understand what goes on with these transforms if you can visualize things in terms of the basis functions. Which in the case of a 32x32 image like above would be http://0x09.net/img/basis.png (warning: eye strain). All the examples above pertain to the DCT, partly because of JPEG and partly so I could avoid getting phase involved, but the principles apply equally to the other transforms in the family. * although recent versions of libjpeg can use other sizes
分享了一个帮助理解傅里叶级数的交互式可视化工具
如果你想看看傅里叶级数如何作用于声音和振荡器，我做了个很酷的交互式可视化：http://bgrawi.com/Fourier-Visualizations/
diger44查看英文原文
If you want to see how the Fourier series work with sound and oscillators, I made a cool interactive visualization here: http://bgrawi.com/Fourier-Visualizations/
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