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The unreasonable effectiveness of the Fourier transform

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Re: The unreasonable effectiveness of the Fourier transform

#61

Post by shihab »

If you are from ML/Data science world, the analogy that finally unlocked FFT for me is feature size reduction using Principal Component Analysis. In both cases, you project data to a new "better" co-ordinate system ("time to frequency domain"), filter out the basis vectors that have low variance ("ignore high-frequency waves"), and project data back to real space from those truncated dimension ("Ifft: inverse transform to time domain").

Of course some differences exist (e.g. basis vectors are fixed in FFT, unlike PCA).

Re: The unreasonable effectiveness of the Fourier transform

#62

Post by flufluflufluffy »

rbonvall wrote: post #47
pizza wrote: post #24

Earlier quoted context omitted.

Lies, Damned lies, and Unreasonable Effectiveness

Lies, Damned lies, and Unreasonable Effectiveness For Fun and Profit

Lies, Damned Lies, and Unreasonable Effectiveness: How Lies in Titles are Damn Near Unreasonably Effective

Re: The unreasonable effectiveness of the Fourier transform

#63

Post by vaenaes »

Chrupiter wrote: post #48
hedgehog wrote: post #33

Once you start looking at the world through the lens of frequency domain a lot of neat tricks become simple. I have some demo code that uses fourier transform on webcam video to read a heartrate off a person's face, basically looking for what frequency holds peak energy.

I don't think pulsing skin (due to blood flow) is visible from a webcam though.

[dead]

Re: The unreasonable effectiveness of the Fourier transform

#65

Post by libraryofbabel »

rcxdude wrote: post #57
libraryofbabel wrote: post #45

Earlier quoted context omitted.

One of the things I admire about many top mathematicians today like Terence Tao is that they are clearly excellent mentors to a long list of smart graduate students and are able to advance mathematics through their students as well as on their own. You can imagine a half-formed thought Terence Tao has while driving to work becoming a whole dissertation or series of papers if he throws it to the right person to work o…

Well, in that time it was more or less how mathematics worked. It was a way of showing off, and often it would be a case of "Hey I've solved this problem, bet no-one else can". It was only later it became a lot more collaborative (and a bit more focused on publishing proofs).

You're correct that the culture of mathematics has changed a lot, and has become much more collaborative. The rise of the modern doctoral training system in Germany later in the 19th century is also relevant. So really Gauss's example points primarily to how much mathematics has changed. But at the same time, I think you could reasonably take Gauss to task even applying the standards of his own era - compare him with Euler, for example, who was much more open with publication and generous with his time and insights, frequently responding to letters from random people asking him mathematical questions, rather like Tao responding to random comments on his blog (which he does). I admire Euler more, and he was born 70 years before Gauss.

Of course, irascible brilliance and eccentricity has an honorable place in mathematics too - I don't want to exclude anyone. (Think Grigori Perelman and any number of other examples!)

Re: The unreasonable effectiveness of the Fourier transform

#66

Post by hahahahhaah »

derektank wrote: post #23

My favorite story about the Fourier Transform is that Carl Friedrich Gauss stumbled upon the algorithm for the Fast Fourier Algorthim over a century before Cooley and Tukey’s publication in 1965 (which itself revolutionized digital signal processing).[1] He was apparently studying the motion of the asteroids Pallas and Juno and wrote the algorithm down in his notes but it never made it into public knowledge. [1] http…

Gauss is gonna Gauss.

Re: The unreasonable effectiveness of the Fourier transform

#67

Post by hahahahhaah »

Jun8 wrote: post #58

A signal cannot be both time and frequency band limited. Many years ago I was amazed when I read that this fact I learned in my undergraduate is equivalent to the Uncertainty Principle! On a more mundane note: my wife and I always argue whose method of loading the dishwasher is better: she goes slow and meticulously while I do it fast. It occurred to me we were optimizing for frequency and time domains, respectively,…

The self loading dishwasher would be the greatest marriage saving invention since car navigation systems.

Re: The unreasonable effectiveness of the Fourier transform

#68

Post by hahahahhaah »

Chrupiter wrote: post #48
hedgehog wrote: post #33

Once you start looking at the world through the lens of frequency domain a lot of neat tricks become simple. I have some demo code that uses fourier transform on webcam video to read a heartrate off a person's face, basically looking for what frequency holds peak energy.

I don't think pulsing skin (due to blood flow) is visible from a webcam though.

I have seen apps that use the principle for HRV. Finger pushed on phone cam.

Re: The unreasonable effectiveness of the Fourier transform

#69

Post by astrange »

cogman10 wrote: post #40
hedgehog wrote: post #33

Once you start looking at the world through the lens of frequency domain a lot of neat tricks become simple. I have some demo code that uses fourier transform on webcam video to read a heartrate off a person's face, basically looking for what frequency holds peak energy.

It's effectively the underpinning of all modern lossy compression algorithms. The DCT which underlies codecs like Jpeg, h264, mp3, is really just a modified FFT.

Inter/intra-prediction is more important than the DCT. H264 and later use simpler degenerate forms of it because that's good enough and they can define it with bitwise accuracy.
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