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What Is the Fourier Transform?

quantamagazine.org

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Re: What Is the Fourier Transform?

#211

Post by astrange »

HarHarVeryFunny wrote: post #205
astrange wrote: post #197

Earlier quoted context omitted.

> large scale detail (corresponding to low frequency FFT components) This isn't true in practice - images are not bandlimited like audio so there aren't really visual elements of images corresponding to low frequency cosine waves. That's why the lowest frequency DCT coefficient in a JPEG image is 16x16 pixels, which is hardly large scale. But you do quantize all components of the DCT transform, not just the highest o…

Sure, but quantization is just another level of lossiness once you've already decided what information to throw away. In terms of understanding how JPEG compression works, and how it relates to human perception, I'd say that in order of importance it's: 1) Throw away fine detail by discarding high frequency components 2) More heavily compress/discard color than brightness detail (using YUV) 3) Quantize the frequency…

> 1) Throw away fine detail by discarding high frequency components

The reason it works is that fine detail is almost completely correlated across colors, so if you only keep the Y plane at full resolution it still stores it.

You couldn't just throw it out in RGB space because eg text would be unreadable.

Re: What Is the Fourier Transform?

#212

Post by alberto_balsam »

pcfwik wrote: post #36

To add another suggestion for understanding the Fourier transform, personally the first explanation that ever clicked with me was the Aho/Hopcroft/Ullman algorithms textbook. Rather than talking about sine and cosine waves, they motivate the Fourier transform entirely in terms of polynomials. Imagine you want to multiply two polynomials (p(x) and q(x)). The key is to recognize that there are two ways to represent eac…

Great explanation, thanks for sharing

Re: What Is the Fourier Transform?

#213

Post by another_twist »

Sesse__ wrote: post #200
another_twist wrote: post #193

Earlier quoted context omitted.

When I first learned Laplace transform in university, it was my goto for differential equations of any kind. I was even naive enough to believe well this is a solved problem now. Eventually found out this wasnt the case after studying PDEs. Its still my favourite transform. Immensely useful not to mention the whole method of moments in random variables is basically laplace transform. I don't like Fourier transform bu…

You know that if you have the Laplace transform, you can just insert s = iω and then you have the Fourier transform, right? :-P (Or jω, if you prefer that notation)

Tell me about it. But I like my numbers real not imaginary !

Re: What Is the Fourier Transform?

#214

Post by schlauerfox »

IAmBroom wrote: post #178
schlauerfox wrote: post #169

All these comments and no Steinmetz mention. The original papers he wrote applying the transforms to electrical engineering work are so genius and simplifying for the craft, and he's such a nicer person for one of the founders of the field of EE. Also the origin of that old story about $10 for hitting the machine with a hammer $9990 knowing where to hit it, from what I can gather.

Steinmetz is an unsung hero of tech, except to well-educated EE's. Was he a dwarf or a hunchback? I seem to recall a physical setback.

The story I heard was he had a genetic hunchback, and never married out of fear of passing it on. It never seemed to stop him, he seemed to live a good life surrounded by friends.
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