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| Bild: Screenshot aus"What is k-space" |
Gedankenstütze für weitere Fragestellungen im Bereich Fourier Transform / MRI:
Kommentar in http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-homer-simpsons-face:
"The idea behind the Fourier Transform is that you can express any periodic wave by summing loads of sines with different frequencies. In other words, that you can decompose any wave in a bunch of sines with different frequencies. Of course, when you hear this kind of thing, you should ask two questions:
1- Is it really possible to decompose any periodic wave in sines?
2- Is there an unique decomposition?
It turns out that you can do this decomposition for a lot of periodic waves (including most practical periodic waves), and, if this decomposition exists, that it is unique. But it's not a trivial result. The basic thing here is that sines with different frequencies are linearly independent. What does this means? Well suppose you have an equation like:
a * 1 + b * 5 = c * 3
You can solve this equation with a and b equal to 1 and c equal to 2. In fact you can solve any equation like that: integers are linearly dependent.
Suppose you had something like this:
a * Very slow sine + b * slightly faster sine + c * slightly faster than the other + d * ... + y * second fastest sine = z * fastest sine
Even with infinite sines on the left side, you are never going to solve this equation. The sum of lots of slow sines is never going to vary as fast as the fastest sine: they will be at most as fast as the second fastest one. Of course, if you can't solve this problem you can't solve this one either...
a * Very fast sine + b * slightly slower sine + c * slightly slower than
the other + d * ... + y * second slowest sine = z * slowest sine
Because it's actually the same problem: just the names have changed!
Now that we have an infinite number of periodic waves that are "very different" (Linearly Independent), it may seem easier to see how they can form any periodic wave, and how this decomposition is unique..."
