This activity, we do a lot of Fourier Transforms. We find out first how they behave so we can use that, because otherwise, it’s not really science.
A. Anamorphic Property
The great thing about the fourier transformn is that we can more or less predict what it will look like given the original dataset. to show this, let’s look at some things
Left: Original Image, Right: FFT
We see that the rectangle’s fft is wider the narrower the dimension is ( a tall rectangle’s fft is wider than that of a wide one). We also see the two bringing two dots together increase the spacing between the bands.
B. Rotation property of the fourier transform.
Let’s start with a sinusoid along the x axis. Since this has a fixed frequency, we should see just dots in its FT
Left, original. Right, FFT
Well, as expected, the FFT are dots, as to why there are two, well it’s because the frequency could both be positive and negative. But here is where it gets interesting. What if we multiply them?
Well,. now we see repetition in the Frequency space. What if we rotate the image?
That’s interesting, the FT rotates as well. Let’s make it more interesting. Let’s take these images,
and add them into
Since, respectively, their FTs are:
and FT is linear, we expect them to just add up
Well. Isn’t that handy? Now we know if we take the FT, we just remove the frequencies we don’t like and they should disappear after and inverse Fourier Transform.
C. Convolution Theorem Redux
We return to convolution. Apparently, aside from simulations of apertures and edge detection, we can use it in another way. Let’s look at the effect of changing distances between two dots.
We see that the resulting bandsa just get closer and closer. Let’s replace the dots with circles
We see that increasing the radius gets a smaller radius in the fft, but it also looks like there’s a sinusoid imposed. Let’s check for another shape
As we increase the width of the squares, the FFT looks more and more like the FFT of a single squatre aperture. but It also still has the sinusoidal FFT of the single dot. Let’s try a gaussian curve
The FFT still looks like that of the Gaussian curve but now there is still that sinusoid.Maybe we can use that.
Let’s use a random pattern
Just a bunch of random 1s in a 200X200 grid. I’ll use a small pattern
And convolve this with the random pattern.
Now, I get it. This is the property of the Dirac delta. In this example, the 1s are random dirac deltas. For more information, we can follow here. This could be really useful for automation in editing softwares. We can just replace regularly occuring objects(say pimples in a face).
D. Fingerprint enhancement
Let’s use what we learned in part C in something real
My fingerprint is kinda hard to determine. there are a lot of blotches from the ink and the ridges are kinda blurry. But I think since this is a repeating pattern, these blotches and ridges must be easier to separate in Frequency space.
Aha. Since the blotches are thick dots, the must contribute to the bright spot in the middle. Which means we can filter it out using
Which hopefully results in something useful for solving cases
Left: Original,. Right Filtered.
Well, it isn’t really clean. But the ridges are deifinitely clearer. Not bad since I used MS paint and didn’t binarize the image.
E. Lunar Landing Images with lines
The following image is one of the greatest testaments to mankind’s ingenuity.
We got to look closely at something that takes light a minute to get to. If only there weren’t any lines. So using the same method as the fingerprint.
We take the FFT, use a mask, multiply the two and take the inverse
Left: FFT, Right: Mask Used
And we get
Left: Original Image, Right: Filtered Image
F. Canvas Weave Removal
Is this painting a good one? I would’t know, at least not with the canvas thread being so obvious. maybe our new tool can help with this.
the FFT looks like
so i used this mask
that i used in MS paint. The result is kinda weird.
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