It turns out when you can approximate any function. There is a lot you can do? Who da thunk?
Generating Coherent Noise Using Fourier Transforms
21–30 of 31 posts
Re: Generating Coherent Noise Using Fourier Transforms
#22What's a situation where noise like this is useful? In any case it's very pretty and nice and I enjoyed the article.
There are scientific applications for this kind of procedure. If an experiment has a noise source with a known frequency distribution, you can simulate the experiment by generating many thousands of realizations of noise superimposed with your (expected) signal. The variance in your measurement introduced by the noise can be used to assess the systematic uncertainty of the experiment.
For example, in ground-based experiments that measure the cosmic microwave background radiation, there is a substantial foreground noise from the atmosphere that can be modeled as a 1/f distribution. And actually the observations themselves are subject to a random variance (see cosmic variance) due to the fact that we get to observe the early universe from only one point in space. So you can use a similar trick to sample many random realizations of the CMB for given physical constants, and decide if our one-off observation is compatible with the theory.
Re: Generating Coherent Noise Using Fourier Transforms
#23What's a situation where noise like this is useful? In any case it's very pretty and nice and I enjoyed the article.
You’d never do a DFT for generating a fBM, but the same technique using a different noise spectrum is how we’ve been generating ocean waves in the VFX industry since forever: https://people.cs.clemson.edu/~jtessen/reports/papers_files/...
Re: Generating Coherent Noise Using Fourier Transforms
#24> The only benefit this has over its contemporaries is that it is tileable. Although, it does repeat making this benefit useless considering that Perlin and Simplex noise are non-repeating and infinite.
Perlin and Simplex are also easily tileable too. Just make your hash function periodic and the resulting noise while tile at the same period.
This is a neat article and a neat technique, but probably not super practical. If you know how synthesizers (like the musical instruments) work, then you can think of Perlin noise as additive synthesize and the article here as subtractive synthesis.
Taking the FFT, modifying frequency amplitudes, and then taking the IFFT is one way to implement a filter. A more direct way is to filter in the time (well, space here) domain using something like a FIR or IIR. In spatial terms, that means applying a convolution filter, which is exactly how most blurring algorithms in programs like Photoshop work.
So, another way to look at this, is that you can generate pretty terrains by taking white noise and blurring it with the right convolution kernel.
Re: Generating Coherent Noise Using Fourier Transforms
#25Quote the article: 1. Generate some White Noise. 2. Perform a Fourier transform on the White Noise. Are the two separate steps necessary? It should be possible to directly generate the Fourier Transform of the white noise, rather than applying FFT to the waveform, right?
Yup! That's basically what Perlin noise does.
Generate a bunch of sine waves at various frequencies ("octaves") and add them together.
Re: Generating Coherent Noise Using Fourier Transforms
#26Earlier quoted context omitted.
Yes integration is a 1/f Fourier multiplier. But if you want to do (1/f)^alpha then it's not so straightforward in the time domain.
In the theoretical PDEs world, non-integer alpha represents a fractional derivative.
Word. I did my time in the Sobolev spaces.
Re: Generating Coherent Noise Using Fourier Transforms
#27Quote the article: 1. Generate some White Noise. 2. Perform a Fourier transform on the White Noise. Are the two separate steps necessary? It should be possible to directly generate the Fourier Transform of the white noise, rather than applying FFT to the waveform, right?
Right (and that would be more efficient, I guess), but the Fourier transform of White Noise yields complex white noise with real and imaginary part such that their std deviation a and b obeys var_x = a^2 + b^2 (This follows from FFT being Unitary or Parveval's theorem), so each frequency component has sqrt(2)/2 times the (non-transformed) signal standard deviation. So simply generate two i.i.d. white noise fields with scaled variance (1/2).
It's important to keep in mind when speaking of noise that we are referring to average values (or statistical values), e.g. although the power spectrum of white noise is on average flat, as we discussed it's really random (the expected amplitude spectrum in fact has average 0 anywhere, since it's also white noise!).
Re: Generating Coherent Noise Using Fourier Transforms
#28Hey guys Author here. Thank you all so much for the reads and suggestions. Just some context because I think the article comes across as me being ignorant of a lot of stuff
1. I am a noob at math and the original paper really did surprise me because I find it weird and interesting that you can morph white noise to be coherent. I am sure to the people who know what they’re doing it seems obvious but it wasn’t to me.
2. The article is simply an explanation of what I think is the reason behind the mechanism as explained in the paper. Of course now I know with the help of all of you that there are better ways of doing things and most of the steps are redundant.
I’ll make a revised post with the improvements sometime in the future. Thank you again :)
Re: Generating Coherent Noise Using Fourier Transforms
#29Earlier quoted context omitted.
3. apply filter 4. apply inverse FT It is equivalent to replace 1, 2 and 3 with a proper stochastic but direct sampling of the 1/f function to get the Fourier amplitudes and a uniform sampling for Fourier phase. This would save processing time by avoiding the calculation of one 2D FFT and the application of the filter (a 2D array multiplication).
Even simpler: create your desired amplitude spectrum to match your desired filtered noise profile. This is trivial for any noise spectrum with a simple linear filter - it's just a linear function with the desired slope. It's only slightly less trivial for more complex spectra. Randomise the phases. (i)FFT. Done.
The 1/f (or any) noise spectrum represents only the *mean* amplitude in frequency domain. In reality, the amplitude at each frequency itself has an associated distribution. Since frequency domain is complex the amplitude represents a radius in 2D space and if the underlying random walk is Gaussian then the amplitude is distributed according to Rayleigh.
For generating artistic images, the artist may ignore this fact and generate only the random phases. Image to image, humans probably won't know the shortcut. But in fact, each image will have exactly the same amplitude spectra and that's not physical for real stochastic processes. Depending on the usage of the images, the shortcut could be fatal.
Eg, if we train a neural net on these generated images we should not expect that NN to perform well on the real data which our generated images were meant to represent.
Re: Generating Coherent Noise Using Fourier Transforms
#30Hey guys Author here. Thank you all so much for the reads and suggestions. Just some context because I think the article comes across as me being ignorant of a lot of stuff 1. I am a noob at math and the original paper really did surprise me because I find it weird and interesting that you can morph white noise to be coherent. I am sure to the people who know what they’re doing it seems obvious but it wasn’t to me. 2…
One thing that gave me pause is calling this "coherent noise".
I'm used to "coherency" being a property between two different "channels" or sources of signal.
Here, I guess it is being used to mean coherency between the X and Y dimensions. Is that right?
If so, I think this is not truly coherent noise but rather mimicry. We see large "patches" in X-Y because the 1/f increases the relative power in lower frequency. It is thus a "natural accident" to have low frequency in X and Y form some patches somewhere. But, it's not a coherent effect.
I'd be curious to learn if I'm misunderstanding the use of the term.