Colour in Fourier Space

As I threatened promised after Anton’s interesting essay on the perception of colour, a couple of days ago, I thought I’d write a quick item about something vaguely relevant that relates to some of my own research. In fact, this ended up as a little paper in Nature written by myself and Lung-Yih Chiang, a former student of mine who’s now based in his homeland of Taiwan.

This is going to be a bit more technical than my usual stuff, but it also relates to a post I did some time ago concerning the cosmic microwave background and to the general idea of the cosmic web, which has also featured in a previous item. You may find it useful to read these contributions first if you’re not au fait with cosmological jargon.

Or you may want to ignore it altogether and come back when I’ve found another look-alike

The large-scale structure of the Universe – the vast chains of galaxies that spread out over hundreds of millions of light-years and interconnect in a complex network (called the cosmic web) – is thought to have its origin in small fluctuations generated in the early universe by quantum mechnical effects during a bout of cosmic inflation.

These fluctuations in the density of an otherwise homogeneous universe are usually expressed in dimensionless form via the density contrast, defined as\delta({\bf x})=(\rho({\bf x})-\bar{\rho})/\bar{\rho}, where \bar{\rho} is the mean density. Because it’s what physicists always do when they can’t think of anything better, we take the Fourier transform of this and write it as \tilde{\delta}, which is a complex function of the wavevector {\bf k}, and can therefore be written

\tilde{\delta}({\bf k})=A({\bf k}) \exp [i\Phi({\bf k})],

where A is the amplitude and \Phi is the phase belonging to the wavevector {\bf k}; the phase is an angle between zero and 2\pi radians.

This is a particularly useful thing to do because the simplest versions of inflation predict that the phases of each of the Fourier modes should be randomly distributed. Each is independent of the others and is essentially a random angle designating any point on the unit circle. What this really means is that there is no information content in their distribution, so that the harmonic components are in a state of maximum statistical disorder or entropy. This property also guarantees that fluctuations from place to place have a Gaussian distribution, because the density contrast at any point is formed from a superposition of a large number of independent plane-wave modes  to which the central limit theorem applies.

However, this just describes the initial configuration of the density contrast as laid down very early in the Big Bang. As the Universe expands, gravity acts on these fluctuations and alters their properties. Regions with above-average initial density (\delta >0) attract material from their surroundings and get denser still. They then attract more material, and get denser. This is an unstable process that eventually ends up producing enormous concentrations of matter (\delta>>1) in some locations and huge empty voids everywhere else.

This process of gravitational instability has been studied extensively in a variety of astrophysical settings. There are basically two regimes: the linear regime covering the early stages when \delta << 1 and the non-linear regime when large contrasts begin to form. The early stage is pretty well understood; the latter isn’t. Although many approximate analytical methods have been invented which capture certain aspects of the non-linear behaviour, general speaking we have to  run N-body simulations that calculate everything numerically by brute force to get anywhere.

The difference between linear and non-linear regimes is directly reflected in the Fourier-space behaviour. In the linear regime, each Fourier mode evolves independently of the others so the initial statistical form is preserved. In the non-linear regime, however, modes couple together and the initial Gaussian distribution begins to distort.

About a decade ago, Lung-Yih and I started to think about whether one might start to understand the non-linear regime a bit better by looking at the phases of the Fourier modes, an aspect of the behaviour that had been largely neglected until then. Our point was that mode-coupling effects must surely generate phase correlations that were absent in the initial random-phase configuration.

In order to explore the phase distribution we hit upon the idea of representing the phase of each Fourier mode using a  colour model. Anton’s essay discussed the  RGB (red-green-blue) parametrization of colour is used on computer screens as well as the CMY (Cyan-Magenta-Yellow) system preferred for high-quality printing.

However, there are other systems that use parameters different to those representing basic tones in these schemes. In particular, there are colour models that involve a parameter called the hue, which represents the position of a particular colour on the colour wheel shown left. In terms of the usual RGB framework you can see that red has a hue of zero, green is 120 degrees, and blue is 240. The complementary colours cyan, magenta and yellow lie 180 degrees opposite their RGB counterparts.

This representation is handy because it can be employed in a scheme that uses colour to represent Fourier phase information. Our idea was simple. The phases of the initial conditions should be random, so in this representation the Fourier transform should just look like a random jumble of colours with equal amounts of, say, red green and blue. As non-linear mode coupling takes hold of the distribution, however, a pattern should emerge in the phases in a manner which is characteristic of gravitational instability.

I won’t go too much further into the details here, but I will show a picture that proves that it works!

What you see here are four columns. The leftmost shows (from top to bottom) the evolution of a two-dimensional simulation of gravitational clustering. You can see the structure develops hierarchically, with an increasing characteristic scale of structure as time goes on.

The second column shows a time sequence of (part of) the Fourier transform of the distribution seen in the first; for the aficianados I should say that this is only one quadrant of the transform and that the rest is omitted for reasons of symmetry. Amplitude information is omitted here and the phase at each position is represented by an appropriate hue. To represent on this screen, however, we had to convert back to the RGB system.

The pattern is hard to see on this low resolution plot but two facts are noticeable. One is that a definite texture emerges, a bit like Harris Tweed, which gets stronger as the clustering develops. The other is that the relative amount of red green and blue does not change down the column.

The reason for the second property is that although clustering develops and the distribution of density fluctuations becomes non-Gaussian, the distribution of phases remains uniform in the sense that binning the phases of the entire Fourier transform would give a flat histogram. This is a consequence of the fact that the statistical properties of the fluctuations remain invariant under spatial translations even when they are non-linear.

Although the one-point distribuition of phases stays uniform even into the strongly non-linear regime, they phases do start to learn about each other, i.e. phase correlations emerge. Columns 3 and 4 illustrate this in the simplest possible way; instead of plotting the phases of each wavemode we plot the differences between the phases of neighbouring modes in the x  and y directions respectively.

If the phases are random then the phase differences are also random. In the initial state, therefore, columns 3 and 4 look just like column 2. However, as time goes on you should be able to see the emergence of a preferred colour in both columns, showing that the distribution of phase differences is no longer random.

The hard work is to describe what’s going on mathematically. I’ll spare you the details of that! But I hope I’ve at least made the point that this is a useful way of demonstrating that phase correlations exist and of visualizing some of their properties.

It’s also – I think – quite a lot of fun!

P.S. If you’re interested in the original paper, you will find it in Nature, Vol. 406 (27 July 2000), pp. 376-8.

About these ads

8 Responses to “Colour in Fourier Space”

  1. Anton Garrett Says:

    “Because it’s what physicists always do when they can’t think of anything better, we take the Fourier transform…”

    Yes quite. At risk of going off at a tangent, physicists know that there are often good reasons to Fourier transform, but there is one situation where it is a lousy thing to do, and that is analysis of sound signals designed for human perception, such as music. Notes played of a given frequency on an instrument build up and die away with time, so that they really have a spread of frequencies around the ‘official’ one, according to an uncertainty-principle type of relation – yet we don’t hear that spread of frequencies. We really need some normal modes that allow a description of music based on the time-varying amplitudes of the frequency-tuned oscillators within the ear.

    Anton

  2. Don’t you just love those automatically generated posts!

  3. telescoper Says:

    Anton,

    Tangents are good!

    Fourier analysis is extremely powerful in many situations, but in messy non-linear problems it often leads to a complete nightmare of coupled equations. Phenomena which are highly localized in real space (such as clusters of galaxies) are spread out all over Fourier space.

    Written music is a essentially a time-dependent discrete Fourier transform, written time going across the page and frequency plotted upwards. As you say, however it doesn’t capture the frequency content of the sound very well. In addition to the point you make, instruments don’t produce pure tones in the first place.

    Perhaps we need an essay on the perception of sound…

    Peter

  4. Anton Garrett Says:

    Peter,

    Not one written by me! But in Sydney I went to a superb lecture – subsequently written up – by the astrophysicist and polymath Tommy Gold about the theory of hearing, and specifically about how the oscillators in the ear are maintained in an ‘active’ state such that their response is very frequency-specific, when their physiological structure would imply a broad frequency response. This had been conjectured, and strong evidence for it was the occasional detection of human ears *emitting* noice at particular frequencies, when the active control went haywire.

    The problem of sound perception is wholly linear – and still inappropriate for Fourier analysis.

    The maximum entropy algorithm has not, to my knowledge, been applied to sound smoothing as it has to image smoothing, because light intensity is positive definite and can act as a proxy for probability in the “p log p” form, whereas sound amplitude oscillates rapidly between positive and negative. Perhaps this difference relates to something as fundamental as the lack of any aether that oscillates when light propagates, whereas sound waves propagate through a medium.

    Anton

  5. How large are the phase correlations, numerically speaking? Is there any hope of observing them?

  6. Michael,

    That depends on how non-linear things are. But they are definitely measurable, in various ways. A Gaussian random field has a bispectrum which is zero. Non-linear gravitational evolution generates a non-zero bispectrum which can certainly be measured. It would remain zero if it were not for the development of phase correlations between the three modes involved in its construction.

    Peter

  7. [...] on the left is much fuzzier. In fact, the sharply delineated features  are produced by mode-mode correlations and are therefore not well described by the power spectrum, which involves only the amplitude of [...]

  8. […] on the right is much fuzzier. In fact, the sharply delineated features  are produced by mode-mode correlations and are therefore not well described by the power spectrum, which involves only the amplitude of […]

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 3,588 other followers

%d bloggers like this: