Information field theory to infer missing data in an astronomical measurement
Statistical mechanics is the application of probability theory to the motion of particles or objects when subjected to a force. It provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that can be observed in everyday life.
A statistical field theory is any model in statistical mechanics where the degrees of freedom (in 3D, there are 6 degrees of freedom associated to the movement of a mechanical particle, 3 for its position, and 3 for its momentum) comprise a field or fields. In other words, the microstates of the system are expressed through field configurations. It is closely related to quantum field theory, which describes the quantum mechanics of fields.
A Feynman diagram is a representation of quantum field theory processes in terms of particle paths. The particle trajectories are represented by the lines of the diagram, which can be squiggly or straight, with an arrow or without, depending on the type of particle. A point where lines connect to other lines is an interaction vertex, and this is where the particles meet and interact— by emitting or absorbing new particles, deflecting one another, or changing type. The Feynman diagrams represent the trajectories of particles in intermediate stages of a scattering process and only the sum of all the Feynman diagrams represent any given particle interaction as particles do not choose a particular diagram each time they interact. The law of summation is in accord with the principle of superposition— every diagram contributes a factor to the total amplitude for the process.
Wiener filter: This is a rule for supplementing incomplete spatially distributed data in cases where data is missing in an astronomical measurement. The Wiener filter applies only under a number of conditions: the noise of the instrument must be independent of the signal’s strength, and the measuring instrument’s response to the signal must increase in a linear fashion, in other words, evenly in line with the increase in strength. And finally, the noise and the signal must follow Gaussian statistics, which are easy to apply mathematically. Torsten A. Ensslin, a physicist at the Max Planck Institute for Astrophysics, explains how this filter works with the help of an analogy, “If you can see a lot of trees, you’re probably standing in a forest, Even if your sight is impaired, you can conclude that there is another tree standing next to all the trees you can see.” The drawback of this filter is that often at least one of these conditions is not met.
Work done by Torsten A. Ensslin, Mona Frommert, Francisco S. Kitaura
Torsten A. Ensslin, Mona Frommert, Francisco S. Kitaura, physicists at the Max Planck Institute for Astrophysics in Garching, have formulated a theory of spatial perception called information field theory. The purpose of this theory is to infer missing data in an astronomical measurement. For e.g. with the help of this theory scientists can add data in places where it cannot be measured: for instance, when they want to take a picture of the universe behind the Milky Way, which telescopes are unable to penetrate (called astronomical blind spots). Basically this theory is a series of rules for reconstituting incomplete and noisy image data. The authors of this theory have also established the various conditions under which the rules should be applied. The Wiener filter – is a simple special case of the information field theory.
The inspiration behind the information field theory: In the absence of a theory that could accurately infer missing data in an astronomical measurement, Ensslin thought of doing something about it. Keeping this in mind he waded through a textbook on quantum field theory and came across a footnote explaining how human visual perception can be described as statistical field theory. This footnote turned out to be the inspiration behind the information field theory. Ensslin said, “This gave me the idea to formulate information field theory, because we have measuring problems especially when researching cosmic microwave radiation and the distribution of matter in the universe. These can be very well described by statistical field theories,” he says. “Someone could have come up with the idea earlier, but quantum physicists do not usually concern themselves with signal recognition and electrical engineers do not read books about quantum field theory”.
To develop this theory Ensslin and his colleagues used as an example the cosmic microwave background – a radiation echo of the Big Bang, behind the Milky Way, where even the most clear-sighted telescope is blind. Based on this theory the scientists partially completed measurements of the cosmic microwave background. Ensslin said, “We add the missing data on the basis of the existing measuring points around the edge of the blind spot. These conclusions are more or less uncertain, of course. Our theory also calculates precisely how uncertain the statements are.” This is done because adding data in a way that appears to make sense is not enough to avoid reaching a wrong conclusion.
Information field theory is based on the responses to two questions, which the system must answer for each unknown point. If the researchers want to reconstruct the microwave background on the basis of measuring data, for example, they first ask: How probable are the measured data? Then they ask: How probable are our assumptions on the microwave background? These two probabilities determine how plausible the respective images of the microwave background are in light of the data and prior knowledge. An optimal reconstruction lies in the middle of the probable images.
The relationship between the signal sensitivity and the noise of the measuring instrument plays a decisive role in answering the first question. The noise disturbs the measurement, and at worst a physical measuring signal can get lost in the noise – like the static that distorts an analogue radio transmission with poor reception.
“The answer to the second question comes from the previous question; in other words, my expectation of a signal resulting from my prior knowledge,” explains Ensslin. The signal corresponds to the reality of the data that the measuring instrument may only be able to reproduce with distortion. Correctly applying the expectation of a signal is a tricky business. “If I really want to see something, I choose a strong prior – but then I’m blind to everything else,” says Ensslin. Up to now, scientists often constructed their expectations of measuring data more or less randomly and equally randomly decided how strongly they should be incorporated into a data point. Information theory, on the other hand, precisely regulates how expectations should be formulated and also what weighting they should carry. “What’s new about our theory is that we can apply information theory to spatially distributed parameters – we call them fields – when we broaden them for the purposes of information field theory,” says Ensslin.
Ensslin and his colleagues have formulated a description of how to proceed in individual cases in the form of Feynman diagrams – schematic drawings consisting of dots, lines and circles, which, if you know how to read them, reveal what mathematical operations need to be carried out.
In fact on the basis of this theory Torsten Ensslin has developed a mathematical algorithm that can be of great help to many – and not only to astrophysicists. Medical practitioners would, in numerous cases, be able to make more precise diagnoses if the imaging procedures took a less limited perspective and also geologists could locate mineral resources where measurements provide an incomplete picture.
Source: http://www.mpg.de/english/illustrationsDocumentation/documentation/pressReleases/2009/pressRelease200911231/
November 23, 2009