Stephen F. Gull

Algorithms and Applications

Maximum entropy, using the Shannon/Jaynes form -Σ p log p, is an enormously powerful tool for reconstructing positive, additive images from a wide variety of types of data. The alternative form Σ log f, due to Burg, is shown to be inappropriate for image reconstruction in general, including radio astronomy, and also for the reconstruction of the profiles of power spectra.

Image restoration by a powerful maximum entropy method

Abstract A powerful iterative algorithm has been developed which produces a maximum entropy solution to the image restoration problem. It has been applied to images containing up to 1024 × 1024 pixels and has been implemented on both mini and mainframe computers. Unlike some methods, the algorithm does not require the point-spread function to have any special symmetry properties. Examples are given of the application of this method both to artificially and to experimentally blurred photographs, and also to an X-ray radiograph, blurred by the size of the radiation source. For comparison, restorations of some of these images by the linear method of constrained least squares are also shown. The maximum entropy algorithm revealed detail on the images not seen in the linear restorations, but which are known to exist. Maximum entropy is also applied to the reconstruction of images from sparse data, which no comparable linear algorithm can handle.

On radar time and the twin `paradox'

In this paper we apply the concept of radar time (popularized by Bondi in his work on k calculus) to the well-known relativistic twin “paradox.” Radar time is used to define hypersurfaces of simultaneity for a class of traveling twins, from the “immediate turn-around” case, through the “gradual turn-around” case, to the “uniformly accelerating” case. We show that this definition of simultaneity is independent of choice of coordinates, and assigns a unique time to any event (with which the traveling twin can send and receive signals), resolving some common misconceptions.

Classic maximum entropy recovery of the average joint distribution of apparent FRET efficiency and fluorescence photons for single-molecule burst measurements.

We describe a method for analysis of single-molecule Förster resonance energy transfer (FRET) burst measurements using classic maximum entropy. Classic maximum entropy determines the Bayesian inference for the joint probability describing the total fluorescence photons and the apparent FRET efficiency. The method was tested with simulated data and then with DNA labeled with fluorescent dyes. The most probable joint distribution can be marginalized to obtain both the overall distribution of fluorescence photons and the apparent FRET efficiency distribution. This method proves to be ideal for determining the distance distribution of FRET-labeled biomolecules, and it successfully predicts the shape of the recovered distributions.

Recent Developments at Cambridge

In recent years at Cambridge University we have had a small but vigorous team working on maximum entropy and related topics. Most of our work to date has concerned image processing in one form or another, and a progress report (to 1981) was given at the first of these meetings [Skilling and Gull, 1985]. Figure 1 depicts a selection of practical results, with examples of maximum entropy data processing taken from radio astronomy, forensic deblurring, medical tomography, Michelson interferometry, and “blind” deconvolution.

State-Space-Based Approach to Quantum Field Theory for Arbitrary Observers in Electromagnetic Backgrounds

Abstract A reformulation of fermionic QFT in electromagnetic backgrounds is presented which uses methods analogous to those of conventional multiparticle quantum mechanics. Emphasis is placed on the (Schrodinger picture) states of the system, described in terms of Slater determinants of Dirac states, and not on the field operator ψ(x) (which is superfluous in this approach). The vacuum state “at time τ” is defined as the Slater determinant of a basis for the span of the negative spectrum of the “first quantized” Hamiltonian H(τ), thus providing a concrete realisation of the Dirac Sea. The general S-matrix element of the theory is derived in terms of time-dependent Bogoliubov coefficients, demonstrating that the S-matrix follows directly from the definition of inner product between Slater determinants. The process of “Hermitian extension,” inherited directly from conventional multiparticle quantum mechanics, allows second quantized operators to be defined without appealing to a complete set of orthonormal modes and provides an extremely straightforward derivation of the general expectation value of the theory. The concept of “radar time,” advocated by Bondi in his work on k-calculus, is used to generalise the particle interpretation to an arbitrarily moving observer. A definition of particle results, which depends only on the observers motion and the background present, not on any choice of coordinates or gauge, or of the particle detector. We relate this approach to conventional methods by comparing and contrasting various derivations. Our particle definition can be viewed as a generalisation to arbitrary observers of the approach of G. W. Gibbons (1975, Comm. Math. Phys.44, 245).

Reconstruction of Calmodulin Single-Molecule FRET States, Dye-Interactions, and CaMKII Peptide Binding by MultiNest and Classic Maximum Entropy.

We analyze single molecule FRET burst measurements using Bayesian nested sampling. The MultiNest algorithm produces accurate FRET efficiency distributions from single-molecule data. FRET efficiency distributions recovered by MultiNest and classic maximum entropy are compared for simulated data and for calmodulin labeled at residues 44 and 117. MultiNest compares favorably with maximum entropy analysis for simulated data, judged by the Bayesian evidence. FRET efficiency distributions recovered for calmodulin labeled with two different FRET dye pairs depended on the dye pair and changed upon Ca2+ binding. We also looked at the FRET efficiency distributions of calmodulin bound to the calcium/calmodulin dependent protein kinase II (CaMKII) binding domain. For both dye pairs, the FRET efficiency distribution collapsed to a single peak in the case of calmodulin bound to the CaMKII peptide. These measurements strongly suggest that consideration of dye-protein interactions is crucial in forming an accurate picture of protein conformations from FRET data.

State-Space Based Approach to Particle Creation in Spatially Uniform Electric Fields

Abstract Our formalism, described recently in (C. E. Dolby and S. F. Gull, Ann. Phys.293 (2001), 189–214.) is applied to the study of particle creation in spatially uniform electric fields, concentrating on the cases of a time-invariant electric field and a so-called adiabatic electric field. Several problems are resolved by incorporating the Bogoliubov coefficient approach and the tunnelling approach into a single consistent, gauge invariant formulation. The value of a time-dependent particle interpretation is demonstrated by presenting a coherent account of the time-development of the particle creation process, in which the particles are created with small momentum (in the frame of the electric field) and are then accelerated by the electric field to make up the bulge of created particles predicted by asymptotic calculations [2, 3]. An initial state comprising one particle is also considered, and its evolution is described as being the sum of two contributions: the sea of current produced by the evolved vacuum and the extra current arising from the initial particle state.

Prior Knowledge Must be Used

Entropy is defined [Jaynes, 1962] as a relative quantity