B. Roy Frieden

Restoring with Maximum Likelihood and Maximum Entropy

Given M sampled image values of an incoherent object, what can be deduced as the most likely object? Using a communication-theory model for the process of image formation, we find that the most likely object has a maximum entropy and is represented by a restoring formula that is positive and not band limited. The derivation is an adaptation to optics of a formulation by Jaynes for unbiased estimates of positive probability functions. The restoring formula is tested, via computer simulation, upon noisy images of objects consisting of random impulses. These are found to be well restored, with resolution often exceeding the Rayleigh limit and with a complete absence of spurious detail. The proviso is that the noise in each image input must not exceed about 40% of the signal image. The restoring method is applied to experimental data consisting of line spectra. Results are consistent with those of the computer simulations.

Optical Transfer of the Three-Dimensional Object*†

A transfer theory is developed that determines the image space, and three-dimensional image spectrum, of a 3–D object. For both incoherent and coherent illumination, the image is found to obey convolution, transfer, and sampling theorems that resemble the familiar results of ordinary 2-D theory. A 3-D transfer function is related to the pupil function of the image-forming optical system. One result of the theory is that with incoherent illumination, the object image space contains no more than 1/(λ3f/no.4) degrees of freedom/unit volume, where λ is the wavelength of light. The transfer theory is based on the existence of volumes of stationarity, termed “isotomes;” into which the object must be partitioned. Isotomicity is shown to be approximated, over sufficiently small volumes, in the diffraction-limited case.

VIII Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions

Publisher Summary This chapter discusses applications, specifically dealing with the performance characteristics of optical systems. The linear prolate functions are a set of band limited functions, which like the trigonometric functions, are orthogonal and complete over a finite interval. However, unlike the trig functions, they are also complete and orthogonal over the infinite interval. Fourier transform of a linear prolate function is proportional to the same prolate function. The mathematical properties of the circular prolates, which make them readily applicable to the analysis of optical imagery, are summarized in the chapter. The optical applications arise from the modern use of the prolates as a convenient set of one-dimensional, orthogonal functions. These applications have been on the general subject of optical systems analysis. Performance characteristics of the laser have been established, along with the ultimate ability of lens and lensless systems to form high quality images. There is one physical phenomenon that unifies these applications—diffraction at a finite aperture—and it can be said that this phenomenon is optimally analyzed by use of the prolate functions.

Fisher information as the basis for the Schrödinger wave equation

It is shown that the assumption that nature acts to make optimum estimates of position maximally in error leads to the Schrodinger (energy) wave equation (SWE). In this way, the SWE follows from a simple statement of uncertainty. The approach grows out of probability estimation theory, in particular the principle of minimum Fisher information (or maximum Cramer–Rao bound). The minimized Fisher information turns out to be proportional to the mean kinetic energy of the particle. This approach is an attractive supplement to conventional ways of introducing quantum mechanics to students, since it avoids the use of imperfect physical models (such as vibrating strings) or the immediate need for complex momentum operators, and follows from a plausible assumption as to how nature works.

Lossless Conversion of a Plane Laser Wave to a Plane Wave of Uniform Irradiance

Interest has recently been shown in methods of converting the plane wave from a laser in the uniphase TEM0,0 mode to a plane wave having (a) uniform irradiance over a required cross section, and (b) all the power of the original beam. Two methods are proposed for accomplishing these aims: one employs two plano-aspheric lenses; the other requires a pair of selectively aberrated lens systems. A computer program has been written which determines the aspherics, and one example is presented. The aberrations required of the second method are expressed algebraically in terms of known quantities. These aberrations could conceivably be designed into a system of spherical lenses, by use of one of the automatic lens design programs now extant.

Band-Unlimited Reconstruction of Optical Objects and Spectra*

A method is derived for digitally reconstructing any two-dimensional, partially coherent, polychromatic object from experimental knowledge of the image and point spread function. In the absence of noise, the reconstruction is perfect. The object must lie wholly within a known region of the object plane. The optics may be generally coated and tilted, and may have any aberrations. As an illustration, the reconstruction process is applied to the problem of resolving double stars. The reconstruction scheme is also used to correct the output of a conventional spectrometer for instrument broadening, and to correct the output of a Fourier-transform spectroscope for finite extent of the interferogram. Practical use of the method requires the calculation of prolate spheroidal wavefunctions and eigenvalues. The effect of noise upon the accuracy of reconstruction is analytically computed. It is shown that periodic noise and piecewise-continuous noise both cause zero error at all points in the reconstruction except at the sampling points, where the error is (theoretically) infinite. Finally, bandwidth-limited noise is shown to be indistinguishable from the object.

Exploratory Data Analysis Using Fisher Information

to Fisher Information: Its Origin, Uses, and Predictions.- Financial Economics from Fisher Information.- Growth Characteristics of Organisms.- Information and Thermal Physics.- Parallel Information Phenomena of Biology and Astrophysics.- Encryption of Covert Information Through a Fisher Game.- Applications of Fisher Information to the Management of Sustainable Environmental Systems.- Fisher Information in Ecological Systems.- Sociohistory: An Information Theory of Social Changef.

Statistical models for the image restoration problem

Abstract Inversion of the image formation equation for its object is an unstable or “ill-conditioned” problem. Severe error propagation tends to result. This can be reduced, however, by building a priori knowledge about the object in the form of constraints into the restoring procedure. In turn, these constraints can be accomplished by modeling the object in a suitable, statistical way. This paper is a survey of statistical models that have led to restoration methods which overcome to various degrees the ill-conditioned nature of the problem.

Degrees of freedom, and eigenfunctions, for the noisy image

An eigenvalue analysis of the noise-prone image leads to (a) an analysis of the eigenfunctions and eigenvalues of the sin2(x)/x2 kernel; and (b) an expression relating an effective number Neff of degrees of freedom directly to the signal-to-noise ratio σ0/σv. The latter are the variances of object and noise, respectively. For the particular case of incoherent, diffraction-limited imagery, Neff is found to be reduced from its noise-free value, the Shannon number, by the factor (1 − σv/σ0). A maximum number Nmax of degrees of freedom is also defined. Comparing one-dimensional objects illuminated alternatively by coherent and incoherent light, we find they have the same number Nmax of degrees of freedom. However, for the corresponding two-dimensional case, the incoherent value for Nmax is double that of the coherent value.

Restoring with maximum entropy. III. Poisson sources and backgrounds

The maximum entropy (ME) restoring formalism has previously been derived under the assumptions of (i) zero background and (ii) additive noise in the image. However, the noise in the signals from many modern image detectors is actually Poisson, i.e., dominated by single-photon statistics. Hence, the noise is no longer additive. Particularly in astronomy, it is often accurate to model the image as being composed of two fundamental Poisson features: (i) a component due to a smoothly varying background image, such as caused by interstellar dust, plus (ii) a superimposed component due to an unknown array of point and line sources (stars, galactic arms, etc.). The latter is termed the “foreground image” since it contains the principal object information sought by the viewer. We include in the background all physical backgrounds, such as the night sky, as well as the mathematical background formed by lower-frequency components of the principal image structure. The role played by the background, which may be separately and easily estimated since it is smooth, is to pointwise modify the known noise statistics in the foreground image according to how strong the background is. Given the estimated background, a maximum-likelihood restoring formula was derived for the foreground image. We applied this approach to some one-dimensional simulations and to some real astronomical imagery. Results are consistent with the maximum-likelihood and Poisson hypotheses: i.e., where the background is high and consequently contributes much noise to the observed image, a restored star is broader and smoother than where the background is low. This nonisoplanatic behavior is desirable since it permits extra resolution only where the noise is sufficiently low to reliably permit it.