QC 351 A7 no. 67 / Given M sampled image values, what can be deduced as the most likely object? We find the most likely object to be represented by a restoring formula that is positive and not bandlimited. Using computer simulation, we test this restoring formula upon noisy images due to star fields and randomly stepped objects. In all test cases involving star fields, the maximum-likelihood restorations have significantly higher resolution than do corresponding restorations by inverse filtering. Further, there is no spurious detail in the maximum-likelihood restorations when the star image has noise of about 5% (or less). This lack of spurious detail does not, however, hold in restoring the randomly stepped objects. In the first test of the method upon experimental data, some spectroscopic data are restored, again with enhanced resolution. For example, what appears as a single line in the data is restored as a doublet.

QC 351 A7 no. 72 / The problem of inferring some unknown distribution (object) from measurements of physical quantities (image data) occurs frequently in scientific investigation. This study is concerned with the numerical estimation of a continuous object distribution from a finite set of noisy image data, where the transformation (degradation) between the object and the noiseless portion of the data is assumed known. So defined, the restoration problem is inherently a statistical one, requiring a priori information to define the "most probable" object and noise associated with a given set of data. With a random particle model, the problem of implementing complex statistical and analytical foreknowledge (positive and bounded objects, signal - dependent noise, multiple -stage imaging, spatial correlation, etc.) is reduced to a few simple restoring formulas. For linear problems, noise impedes the ability to restore those object modes (statistically orthogonal components) that have a low power transmission through the imaging process. Linear applications of Fourier transform techniques are dis- cussed, where modifications to the standard Wiener filter are required for under - sampling a bandlimited image. The use of nonlinear object formulas tends to reduce the effects of noise and may extend restored resolution to well beyond the Rayleigh limit. This enhancement occurs in relatively isolated regions in an extended object, where the average restored information can never exceed that in the image. Particular attention is given to developing numerical algorithms for efficient use in digital computers. A positive or bounded object estimate is found through a series of linear matrix solutions, with an example of "superresolving" two impulses separated by half the Rayleigh limit. For 2 -D problems with stationary imaging, a purely iterative algorithm is developed, based on a series of Fourier transform operations. For restoring a 64 X 64 data array with nonlinear constraints, computation time may require only seconds (CDC 6600) as compared with hours using direct matrix methods. The iterative transform method is then applied to experimental absorption spectra, resulting in considerable resolution enhancement. Included are brief discussions of restoring photon -limited images, multiple -stage imaging problems, estimation of the imaging response, use of a finite object extent, the problem of systematic errors, and possible applications of the restoring techniques.

QC 351 A7 no. 73 / We report here the first experiments in a long-range program for investigating the effectiveness of image-retrieval or image-enhancement procedures. We employed a signal-detection mode of observer response, and our stimuli were computer-generated, pointed, abstract forms that we call quadrigons. Four values of signal-to-noise ratio were provided by varying the amount of roundedness of the interior and exterior angles of these forms. Linear blur, grain magnification, and figure-surround contrast ratio were the other independent variables in our factorial design. For each quadrigon, observers gave scaled expressions of confidence that the photographed object was originally pointed (the "signal present" condition). Scores from 12 observers, who were each exposed twice to a set of 500 quadrigons, yielded receiver operating characteristics (ROC functions) that are sensitive, quantitative indicators of the discriminabilities of the stimuli. By this method, also, an observer's criterion state (his degree of willingness to guess that a signal was present) was evaluated and removed as a contaminating factor. Signal-to-noise ratio had a strong and systematic effect upon signal detection accuracy when the effects of all other variables were combined. Linear blur, grain size, and contrast ratio each affected observer performance greatly. All two-way, three-way, and four-way interactions among the independent variables were highly significant; each source of image degradation had differential effects upon every other variable, and combinations of the variables had additional effects. The results are complex, but they provide useful implications for image processing that is designed to enhance information retrieval. We can, for instance, define the effects upon signal detectability when linear blur is reduced by specified amounts, or grain size is reduced, or contrast conditions are improved. We can, moreover, specify certain optimum combinations of values for these variables.

QC 351 A7 no. 76 / In recent years, many new image detectors have been developed that are sensitive to extremely low light levels. These devices have given new importance to a performance criterion called detective quantum efficiency, or DQE. The subject of this investigation is to develop the basic experimental technology necessary to measure DQE of image detectors, and to apply these techniques specifically to photographically recorded image intensifiers and to unaided photographic emulsions. DQE is defined as the square of the ratio of output S/N to input S /N. Input S/N is determined by the input photon statistics. Output S/N is determined by measuring the recorded image. Two ways of measuring DQE are the single-level gradient method and the two-level AD method. The AD method gives a more appropriate measure of the performance of image intensifiers that have light- induced background. A special two -channel projector was built to measure DQE by the AD method. Its function is to superimpose the image of a signal target upon a uniform background irradiance. The same instrument can be used to obtain DQE measurements using the gradient method. DQE was measured for two image intensifiers (a high -gain TSE tube, model P829D, made by EEV, and a two -stage cascade intensifier, model C3301 1, a Carnegie tube made by RCA) and two unaided photographic emulsions (Kodak IIa -O and Kodak Medium Contrast Projector Slide). The results show that different image detectors may have considerably different DQE performance. For example, at 425 nm, DQE values for the Carnegie tube were more than 20% (the quantum efficiency of the input photocathode is 28 %) whereas both of the unaided photo- graphic emulsions gave peak DQE values no greater than 0.8 %. In addition, the peak DQE of the Carnegie tube occurred at roughly 10' photons /cm2 whereas the peak DQE of unaided IIa-O occurred at 109 photons /cm2 (both at 425 nm). Furthermore, single photon event detectors (such as high -gain image intensifiers) have a peak DQE near zero exposure with DQE decreasing as exposure increases whereas multiple photon event detectors (such as unaided photographic emulsions) have zero DQE near zero exposure, a peak DQE at an exposure which yields a corresponding output density of about 0.2 above fog, and a decreasing DQE for further exposure increases. Also measured were granularity, light- induced background, system modulation transfer functions, and relative system speeds.

QC 351 A7 no. 79 / In this report we describe the work carried out under contract F19628-72-C-0047 entitled "Geodetic Optics Research" for the Air Force Cambridge Research Laboratories (AFCRL). The work consisted mainly of the fabrication of the optical components for a telescope with a 152 cm-diam (60-in.) primary mirror masked down to 147-cm diam for use by the AFCRL for a lunar ranging experiment. Among the noteworthy achievements of this contract were the following: (a) Completion of the primary and secondary mirrors for a high -quality 147 -cm- diam telescope system in eight months from the start of edging the primary. (b) Manufacture and testing of a unique center mount for the primary according to an AFCRL design that allowed for a thin-edged and therefore less-massive mirror. (c) Development of a quantitative analysis of the wire test for calculating the departure of the mirror figure from the design figure quickly and accurately after each polishing step. This analysis method in conjunction with a knowledge of polishing rates for given weights and diameters of tools, mirror, and polishing materials should considerably reduce the polishing time required for future large mirrors. The emphasis in this report is on these three items; however, considerable work was also undertaken in telescope design; null lens design and mounting; tracking optics design, fabrication, and mounting; and special thin -film coatings for the laser send and receive optics.

QC 351 A7 no. 83 / Diffraction of light by a sinusoidal sound wave is discussed in detail. Assuming that the sound column modulates only the phase of the incident light in both time and space, the frequencies, wavevectors, and intensities of the diffracted waves are obtained for normal incidence. A transition length (width of sound beam) is defined, above which all diffraction effects disappear due to destructive interference. Constructive interference is obtained, however, provided the light is incident at the Bragg angle, in which case the diffracted beam appears to be reflected from the acoustic wavefronts. The transition length thus separates the region of multiple -order (Raman -Nath) diffraction from the region of single -order (Bragg) diffraction. It is found to be directly proportional to the square of the acoustic wavelength and inversely proportional to the optical wavelength. In the case of Bragg diffraction, the energy is exchanged sinusoidally between the diffracted and undiffracted beams. Owing to the finite width of the sound beam, the Bragg condition is relaxed, and the effect can be used to control the direction and phase of the diffracted beam or to determine the angular distribution of the acoustic power. Next, a particle picture of diffraction in terms of photons and phonons is given. The diffraction process is described as a single as well as a multiple three -particle interaction. The effects of finite optical and acoustic beamwidths and variation of acoustic frequency are considered in terms of momentum conservation. Finally, an analysis based on Maxwell's equations for an arbitrarily polarized light beam propagating in an arbitrary direction is given using the partial -wave approach. A set of coupled difference- differential equations for the diffracted amplitudes is derived from the optical wave equation and analytic solutions are obtained in the Raman-Nath and Bragg regions of diffraction. The results for normal and Bragg incidence are obtained as special cases. Limits of the two regions are defined, thus giving a transition region in which numerical solutions can be obtained.

QC 351 A7 no. 84 / The problem: to what extent do cortical evoked potentials correlate with differences in complex visual stimuli and with differences in observer confidence that a correct judgment has been made? Electroencephalographic records obtained in response to complex visual stimuli were examined in two studies involving thousands of stimulus presentations and a variety of stimulus characteristics. Six observers made responses to subsets of the stimuli. A signal detection paradigm governed the experimental procedures and analyses. Even when stimulus differences were extremely small, it was often possible to make accurate assignments of sets of averaged EEG's to the appropriate stimulus conditions and to an observer's degree of confidence in the correctness of his judgment.

QC 351 A7 no. 13 / This paper presents a transfer theory for determination of the image space and image spectrum of a three-dimensional object. The theory assumes the existence of volumes of stationarity, called "isotomes," into which the object must be divided. Isotomicity is approximated, over sufficiently small volumes, in the diffraction- limited case. The main development assumes the object to be radiating incoherently; the results are as follows: The image (irradiance) distribution i(x,y,z) is the three-dimensional convolution of the point spread function s(a,ß,y) with the object distribution o(x',y',z'). The image spectrum I(wl,w2'w3) is defined as the three -dimensional Fourier transform of i(x,y,z). It is found that I obeys a transfer theorem, I = F.0, where F(wl,w2,w3) is the three -dimensional Fourier transform of s(ct,ß,y) and 0(wl,w2'w3) is an integral transform of o(x',y',z'). This transfer theorem establishes the value of using F as a criterion of optical design. In the Fraunhofer approximation, F may be represented as a line integral across the pupil U. This shows that F contains a simple pole at col = w2 = O. Nevertheless, all integrals involving F are convergent. The pupil representation for F also shows that F is zero outside a restricted volume E of (wl,w2,w3)- space. Because F is bandwidth- limited, F, I, s, i and T (the optical transfer function) individually obey sampling theorems. These theorems imply that if each point in image space is regarded as an independent degree of freedom, there can be no more than 1/0f4 degrees of freedom/volume in image space. For coherent object radiation, analogous theorems of convolution, transfer, and sampling can be constructed. In addition, the "amplitude" transfer function W(wl,w2,w3), defined as the Fourier transform of the point amplitude distribution u(a,ß,y), is proportional to the pupil func- tion and to S[w3 - (w2 + w2) /2k], where 6 is the Dirac delta function and k = 2r/a. This relation is used to establish sampling theorems for u and for g (the image amplitude) and to express g(x,y,z) as a double integral over U and O.

QC 351 A7 no. 18 / This paper derives a method 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 be aberrated to any extent. 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. Bandwidth -limited noise is shown to be indistinguishable from the object.

QC 351 A7 no. 68 / A tilted component telescope (TCT) is one that features no obstructions in the light path yet is appreciably simpler to build than conventional off -axis instruments. The principles of TCT design are applicable to scanning and image-stabilized optics and should allow improvements in that field. The author has collected and computer-evaluated designs representative of existing art: Schiefspiegler, Yolo, catadioptric Herschelian, and Schupmann. It is expected that these evaluations will enable optical scientists to appraise the merits of the TCT approach and will stimulate the development of second -generation designs.