Showing papers on "Point spread function published in 1969"

Properties of a Defocused Optical System

Abstract: The optical properties of an aberration-free defocused optical system used to image incoherently illuminated objects are analyzed. The exact diffraction optical transfer function (OTF) and point spread function (PSF) are compared to the geometrical OTF and PSF, respectively, for both small and large amounts of defocusing. It follows that geometrical optics does not describe diffraction optics very well for any reasonable amount of defocusing. Calculation of the diffraction OTF is complicated and time consuming, even with a large computer. An empirically derived analytic approximation to the diffraction OTF which is much easier to calculate, is formulated. It is also shown, and examples are presented to demonstrate, that the exact OTF and PSF are different in planes at equal distances on the two sides of the in-focus plane.

Detection and Resolution of Incoherent Objects by a Background-Limited Optical System*

Abstract: The detection of incoherent objects, light from which is received in the presence of a spatially and temporally white background, is treated by signal detection theory. For a given total energy received from the object, the detectability decreases with a decrease of the degree of coherence, at the aperture of the optical system, of the light from the object. Resolution of two identical objects depends on deciding whether two are present in the field of view or only one, and their resolvability is measured by the probability of making this decision correctly. Its dependence on the area of the aperture and the size and separation of the objects is worked out. A conceptual scheme for realizing optimum detection is presented. Cramer–Rao bounds are given for the variances of unbiased estimates of object parameters such as radiance and position.

Bandlimited Image Restoration by Linear Mean-Square Estimation

Abstract: A linear mean-square estimator, optimum for image data available only on a finite interval, is derived for the restoration of images degraded by a system with a bandlimited spread function. The analysis is carried out in one dimension using a prolate-spheroidal-wavefunction expansion of the image data. When the noise is bandlimited to the same bandwidth as the spread function, the expansion represents the image data with zero mean-square error on the entire interval, and the mean-square reconstruction error is equal to that of the optimum linear estimate for image data on the infinite interval. The rate at which the series representation of the estimate converges is discussed and an example presented.

Atmospheric Degradation of Holographic Images

Abstract: The resolution of images formed from turbulence-degraded lensless Fourier-transform holograms is analyzed for gaussian and exponential refractive-index structure function models, first by a geometrical-optics method and then by Schmeltzer’s series-expansion method, under the assumption that the random log-amplitude and phase perturbations across the entrance pupil of the recording apparatus are locally stationary processes with gaussian statistics. Both long and short exposures are treated.The resolution of turbulence-degraded holographic images is governed primarily by a function m2(r1), for which analytical expressions are derived and results of experimental measurements are given. The analytical results obtained for m2(r1) by the geometrical-optics method are identical with those obtained by Schmeltzer’s series-expansion method. The measurements were made by an interferometric technique for horizontal atmospheric propagation paths of 86 and 542 m. In addition, holographic images of an extended object were obtained for the 86-m path.

Detection and Resolution of Incoherent Objects Seen through a Turbulent Medium

Abstract: A background-limited optical system is to view an object through a turbulent medium. The threshold systems for detecting and resolving specific details of the object are derived and their probabilities of correct detection and resolution evaluated. The variance of a maximum-likelihood estimate of the position of the object is calculated for large signal-to-noise ratio. Specific numerical results are given for turbulence that produces a gaussian mutual-coherence function in light from a point object. A model of such turbulence, depicting it as concentrated in one or more thin, homogeneous phase screens, is proposed.

On the Resolution of Imaging Systems

Abstract: The angular resolution of imaging systems is defined as the capability of the system to estimate the position of a point source object which is observed in a background of gaussian noise. The application of statistical estimation theory gives the resolution in terms of the noise characteristics and the modulation transfer function of the system. The resolution is also considered as the ability to estimate the separation of two point sources. Examples of estimation of both position and source intensity are given for a gaussian-shaped spread function.

Optical Imaging with Partially Coherent Non-thermal Light

Abstract: The reconstruction of an object from its image is studied from the point of view of the coherence theory including detection in the process of imaging. Finite-size objects permit the use of the theory of entire functions to prove that if the transformation from object to image is linear (detection is not included in imaging) the object can be reconstructed uniquely. Non-linear transformations from object to image (detection is included in imaging) for weak visibility objects and for a particular choice of the diffraction function are also studied. In the first case a finite-size object is determined uniquely, in the second case the reconstructed object is determined apart from the phase, which can sometimes be determined from the dispersion relations. Further, the similarity between an object and its image is investigated. It is shown that partial coherence creates typical non-linearity in imaging which results in an interval of eigenvalues (similarity coefficients) and that there may arise the branching ...

Detection of Incoherent Objects by a Quantum-Limited Optical System*

Abstract: The detection of an incoherent object, light from which is received in the presence of thermal radiant energy, is treated by quantum detection theory under the assumption that the product of the observation time and the bandwidth of the object light is large. The threshold detector measures a quantum-mechanical operator that is a quadratic functional of the field at the aperture of the receiver. The statistics of the outcome of this measurement are approximately Poisson, and the detectability of the object depends—as under a background limitation—on the number of effectively independent spatio-temporal modes of the field. The detectabilities of objects with a rectangular and a Lorentz spectrum are compared under the assumption that the light from each possesses first-order coherence at the aperture.

Restoration of Linearly Smeared Transparencies

Abstract: Photographic distortion caused by one-dimensional linear motion of the film during exposure is treated in this paper The distortion (point spread)function is obtained by making a photographic record of the pulse shape This data is in turn used by a digital computer to produce the Fourier transform of the point spread function Optical spatial filtering of the distorted image with an inverse filter using a coherent, monochromatic optical processor has been used Images which have been smeared by up to three times the minimum resolution length have been restored The filtering technique consists of manipulating both the phase and amplitude of the distorted scene In general, the theory of optimal filtering has considered additive noise in the form of signal-to-noise ratios The signal-to-noise ratio has been treated by considering it as a constant or some function of the spatial frequency In the latter case, improvement in both restoration and cosmetics was obtained Film linearity of both the input and output imagery is controlled by processing the film over the linear portion of the Ta vs E curve MTF curves of the frequency response before and after filtering are presented Theoretical error analysis was performed on the restored imagery and good agreement between theory and experiment was obtained© (1969) COPYRIGHT SPIE--The International Society for Optical Engineering Downloading of the abstract is permitted for personal use only

MTF of a Gap

Abstract: Subject to reasonable assumptions of geometrical optics, the point spread function is calculated for a point source in a plane radiating to a nearby parallel plane. The calculation includes the case in which the gap between the planes attenuates the signal. The modulation-transfer function (MTF) is calculated from the point-spread function. If the source is lambertian, and the receiving plane (e.g., photographic emulsion) absorbs as a more or less typical diffuse absorber, and there is a response which can be expressed as a linear function of the absorbed input, the MTF can be expressed as an integral that can be evaluated numerically. The computed curves are normalized in terms of the gap width, which is taken to be unity. The method may be applicable to a fairly wide class of image-evaluation problems.

An Intensity MTF Theory for Coherent Imaging

Abstract: The effect that any parameter of a holographic or coherent imaging system would have on the resultant image is normally described by an amplitude modulation transfer function. We show here that an intensity modulation transfer function (MTF) can be used to analyze the coherent systems if diffuse targets are employed and if the usable area of the detector is much larger than the least resolvable spot in the image.

Geometric optical image aberrations of single and double mirrors.

Abstract: Single and double mirror systems geometrical optical image aberrations noting focal number, angle of field, secondary magnification and mutual position tolerances