Showing papers on "Point spread function published in 1974"

Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses

Abstract: A self-scanned 1024 element photodiode array and minicomputer are used to measure the phase (wavefront) in the interference pattern of an interferometer to lambda/100. The photodiode array samples intensities over a 32 x 32 matrix in the interference pattern as the length of the reference arm is varied piezoelectrically. Using these data the minicomputer synchronously detects the phase at each of the 1024 points by a Fourier series method and displays the wavefront in contour and perspective plot on a storage oscilloscope in less than 1 min (Bruning et al. Paper WE16, OSA Annual Meeting, Oct. 1972). The array of intensities is sampled and averaged many times in a random fashion so that the effects of air turbulence, vibrations, and thermal drifts are minimized. Very significant is the fact that wavefront errors in the interferometer are easily determined and may be automatically subtracted from current or subsequent wavefrots. Various programs supporting the measurement system include software for determining the aperture boundary, sum and difference of wavefronts, removal or insertion of tilt and focus errors, and routines for spatial manipulation of wavefronts. FFT programs transform wavefront data into point spread function and modulus and phase of the optical transfer function of lenses. Display programs plot these functions in contour and perspective. The system has been designed to optimize the collection of data to give higher than usual accuracy in measuring the individual elements and final performance of assembled diffraction limited optical systems, and furthermore, the short loop time of a few minutes makes the system an attractive alternative to constraints imposed by test glasses in the optical shop.

Quantum noise in Fresnel zone plate imaging

Abstract: A Fresnel zone plate, used as a coded aperture, offers a great advantage in geometric collection efficiency over a conventional pinhole or collimator. We present a detailed analysis of the signal-to-noise ratio (SNR) of a quantum-limited zone plate camera. The magnitude and spatial distribution of the noise field and its dependence on the source distribution and the characteristics of the optical processing system are derived. It is shown that the largest SNR advantage occurs for a point source, while for very large, uniform sources there may be a slight net disadvantage to using a zone plate. It is also shown that optical processing does not give the highest possible SNR.

Image restoration by discrete convolution of minimal length

Abstract: A conventional method of restoring blur-degraded image data is inverse filtering. This technique is usually implemented with 32 or 64 image data points contributing, via two Fourier transforms or one convolution, to each restored output point. We show here that actually about seven data points (in a seven-term convolution) suffice for producing about the same resolution and side-lobe characteristics as for inverse filtering. We calculate, by use of a computer search routine, the necessary weights for use in a convolution-type restoring formula of 5, 7, 11, or 15 terms. The criterion for weight selection is a required first-zero position in the output point spread function and, simultaneously, a minimum value of the largest side lobe in the point spread function. The 15-term case is further constrained to have weights selected from the values −1, 0, and +1, only. This defines a method of restoring by addition and subtraction of image values. Experimental results using diffraction-blurred edge data are used to test the methods.

Measurements Of Optical And Noise Properties Of Screen-Film Systems In Radiography

Abstract: The optical and noise properties of radiographic screen-film systems have often been evaluated by concepts and techniques related to Fourier analysis. Figure 1 illustrates the physical characteristics of these properties. The point spread function (PSF), which is the two dimensional representation of the optical property of the screen-film system, has not been applied experimentally for evaluation. The screen-film system is generally considered to be isoplanatic and isotropic; therefore, the line spread function (LSF) in an arbitrary direction can be used. This function is much simpler to measure than the PSF. In the spatial frequency domain, the optical property is evaluated by the modulation transfer function (MTF), which is the absolute value of the Fourier transform of the LSF. Other optical characteristics such as the edge response are usually related to the LSF or the MTF.© (1974) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Trade Off Between Resolution and Noise in Restoration by Superposition of Images

Abstract: The inversion technique of Backus and Gilbert is applied to the problem of restoration of optical objects by superposition of images. The problem is formulated in the general case of space-variant systems. Curves for the trade off between resolution and noise in restoring objects degraded by simple diffraction and susceptible to uncorrelated noise are obtained. The technique is used to restore a test object whose image is sampled at the Nyquist rate and to which noise (simulated on a computer) is added. Restorations corresponding to several points on the trade-off curve are obtained.

On the electron optics of magnetically focused image tubes: I. Chromatic aberration in uniform fields

Abstract: Methods are devised for obtaining the point spread function, line spread function and modulation transfer function for magnetically focused image tubes. These are applied to a variety of synthetic electron distribution functions which imitate the behaviour of real photocathodes. These distributions lead to analytical solutions for the intensity in the gaussian image plane, but for other planes numerical methods must be adopted. The maximum possible resolution and depth of focus can then be derived. Other limitations on the resolution are discussed.

Imaging of Extended Objects Through a Turbulent Atmosphere

Abstract: The extended Huygens-Fresnel principle is used to derive an explicit expression for the image plane illuminance distribution of an extended object with an arbitrary luminance emittance distribution. The combined simultaneous effects of attenuation, background luminance, and atmospheric turbulence are given. A quantitative comparison of these effects is made, and their contribution to the over-all loss in resolution is given. In particular, we derive a quantitative expression for the point spread function of the combined atmospheric-optical system. Explicit expressions are derived for both the atmospheric-optical system modulation transfer function (MTF) and image plane modulation. Numerical results for the image plane modulation are presented for imaging both up and down along an atmospheric path under various viewing conditions.

Digital fourier transforms as means for scanner evaluation.

Abstract: This brief paper is presented as an illustration of the possible use of the Fourier transform as an aid in evaluating the performance of digital image scanners. A discussion is presented on the interpretation of Fourier energy along the horizontal and vertical frequency axes, and it is suggested that such perfectly aligned energy is most often man-induced (purposefully or unknowingly). Windowing, scanner jitter, and aperture effects are described; specific examples are presented illustrating all these points for a laser scanner system on some scanned imagery. In addition to the main theme of scanner evaluation, a peripheral example of the use of a priori knowledge for image enhancement is presented. A nonlinear space domain transformation (amplifier) is described, based on a priori knowledge, as a tool for noise reduction in the above-mentioned imagery examples.

Rejection of Out-of-Field Optical Sources

Abstract: Requirements sometimes exist for the measurement and/or detection of weak optical sources in the vicinity of bright sources. Capability of observing weak optical sources is dependent on the system sensitivity which is limited by the detector D*, processing electronics noise, mechanically coupled scan or jitter noise, thermal effects on the detector sensitivity, optical emissions of sensor housing and imaging optics, and the point spread function. The point spread function is mainly governed by the optical element size, shape, and aberrations. When a strong source is in the proximity of the weak signal, the wings or roll offs of the point spread function play an important role in the system detection capability. Two factors dominate the roll offs, namely diffraction and scattering of the optical train. This paper discussed the point spread function requirements or conversely, the out-of-field rejection requirements needed to suppress the unwanted sources, small or large in angular extent, to a level tolerable to the system. Stray light sources are discussed and suppression designs considered.

Maximum-likelihood restoration of nonstationary imagery

Abstract: A statistical method of extrapolating the bandwidth of nonstationary imagery is outlined. The technique combines the information that is passed by a linear optical systems, an estimate of the modulus of Fourier components in the remainder of the object spectrum, and a priori knowledge about the object to estimate the most-likely values of spectral components beyond the image bandwidth. It can be made to tolerate noise within the image bandwidth. The method is applied to images produced by objects that are finite in extent and by objects that are modulated by periodic gratings.

Angular spectrum and coherent imaging

Abstract: The predicted sharp cutoff of the transfer function of a coherent imaging system is not seen experimentally. The experiment suggests that this is due to the neglect of the diffraction, caused by the finite lens, of the individual plane-wave components of the angular spectrum of the object wave. It is shown that this omission is equivalent to dropping a quadratic phase term in the description of the image obtained with the Fresnel–Kirchhoff diffraction formula.

Optical Interpretation of the Merit Function in Grey's Lens Design Program.

Abstract: A ray and wavefront analysis is the basis of an interpretation of the aberration balancing merit function in Grey's lens design program. From each of several field angles, this program traces a coarse grid of rays through an axially symmetric optical system. A polynomial in a region around each ray describes the wave error. This polynomial includes terms (through second order) in aperture, field angle, and wavelength variables. The coefficients of the terms in the polynomial are calculated from real and differential ray trace data. The merit function is the variance of the wave error, obtained by squaring and integrating the polynomials over appropriate ranges of the variables. Distortion is removed from the wave error polynomials and is included in the merit function in a unique manner.

Rapid Evaluation Of Extended-Field Photographic Quality Of Lenses

Abstract: Monitoring of high-volume lens manufacture for Polaroid Land photog-raphy requires an acceptance criterion which is relevant to picture quality and suitable for automatic testing. In a hardware test based upon the Modu-lation Transfer Function, the problem is to choose what not to measure, while obtaining enough information about a lens to assign it a useful figure of merit. The predominant defects of manufacture are field tilt, resulting from tilting or decentering of the components, and field curvature, from small errors in spacing, thickness, radius, or index. Field tilt and curvature are not "local" defects of the image surface, insofar as they may be removed by a local refocus. We therefore need a test procedure which can (perhaps simultaneously) examine image quality over an extended field, but which is principally called upon to evaluate focus differences. At multiple field locations a single-frequency modulation measurement adequately defines the MTF for the low spatial frequencies of interest, and can be interpreted easily in terms of the root-mean-square blur of the Point Spread Function. We will discuss the logic of 100% lens acceptance testing and two types of instrumentation we have used.

Application of a sunburst aperture to diffraction suppression

Abstract: A closed-form solution to a two-dimensional Fourier transform of a toothed wheel or sunburst is presented. The solution was obtained by decomposing the sunburst aperture into triangles and applying the superposition principle to sum the Fourier transforms of the subapertures. This technique is applicable to analyzing and designing other complicated apertures. Computer plots are presented that show the spatial transforms along the radial and angular directions. For small radial distances, the transform resembles that of a circle, whereas for large radial distances, it has preferred angular directions that are perpendicular to every edge of the aperture. The power spectrum or the magnitude squared of the transform is a close approximation of the Fraunhofer diffraction pattern of the aperture. Contour shaping of this type is an effective means of apodizing diffraction of unwanted sources of small angular subtense but becomes ineffective when the angular subtense of the unwanted sources are large.