Showing papers on "Point spread function published in 1985"

Three-dimensional imaging by a microscope

Abstract: In transmission microscopy, many objects are three dimensional, that is, they are thicker than the depth of focus of the imaging system. The three-dimensional (3-D) image-intensity distribution consists of a series of two-dimensional images (optical slices) with different parts of the object in focus. First, we deal with the fundamental limitations of 3-D imaging with classical optical systems. Second, a transfer theory of 3-D image formation is derived that relates the 3-D object (complex index of refraction) to the 3-D image intensity distribution in first-order Born approximation. This theory applies to weak objects that do not scatter much light. Since, in a microscope, the illumination is neither coherent nor completely incoherent, a theory for partially coherent light is needed, but in this case the object phase distribution and the absorptive parts of the object play different roles. Finally, some experimental results are presented.

Fluorescence imaging system

Abstract: A fluorescence imaging system. A light source irradiates an object to be viewed. The irradiated object is imaged through a beam-splitting system to form individual displaced images of the object. The displaced images are further filtered in a filter having a frequency passband different from the other filters. The filtered images are imaged on a detector. The same image point of each image is detected and converted into an electrical signal. A circuit means is provided to receive the signals of each common image point, and weight the same. The weighted signal represents an image point pixel. The weighted pixels are displayed as an image of the object having improved contrast.

A simplified approach for modulation transfer function determinations in computed tomography.

Abstract: In order to determine the modulation transfer functions (MTF's) for x-ray computed tomography (CT) scanners, a measurement must be performed to obtain either the point spread function (PSF) or the line spread function (LSF). Thereafter, the usual procedure is to interpolate between the measured points and to determine the Fourier transforms numerically in order to obtain the MTF. Since this must usually be done many times to evaluate various reconstruction kernels and scan modalities, the process is tedious. Fortunately, it can be greatly simplified by utilizing a mathematical function to describe the PSF or LSF. Measured data for five CT scanners indicates that the PSF can usually be described by a Gaussian function. Hence, the MTF can be written in a generalized form eliminating the necessity of performing Fourier transformations each time. The MTF is determined directly from a single performance characteristic related to the full width at half maximum. The accuracy of the approach is compared with detailed MTF calculations for five CT scanners and it is shown to agree favorably with this data.

NMR imaging in the presence of magnetic field inhomogeneities and gradient field nonlinearities

Abstract: The relative merits of the spin‐warp method and the multiple‐angle projection–reconstruction method are assessed for nuclear magnetic resonance imaging in the presence of magnetic field inhomogeneities. The results of computer modeling studies demonstrate the superiority of the spin‐warp method under these conditions. Because the form of the point spread function for the spin‐warp approach is insensitive to magnetic field irregularities, simple correction algorithms have been developed to reduce artifacts associated with bias field inhomogeneities. In addition, the formalism developed for magnetic field inhomogeneities can be easily extended to correct for gradient field nonlinearities.

Removal of image intensifier veiling glare by mathematical deconvolution techniques

Abstract: X-ray images acquired with an image intensifier detector system suffer from veiling glare, a low-frequency degradation described by a point spread function (PSF). The PSF has two experimentally determined parameters unique to a given image intensifier. This information is utilized to deconvolve the degradation from digitally acquired images. Results demonstrate a significant increase in contrast ratio of high-contrast objects after deconvolution and image restoration.

Image restoration by Wiener deconvolution in limited-view computed tomography

Abstract: In many applications of computed tomography, we cannot acquire the projection data at all angles evenly spaced over 360°. In such cases, the computed tomography images reconstructed using a limited number of projections, measured over a narrow angle range, are characterized by approximately elliptical distortion along the view angles used and poor contrast at angles not used (anisotropic resolution). This systematic geometric distortion is caused by the 2-D point spread function of the reconstruction process. In this paper, we show that such geometric distortion and other artifacts introduced in the reconstruction process can be reduced substantially by deconvolution performed via Wiener filtering using a priori knowledge derived from the given projections. The 2-D system transfer function used in the deconvolution is obtained from the reconstruction of a test image by the same reconstruction algorithm which has been used for reconstructing the unknown object.

Atmospheric transmittance from spacecraft using multiple view angle imagery

Abstract: A procedure for determining atmospheric transmittance using surface images acquired at several view angles from space is presented. Fourier filtering of the image data is used to mitigate the effects of atmospheric path radiance and instrument noise. Although it is essential for some contrast to be present in the images, knowledge of the atmospheric scattering phase function or the surface bidirectional reflectance function is not required. The method appears to provide absolute transmittances with an accuracy of better than 10 percent, which is useful in some applications.

Reflection and refraction optical system

Abstract: An off-axis imaging optical system, includes: a first imaging optical system including a reflection optical element and a refraction optical element; a second imaging optical system including a reflection optical element and a refraction optical element, wherein the second imaging optical system is disposed with respect to the first imaging optical system so as to reimage an image formed by the first imaging optical system; wherein one of the first and second imaging optical system has unit magnification or enlarging magnification imaging power, and the other imaging optical system has a reducing magnification imaging power, so as to provide a reducing magnification power as a whole.

FLIR imager with hybrid optical/electronic processor

Abstract: A FLIR imager is modified to include a hybrid optical/electronic processor for automatic local area dynamic range normalization. FLIR imagers have an objective lens for focusing IR energy emanating from a scene on a detector array. The detector array generates electrical signals representative of the scene. LEDs generate a visible picture of the thermal image of the scene for a video processor for formatting the signals for a particular type display (TV). The hybrid optical/electronic processor is inserted between the LEDs and the video processor; it includes a beamsplitter for directing the image to a pair of CCD cameras. The image at one camera is set in sharp focus to preserve all of the spatial frequency content (allpass) through the image reconstruction optics. While the image at the second camera is slightly defocused for averaging image information over small local regions the size of the defocused point spread function (spatial lowpass filtered). The electrical signal outputs of the CCD cameras are connected to a differential amplifier for subtracting the lowpass image from the allpass image to produce the highpass image and amplification to bring the resulting signal level above the noise level of the CRT display; while the local average image output of the second camera is connected to a limiter for limiting the local average variations. A summing amplifier is used to bring the signal back together for restoring the visual impression of hot and cold objects without saturation or blanking and provide enhancement of image detail.

A Simplified EM Reconstruction Algorithm for TOFPET

Abstract: A simplified expectation-maximization (EM) algorithm for image reconstruction in positron emission tomography with time-of-flight information (TOFPET) has been developed. This new method requires substantially less computation time than the estimated posterior-density weighting (EPDW) method developed by Snyder and Politte [1,2]. Mathematically, the integration of TOFPET images at different angles yields a 2-D image which is the convolution of the true image and a rotationally symmetric point spread function (PSF). The new method uses this PSF to construct the probability functions and employs the summed TOFPET image as the incomplete data set in the expectation step of the EM algorithm; thus, the time-costly angle-by-angle operations required in EPDW are reduced to a single operation, cutting the processing time by a factor of approximately l/M for an image reconstruction using M projection angles. Results from computer simulation studies suggest that this new method may offer image quality superior to that produced by other algorithms.

Image formation by a general optical system. 1: General theory

Abstract: The most general optical system is considered, in which the object and image may also be curved surfaces. For each object point a base ray is chosen, whose intersection with the image surface defines the geometrical image point. It is shown that, with suitable choices of axes for the object and image spaces, principal azimuths always exist around the object space and image space parts of the base ray. These azimuths are such that, in a paraxial type approximation, rays entering with coordinates (XS,O) and (O,yT), respectively, emerge with coordinates (X′S,O) and (O,y′T). This theorem enables the definition of canonical entrance and exit pupil variables, (xS,yT) and (x′S,y′T), and object and image space variables, (GS,HT) and (G′S,H′T), such that the geometrical image of (GS,HT) is at G′S=GS,H′T=HT, and (for an isoplanatic image point) any finite aperture ray entering with coordinates (xS,yT) emerges with x′S=xS,y′T=yT. The analysis also leads to formulas for the two principal local magnifications of the image and to the sine condition for a general optical system. Both the geometrical and diffraction theory of image formation for a general optical system are in this manner shown to reduce to exactly the same forms as for an axially symmetric system. In Part 2, the necessary computing methods are obtained for the practical application of the theory.

Maximum bounded entropy: application to tomographic reconstruction

Abstract: A new restoring algorithm, maximum bounded entropy (MBE), has been investigated. It incorporates prior knowledge of both a lower and upper bound in the unknown object. Its outputs are maximum probable estimates of the object under the following conditions: (a) the photons forming the image behave as classical particles; (b) the object is assumed to be biased toward a flat gray scene in the absence of image data; (c) the object is modeled as consisting of high-gradient foreground details riding on top of a smoothly varying background that is not to be restored but rather must be estimated in a separate step; and (d) the image noise is Poisson. The resulting MBE estimator obeys the sum of maximum entropy for the occupied photon sites in the object and maximum entropy for the unoccupied sites. The result is an estimate of the object that obeys an analytic form that functionally cannot take on values outside the known bounds. The algorithm was applied to the problem of reconstructing rod cross sections due to tomographic viewing. This problem is ideal because the object consists only of upper- and lower-bound values. We found that only four projections are needed to provide a good reconstruction and that twenty projections allow for the resolution of a single pixel wide crack in one of the rods.

Multidimensional state-space model Kalman filtering with application to image restoration

Abstract: In this paper a set of three-dimensional (3-D) state-space models based on Roesser's model is employed to restore the degraded image by Kalman filtering. The 3-D models extend the regions of the correlation of image pixels and of the point spread function (PSF) of blur to a nonsymmetric half-plane (NSHP). In addition, the correlations of both models may be inseparable in vertical and horizontal directions, so that these models are more compatible with the innate characters of image and blur processes. Furthermore, these two models (image process and PSF of blur) may be reduced to one by merging their signal flow graphs, thus lowering the order of states and simplifying the computational algorithm. A state-space model for strip filtering can then be derived from this merged 3-D model. A numerical example is presented below to illustrate this idea, and a strip filtering model with two scan lines is derived from it for the image restoration. As can be seen from the restored images resulting from the simulation experiment, this 3-D model has been very effective.

Rotation-variant optical data processing using the 2-D nonsymmetric Fourier transform.

Abstract: In this paper, we consider the properties of the nonsymmetrical Fourier transformation which is space-variant in both rectangular and polar coordinates A coherent optical processor composed of two nonsymmetrical Fourier transformers is introduced This processor allows rotation-variant linear filtering operations and matched filtering Two configurations for such a processor are proposed For certain parameters of both nonsymmetrical Fourier transformers it is possible to obtain a space-invariant processor with both lateral magnifications equal to unity However, introducing any filter operation results in a rotation-variant performance

Limited Angle Reflection Mode Computerized Tomography

Abstract: A short time after computerized tomography (CT) was introduced for X-ray diagnostic purposes, similar concepts were investigated using ultrasound. Transmission mode ultrasound CT-concepts have the disadvantage of poor resolution because the dimensions of the object details have the same order of magnitude as the acoustic wavelength. Therefore, since 1977 several groups have been working on alternative methods using reflection data for CT-reconstruction (/1/,/2/,/3/,/4/). Different concepts of data collection and of signal processing were investigated, for example single transducer scanning and linear array application, as well as image reconstruction using RF echo signals and demodulated echo signals. There are some advantages in using conventional echo techniques, and therefore all considerations in this paper are based on the imaging apparatus shown in Fig. 1a. A linear array of a conventional B-scanner receives B-scan images of a certain cross-sectional area from different angular positions. The array is rotatable and takes those images from a finite number of equidistant angles by moving around the object. All B-scan images are added with respect of their angular orientation.

Three-Dimensional Object Representation In Imaging Systems

Abstract: In image-forming optical systems the image of a three-dimensional object consists of a superposition of focused and defocused object layers. For a quantitative evaluation of the object it is necessary to decompose the superposi-tion image into different images corresponding to single object layers. For this purpose the object radiation is measured with different optical transfer func-tions of the imaging system, for example, by simply changing the focus plane. Each image contains focused and defocused parts of the object and can be described as a linear equation of the object layers, assuming linear space-invariant imaging properties. From these images the real object distribution can be calculated by the evaluation of the resulting linear system of equations in the Fourier domain. Due to noise in the detected images it is only possible to get an estimate of the true object distribution. In our case this estimate is based on an integral minimal mean square error in the reconstructed object. The algorithm is presented and demonstrated by simulation experiments and reconstructions of real human cell images in optical microscopy.

Point-spread-function computation: quasi-digital method

Abstract: Starting with an expansion of a pupil function into azimuthal Fourier harmonics, a general formula that represents the point-spread function as a weighted sum of successive Hankel transforms is derived. The corresponding transforms can rapidly be computed by using the quasi-fast Hankel transform algorithm. The method appears to be far more useful than purely digital two-dimensional fast-Fourier-transform techniques, especially for symmetrical systems.

Imaging device using a roof mirror lens array

Abstract: As a unit magnification imaging device, we investigated a roof mirror lens array. A prototype of the unit magnification imaging device was made in an integrally elongated form by plastic molding. With this prototype, a high resolution imaging device with a F/2 range was attained, which can be employed for both area scanning and line scanning.

Some practical aspects of moving object deblurring in a perspective plane.

Abstract: An image restoration problem commonly encountered in many branches of science and engineering is that of removing the effects of uniform object motion. If the direction of object motion is not orthogonal to the line of sight of the imaging device, an image with both nonuniform motion blur and perspective may be produced so that the usual assumption of a space-invariant linear operation is no longer valid. In this paper we show that a proper initial geometrical transformation reduces the problem to one which can be handled by standard methods for removing uniform motion blur. As will be demonstrated, the process is strictly valid for planar (2-D) objects only. For objects with depth, residual space-variant blurs still exist. We highlight through examples several practical aspects of digitally deblurring such objects, including proper geometrical transformations, initial blur estimation, and residual space variance.

Reply to "Comments on Geometric Deconvolution: A Meta-Algorithm for Limited View Computed Tomography"2

Abstract: We have developed two new "meta-algorithms" for computed tomography that give significantly improved images through deconvolution of the two-dimensional point spread function of standard, quasi-linear algorithms. In geometric deconvolution the projections of the point spread function provide the basis for a set of one-dimensional deconvolutions. In two-dimensional Wiener deconvolution, the two-dimensional point spread function is deconvoluted directly. The criticism that there is no data available for these deconvolutions is met here by showing that the "missing data" is partly provided by incorporation of a priori information, as is the practice in other superresolution work.

Roll Deconvolution Of Space Telescope Data: Inverse Filtering Of Two Speckle Interferograms

Abstract: Roll deconvolution is a speckle method that can improve the resolution of the 2.4m Hubble Space Telescope (HST) at short UV wavelengths. A diffraction-limited image can digitally be reconstructed from two degraded images recorded at two different roll angles of the HST. The reconstruction is performed by complex inverse filtering of the two degraded images. In digital simulations we have investigated how the signal-to-noise ratio of the reconstruction depends on photon noise (104 to 50 photon counts per pixel), on the structure and size of the point spread function (caused by aberrations) and other parameters.

Monochromatic x-ray imaging with a metal multilayer camera.

Abstract: A monochromatic x-ray camera consisting of a slit and two metal multilayers as the imaging elements has been built and tested. The principles by which we obtained the unity magnification and 100-μm resolution are presented. We also describe the setup for performing the throughput and resolution measurements using a Henke copper anode as the test source. Finally, several images obtained with this camera are presented and are compared with polychromatic images from a conventional x-ray pinhole camera.

Circular harmonic analysis of PSF's corresponding to separable polar-coordinate frequency responses with emphasis on fan filtering

Abstract: A method based on radius-dependent angular Fourier coefficients (circular harmonics) is used to analyze the point spread function (PSF) of any 2-D filter whose frequency response is separable in the polar-coordinate variables. The approach relies on the ability to express the PSF as the angular cyclic convolution of two functions related to the angular and radial characteristics of the frequency response, respectively. A general theory is developed and is subsequently specialized to the ideal, but important, case of fan filtering. In the general case, the exponential and Gaussian radial profiles are considered in detail, both because of their mathematical tractability and because of their usefulness for discussing fan filtering (uniform radial profile) through limiting arguments. Besides providing insight into the structure of the PSF for separable and fan filters, the theory leads to a synthesis procedure via FFT's, which is demonstrated in situations of interest.

Phase Retrieval: A Practical Application For The Space Telescope

Abstract: A phase retrieval algorithm, developed and implemented by Perkin-Elmer, is used as a technique for in-orbit alignment of the optical system in NASA's Space Telescope. Reasonably accurate estimates of wavefront aberrations are obtained from measurements of the system point spread function (PSF). Further, the accuracy of the estimates increases as the magnitude of the aberrations decreases. Therefore, even if initial aberrations are large and result in crude estimates, a partial correction of the system can be made. A reapplication of the algorithm to the improved PSF will yield more accurate estimates of the uncorrected aberrations. In this fashion, a few iterations of the phase retrieval algorithm will allow perfect system correction from an aberrated state. The algorithm consists of initially defining an error function. The goal is to minimize the error function with respect to a set of parameters. Using Zernike polynomials to describe the phase in the pupil, we are able to write an analytic expression representing the PSF. An error function that can be minimized with respect to the coefficients of the Zernike polynomials is then calculated.

Fourier Theory in Modern Imaging

Abstract: Fourier theory is a powerful tool to the understanding of imaging systems. But imaging has evolved in a number of ways and the theory of these imaging systems has developed in a way far removed from the classical optical system. Much of the innovation arose in radioastronomy where the millionfold longer wavelength than in optics meant new approaches were needed before resolution matched the optical case. Radiotelescopes now surpass optical telescopes in resolution by the use of new techniques. These techniques illustrate the limitations of Fourier approaches. There are strong parallels with medical imaging where image synthesis, iterative processing and other digital techniques are used in place of a direct Fourier approach to image generation.

Design And Analysis Of Spectral Slicing X-Ray Telescope Systems

Abstract: Layered Synthetic Microstructure (LSM) x-ray optics can he used to couple a conventional Wolter Type I or Wolter-Schwarzschild x-ray mirror to a high sensitivity/broad wavelength response detector. Since the LSM mirror can effectively reflect only a narrow spectral slice of the incident radiation, this new instrument is referred to as a "Spectral Slicing X-Ray Telescope." By the use of figured multilayer (LSM) optics, it is possible to alter the plate scale of the primary mirrors to allow improved spatial resolution to he realized with solid state detectors. We will present the results of theoretical design and analysis studies of several spectral slicing x-ray telescope systems utilizing LSMs figured as hyperboloids, spheres, ellipsoids, and constant optical path aspheric elements. The RMS spot size and point spread function calculations will be presented for systems in which the LSM optics are figured to yield magnifications of 2X, 6X, and 8X, and it will be demonstrated that these systems yield superior off-axis performance over the primary optic alone.

Detection of moving point objects against a moving background.

Abstract: The temporal spectral characteristics of a dim moving point object and a moving background, as observed by a sensor array, are analyzed. This type of problem occurs in remote sensing, machine vision, and many other applications. The diffraction limitation of the sensor optics ensures that the temporal spectrum of the background moving with a finite velocity has a finite maximum bandwidth, regardless of background structure. Because the outputs of the sensor array are time sampled, its spectrum is infinitely replicated over an interval of temporal frequency equal to the reciprocal of the sampling time. If this interval is at least twice as large as the maximum background temporal frequency, there is a region with no background components in the middle of each interval. However, because the point object temporal spectrum in the sampled sensor array output is continuously distributed, this region will contain part of the point object signal. Thus, a criterion for the existence of an effective background suppression filter is that the point object fundamental frequency must be greater than the maximum background temporal frequency. When this criterion is satisfied, the amount of background leakage in the filter depends on the sharpness of its passband response and its stopband characteristics. In general, higher-order filters have sharper response and hence better performance. If the criterion is not met, all types of filter lose their effectiveness since the background signal will leak through the passband of the filter. The fundamental concepts developed here were examined for some typical parameter values. It is shown that for this system the point object can be effectively discriminated. In some cases the point object and background temporal spectral responses vary significantly with spatial position within the field of view. Because the filter's center frequency must match the point object temporal fundamental frequency, it is necessary to use an adaptive filter in these situations.