Estimation of point-spread functions and modulation-transfer functions of optical devices by statistical properties of randomly distributed surfaces

The point-spread function a(PSF) and the modulation-transfer function (MTF) are important tools to characterize the information transfer through optical devices. They give useful information about the resolution. Several methods have already been achieved to calculate the PSF and the MTF from theoretical aspects of wave propagation or from experimental results. I present a novel way of estimating these two functions. It deals with statistical considerations for a randomly distributed surface involving a statistical determination of the PSF and the MTF. Indeed, in this case the theoretical shape of the autocorrelation function of such surface profiles is known. It is a decaying exponential function α[exp(-β|x|)]. Comparingthe theoretical autocorrelation-function profile with the experimental one and deconvolving in Fourier space leads to an estimation of the MTF of the imaging device. Applying the inverse Fourier transform to the MTF involves the computation of the PSF, assuming that the latter has no imaginary part and is symmetrical. The two-dimensional images are regarded as an iteration of one-dimensional ones according to the orthogonal direction. The MTF's and PSF's are therefore one-dimensional. Different results are presented. The first result proceeds from investigation with scanning near-field microscopy and illustrates the method step by step. The tunneling effect is detected assuming that the information transfer is linear. The last result concerns an optical profilometer, and the influence of the microscope objective is studied.     

Measurement of the modulation transfer function of paper

In recent years there has been a renewed interest in modeling the halftone microstructure to better control the colors produced in a halftone image. Diffusion of light within the paper has a significant effect on the halftone color; this effect is known as optical dot gain or the Yule-Neilsen effect. Because of diffusion, a photon may exit the paper from a different region of the halftone microstructure than that into which it entered the paper. To account rigorously for this effect requires knowledge of the paper's point-spread function or, equivalently, the paper's modulation transfer function (MTF). A new technique for measuring the MTF of paper-the series-expansion bar-target technique-is introduced. The method uses a bar target, but the analysis more closely resembles that of the edge-gradient technique. In the series-expansion method, bar-target image data are expanded into a Fourier series, and the paper's MTF is given by the series-expansion coefficients. It differs from the typical bar-target analysis in that the typical method plots the amplitude of the fundamental frequency component for several targets of varying frequency, whereas the series-expansion method plots the amplitude of the fundamental and its harmonics for a single target. Two possible techniques for measuring the MTF with the bar-target series-expansion method are considered. In the first, the image of the bar target is projected onto the paper, and in the second, the bar target is placed directly on the paper, in close contact.     

Calculation of the modulation transfer function for object brightness distribution function oriented along any direction in axis-symmetrical optical systems

The literature deals with the modulation transfer function (MTF) only for object brightness distribution functions (OBDFs) oriented along the meridional and sagittal directions. This paper addresses computation of the geometrical MTF for an off-axis source point when the OBDF is oriented along any arbitrarily defined direction. This study finds that the MTF is not a monotonic increasing or decreasing function when the direction of the OBDF is changing. Consequently, the extreme MTF values may occur when the OBDF is aligned at any direction between the meridional and sagittal directions. Four theorems are provided for the MTF and the phase shift variations that take place when the OBDF is translated or rotated. It is found that the MTF and the phase shift are symmetrical or antisymmetrical about certain directions. Thus, to observe all possible changes in the MTF and the phase shift, it is sufficient to rotate the OBDF through a range of just 90°. The presented method is based on a recent irradiance method for MTF computation that does not rely on counting the number of ray hits on a mesh, making the method immune to effects of grid size and thus improving traditional accuracy.     

Drift-controlled design of reinforced concrete frame structures under distant blast conditions—Part II: Implementation and evaluation

The model for the calculation of an equivalent static force (ESF) and the design procedure with ESF for single-degree-of-freedom (SDOF) systems presented in the first part of the two-part paper are extended into the design for a reinforced concrete (RC) frame structure under distant blast conditions. An empirical formula for the ESF factor involved in the ESF model is presented based on sample points obtained by comparing the nonlinear dynamic responses of frame structures under the blast loading with the corresponding nonlinear pushover analysis of the structure due to an ESF. The use of the method is demonstrated with two six-storey RC frame structures. Numerical verification of the method indicates that the maximum inter-storey drift ratios (MIDR) of the two designed frame structures in comparison to their respective targets are conservative to some extent. The reasons that may lead to the conservative designs are discussed.     

Influence of the type of illumination on the measurement of the modulation transfer function in the living human eye: A theoretical study

Abstract

Applications such as refractive surgery demand an objective appraisal of the retinal image quality. The modulation transfer function (MTF) provides that information when measured directly. Moreover, the MTF obtained using a simple and objective method such as that described in this paper allows the neural contrast sensitivity function (CSF) to be obtained from the global CSF and the MTF. When calculating the MTF it must be borne in mind whether the applicable theory is coherent or incoherent. In the literature, the developed theory presents some approximations and incongruities. Also, it is interesting to note that the method of recording the MTF (short or long time of integration, sum of images, etc.) and the source used (laser or others) can essentially change the nature of the problem. This theory is not directly applicable to any experiment. This work comprises a thorough theoretical study of the possible cases according to the experimental method and therefore as a function of the coherence degree. Two different solutions are found: the MTF calculated as the square root of the Fourier Transform (FT) of the recorded intensity for any light source or the MTF calculated as the FT square root of the image.     

Wide field-of-view imaging spectrometer using imaging fiber bundles

A field-of-view-folding approach is proposed to extend the field of view (FOV) of a dispersive imaging spectrometer after introducing several linear arrays of imaging fiber bundles to which to replace the slit. The fiber bundles can flexibly connect fore-optics with a spectrometer to yield an imaging fiber-optic spectrometer (IFOS). The technology of FOV segmenting and folding, which can decrease simultaneously the dimension and spectral distortion of the imaging spectrometer, is described in detail. Because of the sampling function of the fiber bundles, the IFOS is a double-sampling imaging system. We analyze the effect of fiber coupling on the modulation transfer function (MTF) and then develop a cascade MTF model to estimate the imaging performance of the IFOS. A spaceborne IFOS example is presented to describe how the method can be used.     

Drift-controlled design of reinforced concrete frame structures under distant blast conditions—Part I: Theoretical basis

Proper control levels of lateral drifts anticipated for reinforced concrete (RC) frame structures within the predefined performance level become crucial when the frame structure is subjected to distant intense surface explosions. For this purpose, a new design method is presented in a two-part paper based on the transformation of a blast loading into an equivalent static force (ESF). The ESF is calculated in such a manner that the same maximum inter-storey drift ratio (MIDR) under the blast loading will be reproduced. The first part of the two-part paper focuses on the computational model of ESF for a single-degree-of-freedom (SDOF) system and the design method based on ESF with the requirement for controlling its maximum displacement response to achieve the specified target displacement. Numerical examples have been included to illustrate the method while the verifications of the dynamic responses of the designed SDOF system are performed with nonlinear dynamic analyzes. The numerical results indicate that the target displacement is well met for the designed SDOF system in resisting a given blast loading. Extension of the computational model of ESF and the corresponding design method with ESF for a SDOF system into a RC frame structure will be further discussed in the companion paper.     

Modulation Transfer Function Measurement Using Three- and Four-bar Targets

Modulation-transfer-function (MTF) measurement often involves the use of three- and four-bar resolution targets. In the conversion of three- and four-bar image data to MTF, biased results can occur when we use series-expansion techniques appropriate for square-wave targets of infinite extent. For systems where the image data are digitally recorded, a convenient and accurate conversion of bar-target data to MTF can be performed using a Fourier-domain method.     

Assessment of radiographic screen-film systems: a comparison between the use of a microdensitometer and a drum film digitiser

A high-end drum film digitiser (Tango, Germany) and a calibrated linear microdensitometer developed by PTB were used to assess the modulation transfer function (MTF) and the noise power spectra (NPS) of 3 mammographic screen film systems at optical density levels of 0.8, 1.5 and 2.5. The use of a drum scanner to assess MTF and NPS data appears to be adequate but requires an appropriate characterisation of the scanner to verify its internal noise level and its MTF. It is further necessary to calibrate the scanner output in terms of visual diffuse optical densities. Processing of two-dimensional digital data of grating images need to be more strictly defined for accurate MTF measurements of screen-film systems. Nevertheless, even now it seems to be feasible to use commercially available high-end and well calibrated scanners to assess screen film systems. This is especially important for quality assurance purposes because important parameters of screen film systems such like MTF and NPS can now be determined without using sophisticated microdensitometers which are not commercially available.     

用CCD测量光学系统OTF的研究

线性光学系统的光学传递函数可由线扩散函数的付立叶变换求出。本文研究了用电荷耦合器件测量光学系统OTF的系统,用CCD作为光电转换器件,对CCD接收到的LSF的光强信号进行视频放大,采样一保持、A/D转换接口到计算机,可方便而精确地测得光学系统的OTF。为了避免采用高速A/D和DMA方式,对CCD的信号处理采用分时处理。用CCD测量MTF的误差为3%,不重复性为1%,能在2分钟之内测试出物镜的10种空间频率的MTF值。     

Modulation transfer function measurement method for electrically addressed spatial light modulators

The modulation transfer function (MTF), when used with amplitude modulation (m(A)) data, is a vital coherent optical performance measure for a spatial light modulator (SLM). A new image plane amplitude MTF (MTF(A)) measurement method is presented for electrically addressed SLMs. It involves digital analysis of the output image of a square-wave pattern written onto the SLM. Modulation-level effects are also addressed. Optical laboratory results are presented for two liquid-crystal SLMs. The need to consider amplitude rather than intensity modulation (when coherent optical processing applications are considered) is noted in terms of SLM biasing.     

Method for the measurement of the modulation transfer function of sampled imaging systems from bar-target patterns

A rigorous and simple method for the determination of the modulation transfer function (MTF) of a sampled imaging system is presented. One calculates the MTF by imaging bar patterns and calculating the reduction in amplitude of the fundamental frequency components. The optimal set of bar-pattern frequencies that reduce errors from aliased frequency components is derived. Theoretical and experimental data are presented.     

Comparative analysis of techniques for measuring the modulation transfer functions of charge-coupled devices based on the generation of laser speckle

Two methods for measuring the modulation transfer function (MTF) of a charge-coupled device (CCD) that are based on the generation of laser speckle are analyzed and compared. The method based on a single-slit aperture is a quick method, although the measurements are limited to values of less than the Nyquist frequency of the device. The double-slit method permits the measurement of values of as much as some 1.8 times the Nyquist frequency, although it is a slower method because of the necessity to move the CCD. The difference between the MTF values obtained with the two methods is less than 0.1 in magnitude; the root-mean-square error between the two curves is 0.046 (4.6%).     

Modulation transfer function assessment for sampled imaging systems: effect of intensity variations in periodic thin-line targets

Different techniques have been proposed to assess the modulation transfer function (MTF) of sampled imaging systems. Some of these are based on the use of periodic targets made up of thin lines or points. The main potential problem of implementation concerning these methods is the fact that, in specific experiments, it may be difficult to ensure a good balance in intensity between the individual lines or points. An analytical model permitting a first estimation of the actual importance of this problem is presented. The error in the MTF assessment for two generic cameras is then estimated, taking into account the experimental process.     

Modal analysis of transport processes in SPRITE detectors

Carrier transport in signal-processing-in-the-element (SPRITE) detectors is an important phenomenon because it determines properties such as the responsivity and the modulation transfer function (MTF). The previous literature has presented approximate solutions to the transport problem that neglect boundary effects, which have long been thought to play a major role in SPRITE behavior. We present a new solution to the problem through the use of modal analysis. This method intrinsically includes boundary conditions and thus is more complete than the previous analysis. Furthermore we use this solution to derive expressions for the MTF. The effects of the boundary conditions on the MTF are studied to determine their optimum values.     

Analytical optimal solution of the extension of the depth of field using cubic-phase wavefront coding. Part I. Reduced-complexity approximate representation of the modulation transfer function

In this paper we derive an approximate analytical representation for the modulation transfer function (MTF) of an imaging system possessing a defocused cubic-phase pupil function. This expression is based on an approximation using the Arctan function and significantly reduces the computational time required to calculate the resulting MTF. We derive rigorous bounds on the minimum and average accuracy of our approximation. Using this approximate representation of the MTF, the analytical solution of the problem of calculating the extension of the depth of field for a circular aperture with a cubic phase mask becomes possible. We also comment on how one can modify our method to construct a lower-bound or an upper-bound approximate analytical expression for the MTF.     

Measurement of the modulation transfer function of digital X-ray detectors with an opaque edge-test device

A variant of the edge method for the determination of the pre-sampled modulation transfer function (MTF) of digital X-ray imaging devices has been developed and accepted as the standard method in the novel DQE measurement standard IEC 62220-1. An opaque tungsten edge-test device accomplishes the ideal step-like profile of the incident X rays. The edge spread function is measured over a large region across the edge transition that enables an accurate MTF measurement including the 'low-frequency drop'. The method has been applied to different state-of-the-art X-ray imaging detectors, a computed radiography, a CsI-based indirect and an Se-based direct flat-panel detector. The MTF measurement results will be presented. In contrast to the opaque edge device, the commonly used semi-transparent edge-test devices produce scatter radiation that deteriorates the incident X-ray profile, which leads to a systematic overestimation of the MTF.     

Modulation transfer function measurement of sparse-array sensors using a self-calibrating fringe pattern

A simple method for the measurement of the pixel modulation transfer function (MTF) of sparse-array (extended MTF) sensors has been developed. We use a phase-shifting Twyman-Green interferometer to generate a series of single spatial-frequency fringe patterns incident on the sensor The resulting signal modulation is measured. We achieve self-calibration by restricting the measured spatial frequencies to multiples of the Nyquist frequency. The aliased patterns at these frequencies are unique and easily identifiable. Spatial frequencies of 480 cycles/mm are generated and measured. This frequency value is more than ten times that of the sensor sampling frequency. The expected MTF shape is obtained at multiples of the sampling frequency. At odd multiples of the Nyquist frequency, the MTF's are affected by the electronic bandwidth and cross talk in the charge-injection device sensor.     

Modulation transfer function of a lens measured with a random target method

We measured the modulation transfer function (MTF) of a lens in the visible region using a random test target generated on a computer screen. This is a simple method to determine the entire MTF curve in one measurement. The lens was obscured by several masks so that the measurements could be compared with the theoretically calculated MTF. Excellent agreement was obtained. Measurement noise was reduced by use of a large number of targets generated on the screen.     

基于插值数浮动的MTF精确测量方法

采用刀口法测量成像系统调制传递函数(MTF)时,刀口装置与探测器矩阵夹角的准确测量是难点和关键点之一.针对这一问题,首先模拟刀口图像倾角对应的理论基准插值,研究了加载噪声情况下插值数浮动与线扩散函数(LSF)波动性变化以及MTF曲线衰减特性的定量关系;接着对实际X射线数字化成像系统获得的特定倾角下的刀口边缘图像进行了基于插值浮动的MTF测量,并将由其获得的MTF值与线对卡结果进行比较,证明了该方法用于成像系统MTF测量的有效性.结果表明:角度测量误差造成实际插值数偏离理想基准插值,导致LSF曲线光滑性变差,MTF衰减程度高于实际衰减.通过插值浮动方法,能够有效地避免角度测量误差引起的MTF测量的误差,降低角度测量的高精密性要求.