Analytic MTF Expression for a Turbid Medium

Several methods are available to measuring the presampled modulation transfer function (MTF) of a digital radiography system by using a slit camera, a slanted edge test device or a bar phantom. Many studies demonstrated the advantages of the edge method compared to the slit method. However, the method of processing to estimate the MTF is slightly different for each publication. The purpose of this study is to compare the effects of various processing steps with an edge image on estimating the presampled MTF. A digital image of a slanted edge test device was acquired and then examined using different image processing such as windowing the edge spread function (ESF), determining a region of interest (ROI) and locally smoothing the ESF data. The results of this study indicated that the impact of determining a ROI on the MTF is relatively modest and that the truncation of the ESF due to a finite window width causes an incorrect inflation error in the MTF estimate and loss of detail at high frequency. Moreover, the influence of ESF smoothing on the MTF is demonstrated, and more rigorous smoothing of the ESF produces a less noisy MTF and, consequently, a high data loss at high-frequency. In conclusion, an ESF processing technique can moderately impact the MTF estimates, and an adequate ROI and window width without a smoothing process is suggested.

Traditional imaging methods often separate the two technical procedures of optical system design and imaging processing, where the lens subsystem is firstly optimized using an optical measure of performance and the image processing subsystem is optimized subsequently, resulting in the failures in efficient cooperation between the optical and digital elements and missing of the optimal solution. Therefore, an innovative approach is presented to combine optical system design and image processing together to obtain high resolution images from the final system. In this paper, the theoretical foundations for this computing optical design method is analyzed by considering the modulation transfer functions (MTF) and aberrations. According to the wave aberration theory and Zernicke formula, the relationship between the point spread function (PSF) and a certain monochromatic aberration can be obtained. Fourier transformation is taken on these PSFs and then modulus value calculation is adopted to get the MTFs. By analyzing the MTFs of idealized optical systems, the influence of each monochromatic aberration on the image can be concluded. Note that some of the MTFs values have zeros and others do not. Considering that imaging processing algorithm cannot recover the information lost at such zeros, thus some aberrations can be easily corrected through imaging processing algorithm, while others not. Aimed at a single monochromatic aberrations respectively, the optical system with single lens is designed in ZEMAX software and the relevant imaging processing algorithm is adopted. From the comparison among the final images, the theoretical foundations for the computing optical design method considering the MTF and aberrations can be verified. The compensation for aberration of imaging processing can be utilized to reduce the pressure of optical system design. The computing optical design method provides a new approach to simplify structure and reduce cost, as well as to gain high resolution images simultaneously, especially for designing certain complex optical system, which shows promising perspective of industrial application.

Purpose: To demonstrate a method of measuring the modulation transfer function (MTF) of digital imaging systems using the common star pattern test tool. The goal of this research is to develop a technique that brings precise and objective evaluation of spatial resolution to routine quality control in radiography and fluoroscopy. Method and Materials: Images of a 2‐degree star pattern were obtained on a variety of digital radiographic imaging systems, spanning a range of spatial resolution performance levels. One‐dimensional profiles from concentric circles around the center of the star were obtained from each star image. The circular profiles were unwrapped to produce a series of square waves, which were analyzed to produce the square wave response function (SWRF). Droege's method was used to compute the MTF from the SWRF. The MTF determined using the star pattern method was compared to published and quality control measurements of the MTF for several digital mammography and digital radiography systems. In implementing this method, important parameters that must be considered include the radial and angular sampling rates, the method for determining the zero‐frequency normalization and correction for beam nonuniformity. Results: The MTF computed from star pattern profile measurements closely matched the MTF obtained using other methods over the range of spatial frequencies between the Nyquist frequency and 1/3 of the Nyquist frequency. The star pattern MTF, which is a system measurement and includes the effects of detector and geometric unsharpness, was within 6–8% of reference values obtained from the scientific literature for two models digital mammography equipment. Conclusion: Images of a 2‐degree star pattern can produce estimates of the two‐dimensional system MTF. This method may offer a convenient means of assessing the spatial resolution of digital imaging systems for acceptance testing and quality control.

The authors examine methods of varying the standard modulation transfer function analysis approach to accommodate sensor sampling. Sampled imaging arrays such as charge-coupled devices do not satisfy the spatial shift-invariance criteria,which is required to apply traditional linear systems analysis techniques. The nonisoplanatic condition in imaging arrays is due to the phase dependency of the image position relative to the fixed pixel locations. A procedure to effectively eliminate the spatial phase dependency by using linear systems procedures to model image sampling and then averaging the image scene over the pixel sampling centers is described. A modulation transfer function expression is formulated which yields a statistical estimate of the spatial frequency response of any type of linear or area imaging array. >

To derive a theoretical model for the modulation transfer function (MTF) of the complete imaging chain of a cardiovascular x-ray system. A sufficiently accurate MTF model may be used in quantitative coronary arteriographic applications to decrease the systematic overestimation of small coronary vessels. The model could be used to restore blurred images in image restoration applications and also may be used for image quality assurance measurements. Digital images of a step wedge phantom were acquired at different modes (5-, 7-, and 9-inch) of the image intensifier and at different kilovolt levels of the x-ray generator. From these images the MTFs that were used to estimate the parameters of the model were assessed. The total MTF of the cardiovascular x-ray imaging chain consisting of image intensifier, video camera, optical elements, and analog/digital converter can be modeled by the simple formula MTF = e-(w/f)R, where w is the frequency in linepairs per millimeter. The constants frequency f and the device index n are parameters that can be derived for an individual x-ray system using the phantom. In this formula, the contribution from the focus of the x-ray source has been excluded. The mean square error (MSE) between the measured values and those assessed from the theoretical model of the MTF for the horizontal direction was found to be equal to 0.00003 for the 5-inch mode of image intensifier, and for the vertical resolution at the 9-inch mode this MSE was equal to 0.00038. In addition, we have demonstrated that the contributions of the video camera and the image intensifier to the total MTF can be replaced by an imaginary electron-optical device. An accurate representation for the MTF of a cardiovascular x-ray system has been derived. This MTF model can be used in x-ray image quality assurance measurements and in quantitative image processing applications.

In this work, the effect of noise on the precision in the modulation transfer function (MTF) calculation using different phantoms is investigated. Three different techniques are studied, based on three different phantoms: edge, bar, and star bar. For each technique, theoretical expressions for the standard deviation and the signal to noise ratio (SNR) of the MTF are derived. In addition, Monte Carlo simulations are carried out to study the accuracy and precision of all the three techniques. Using the analytic expressions for the MTF calculation, statistical fluctuations of noise in the phantom images are propagated to MTF values. Then, for each frequency the standard deviation and the SNR of the MTF are calculated. Monte Carlo simulations are carried out for each phantom image and for different noise level. Images are artificially generated following a procedure that simulates sampling, blurring, and noise present in a real phantom image and takes into account the attenuation of each phantom. Then, oversampling procedures are applied to these images to obtain presampled MTFs. In all three techniques the standard deviation of the MTF is proportional to that of the noise in the image. For the edge image, the standard deviation is proportional to the frequency f, whereas for the bar and star bar techniques it is proportional to the square root of f. Regarding the precision, theoretical expressions and simulations results agree that the bar technique shows the best precision for all the frequencies and all the noise levels analyzed in this study. In addition, simulation results show how, for large levels of noise, the edge technique gives rise to large bias errors at high and medium frequencies. Finally, the precision of the star bar technique is similar to that of the bar technique for isotropic systems, but worse for nonisotropic ones. The uncertainties in the MTF calculation with the edge technique are larger than with the bar technique, and the differences increase with the frequency. On the one hand, the stronger dependency on the frequency shown by the standard deviation of the MTF makes the precision gets worse faster in the edge technique. On the other hand, in the case of the edge and for high levels of noise, differentiation produces line spread functions (LSF) with a poor SNR, increasing erroneously the MTF and worsening accuracy.

Modulation transfer function (MTF) is an important parameter for one image system. In order to obtain the more imaging quality performance of the modified super-wide-angle Sagnac imaging interferometer (MSASII) put forward by our group, and also have a qualitative analysis of its advantages and disadvantages in the process of application. Its dynamic MTF is focused on in this paper. The dynamic MTF expression of the MSASII is calculated. According to earlier research parameters of the MSASII, the dynamic MTF curve is described and corresponding corrective measures in theoretical is given. The result is demonstrated that the imaging quality of MSASII could satisfy the need of detecting upper atmospheric wind field.

CMOS imagers based on Active Pixel Sensors (APS) are very important among others because of their possible technical innovations leading to ultra-low power image acquisition or efficient on-chip image preprocessing. Implementation of the image processing tasks (focal plane preprocessing and subsequent image processing) can be done effectively only with the consideration of known transfer characteristics of the imager itself. Geometrical Point Spread Function (PSF) depends on the certain geometric shape of active area in the particular design of CMOS APS. In this paper, the concept of Modulation Transfer Function (MTF) analysis is generalized to be applicable to the sampled structures of CMOS APS. Recalling theoretical results, we have analytically derived the detector MTF in the closed form for some special active area shapes. The paper also deals with the method based on pseudorandom image pattern with uniform power spectral density (PSD). This method allows to evaluate (in contrast to other methods) spatially invariant MTF including sampling MTF. It is generally known that a signal acquired by image sensor contains different types of noises. The superposition of these noises produces noise with a Gaussian distribution. The denoising method based on Bayesian estimator for implementation into the smart imager is presented.

Binary pseudo-random gratings and arrays for calibration of modulation transfer functions of surface profilometers

The presampling modulation transfer function (MTF) of a digital imaging system is commonly determined by measuring the system’s line spread function (LSF) using a narrow slit or differentiating the detector’s edge spread function (ESF) with an edge device. The slit method requires precise fabrication and alignment of a slit as well as a high radiation exposure. The edge method [3] is a complicated image processing procedure, requiring determination of the edge angle, reprojection, sub-binning, smoothing and differentiating the ESF, and spectral estimation. In this paper, a simple method is employed to evaluate the MTF using an edge device. The image processing procedures required by this method involve simply the determination of the over-sampling rate and the Fourier transform of the modified ESF. Differentiation and signal to noise ratio (SNR) improvement are jointly applied in the Fourier domain. The MTFs obtained by this simple method are compared to the theoretical MTF and the previously proposed more complicated edge method. The experimental results show that the proposed method provides a simple, accurate and convenient measurement of the presampling MTF for digital imaging systems.

The modulation transfer function (MTF) is a well established method in the field of image processing, to characterize the spatial resolution of display systems. For the display of video material, the temporal, or 'dynamic', resolution of a display is at least equally important. Non-ideal temporal characteristics can lead to a variety of artifacts, such as motion blur, dynamic false edges and colour break-up. Although, typically, many individual properties of the display influence the temporal performance, they can be combined into the temporal equivalent of the spatial MTF. This paper deals with the temporal MTF, and its relation to motion blur. The temporal MTF is used to discuss some recent developments in the field of display processing and driving, aimed at improving the dynamic resolution.

Many properties of the atmosphere affect the quality of images propagating through it by blurring and reducing their contrast. The atmospheric path involves several limitations such as scattering and absorption of the light and turbulence, which degrade the image. The recently developed atmospheric Wiener filter, which corrects for turbulence blur, aerosol blur, and path radiance simultaneously, is implemented here in digital restoration of Landsat Thematic Mapper (TM) imagery over seven wavelength bands of the satellite instrumentation. Turbulence MTF (Modulation Transfer Function) is calculated from meteorological data or estimated in no meteorological data were measured. Aerosol MTF is consistent with optical depth. The product of the two yields atmospheric MTF, which is implemented in the atmospheric Wiener filter. Restoration improves both smallness of size of resolvable detail and contrast. Restorations are quite apparent even under clear weather conditions. Different restoration results are obtained by trying to restore the degraded image. A way to determine which is the best restoration result and how good is the restored image is presented here, by examining mathematical criteria such as MSE (Mean Square Error), ROH (Richness of Histogram), and SOH (Similarity of Histogram), to obtain an improved image and consequently better visual restoration results.

Solution for atmospheric optical transfer function using spherical harmonics method

Using a Monte Carlo method, we investigate the effect of a turbid medium on image transmission by means of the modulation transfer function approach. We present results that refer to a medium that consists of a random distribution of water spherical particles in air. We analyze the effect of geometric conditions (medium width and position) and source characteristics (Lambertian, beam emission). We present results for small spheres (Rayleigh scattering) and spheres (1.0-microm diameter) that are not small in comparison with the wavelength lambda = 0.6328 microm. Numerical data show a large modulation transfer function dependence on the source emission aperture and a substantial independence of the medium width for a fixed value of the optical depth. In accordance with reciprocity principles, we test an inverse scheme of Monte Carlo calculation, the advantage of this scheme being a substantial reduction in calculation time.

In order to evaluate and characterize the spatial resolution properties of a direct digital radiography (DDR) system based on a four CCD array detector, we measure the modulation transfer function (MTF) of the imaging system from its edge spread function (ESF). Different from the traditional algorithm for fine sampling of the ESF of an edge or a slit image with a slight angle to the image coordinate, in this paper we propose a new iterative algorithm. The novel algorithm finely samples the ESF along the column almost parallel to the edge, but not the row perpendicular to the edge. The samples of the first column and the second column are combined and averaged to get a segment of fine-sampled ESF. Then the segment of fine-sampled ESF is further combined with the next column and averaged to get a slight longer segment of ESF, and so forth. A complete fine-sampled ESF can be obtained after repeating the procedure for all columns of the edge image. An adaptive filtering is used for smoothing the ESF so that the sharpness at the edge of ESF can be protected. The MTF of the DDR is obtained from a series of image processing and data processing. The image processing includes the automatic determination of the angle of the edge to the pixel array of the edge image and fine sampling of edge spread function (ESF); data processing includes smooth filtering of ESF, numerical differential and fast Fourier transformation (FFT).