Characterization of the point spread function and modulation transfer function of scattered radiation using a digital imaging system

A digital radiographic system was used to measure the distribution of scattered x radiation from uniform slabs of Lucite at various thicknesses. Using collimation and air gap techniques, [primary + scatter] images and primary images were digitally acquired, and subtracted to obtain scatter images. The scatter distributions measured using small circular apertures were computer fit to an analytical function, representing the circular aperture function convolved with a modified Gaussian point spread function (PSF). On the basis of goodness of fit criterion, the proposed Gaussian function is a very good model for the scatter PSF. The measured scatter PSF's are reported for various Lucite thicknesses. Using the PSF's, the modulation transfer functions are calculated, and this spatial frequency information may have value in analytical scatter removal techniques, grid design, and air gap optimization.     

A method for modulation transfer function determination from edge profiles with correction for finite-element differentiation

In this paper we describe a technique for determining the modulation transfer function (MTF) of an imaging system from an experimentally obtained edge profile. The technique includes an exact correction for the frequency passband of the finite-element differentiation required to obtain the line spread function from the edge spread function. This correction has been ignored by investigators in the past and is required whenever finite-element differentiation is used rather than analytic differentiation of a model fitted to the edge response data. The magnitude of the MTF correction is approximately 11% at f = fc/2 and approximately 57% at f = fc, where fc = fs/2 is the maximum frequency reproducible without aliasing with a sampling rate of fs. The correction is performed in the spatial frequency domain by multiplying the uncorrected MTF by 1/sinc (pi f/2fc). A computer simulation is presented to demonstrate the effect and the correction procedure. An experimental MTF of an x-ray image intensifier system obtained using this technique is found to be consistent with an MTF obtained using a bar pattern test phantom.     

Signal and noise in modulation transfer function determinations using the slit, wire, and edge techniques.

The modulation transfer function (MTF) of an idealized imaging system can be determined from the Fourier transform of the system's line-spread function (LSF). Three techniques of experimentally determining the LSF require imaging either a slit, wire, or edge. In this paper, these three techniques are modeled theoretically to determine the noise in the calculated MTFs as a function of spatial frequency resulting from both quantum fluctuations and stochastic detector noise. The techniques are compared using the signal-to-noise ratio (SNR) in the MTF, defined as the ratio of the MTF value to the standard deviation in an ensemble of MTF determinations from independent measurements. It is shown that for a specified photon fluence, the edge method MTF has the highest SNR at low spatial frequencies, while that of the slit method is superior at high frequencies. The wire method SNR is always inferior to that of the slit technique. This suggests that the edge method is preferable for measuring parameters such as the low-frequency drop, and the slit method is preferable for determining high-frequency response. The cross-over frequency at which the slit and edge methods are equal (f(e)) for quantum-noise limited systems is a function of the slit width and the length over which the LSF is measured. For detector-noise limited systems, f(e) is dependent on the slit width only. The SNR in all but the quantum-noise limited slit method can therefore be increased by decreasing the length over which the LSF is measured, smoothing the tails of the LSF, or by fitting the tails to an analytic expression.     

Accuracy of a simple method for deriving the presampled modulation transfer function of a digital radiographic system from an edge image

Several methods for accurately deriving the presampled modulation transfer function (MTF) of a pixelated detector from the image of a slightly slanted edge have been described in the literature. In this paper we report on a simple variant of the edge method that produces sufficiently accurate MTF values for frequencies up to the Nyquist frequency limit of the detector with little effort in edge alignment and computation. The oversampled ESF is constructed in a very simple manner by rearranging the pixel data of N consecutive lines corresponding to a lateral shift of the edge by one pixel. A regular subsampling pitch is assumed for the oversampled ESF, which is given by the original pixel sampling distance divided by the integer number N. This allows the original data to be used for further computational analysis (differentiation and Fourier transform) without data preprocessing. Since the number of lines leading to an edge shift by one pixel generally is a fractional number rather than an integer, a systematic error may be introduced into the presampled MTF. Simulations and theoretical investigations show that this error is proportional to 1/N and increases with spatial frequency. For all frequencies up to the Nyquist limit, the relative error delta MTF/MTF is smaller than 1/(2N). It can thus be kept below a given threshold by suitably selecting N, which furnishes a certain maximum edge angle. The method is especially useful for applications where the presampled MTF is needed only for frequencies up to the Nyquist frequency limit, such as the determination of the detective quantum efficiency (DQE).     

Determination of the modulation transfer function for a time-gated fluorescence imaging system

The use of fluorescence for cancer detection is currently under investigation. Presently, steady-state fluorescence detection methods are in use, but have limitations due to poor contrast between the fluorescence of the tumor and background autofluorescence. Improved contrast can be obtained with time-resolved techniques because of the differing lifetimes between autofluorescence and exogenous photosensitizers that selectively accumulate within tumor tissue. An imaging system is constructed using a fast-gated (200-ps) charge-coupled device (CCD) camera and a pulsed 635-nm laser diode. To characterize the ability of the system to transfer object contrast to an image, the modulation transfer function (MTF) of the system is acquired by employing an extended knife-edge technique. A knife-edge target is assembled by drilling a rectangular well into a block of polymethyl methacrylate (PMMA). The imaging system records images of the photosensitizer, chloroaluminum phthalocyanine tetrasulfonate (AlPcTS), within the well. AlPcTS was chosen to test the system because of its strong absorption of 635-nm, high fluorescence yield, and relatively long fluorescence lifetime (approximately 7.5 ns). The results show that the system is capable of resolving 10(-4) M AlPcTS fluorescence as small as 1 mm. The findings of this study contribute to the development of a time-gated imaging system using fluorescence lifetimes.     

Determination of the modulation transfer function of digitally reconstructed radiographs in radiotherapy treatment planning using a point phantom

Digitally reconstructed radiographs (DRRs) have become an important tool in radiotherapy treatment planning and treatment verification but their use can be limited by poor image quality. A point phantom comprised of a 1 mm diameter tungsten carbide ball bearing in a wax block has been manufactured as a means of determining a quantitative measure of DRR spatial resolution by calculating the modulation transfer function (MTF) of the image. This method is an alternative to the more often used bar pattern method. The effect of varying CT slice thickness upon DRR spatial resolution has been assessed and a power law relationship between the two variables has been established.     

Contrast enhancement in dense breast images using the modulation transfer function

This work proposes a method aimed at enhancing the contrast in dense breast images in mammography. It includes a new preprocessing technique, which uses information on the modulation transfer function (MTF) of the mammographic system in the whole radiation field. The method is applied to improve the efficiency of a computer-aided diagnosis (CAD) scheme. Seventy-five regions of interest (ROIs) from dense mammograms were acquired in two pieces of equipment (a CGR Senographe 500t and a Philips Mammodiagnost) and were digitized in a Lumiscan 50 laser scanner. A computational procedure determines the effective focal spot size in each region of interest from the measured focal spot in the center for a given mammographic equipment. Using computational simulation the MTF is then calculated for each field region. A procedure that enlarges the high-frequency portion of this function is applied and a convolution between the resulting new function and the original image is performed. Both original and enhanced images were submitted to a processing procedure for detecting clustered microcalcifications in order to compare the performance for dense breast images. ROIs were divided into four groups, two for each piece of equipment-one with clustered microcalcifications and another without microcalcifications. Our results show that in about 10% of the enhanced images more signals were detected when compared to the results for the original dense breast images. This is important because the usual processing techniques used in CAD schemes present poor results when applied to dense breast images. Since the MTF method is a well-recognized tool in the evaluation of radiographic systems, this new technique could be used to associate quality assurance procedures with the processing schemes employed in CAD for mammography.     

Normalization of the modulation transfer function: the open-field approach

The modulation transfer function (MTF) is widely used to describe the spatial resolution of x-ray imaging systems. The MTF is defined to have a zero-frequency value of unity, and it is common practice to ensure this by normalizing a measured MTF curve by the zero-frequency value. However, truncation of the line spread function (LSF) within a finite region of interest (ROI) results in spectral leakage and causes a reduction in the measured MTF zero-frequency value equal to the area of truncated LSF tails. Subsequent normalization by this value may result in inflated MTF values. We show that open-field normalization with the edge method produces accurate MTF values at all nonzero frequencies without need for further normalization by the zero-frequency value, regardless of ROI size. While both normalization techniques are equivalent for a sufficiently large ROI, a 5% inflation in MTF values was observed for a CsI-based flat-panel system when using a 10 cm ROI. Use of open-field normalization avoids potential inflation caused by zero-frequency normalization.     

A moving slanted-edge method to measure the temporal modulation transfer function of fluoroscopic systems

Lag in fluoroscopic systems introduces a frame-averaging effect that reduces measurements of image noise and incorrectly inflates measurements of the detective quantum efficiency (DQE). A correction can be implemented based on measurements of the temporal modulation transfer function (MTF). We introduce a method of measuring the temporal MTF under fluoroscopic conditions using a moving slanted edge, a generalization of the slanted-edge method used to measure the (spatial) MTF, providing the temporal MTF of the entire imaging system. The method uses a single x-ray exposure, constant edge velocity, and assumes spatial and temporal blurring are separable. The method was validated on a laboratory x-ray image intensifier (XRII) system by comparison with direct measurements of the XRII optical response, showing excellent agreement over the entire frequency range tested (+/- 100 Hz). With proper access to linearized data and continuous fluoroscopy, this method can be implemented in a clinical setting on both XRII and flat-panel detectors. It is shown that the temporal MTF of the CsI-based validation system is a function of exposure rate. The rising-edge response showed more lag than the falling edge, and the temporal MTF decreased with decreasing exposure rate. It is suggested that a small-signal approach, in which the range of exposure rates is restricted to a linear range by using a semitransparent moving edge, would be appropriate for measuring the DQE of these systems.     

Measurement of the presampling modulation transfer function of film digitizers using a curve fitting technique

A curve fitting technique combined with an angulated slit image has been developed for the measurement of the presampling modulation transfer function (MTF) of film digitizers. The noisy line spread functions (LSFs) acquired from an angulated slit image are fitted using a combination of two functions by means of a nonlinear least-square fitting technique. The parameters in the model function for each LSF are obtained by minimizing the residual root mean square (RMS), and then averaged over all the LSF fittings. The resulting analytical function is representative of the continuous presampled LSF. We have found that a combination of Gaussian and exponential functions provides a good fit to the LSFs obtained with film digitizers. The corresponding analytical Fourier transformation of the model function yields the presampling MTF, without Nyquist frequency limitation. Measurements of spatial resolution properties using this method were performed for two laser scanners and an optical drum scanner.     

Conditioning data for calculation of the modulation transfer function

A method for conditioning data used in the measurement of the modulation transfer function (MTF) is discussed. This method is based upon imposing the constraint that the edge spread function (ESF) is monotonic. The advantages of this technique, when applicable, are demonstrated with simulated examples for which the true MTF is known. The application of this technique in the measurement of the MTF of a digital detector in clinical use is also demonstrated.     

New criteria for estimating baroreflex sensitivity using the transfer function method

Computer simulations were carried out to appraise three new criteria for the estimation of baroreflex sensitivity (BRS) using the tranfer function method. The major goal was to identify a computation procedure able to overcome the intrinsic limitations of the classical coherence criterion. Four representative shapes of the gain function and three different average agains (2, 5 and 8 ms(mmHg)⁻¹) in the lowfrequency (LF) band (0.04–0.15 Hz) were considered. The signal-to-noise ratio was made to vary so that the peak coherence in the LF band changed from 0.15 to 0.9 All simulation parameters were derived from previous observations in healthy subjects and heart disease patients. The error of the estimated gain function was obtained from its confidence interval. BRS was computed as average gain in the LF band: (a) including in the average only those points having error≤ threshold (criterion 1, C1); (b) calculating the mean error in the band and accepting BRS measurements only when this error was ≤ threshold (criterion 2, C2); (c) including in the average all points, regardless of the error (criterion 3, C3). The three criteria were compared in terms of measurability (percentage of measured BRS) and accuracy (bias and SD of BRS). Using C1 and C2, measureability dropped to 10% when the peak cohrence in the LF band decreased, respectively, to 0.18–0.41 and to 0.26–0.53, depending on the shape and strength of the gain. In this condition (lower bound of measureability), worst bias and SD (average gain: 8 ms(mmHg)⁻¹) were, respectively, 0.8 ms(mmHg)⁻¹ and 3.3 ms(mmHg)⁻¹ (C1), and 0.1 ms(mmHg)⁻¹ and 1.0 ms(mmHg)⁻¹ (C2), C3, by definition, always ensured 100% measurability and showed bias and SD comparable with, or even lower than, C1 and C2, within the common range of measurable BRS. In the extreme condition of 0.15 coherence, bias and SD were, respectively, 1.7 ms(mmHg)⁻¹ and 2.3 ms(mmHg)⁻¹ (average gain: 8 ms(mmHg)⁻¹). Hence, error checking (C1 and C2) dramatically reduced measurability and did not improve accuracy of BRS measurements compared with performing no error check (C3). In conditions of low signal-to-noise ratio and/or impaired baroreflex gain, leading to markedly reduced coherence, the simple average of the gain function in the LF band allows BRS to be estimated with accuracy adequate for clinical purposes.     

Novel approach to stationary transmission scanning using Compton scattered radiation

Transmission scanning-based estimation of the attenuation map plays a crucial role in quantitative radionuclide imaging. X-ray computed tomography (CT) reconstructs directly the attenuation coefficients map from data transmitted through the object. This paper proposes an alternative route for reconstructing the object attenuation map by exploiting Compton scatter of transmitted radiation from an externally placed radionuclide source. In contrast to conventional procedures, data acquisition is realized as a series of images parameterized by the Compton scattering angle and registered on a stationary gamma camera operating without spatial displacement. Numerical simulation results using realistic voxel-based phantoms are presented to illustrate the efficiency of this new transmission scanning approach for attenuation map reconstruction. The encouraging results presented in this paper may suggest the possibility of proposing a new concept for emission/transmission imaging using scattered radiation, which has many advantages compared to conventional technologies.     

Experimental verification of a technique for predicting scattered radiation transfer: application to low photon energies

It is shown that predictions can be made of scatter-to-primary ratios for a variety of mammographic configurations. The different configurations can be produced by changes in source-detector distance; source-phantom distance; air gap; photon energy; phantom composition, thickness, and cross-sectional area. A detailed analysis of the effect of a grid is also considered. The predictions are not computationally intensive. Experimental verifications of the predictions for varying phantom cross-sectional dimensions, source-detector distance, and air gaps with and without a grid have been carried out. Variations with phantom thickness were not considered in this paper. Detailed comparisons between experiment and theory indicate that scatter-to-primary ratios were predicted within one standard deviation and the coefficient of variation is within 3% for most of the data points with the worst case coefficient of variation about 6%. Measurements of grid secondary transmission have also been compared to theoretical predictions and agreement is within the coefficient of variation of about +/- 15%.     

Method for estimating the intensity of scattered radiation using a scatter generation model

An analytical formula for estimating the intensity of scattered radiation in an x-ray image under various exposure conditions has been developed. The formula was derived using measured data of scatter and primary intensity for various exposure conditions. To simplify the formula, a scatter generation model was constructed mathematically which assumes that the scattered fluence in a material depends on three processes: (1) scattering of the primary photons; (2) scattering of previously scattered photons; and (3) attenuation of the scattered photons. Using this model, the dependence of scatter-to-primary ratio on phantom thickness and the tube voltage was expressed by a simple equation. Parameters included in the model were determined from the analysis of measured data. Based on empirical data, it was assumed that the dependence of scatter-to-primary ratio on air gap and field size is not affected by variations of phantom thickness and tube voltage. The final formula, which does not include the term of phantom thickness, gives an estimate of the intensity of the scattered radiation from exposure conditions. The scatter intensity estimated using the formula was compared with measured data for various phantom thicknesses, tube voltages, air gaps, and field sizes; the results show that the intensity of scattered radiation can be estimated within about +/- 10% using predetermined parameters.     

人工晶状体的调制传递函数研究

目的将眼球屈光系统看成是一种传递光学信息的低通滤波器,运用调制传递函数(MTF)理论,研究人工晶状体及人工晶状体植入人眼后的有关光学问题;对人工晶状体植入后的眼球光学系统的成像质量进行科学的评价。方法使用MTF测定仪测定正常人眼屈光系统MTF的正常值;在精密模型眼中植入各种不同光学结构的人工晶状体,建立研究用模型;运用Zemax光学计算软件对研究模型进行计算,分别用光线追迹办法计算了几何象差、MTF。结果双凸结构的MTF值最佳,凸平及凸凹形也有较好的成像质量,较差的为平凸,最差的为凹凸形。结论笔者研究对于眼科临床及人工晶状体的设计、生产,以及开发新一代的人工晶状体具有重要的实际价值。     

Reduction of aliasing in modulation transfer function measurements

The images formed by many radiological systems are difficult to sample at spatial intervals small enough to avoid aliasing in the calculation of the system's modulation transfer function. However, if a system's response can be assumed to be symmetrical, this assumption can be used to effectively double the sampling density and to double the frequency limit before aliasing occurs. To accomplish this, a more complex algorithm is required. In this work, the formula for the calculation of the modulation transfer function from a symmetrical, one-dimensional line spread function is derived and a similar result for a symmetrical, two-dimensional point spread function is presented. The effect of noisy data and errors in the estimation of the offset of the center of the line spread function from a sampling point are investigated by simulation studies. For low noise (relative standard deviation of 1%) and an offset error of no more than 2% or 3% of a sampling interval, reasonable precision is obtained. These conditions appear to be achievable, especially when the noise is Poisson distributed.     

Ring filter to locate external boundaries in SPECT using scattered radiation

A novel form of filter for SPECT is described, in which, after back projection and summation, the reconstructed signal is a measure of the total activity within a ring of specified radius, centre and width. The filter is applied to the problem of using Compton scattered radiation to locate external boundaries. In the simple case of the determination of the radius of a circular scattering body of known centre, the filter output would identify a transition region and define an appropriate threshold as the boundary was crossed. However it can also be applied to locate the boundaries seen in individual SPECT projections and hence trace out the envelope of the scattering body. Monte Carlo simulation based on 99mTc is used to test the performance of the filter in a range of situations, with encouraging results.     

Characterization of scattered radiation in kV CBCT images using Monte Carlo simulations

Kilovoltage (kV) cone beam computed tomography (CBCT) images suffer from a substantial scatter contribution. In this study, Monte Carlo (MC) simulations are used to evaluate the scattered radiation present in projection images. These predicted scatter distributions are also used as a scatter correction technique. Images were acquired using a kV CBCT bench top system. The EGSnrc MC code was used to model the flat panel imager, the phantoms, and the x-ray source. The x-ray source model was validated using first and second half-value layers (HVL) and profile measurements. The HVLs and the profile were found to agree within 3% and 6%, respectively. MC simulated and measured projection images for a cylindrical water phantom and for an anthropomorphic head phantom agreed within 8% and 10%. A modified version of the DOSXYZnrc MC code was used to score phase space files with identified scattered and primary particles behind the phantoms. The cone angle, the source-to-detector distance, the phantom geometry, and the energy were varied to determine their effect on the scattered radiation distribution. A scatter correction technique was developed in which the MC predicted scatter distribution is subtracted from the projections prior to reconstruction. Preliminary testing of the procedure was done with an anthropomorphic head phantom and a contrast phantom. Contrast and profile measurements were obtained for the scatter corrected and noncorrected images. An improvement of 3% for contrast between solid water and a liver insert and 11% between solid water and a Teflon insert were obtained and a significant reduction in cupping and streaking artifacts was observed.