Simple method for modulation transfer function determination of digital imaging detectors from edge images

Transcription

1 Simple method for modulation transfer function determination of digital imaging detectors from edge images Egbert Buhr *a, Susanne Günther-Kohfahl b, Ulrich Neitzel b a Physikalisch-Technische Bundesanstalt, Bundesallee 100, D Braunschweig, Germany b Philips Medical Systems, Röntgenstraße 24, D Hamburg, Germany ABSTRACT A simple variant of the edge method to determine the presampled modulation transfer function (MTF) of digital imaging detectors has been developed that produces sufficiently accurate MTF values for frequencies up to the Nyquist frequency limit of the detector with only a small amount of effort for alignment and computing. An oversampled edge spread function (ESF) is generated from the image of a slanted edge by rearranging the pixel data of N consecutive lines that correspond to a lateral shift of the edge of one pixel. The original data are used for the computational analysis without further data preprocessing. Since the number of lines leading to an edge shift of one pixel is generally a fractional number rather than an integer, a systematic error may be introduced in the MTF obtained. Simulations and theoretical investigations show that for all frequencies up to the Nyquist limit the relative error MTF/MTF is below 1/(2N) and can thus be kept below a given threshold by a suitable choice of N. The method is especially useful for applications where the MTF is needed for frequencies up to the Nyquist frequency limit, like the determination of the detective quantum efficiency (DQE). Keywords: Modulation transfer function, edge image analysis, digital radiography, digital x-ray detector 1. INTRODUCTION The modulation transfer function (MTF) is a basic performance measure of an imaging system describing the signal transfer characteristics of the system as a function of spatial frequency. 1 For digital imaging systems with a discrete image sampling a characteristic difficulty arises because the response of the detector to a signal pattern may not only depend on the imaging properties of the detector itself but also on the signal pattern and its location relative to the sampling grid of the detector. The signal transfer is thus influenced by two parts and can be considered as a two-step process: 2 the analog transfer of the signal, described by the presampled MTF, 3 and the stage of sampling. Whereas the effects caused by the sampling depend on both the sampling grid and the signal pattern, the presampled MTF is a physical quantity of the imaging detector alone and independent from the signal pattern. Various methods have been proposed to determine the presampled MTF of digital radiography systems, based on slit, edge, or bar pattern images While the application of a periodic bar pattern is sometimes considered problematic because of difficulties in determining the modulations in digital bar pattern images, 3 the use of slits or edges is very common. A comparison of slit and edge techniques has shown that the slit method is superior to the edge method when determining the MTF at high spatial frequencies, while the edge method offers more accurate MTF results for low spatial frequencies. 8 The edge method is thus especially suited for applications where the MTF is needed for low frequencies only. An important example is the determination of the detective quantum efficiency (DQE), which is defined only for frequencies up to the Nyquist frequency limit of the detector. Common to all edge methods is that an edge slightly slanted with respect to the detector grid is imaged with the detector. From the digital edge image a composite, oversampled edge spread function (ESF) is generated, which is differentiated to obtain the line spread function (LSF), and finally the MTF is calculated by a Fourier transform of the LSF. 6,7,11 In order to make use of standard software tools for processing the ESF, it is desirable that the ESF is available * egbert.buhr@ptb.de, Medical Imaging 2003: Physics of Medical Imaging, M. J. Yaffe, L. E. Antonuk, Editors, Proceedings of SPIE Vol (2003) 2003 SPIE /03/$

2 on a regular subsampling grid. For this purpose often a certain, to some extent arbitrarily chosen subsampling pitch is used which requires high-precision edge angle determination and special data pre-processing methods like resampling, smoothing or curve fitting. 6,7 In this paper we report on a new, simplified method to determine the presampled MTF of a digital radiographic system from an edge image. The method described here differs from previous methods in that it derives the ESF directly from the measured. i.e. sampled image data and avoids curve fitting or resampling. In addition, it does not require knowledge of the actual edge angle to a high precision. This method uses the natural subsampling pitch given by the lateral distance between two adjacent pixels p divided by the number of lines N corresponding to a lateral shift of the edge of one pixel. This subsampling pitch has been applied by Fujita et.al. 3 for the analysis of LSF s, but the implications for edge analysis are different and will be studied in this contribution. Especially the systematic bias errors of the MTF produced by this method are analysed and measures to keep them below a given threshold are described. 2. METHOD The algorithm for the determination of the presampled MTF generally consists of the following steps: (1) acquisition of an image of a tilted edge, (2) construction of an oversampled edge spread function, (3) differentiation of the ESF to obtain the line spread function (LSF), and finally, (4) calculation of the Fourier transform of the LSF to obtain the MTF. 2.1 Input data As starting point for the application of the method described here it is assumed that the digital image of an edge, slightly tilted with respect to the horizontal (x) or vertical (y) direction of the image matrix is available. It is further assumed that the pixel data are linear with respect to incident exposure which is usually achieved by an appropriate linearisation procedure. Details of the image acquisition procedure have been described previously by several authors 6,12 and are not in the scope of this paper. Fig. 1 gives a graphical representation of the situation. The sampling distance for neighbouring pixels is given by p. Equal sampling distances in x- and y-direction are assumed here, although this is not a prerequisite. It is assumed in this paper that the edge is slightly tilted with respect to the y-direction. The tilt angle α should be in the range of 1.5 to 3.0. The pixel values in each pixel line represent the edge spread function or edge profile sampled at a pitch of p. From line to line a slight shift x of the sampling positions with respect to the edge position exists due to the tilt of the edge, x = p tanα (1) The average number of lines resulting in a shift of the edge of one pixel is thus given by N ave = p / x =1/ tanα (2) Figure 1. Orientation of edge relative to pixel matrix (schematically, edge angle exaggerated). Figure 2. Outline of the construction of the oversampled ESF on a regular subsampling grid. 878 Proc. of SPIE Vol. 5030

3 2.2 Construction of the oversampled edge spread function The basic idea of our approach is similar to the one applied by Fujita et.al. for the analysis of line spread functions. It uses a group of N consecutive lines to construct the oversampled ESF, where N is determined by the condition that the total shift of the edge transition from the first line to the N-th line is as close as possible to one pixel in x-direction. When applying the method proposed here, it is not a prerequisite that the edge angle is finely adjusted relative to the detector grid. Consequently, a lateral shift of the edge of exactly one pixel is, in general, not obtained by an integer number of lines. The main point of this edge analysis method is that the nearest integer number is chosen for N, i.e. ( p x / 2) < N x ( p + x / 2) ( 3) The first step of the algorithm is therefore the determination of this integer number N. Different methods can be applied for this purpose. One suitable method is a four step procedure: 1. Estimation of the edge location with sub-pixel accuracy in each line using a simple linear interpolation between the pixel data. 2. Use of a linear regression to fit a straight line through the individual edge estimates. 3. Calculation of the average number N ave of lines necessary for a lateral shift of the edge of 1 pixel from the slope b of the regression line (N ave = 1/b). 4. Rounding of N ave to the next integer yields N. Alternatively, N can also be determined from the edge angle α according to N = round( p / x) = round(1/ tanα) (4) Eq. 4 shows that the edge angle α needs to be known only as accurate as to distinguish between integers for N, i.e. α/α < ±1/(2N) which is about ± 0.07 for N=20. The second step of the algorithm is to interleave N consecutive lines to construct the oversampled edge profile. This is done in the following manner: pixel 1 of line 1 is followed by pixel 1 of line 2, and so on until pixel 1 of line N; then follows pixel 2 of line 1, pixel 2 of line 2 and so on until pixel 2 of line N; this procedure is repeated for the third pixels of lines and so on until the last pixel of line N. The construction of the oversampled edge profile can also explained with reference to Fig. 2. In this figure each line has been displaced by x with respect to the preceding line. Thus, the edge locations in each line profile line up in a vertical direction, i.e. all profiles are in phase with respect to the edge. The oversampled profile is obtained by projecting the pixel samplings of N lines onto each other. As can be seen from Fig. 2 the sampling distances of the true subsampling grid will usually be non-uniform: the sampling distance is, e.g., somewhat larger than p/n for the first N-1 consecutive samples and smaller for the N th sample. Only if the edge position is displaced by exactly one pixel after N lines, i.e. if x is an integer fraction of the original sampling distance p, the sampling distance in the oversampled profile is uniform and equal to p/n. This situation can only be achieved by a high-precision adjustment of the edge angle, which is unacceptable for routine application. The third step is to assume as an approximation that the data points belonging to the oversampled edge profile have been sampled at a regular subsampling grid rather than at the true subsampling grid. A constant sampling distance of p/n for the oversampled profile is used, irrespective of the actual, generally non-uniform subsampling distances (see Fig.2). Consequently, certain phase error effects will in general appear in the oversampled ESF. The influence of these phase errors on the final MTF will lead to certain systematic errors which will be discussed in section 3. The process of constructing the oversampled ESF as described above can be repeated for consecutive groups of N lines across the edge. Each group results in an additional representation of the oversampled ESF. These multiple ESF s can be used to reduce the noise in the data and the resulting MTF. The most straightforward way would be to determine the MTF for each individual oversampled ESF and to average the MTF s. Alternatively, the average of all available oversampled ESF s can be calculated and the MTF can be determined from this average ESF. However, as the edge position in the various edge profiles will be different, determination of the edge location to subsampling pitch precision is necessary for in-phase superposition of the individual ESF s. A drawback of averaging the MTF s rather than the ESF s is that for noisy edge images the individual MTF s may be subject to bias errors caused by the noise. 13 Hence, averaging the ESF s prior to evaluate the MTF is preferred. Proc. of SPIE Vol

4 2.3 Determination of the line spread function The data set describing the oversampled line spread function (LSF) is derived from the oversampled ESF by a finiteelement differentiation using e.g. a convolution filter with a [-0.5, 0, 0.5] kernel. The spatial filtering effect of this finite-element differentiation can be corrected since the transfer function of the derivation filter is known. The pixel value array containing the LSF data is truncated to a length of M, with the edge transition lying approximately in the middle of this range. 2.4 Calculation of the MTF To calculate the MTF from the LSF a fast Fourier transform (FFT) is applied to the oversampled ESF data. The discrete spatial frequencies u are determined by the total length of the LSF, M p/n, i.e. u = k N/(2Mp) with k=0,1,... M/2. The magnitude of the FFT is normalised to 1 for the first (zero frequency) value. Since the distances between the subsampling grid points and the edge are obtained along the x-axis and not in a direction perpendicular to the edge, the actual distances between the subsampling grid points and the edge are smaller by factor of cos(α). This effect can be compensated for by a corresponding frequency scaling factor. 3. ACCURACY OF THE METHOD The presampled MTF obtained using the method described here may deviate from the true MTF since a regular subsampling grid for the oversampled edge profile is used rather than the true subsampling grid. A detailed analysis about the quantitative effects of this approximation on the MTF obtained will be published elsewhere. Here, only the main findings will be summarised and discussed. 3.1 Influence of phase effects on ESF and LSF As a consequence of the approximation of using a regular subsampling grid, the oversampled edge profile will show periodic spatial phase effects: For N-1 subsequent data points the edge profile runs faster (or slower) than the true one, and this accumulated phase error is compensated in the N-th subsampling data point. Fig. 3 shows an example of an oversampled edge profile where these phase effects lead to kinks which appear with the periodicity of the original pixel sampling p. The magnitude of the mismatch between the true and the regular subsampling grid positions depends on the average number of lines N ave leading to a lateral shift of the edge of one pixel. The maximum mismatch is always smaller than p/(2n) and is expected if N ave lies half-way between two integer numbers (N ave -N=0.5). The LSF, calculated by differentiating the ESF, shows visible spikes at those locations where the ESF has the kinks. An example LSF is shown in Fig. 4, which is the obtained by differentiating the ESF shown in Fig.3. The spatial distance between these spikes is given by the original pixel pitch p. The position of these spikes in the LSF is determined by the position of the edge relative to the pixel grid. The width of the spikes is proportional to p/n. 3.2 Difference between true and obtained MTF One can expect implications of the phase effects described above on the MTF obtained. An example of an MTF determined from a disturbed LSF is shown in Fig.5 (the MTF shown is derived from the LSF of Fig. 4). The observed differences between true and obtained MTF can be qualitatively understood as follows: Since the spikes in the LSF occur regularly with a distance given by the sampling distance p, they will produce spectral components in Fourier space at frequencies k/p with k=±1, ±2, ±3 etc. The relative height of an individual spike is determined by the slope of the ESF at the location of the spike. Since the slope of the ESF is proportional to the LSF, the height of the spikes is determined by the value of the LSF at the position of the spikes: one may say that the spikes are windowed by the LSF itself. In frequency space, windowing is equivalent to a convolution with the Fourier transform of the window, which in this case is the Fourier transform of the LSF, i.e. the MTF itself. Hence, the additional spectral components can be described as the convolution of the MTF with the Fourier transform of the train of spikes: The additional spectral components are apart from a certain amplitude factor - replicas of the MTF located at the frequencies k/p with k=±1, ±2, ±3 etc. 880 Proc. of SPIE Vol. 5030

5 A quantitative analysis yields details about the difference (or systematic error ) between true and obtained MTF. Here we discuss only the important case of determining the MTF for frequencies below the Nyquist frequency, i.e for frequencies u<u Nyq =1/(2p). The quantitative findings can be summarised as follows: 1. The magnitude of the difference ( MTF) between the true and the obtained MTF rapidly increases with increasing frequency and reaches its maximum at the limit of the frequency range considered here, i.e. at the Nyquist frequency u Nyq (see Fig.6). 2. The magnitude of the systematic error MTF depends on how large the average number of lines needed for a shift of the edge of one pixel (=N ave ) deviates from the nearest integer N: MTF shows a saw-tooth-like behaviour as a function of N ave and reaches maximum values for N ave -N=0.5. No error occurs if the number of lines leading to a lateral shift of the edge of one pixel is an integer number, because then the regular subsampling grid coincidences with the true subsampling grid and no phase error is expected in the edge profile and thus no error is obtained in the MTF. 3. The difference between true and obtained MTF scales with 1/N (see Fig.6). The reason for this is that the relative width of the erroneous spikes showing up in the LSF - and thus the energy contained in these spikes - is proportional to 1/N. Under the assumption that the presampled MTF is negligible for frequencies lying above 2u Nyq ( low MTF case), the maximum relative error of the MTF for frequencies up to the Nyquist frequency is below 1/(4N). In the high MTF case, i.e. when the presampled MTF is non-negligible even beyond 2 u Nyq, the situation is more complex. Model calculations using a sinc function for the MTF (assuming a fill factor of 1) show that the maximum relative error of the MTF is twice as high as for the low MTF case. The maximum error can be controlled by choosing the edge angle: E.g., the maximum error is below 1 to 2 % for N=25 corresponding to an edge angle of about MTF depends on the actual position of the edge relative to the pixel grid of the detector, which can be described by means of a phase ϕ (a lateral shift of the edge by one pixel is equivalent to phase shift of 2π). In the low MTF case a sinusoidal relationship between MTF and ϕ is expected. As a consequence, the difference between the true and the obtained MTF may have a negative or positive sign or may even vanish. In a measurement, the phase ϕ depends on the group of N consecutive lines across the edge actually selected from the edge image. Consequently, two overlapping groups of N consecutive lines which overlap by N/2 lines will show a phase difference of π and consequently the two groups will produce a systematic error MTF of equal magnitude but different sign: An important improvement of the method is possible if overlapping groups of lines are analysed and the corresponding MTF s are averaged. Figure 3. Example showing the effect of phase error on the obtained edge profile (average number of lines N ave leading to a shift of the edge of one pixel in this example: 10.5). Right: zoom showing one of the kinks in the profile which appear with the periodicity of the original pixel sampling. For comparison, the true (continuous) ESF is shown. Proc. of SPIE Vol

6 1,0 0,8 Obtained LSF True LSF 1,000 0,500 Obtained MTF True MTF LSF 0,6 0,4 0,2 0, Position (subsampled pixel) Figure 4. Example showing the effect of phase error on the line spread function LSF which is obtained by differentiating the ESF shown in Fig.3. For comparison, the true LSF is shown. MTF 0,100 0,050 0,010 0, Frequency in 1/pixel pitch Figure 5. Example showing the effect of phase error on the MTF (obtained from the LSF shown in Fig. 4). For comparison, the true MTF is also shown. The Nyquist frequency is at u Nyq = 0.5 / pixel pitch. 15 MTF/MTF x N ave =10.5 N ave =20.5 N ave = ,25 0,5 0,75 Frequency in 1/pixel pitch Figure 6. Maximum relative difference (in percent) between the obtained MTF and the true MTF for different values of N ave (10.5, 20.5, and 40.5, corresponding to edge angles of 5.44, 2.79, and 1.41, respectively). 4. MEASUREMENT EXAMPLE The method described here has been applied to measure the presampled MTF of a digital x-ray detector. A slanted semi-transparent edge (edge angle about 2.5 ) was imaged with the detector. The part of the digital image containing the edge had a size of 512 x 512 pixel. The average number of lines (=N ave ) leading to a lateral shift of the edge of one pixel was about Therefore, groups containing 23 lines each were used to construct the oversampled ESF, i.e. the subsampling pitch in the ESF was 1/23 of the original pixel pitch. Along the edge, in total 22 non-overlapping groups were selected from the 512 x 512 image resulting in 22 ESF s. Fig. 7 shows, as an example, the edge profile obtained for one of these groups of lines. From the average ESF the LSF was calculated (see Fig. 8). The noise which is present in the ESF becomes more visible in the LSF due to the derivation process. However, most of this noise is high frequency noise with frequency components lying beyond the Nyquist frequency of the detector. If one is interested in the 882 Proc. of SPIE Vol. 5030

7 MTF for frequencies up to the Nyquist frequency, these high frequency noise components will therefore not affect the MTF (see Fig. 9). ESF (a.u.) 1,0 0,8 0,6 0,4 0,2 0, Position in mm LSF (a.u.) 1,0 0,8 0,6 0,4 0,2 0, Position in mm Figure 7. Example of an edge profile determined from the edge image. Figure 8. LSF obtained from the ESF by differentiation. The noise levels on the two sides of the LSF differ, because the noise in the edge image depends on the signal level. 1,0 0,8 MTF 0,6 0,4 0,2 0, Frequency in lp/mm Figure 9. MTF obtained from the LSF by Fourier transform. Nyquist frequency is at 3.5 lp/mm. 5. CONCLUSIONS A simplified method to determine the presampled MTF of a digital radiographic system from an edge image has been described. The method requires only rearranging the original pixel data to construct the oversampled edge spread function and avoids high precision edge angle determination, and data resampling or curve fitting. The algorithm can easily be implemented using standard software tools. The systematic errors inherent to this method can be kept below any given threshold for all frequencies up to the Nyquist frequency limit by a suitable selection of the edge angle. The systematic error can be further reduced by using at least two, half-overlapping groups of N lines for additional representations of the supersampled edge profile. The simplicity and robustness of the method makes it well suited for routine application in cases where the presampled MTF shall be determined for frequencies below the Nyquist limit only, an important example being the determination of the detective quantum efficiency. Additional applications include the measurement of the MTF for other digital imaging systems such as CCD cameras or scanners. Proc. of SPIE Vol

8 REFERENCES 1. T. L. Williams, The Optical Transfer Function of Imaging Systems, Institute of Physics Publ., Bristol, W. K. Pratt, Digital Image Procesing, 2nd ed., Wiley, New York, J.T. Dobbins III, Image Quality Metrics for Digital Systems, in: Handbook of Medical Imaging, vol. 1: Physics and Psychophysics, edited by J. Beutel, H.L. Kundel, R.L. Van Metter, SPIE, Bellingham, M.L. Giger, K. Doi, Investigation of basic imaging properties in digital radiography. I. Modulation transfer function, Med. Phys. 11, (1984) 5. H. Fujita, D.-Y. Tsai, T. Itoh, K. Doi, J. Morishita, K. Ueda, A. Ohtsuka, A simple method for determining the modulation transfer function in digital radiography, IEEE Trans. Med. Imaging 11, (1992) 6. E. Samei, M. J. Flynn, D. A. Reimann, A method for measuring the presampled MTF of digital radiographic systems using an edge test device, Med. Phys. 25, (1998) 7. P. B. Greer and T. van Doorn, Evaluation of an algorithm for the assessment of the MTF using an edge method, Med. Phys. 27, (2000) 8. I. A. Cunningham, B.K. Reid, Signal and noise in modulation transfer function determinations using the slit, wire, and edge techniques, Med. Phys. 19, (1992) 9. W. Hillen, U. Schiebel, T. Zaengel, Imaging performance of a digital storage phosphor system, Med. Phys. 14, (1987) 10. F. Rogge, D. Vandenbroucke, L.Struye, H. Bosmans, P. Willems, G. Marchal, A practical method for detected quantum efficiency (DQE) assessment of digital mammography systems in the radiological environment, Proc. SPIE Medical Imaging 2002 Vol. 4682, (2002) 11. S. E. Reichenbach, S. K. Park, R. Narayanswamy, Characterizing digital image acquisition devices, Opt. Eng. 30, (1991) 12. P.R. Granfors, R. Aufrichtig, Performance of a 41x41cm2 amorphous silicon flat panel x-ray detector for radiographic imaging applications, Med.Phys. 27, (2000) 13. M.G. Fisher, MTF Noise Power and DQE of Radiographic Screens, Phot.Sci.Eng. 26, (1982) 884 Proc. of SPIE Vol. 5030