A Method to Measure the Presampling MTF in Digital Radiography Using Wiener Deconvolution

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A method to measure the presampling MTF in digital radiography using Wiener deconvolution

Zhongxing Zhoua,b, Qingzhen Zhua, Feng Gaoa,b, Huijuan Zhaoa,b, Lixin Zhanga,b ,Guohui Li*c aSchool of Precision Instrument and Optoelectronics Engineering, Tianjin University, Tianjin, China 300072; bTianjin Key Laboratory of Biomedical Detecting Techniques and Instruments, Tianjin, China 300072; cTEDA ORKING Hi-Tech Co.Ltd , Tianjin 300072

ABSTRACT

We developed a novel method for determining the presampling modulation transfer function (MTF) of digital radiography systems from slanted edge images based on Wiener deconvolution. The degraded supersampled edge spread function (ESF) was obtained from simulated slanted edge images with known MTF in the presence of poisson noise, and its corresponding ideal ESF without degration was constructed according to its central edge position. To meet the requirements of the absolute integrable condition of Fourier transformation, the origianl ESFs were mirrored to construct the symmetric pattern of ESFs. Then based on Wiener deconvolution technique, the supersampled line spread function (LSF) could be acquired from the symmetric pattern of degraded supersampled ESFs in the presence of ideal symmetric ESFs and system noise. The MTF is then the normalized magnitude of the Fourier transform of the LSF. The determined MTF showed a strong agreement with the theoritical true MTF when appropriated Wiener parameter was chosen. The effects of Wiener parameter value and the width of square-like wave peak in symmetric ESFs were illustrated and discussed. In conclusion, an accurate and simple method to measure the presampling MTF was established using Wiener deconvolution technique according to slanted edge images. Keywords: modulation transfer function(MTF), line spread function(LSF), edge spread function(ESF), Wiener deconvolution, digital radiography system

1. INTRODUCTION The modulation transfer function (MTF) is a basic measure of the resolution properties of an imaging system. The presampling MTF can be determined by imaging an object whose spatial frequency content is known, such as a narrow slit[1] or an edge[2-5]. The narrow slit method requires very precise fabrication and alignment of the device in the radiation beam, and a high amount of radiation exposure to allow ample transmission through the slit. The edge method obtains the edge spread function (ESF) using an opaque object with a straight edge. The ESF is then differentiated to obtain the line spread function (LSF). The MTF is then the normalized magnitude of the Fourier transform of the LSF. To obtain the presampling MTF, all edge methods require that an edge imaged with the detector should be slightly slanted with respect to the detector grid. However, the accuracy of measurement has not been well accredited on the effect of edge angle determination error and the noise in the imaging system. David and his colleagues suggested an alternative method for determining the MTF of a radiographic system by using the Wiener deconvolution technique[6]. In their work, they recovered the two-dimensional MTF from a simulated degraded image of a circular region object according to the pre-known ideal image. However, this method can't be available on a subsampling grid, and thus prevents the acquisition of the presampling MTF. In this paper we report on a new, simplified method based on Wiener deconvolution to determine the presampling MTF of the digital radiographic system from the degraded edge image. This method does not need to know the actual edge angle. It uses the center-of-gravity analysis[7] to determine the central position of the supersampled ESF and construct the ideal ESF according to this edge position. Then the supersampled ESF and the ideal ESF were mirrored and combined with the original one respectively to form the symmetric pattern of ESFs. Thereafter, the presampling MTF of the digital radiographic system was recovered from the symmetric pattern of degraded supersampled ESF in the presence of system noise and ideal symmetric ESF. *[email protected];

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2. THEORY AND METHODS 2.1 Simulated edge model In X-ray imaging system, the acquired image can be modeled by a degradation which accounts for loss of spatial resolution and additive noise. In this work, a simpler one-dimensional model is considered. For the degraded slanted edge image, the supersampled ESF y(x) acquired from the edge image can be modeled as () ()⊗= ()+ (xnxhxfxy ) (1) where ⊗ is the convolution operator, x is the supersampled spatial displacement, (xf ) is the ideal ESF with no degradation corresponding to y(x), ()xh denotes the blurring mask on account of the detector LSF, and (xn ) is the system noise. Theoretically, a deconvolution procedure could recover the detector LSF (xh ) in the presence of noise ()xn and the ideal ESF ()xf . To test the feasibility of the deconvolution method in this work, digital images containing edge transitions with known profiles were simulated as follows. Digital slanted edge images were simulated by stacking lines of pixels, each line containing the same mode edge profile. To simulated an edge with an edge angle of α with respect to the pixel matrix, the adjacent lines were shifted with =Δ wx tanα , where w is the pixel size. A perfect 512×512 edge image was formed, assuming a 0.1 mm pixel size, a maximum pixel value of 13000, a minimum pixel value of 300, and an edge angle of arctan(1/30).The edge image was filled line-by-line using the edge profile: ⎧ e−rx FB *)( +− F xfor ≥ 0 ⎪ 2 ()xESF = ⎨ (2) 2 − erx ⎪ − FB *)( ()+ xforF < 0 ⎩⎪ 2 Where F is the foreground intensity inside the region shielded by the edge, B is the background intensity outside the region shielded by the edge, and r is a parameter describing the steepness of the edge transition. We adopted the edge parameter r=1/w, F=300, and B=13000 in this work. The normalized MTF of this model can be evaluated by a Lorentzian-type function: r 2 ()uMTF = (3) 2 + ()2πur 2 With above settings, the average number of lines resulting in a shift of the edge by one pixel is 30, which can be given by

ave = ΔxwN = tan1/ ()α (4) In order to characterize the impact of image noise on the accuracy of the MTF estimates, additional version of the simulated edge images were thus created by adding uncorrelated Poisson noise to the simulated images. Poisson noise was added by replacing each pixel by sample from a Poisson distribution with the pixel’s intensity as the Poisson parameter.

The supersampled ESF y(x) was constructed by “interlace” Nave consecutive lines in the simulated edge image[3]. Then the ideal ESF ()xf corresponding to y(x), which was an ideal step function, was constructed according to central position of the supersampled ESF. The actual edge position of the supersampled ESF can be obtained according to center-of- gravity analysis: (a) Obtain the mean edge image by averaging all slanted edge images(15 edge images with Poisson

noise were used in this work);(b) Obtain the supersampled ESF from Nave consecutive lines across the edge in the

averaged edge image, repeat this step for all other nonoverlapping groups of Nave consecutive lines across the edge, then determine the mean supersampled ESF by averaging all individual ESFs; (c) calculate Vmin from the mean

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supersampled ESF according to the average value of pixels shielded by the edge, and Vmax according to the average value of the remaining pixels; (d) calculate the mean pixel value of those pixels which have values lying between

()max + VV min 43 and ()3 max +VV min 4 , and determine the edge position corresponding to this mean pixel value. 2.2 Wiener deconvolution In the simplest possible case, without noise term, it would be sufficient to Fourier transform both terms of equation (1), divide by the Fourier transform of ()xf , and then inverse transform the result to obtain the desired term ()xh . The MTF is then the normalized magnitude of the Fourier transform of the LSF (xh ). This method was called direct deconvolution technique. However, system noise can not be ignored in the practical case, so that direct deconvolution method would amplify the high-frequency oscillations of system noise and lead to poor results. Therefore, with the presence of noise term, our goal is to find some φ(x) so that we can estimate ()xh as follows: ˆ()φ ()⊗= ()xyxxh (5)

where ˆ()xh is an estimate of (xh ) that minimizes the mean square error.

The Wiener deconvolution filter provides such aφ(x). In mathematics, Wiener deconvolution is an application of the Wiener filter to the noise problems inherent in deconvolution. It works in the frequency domain, attempting to minimize the impact of deconvoluted noise at frequencies which have a poor signal-to-noise ratio.

2 *( )()HF ωω F *()ω ()ω =Φ = (6) ()2 ()2 + NHF ()ωωω 2 ()2 + ()2 HNF ()ωωω 2

where Φ()ω , F()ω , H ()ω and N()ω are the Fourier transforms of φ(x), (xf ) , (xh ) and ()xn , respectively at spatial frequency ω . Because H ()ω and N()ω are always unknown for practical systems, a simple approximation of the 2 Wiener filter can be obtained by replacing the system dependent term ()2 HN ()ωω with a constant value C[8]:

F * (ω) ()ω =Φ (7) ()ω 2 + CF

The optimal C value depends on the noise level of the acquired image: high C values suppress the noise but can substantially affect the signal, whereas low values provide a more precise reconstruction of LSF but at the price of an increased noise level. Strictly speaking, the Fourier transformation F(ω) of (xf ) did not exist because the ideal ESF ()xf was an ideal step function which could not meet the absolute integrable condition of Fourier transformation. To meet the requirements of frequency domain Wiener deconvolution, we made a modification to the above simulated model. Assuming the supersampled ESF ()xy was mirrored as (− xy ), with the properties of convolution theory, ()− xy can be modeled as

() () ()()−+⊗−=− xnxhxfxy (8)

where ()− xf and ()− xn were the mirrored signal of (xf ) and (xn ) separately.

Concatenate the supersampled ESF ()xy and its mirrored one (− xy ) , we can obtain a novel model according to equation (1) and (8) ( )+− ()= ( )⊗− ()+ (− )+ ( )⊗ ( )+ (xnxhxfxnxhxfxyxy ) (9) []() () ()[] () (−++⊗+−= xnxnxhxfxf )

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t t() () ()+⊗= t ()xnxhxfxy (10) t where t() ( )+−= ()xyxyxy , () ( )+−= (xfxfxf ), and t( ) = ( )+ (− xnxnxn ) . t According to equation (10), t(xy ) and ()xf were square-wave-like function, they met the absolute integrable condition of Fourier transformation. With this novel model, the desired term (xh ) could be obtained from the symmetric pattern of t supersampled degraded ESF t()xy in the presence of ideal symmetric ESF (xf ) and system noise t()xn based on Wiener deconvolution.

3. RESULTS Figure 1(a) gives out the simulated slanted edge image without noise term according to equation (2), the supersampled ESF constructed from this simulated edge image was shown in figure 1(b).

14000

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Intensity of the oversampled profile oversampled the of Intensity 0 0 2000 4000 6000 8000 10000 Position in pixel pitch (a) (b) Figure 1. (a) simulated edge image with edge angle of arctan(1/30); (b) supersampled ESF constructed from 30 consecutive lines across the edge in the edge image Based on center-of-gravity analysis, the actual edge position of the supersampled ESF (figure 1(b)) was obtained for ideal ESF construction. Then a central region of supersampled ESF and the ideal ESF were mirrored and combined with the original one respectively to form the symmetric pattern of ESFs. The symmetric ESF were normalized by its maximum value. Figure 2(a) and figure 2(b) separately gives out the normalized symmetric supersampled ESF and the corresponding normalized symmetric ideal ESF, 3000 subpixels in the central region of the supersampled ESF were selected (from pixel position 4001 to 7000 in figure (b)). Without noise term as shown in figure 2, the desired LSF (xh ) could be obtained by Wiener deconvolution with the parameter C set as 0 (see figure 3). The true LSF of the simulation model was also given out for comparison. In order to characterize the impact of image noise on the accuracy of the MTF estimates, uncorrelated Poisson noises were separately produced and added into the simulated edge images. Then a mean supersampled ESF is determined by averaging the individual ESFs from the mean edge image of all simulated edge images. The symmetric supersampled ESF was constructed by combining the averaged ESF and its mirrored one (see figure 4(a)).

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1 1

0.8 0.8

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0.4 0.4 Relative Intensity Relative Intensity

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0 0 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000 Position in pixel pitch Position in pixel pitch (a) (b) Figure 2. (a) normalized symmetric supersampled ESF by combining the supersampled ESF and its mirrored one; (b) normalized symmetric ideal ESF by combining the ideal ESF and its mirrored one.

0.9 True LSF Obtained LSF 0.8

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0 0 100 200 300 400 500 600 Position in pixel pitch Figure 3 Comparison between the true LSF and the obtained LSF by Wiener deconvolution

Wiener deconvolutions were performed on the noisy symmetric supersampled ESF with the parameter C set as 0.5, 1.5, and 5.0 seperately. The corresponding results were shown in figure 4(b), figure(c), and figure 4(d) respectively. The results showed that higher C value suppressed the noise in LSF curve but broadened the width of LSF. Figure 5(a) shows the theoretical true MTF and the MTFs calculated with the LSFs obtained by Wiener deconvolution (see figure 4(b), figure 4(c), and figure 4(d)). The differences between the theoretical MTF and calculated MTFs were shown in figure 5(b). The graphs showed strong agreement between the theoretical MTF and the calculated MTF obtained by Wiener deconvolution with the Wiener parameter C set as 1.5. The maximum absolute difference between the theoretical MTF and Wiener deconvolution result (C=1.5) was only 0.71%.

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1 0.8

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Relative Intensity 0.4 0.2 Relative Intensity

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0 0 1000 2000 3000 4000 5000 6000 0 100 200 300 400 500 600 Position in pixel pitch Position in pixel pitch (a) (b)

1 1

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0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Position in pixel pitch Position in pixel pitch (c) (d) Figure 4 (a) symmetric supersampled ESF with noise term; (b) the obtained LSF by Wiener deconvolution (C=0.5); (c) the obtained LSF by Wiener deconvolution(C=1.5); (d) the obtained LSF by Wiener deconvolution(C=5)

1 0.02 0.9 True MTF C=0.5 0.015 C=0.5 C=1.5 0.8 C=1.5 0.01 C=5.0 C=5.0 0.7 0.005 0.6 0 0.5 -0.005 MTF 0.4 -0.01

0.3 Relative error -0.015

0.2 -0.02

0.1 -0.025

0 -0.03 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 -1 Spatial frequency(mm-1) Spatial frequency(mm ) (a) (b) Figure 5 (a) Comparison between theoretical MTF and calculated MTFs by Wiener deconvolutions; (b) Relative error between theoretical MTF and calculated MTFs by Wiener deconvolutions

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In the above case, the width of the square-like wave peak for the symmetric supersampled ESF or symmetric ideal ESF was 2000 subpixels (from subpixel position 2001 to 4000, see figure 2). To illustrate the effects of the width of square- like wave peak on measuring MTF, different central regions of supersampled ESF and the ideal ESF were selected to form symmetric ESFs. Figure 6 gives out the symmetric supersampled ESF with a central square-like wave peak of 4000 subpixels width.

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Position in pixel pitch Figure 6. Symmetric supersampled ESF with noise term (4000 subpixels width of central square-like wave peak) Figure 7(a) shows the theoretical MTF and the MTFs calculated with the LSFs obtained by Wiener deconvolution according to the symmetric ESF as shown in figure 6. The differences between the theoretical MTF and calculated MTFs were shown in figure 7(b). The optimum MTF was also obtained by Wiener deconvolution with Wiener parameter C=1.5, and in this case the maximum absolute difference from the true MTF was 0.83%.

1 0.03 True MTF C=0.5 0.9 C=0.5 0.02 C=1.5 C=1.5 0.8 C=5.0 C=5.0 0.7 0.01

0.6 0 0.5 MTF -0.01 0.4 Relative error 0.3 -0.02

0.2 -0.03 0.1

0 -0.04 1 2 3 4 5 6 7 8 9 10 10 20 30 40 50 60 70 80 90 100 -1 -1 Spatial Frequency(mm ) Spatial frequency(mm ) (a) (b) Figure 7 The central square wave peak of the symmetric ESF with 4000 subpixels width was used for MTF measurement (a) Comparison between theoretical MTF and calculated MTFs by Wiener deconvolutions; (b) Relative error between theoretical MTF and calculated MTFs by Wiener deconvolutions The central square wave peak of the symmetric ESF was further broadened to 8000 subpixels width. According to this case, the theoretical MTF and the MTFs calculated by Wiener deconvolution was shown in figure 8(a). The corresponding differences between the theoretical MTF and calculated MTFs were shown in figure 8(b). The relative MTF errors in low frequency range were all worse that 4% with Wiener deconvolution parameter C set as 0.5, 1.5, or 5.0. In this case, higher C value suppressed more low frequency MTF error. Therefore, we chose higher parameter C as 15

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for Wiener deconvolution. However, although Wiener deconvolution with parameter C=15 further suppressed the low frequency MTF error, the error was still worse than 4%. Moreover, the MTF errors in high frequency range were greatly worsened when a higher parameter C was chosen.

1 0.02 True MTF 0.9 C=0.5 0 C=1.5 0.8 C=5.0 0.7 C=15.0 -0.02

0.6 -0.04 0.5

MTF -0.06 0.4 C=0.5 Relative error 0.3 -0.08 C=1.5 C=5.0 0.2 C=15.0 -0.1 0.1

0 -0.12 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 -1 -1 Spatial frequency(mm ) Spatial frequency(mm ) (a) (b) Figure 8 The central square wave peak of the symmetric ESF with 8000 subpixels width was used for MTF measurement (a) Comparison between theoretical MTF and calculated MTFs by Wiener deconvolutions; (b) Relative error between theoretical MTF and calculated MTFs by Wiener deconvolutions

4. DISCUSSION AND CONCLUSION A novel method to determine the presampling MTF from edge images based on Wiener deconvolution has been described along with results from simulation experiments. The method requires mirroring of the oversampled edge spread function to form the symmetric ESF and avoids edge angle determination. The effects of Wiener parameter selection were investigated according to the relative error between the true MTF and calculated MTFs. The empirical results indicate that this method can obtain strong agreement between the theoretical MTF and the calculated MTF when suitable Wiener parameter is selected. The width of the square-like wave peak for the symmetric supersampled ESF is also important in the quality of the measured MTF. If the width is made too large with respect to the field of view, the MTF error from the true MTF will be greatly increased. In our experiment, the central square wave peak of the symmetric ESF with 2000 subpixels width (1/3 the image field of view) or 4000 subpixels width (1/2 the image field of view) can provide a decent result. However, an 8000 subpixels width having a width of 2/3 the image field of view failed to obtain the decent MTF. Keep in mind that the Fourier transform edges will be the sinc function and associated "ringing" will be introduced in the frequency domain. As the width of the square-like wave peak for the symmetric supersampled ESF increases, the main lobe of the sinc function narrows and the side lobes of the sinc function enter into the effective band of MTF such as frequencies within 10 mm-1. The “ringing" effect and zeros crossings of these side lobes may lead to the aliased MTF curve. Detailed researches into the effects of the width of the central square wave peak of the symmetric ESF on measured MTF are planned. Experimental results show that, the systematic error Δ / MTFMTF is less than 1% for all frequencies up to 10 mm-1. The accuracy and simplicity of the method makes it well suited for routine application in MTF measurement.In summary, our results demonstrate the ablity to use Wiener deconvolution method to recover the presampling MTF of an imaging system.

5. ACKNOWLEDGMENTS The authors acknowledge the funding supports from the National Natural Science Foundation of China (30870657, 30970775, 81101106, 61108081), Chinese National Programs for High Technology Research and Development (2009AA02Z413), Tianjin Municipal Government of China (09JCZDJC18200, 10JCZDJC17300), Tianjin Natural

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Science Foundation of China (12JCQNJC09400) and Ph.D. Programs Foundation of Ministry of Education of China (20100032120064 and 20110032120069). Contact Author: Guohui Li Email:[email protected]

REFERENCES [1] Fujita, H., Tsai, D-Y., Itoh, T., Doi, K., Morishita, J., Ueda, K. and Ohtsuka, A., “A simple method for determining the modulation transfer function in digital radiography”, IEEE transaction on medical imaging, 11(1), 34-39 (1992) [2] Cunningham, I.A. and Fenster, A., “A method for modulation transfer function determination from edge profiles with correction for finite-element differentiation”, Medical Physics, 14(4), 533-537 (1987) [3] Buhr, E., Günther-Kohfahl, S. and Neitzel, U.,“Accuracy of a simple method for deriving the presampled modulation transfer function of a digital radiographic system from an edge image”, Medical Physics, 30(9), 2323- 2331 (2003) [4] Samei, E., Ranger, N.T., Dobbins III, J.T. and Chen, Y., “Intercomparison of methods for image quality characterization. I. Modulation transfer function”, Medical Physics, 33(5), 1454-1465 (2006) [5] Carton, A.-K., Vandenbroucke, D., Struye, L., Maidment, A.D.A., Kao, Y.-H., Albert, M., Bosmans, H. and Marchal, G., “Validation of MTF measurement for digital mammography quality control”, Medical Physics, 32(6), 1684-1695 (2005) [6] Reimann, D.A., Jacobs, H.A., and Samei, E., “Use of Wiener filtering in the measurement of the two-dimensional modulation transfer function”, Proceedings of SPIE 3977, 670 - 680 (2000) [7] Greer, P.B., Doorn, T. Van., “Evaluation of an algorithm for the assessment of the MTF using an edge method”, Medical Physics, 27(9), 2048-2059 (2000) [8] Pratt, W.K., [Digital Image Processing], Wiley, New York, (1978)

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