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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 中国科技论文在线 http://www.paper.edu.cn 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]; Design and Quality for Biomedical Technologies VI, edited by Ramesh Raghavachari, Rongguang Liang, Proc. of SPIE Vol. 8573, 85730R · © 2013 SPIE CCC code: 1605-7422/13/$18 · doi: 10.1117/12.2000580 Proc. of SPIE Vol. 8573 85730R-1 转载 中国科技论文在线 http://www.paper.edu.cn 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 y() x= f () x ⊗ h () x+ n( x) (1) where ⊗ is the convolution operator, x is the supersampled spatial displacement, f( x) is the ideal ESF with no degradation corresponding to y(x), h() x denotes the blurring mask on account of the detector LSF, and n( x) is the system noise. Theoretically, a deconvolution procedure could recover the detector LSF h( x) in the presence of noise n() x and the ideal ESF f() x . 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 Δx = wtanα , 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 ()*BF− + F for x ≥ 0 ⎪ 2 ESF() x = ⎨ (2) 2 − erx ⎪()*BF− ()+ F for x < 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 MTF() u = (3) r2 + ()2π u 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 Nave = w/Δ x = 1 tan()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 f() x 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 Proc. of SPIE Vol. 8573 85730R-2 中国科技论文在线 http://www.paper.edu.cn 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 ()VVmax + 3min 4 and ()3VVmax + 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 f() x , and then inverse transform the result to obtain the desired term h() x . The MTF is then the normalized magnitude of the Fourier transform of the LSF h( x). 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 h() x as follows: hˆ() x=φ () x ⊗ y () x (5) where hˆ() x is an estimate of h( x) 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 FH*(ω)() ω F *()ω Φ()ω = = (6) FHN()ω2 () ω2 + () ω 2 FNH()ω2 + () ω2 () ω 2 where Φ()ω , F()ω , H ()ω and N()ω are the Fourier transforms of φ(x), f( x) , h( x) and n() x , 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 NH()ω2 () ω with a constant value C[8]: F * (ω) Φ()ω = (7) FC()ω 2 + 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.
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