A pitfall of using the circular‐edge technique with image averaging for spatial resolution measurement in iteratively reconstructed CT images

Abstract The circular‐edge technique using a low‐contrast cylindrical object is commonly used to measure the modulation transfer functions (MTFs) in computed tomography (CT) images reconstructed with iterative reconstruction (IR) algorithms. This method generally entails averaging multiple images of the cylinder to reduce the image noise. We suspected that the cylinder edge shape depicted in the IR images might exhibit slight deformation with respect to the true shape because of the intrinsic nonlinearity of IR algorithms. Image averaging can reduce the image noise, but does not effectively improve the deformation of the edge shape; thereby causing errors in the MTF measurements. We address this issue and propose a method to correct the MTF. We scanned a phantom including cylindrical objects with a CT scanner (Ingenuity Elite, Philips Healthcare). We obtained cylinder images with iterative model reconstruction (IMR) algorithms. The images suggested that the depicted edge shape deforms and fluctuates depending on slice positions. Because of this deformation, image averaging can potentially cause additional blurring. We define the deformation function D that describes the additional blurring, and obtain D by analyzing multiple images. The MTF measured by the circular‐edge method (referred to as MTF') can be thought of as the multiplication of the true MTF by the Fourier transformation (FT) of D. We thus obtain the corrected MTF (MTFcorrected) by dividing MTF' by the FT of D. We validate our correction method by comparing the calculated images based on the convolution theorem using MTF' and MTFcorrected with the actual images obtained with the scanner. The calculated image using MTFcorrected is more similar to the actual image compared with the image calculated using MTF', particularly in edge regions. We describe a pitfall in MTF measurement using the circular‐edge technique with image averaging, and suggest a method to correct it.


| INTRODUCTION
Iterative reconstruction (IR) algorithms have been widely implemented for clinical computed tomography (CT). IR methods can reduce image noise, which is mainly caused by radiation quantum fluctuation in the CT projection data, while maintaining (or improving) the spatial resolution. [1][2][3][4] Most IR algorithms incorporate statistical models (photon and noise statistics) and the scanner geometry and optics, and have nonlinear properties. 5,6 It has been reported that the nonlinear properties cause spatial resolution variability depending on image noise levels and object contrast. [7][8][9] Therefore, the modulation transfer function (MTF), one of the most comprehensive metrics for spatial resolution, measured using traditional approaches with high contrast wires or beads, is not applicable for characterizing the spatial resolution of clinical IR images. Richard et al. developed a new MTF measurement approach, called the "circular-edge technique," using a low-contrast cylindrical object. 7 This technique has been widely used for MTF measurements of IR images. Most of the studies using this technique computed the average of the consecutive cross-sectional images of the cylinder and/or the average of many images acquired from repeated scans to improve the signal-to-noise ratio. 1,[9][10][11][12] Because of the intrinsic nonlinearity of IR algorithms, the resulting image properties are complicated compared with those of filtered back projection (FBP) images. Leipsic et al. 13 reported that in cardiac CT angiography, reconstructions obtained using adaptive statistical iterative reconstruction (ASIR) differ in appearance from traditional FBP images, exhibiting a different noise texture and smoothed borders. Singh et al. 14 observed a step-like artifact at tissue interfaces (such as the margins of the liver, spleen, and blood vessels) in abdominal CT images reconstructed using ASIR. The imaging at border/edge regions using IR algorithms is potentially sensitive to slight fluctuations in the CT projection data, including noise. In a phantom study, Li et al. 9 obtained multiple IR images using repeated scans, and assessed the standard deviation of CT values locally in the edge regions of circular objects. They considered this standard deviation as "edge-noise," and found that the edge-noise was greater than the standard deviation computed for uniform regions; thereby suggesting a specific anomaly in edge regions. The object edge shape depicted in IR images may deform slightly with respect to the ideal shape (circle) and fluctuate in repeated scans; this effect is one potential reason for the increased edge-noise. Averaging multiple images can reduce the image noise, but does not effectively improve the deformation and fluctuation of the object edge shape depicted in the IR images. When applying image averaging with the circular-edge technique, the occurrence of edge shape deformation may adversely affect MTF measurements.
The aim of this study is to address this issue and propose a method to correct the MTF measured using the circular-edge technique. To verify the validity of the proposed method, we compared the computed images obtained by applying the convolution theorem using the corrected MTF with the true images obtained by the CT scan.

2.A | Equipment and imaging parameters
We used the sensitometry module (CTP404) included with the Catphan 600 phantom (The Phantom Laboratory, Salem, NY). The module consists of eight cylindrical objects; we used two objects made from Delrin (approximately 350 HU at 120 kVp) and polystyrene (PS) (approximately −30 HU at 120 kVp). The background CT value was approximately 100 HU at 120 kVp. We placed the phantom in the center of the scanner field of view (FOV) such that the cylinder was parallel to the z direction, and therefore perpendicular to the x-y scanning plane. We scanned the phantom with a multidetector row CT scanner (Ingenuity Elite, Philips Healthcare, Netherland) at 120 kVp, 100 mA, with a one-second/ rotation, a pitch of 1.17, and detector configuration of   A CT image is characterized by the spatial resolution of the system. When considering a CT image of a uniform cylindrical object placed parallel to the z direction (perpendicular to the x-y scanning plane), the resulting image is expressed as follows: [15][16][17] NARITA AND OHKUBO where O x; y ð Þ is an object function of a circular shape with uniform density, and PSF x; y ð Þ is the two-dimensional (2D) point spread function (PSF). The operator * is the 2D convolution. Because of the uniform circular shape of the cylinder, the cross-sectional image I x; y ð Þ does not change with the slice position along the z-axis. However, we observed differences between consecutive slice images reconstructed using IMR (Fig. 1). To describe a practical image generation system that includes the object-shape deformation present in IMR images, we make several assumptions and modify Eq. (1) as follows.
First, we include a deformation between each of the cross-sectional images and the original circular shape in the object function.
Thus, we write Eq. (1) as follows: where I where I where D x; y ð Þ is a blurring function whereby the blurring is originated from the deformations in O 0 i x; y ð Þ. Therefore, we refer to D x; y ð Þ as the deformation function. By applying Eqs. (4) to (3), we The circular-edge technique is based on Eq. (1) assuming the ideal circular shape of O x; y ð Þ and the isotropy of the in-plane resolution; this provides an MTF that is equivalent to the Fourier transformation of PSF x; y ð Þ. That is, the resultant MTF is written as follows: where F is the Fourier transform, u and v are the spatial frequency coordinates in the x and y directions, respectively, and w is spatial frequency in the radial direction. When applying image averaging with the circular-edge technique, we assume Eq.   We applied the circular-edge technique to obtain the frequency characteristics of the deformation function D x; y ð Þ, which is the  (2) as follows: 19 the frequency characteristics of D x; y ð Þ (Fig. 3). The difference between MTF 0 w ð Þ and MTF corrected w ð Þ obtained for the IMR Body    We calculated the RMSEs for these comparisons for all 200 slice images, and the average RMSEs are shown in Table 2. The average

2.C.2 | MTF correction method
RMSEs corresponding to MTF corrected w ð Þ were smaller than those corresponding to MTF 0 w ð Þ under all conditions.

| DISCUSSION
We considered that images reconstructed using the IMR algorithm potentially deform from the ideal object shape, depending on the noise in the edge regions (Fig. 1). This deformation is caused by the intrinsic nonlinearity of the IMR algorithms, and was not observed in  Fig. 3(e). The MTF' and MTF corrected used for calculating image (g) and (i) are shown in Fig. 3(f).
Large deformations were observed in the images reconstructed using the algorithm Body SharpPlus compared with those reconstructed using Body Routine (Fig. 1). The deformations caused blurring in an average image, reducing the frequency characteristics of D x; y ð Þ. As shown in Fig. 3 Fig. 2), even when using a lower-contrast object (the contrast of the PS to the background was   Fig. 3 approximately −130 HU). However, when using a considerably lower-contrast object, improvements of generating the effective object function might be necessary.

| CONCLUSION
We demonstrate a pitfall in the circular-edge technique accompanied with image averaging for MTF measurement, particularly when using an edge-enhancement type IMR algorithm. To address this issue, we made several assumptions, modified the equation for the image generating system, and proposed a method to correct the MTF. We confirmed the validity of the proposed method by comparing the calculated images based on the corrected MTF with the actual (true) images. When using an edge-enhancement type IMR algorithm, the MTF correction method improves the results obtained using the circular-edge technique.

This work was supported by JSPS KAKENHI Grant Number
JP17K09059. We thank Irina Entin, M. Eng., from Edanz Group (www.edanzediting.com/ac) for editing a draft of this manuscript.

CONF LICT OF I NTEREST
The authors have no relevant conflict of interest to disclose.