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We investigate the detection performance of transverse and longitudinal planes for various signal sizes (i.e., 1 mm to 8 mm diameter spheres) in cone beam computed tomography (CBCT) images. CBCT images are generated by computer simulation and images are reconstructed using an FDK algorithm. For each slice direction and signal size, a human observer study is conducted with a signal-known-exactly/background-known-exactly (SKE/BKE) binary detection task. The detection performance of human observers is compared with that of a channelized Hotelling observer (CHO). The detection performance of an ideal linear observer is also calculated using a CHO with Laguerre-Gauss (LG) channels. The detectability of high contrast small signals (i.e., up to 4-mm-diameter spheres) is higher in the longitudinal plane than the transverse plane. It is also shown that CHO performance correlates well with human observer performance in both transverse and longitudinal plane images.
© 2016 Optical Society of America
1. Introduction
Introduction of X-ray flat panel detectors has resulted in widespread use of cone beam computed tomography (CBCT) and has accelerated the utilization of three-dimensional (3-D) images for clinical diagnosis [1,2]. Diagnostic accuracy is closely related to imaging performance; thus, optimization of imaging systems and image processing algorithms has been an important issue. CT imaging performance can be quantitatively characterized in terms of Fourier-based metrics such as modulation transfer function (MTF), noise power spectrum (NPS), noise equivalent quanta (NEQ), and detective quantum efficiency (DQE) [3–13]. While these metrics provide essential information about imaging performance, they are insufficient when the system produces non-linear and non-stationary images.
CT images are always taken for a specific diagnostic purpose, such as lesion detection; thus, image quality should be judged by the ability to detect a given task, known as task-based assessment [14]. The most desirable way to perform task-based image quality assessment is to conduct a human observer study because a human observer is the one to make the diagnostic decision. However, human observer studies are time-consuming and expensive, and results can vary depending on experimental conditions. Therefore, it is impractical to use human observers to evaluate image quality for a variety of image types and tasks [15]. To resolve this limitation, a mathematical observer called a channelized Hotelling observer (CHO) has been proposed as a surrogate for human observers [16]. Several studies have shown that a CHO with anthropomorphic channels, such as Difference-of-Gaussians (DOG), can predict human observer performance, and therefore the CHO has been widely used for task-based image quality assessment and optimization of imaging systems [17–21]. Alternatively, CHO models using efficient channels, such as Laguerre-Gauss (LG), can approximate the performance of the Hotelling observer, providing upper bounds for ideal linear observer performance [22, 23].
Much work has been conducted to assess the image quality of CT images using human and mathematical observers. For 3-D CT images, the detection performance of different reconstruction algorithms (e.g., linear vs non-linear) was evaluated, [24–26] and the effects of various imaging conditions on signal detection (e.g., lesions, microcalcifications, and lung nodules) were also explored [27–29]. These prior studies used only transverse planes to evaluate detection performance. In [30], detection performance of an object in transverse and longitudinal planes was compared, and it was shown that the transverse plane provided better detection performance than the longitudinal plane for the detection task of a 6.4-mm-diameter sphere. However, the 3-D NPS of a CBCT image is highly asymmetric; thus, depending on signal size, objects in transverse and longitudinal planes might have different detectabilities.
In this work, we compared the detection performance of various signal sizes in the transverse and longitudinal planes of CBCT images with isotropic 3-D spatial resolution. The main finding of this work is that using longitudinal planes can be more advantageous in performing detection tasks of small lesions in CBCT images. For the evaluation, we focused on a signal-known-exactly/background-known-exactly (SKE/BKE) binary detection task. The signal size ranged from 1 mm to 8 mm in diameter, which corresponds to small lesions or early stage cancers [31–37]. For each signal size and slice direction, the detection performance of human observers and a CHO with DOG was evaluated. The detection performance of an ideal linear observer was represented by a CHO with LG.
The rest of this paper is organized as follows. In section 2, we briefly review the theory of a binary detection task and describe experiments using human observers and mathematical observers, respectively. In section 3, we compare human and mathematical observer performance. In section 4, we discuss our findings and present conclusions.
2. Methods
2.1. Binary detection task
We consider a SKE/BKE binary detection task which determines whether an image contains a signal (lesion) or not. Two hypotheses for the binary detection task (i.e., H0 for signal-absent hypothesis and H1 for signal-present hypothesis) are given by
where vectors s and b are the reconstructed noiseless signal and background image of the CBCT system, respectively, n is the noise stemming from the Poisson noise added to the projection data, and g is the resulting data vector. Because the purpose of our study is to compare detection performance between transverse and longitudinal planes of reconstructed 3-D CBCT images, 2-D transverse and longitudinal planes are used as g.2.2. Image generation
Eight spherical objects were used to compare the detection performance of transverse and longitudinal planes of 3-D CBCT images. Spheres were centered at the iso-center in each slice. The diameters of the spheres were [1, 2, 3, 4, 5, 6, 7, 8] mm, and the corresponding attenuation coefficients were [0.075, 0.035, 0.025, 0.0175, 0.015, 0.01, 0.0095, 0.008] cm−1, which were determined by trial and errors using the transverse plane to produce a percent correct of human observers around 0.9 [38] with the presence of quantum noise. As a result, larger spheres have smaller attenuation coefficients. Note that the contrasts of spheres in CT number were [75, 35, 25, 17.5, 15, 10, 9.5, 8]. The analytic projection data of spheres were generated using 17 × 17 source and detector lets, which modeled the finite X-ray focal spot and detector cell. We assumed that the background was uniform air, and thus its attenuation coefficient was 0 cm−1. For quantum noise simulation, uniform Poisson noise for a detected signal of 200 photons per detector cell was generated. Log normalization was performed on the generated Poisson noise and then added to the noiseless projection data. The projection data were reconstructed using the FDK algorithm [39]. A Hanning weighted ramp filter was used as a reconstruction filter, and voxel-driven back-projection was performed using linear interpolation. To avoid noise aliasing, a small voxel size (0.119 mm in all directions) and a detector quarter offset were used. Simulation parameters are summarized in Table 1. Figure 1 shows the reconstructed noiseless sphere images used for the calculation of task SNR of the CHO, and Fig. 2 shows the reconstructed noise only images of the transverse and longitudinal planes.
Table 1. Simulation parameters
Fig. 1 Reconstructed sphere images from 1 mm diameter (left) to 8 mm diameter (right) in the (a) transverse and (b) longitudinal planes. Unit of the reconstructed values is cm−1 and the display window is [00.05].
Fig. 2 Noise structures in the (a) transverse and (b) longitudinal planes. Unit of the reconstructed values is cm−1 and the display window is [−0.2 0.2].
2.3. Human observer study
The detection performance of spheres in transverse and longitudinal planes was evaluated by a two-alternative forced-choice (2AFC) detection task [40]. Five human observers participated in these experiments. For the 2AFC detection task, a signal-absent image and signal-present image were displayed simultaneously on the same monitor. Because the detection task was a SKE/BKE, signal only image was shown to the human observers during the test. A dark-gray background was used around images in order to minimize human observer distraction (Fig. 3). The display monitor was a 21.3 inch Nio 3MP LED monitor with a maximum resolution of 2048 × 1536 pixels (Barco, Kortrijk, Belgium) with a pixel pitch of 0.2155 mm. In each trial, signal-present and signal-absent images were switched randomly and the observers were asked to select the signal-present image with instant feedback (correct/incorrect). The experiment was performed in a completely darkroom, and the viewing distance and decision time were not limited.
All observers participated in a training session composed of 100 trials for each signal size and slice direction. After training, each observer ran two sessions per experimental condition. For each signal size and slice direction, the test image set of each session consisted of 100 128 × 128 pixel (15.2 × 15.2 mm2) images; thus, each observer participated in 3200 trials in total. The detection performance of human observers was recorded using percent correct Pc as an estimate of area under the curve (AUC) [41]
where Nt is the number of trials. If an observer made the correct decision for the i-th trial, oi is set to 1. Otherwise, oi is set to 0. Variance of Pc was estimated using multireader multicase (MRMC) variance analysis for binary data [42, 43].2.4. Mathematical observer study
2.4.1. LG channels
A CHO with rotationally-symmetric LG channels can approximate the Hotelling observer performance when the signal is rotationally-symmetric in a known location, and the background is stationary [22, 23, 44, 45]. Because our signal was a sphere embedded in a uniform background and the background noise was stationary within a field of view (FOV), we used a CHO with LG channels to estimate Hotelling observer performance in transverse and longitudinal planes. LG channels are modelled by LG functions defined by the product of Laguerre polynomials and Gaussian functions
where r represents a 2-D spatial coordinate and au is the width of the Gaussian function. La-guerre polynomials Lp(x) are defined as where p is a polynomial order.Detection performance of LG CHO depends on the number of channels Nc and the value of au. Because the optimal value of au that maximizes detection performance depends on signal size [23], au should be determined adaptively for each signal size. We selected the value of au equal to the diameter of signal in pixels [45]. Table 2 summarizes the selected au values for the different signal sizes. We computed the detection performance of each signal with the selected au by increasing Nc from 3 to 13. As described in section 3, Nc was selected to be 7 when the detection performance saturated. Note that the detection performance of the LG CHO was measured by the task SNR of CHO as described in section 2.4.3. Example images of LG channels with Nc = 7 and au = 8 are provided in Fig. 4.
Table 2. au for each signal size (pixels). Pixel size is 0.119 mm
Fig. 4 128 × 128 LG channels with Nc=7 and au=8 from p=0 (left) to p=6 (right). The Gaussian in the first channel is similar to 1 mm diameter signal.
2.4.2. DOG channels
To predict the detection performance of human observers, we used CHO with DOG channels because it is known to be a good predictor of human observer performance for the types of background statistics and signals used in this study [18]. DOG channels are defined in the frequency domain as
where ρ is radial frequency, Q is a multiplicative factor, and σj is the standard deviation of the j-th channel defined as σj = σ0αj. We used the following DOG channel parameter values: σ0 = 0.005,α = 1.4 and Q = 1.67 [18]. We used seven DOG channels since it produced the best correlation with the human observer when internal noise was added. Example images of DOG channels are provided in Fig. 5.2.4.3. Figures of merit
To quantify the detection performance of the mathematical observer, the task SNR of the CHO was computed [18] as
where t is a transpose operator, Δg is the mean difference image between signal-present and signal-absent images, T is the channel matrix, Kg is the covariance matrix of image noise, and Kε is the covariance matrix of internal noise. For LG channels, T comprised discrete sampled values of Eq. (4), and for DOG channels, T comprised discrete sampled values of the inverse Fourier transform of Eq. (6). Because the noise in the transverse and longitudinal planes had a zero mean, Δg was a signal only image, which was acquired from the noiseless reconstruction. The channelized noise covariance matrix, Kv = TtKgT, was estimated using 3000 training data sets [46].To compare observer performance, Pc of the human observer had to be converted to task SNR. Because the Pc in a 2AFC experiment is an estimate of the AUC [41], the task SNR of the human observer was calculated as [47]
where er f is a Gauss error function.2.4.4. Internal noise
To match the task SNR of the DOG CHO with that of human observers, internal noise was added. We used a channelized internal noise model that was proportional to the diagonal components of Kv [48].
where p is a proportional constant. Note that, in this work, p was optimized for the smallest signal (i.e., 1-mm-diameter sphere).2.4.5. Human efficiency
Human efficiency relative to that of the LG CHO represents how much available information is used by the human observer when detecting signals [49]. Human efficiency relative to that of the LG CHO is defined [20, 49] as
2.5. Relative dose efficiency of the two planes
Noise variance of a reconstructed CT image is inversely proportional to the dose [50], and its effect on detectability is reflected in the noise covariance matrix of the mathematical observer. Because the CBCT system produces different noise structures in transverse and longitudinal planes, the relative dose efficiency between transverse and longitudinal planes can be calculated by the SNR2 ratio of the two slice directions. In this work, we calculated the relative dose efficiency between transverse and longitudinal planes of the human observer as
The relative dose efficiency was compared for each signal size.
3. Results
Figure 6 shows the averaged Pc and task SNR in transverse and longitudinal planes with 95% confidence intervals, and the ratio of task SNR values between transverse and longitudinal planes. It is observed that the length of confidence interval representing both case and inter-observer variability is similar for all signal sizes. Note that the averaged Pc was converted to a task SNR using Eq. (8). The averaged Pc values of the transverse and longitudinal planes show a similar trend as a function of signal size owing to the nature of Pc saturating to one. But, the task SNR shows evidence that the task SNR values of small signals (i.e., 1 mm to 4 mm diameter signals) are higher in the longitudinal planes. The main cause of this behavior is the different noise structures in transverse and longitudinal planes.
Fig. 6 (a) Averaged Pc with 95% confidence interval in the transverse and longitudinal planes. (b) Comparison of task SNR in the transverse and longitudinal planes. (c) The ratio of averaged task SNR between the transverse and longitudinal planes.
In FDK reconstruction, the reconstruction filter (i.e., Hanning weighted ramp filter) is applied only in the transverse direction and thus the NPS of the transverse and longitudinal plane shows different patterns (i.e., symmetric in the transverse plane and asymmetric in the longitudinal plane), as shown in Fig. 7(a). The noise structure of the transverse (longitudinal) plane is reflected as negative (positive) off-diagonal terms of covariance matrices in a Hotelling observer, as shown in Fig. 7(b), thereby revealing different correlation structures of the two planes. Note that the covariance matrices are estimated from 3000 independent noise realizations. In Fig 7(c), comparison of the profiles of the transverse and longitudinal NPS estimates shows that the noise power of the transverse plane is higher around the mid-frequencies, but lower near the DC component compared to the longitudinal plane. These NPS profiles are consistent with the finding that the detectability of small diameter spheres is higher in the longitudinal plane. Although we could relate the overall trend of Fig. 6 to the NPS profiles, when the noise structure is complex like the longitudinal plane, a complete characterization of the detectability should be performed using task SNR of LG CHO. Note that task SNR of LG CHO provides an upper bound of detection performance, and thus their trends may not match the human study results exactly. The NPS in Fig. 7 is unitless, since it is computed from a single transverse (longitudinal) plane. Since the NPS is equivalent to the projection of 3-D NPS [4], the NPS has a non-zero DC component.
Fig. 7 (a) 2-D NPS of transverse plane and longitudinal plane, (b) the first 2000 × 2000 zoomed-in version of the 16384 × 16384 full covariance matrix in the transverse and longitudinal planes, (c) central profile of corresponding NPS.
For a qualitative comparison, sampled transverse and longitudinal plane images with different sphere sizes are shown in Fig. 8. To more clearly demonstrate the different detectabilities of the spherical objects, we adjusted the attenuation coefficients of the spheres as [0.25, 0.2, 0.15, 0.1, 0.09, 0.08 0.07 0.06] cm−1 with increasing sphere size, which is much higher than the attenuation coefficients used in human and mathematical observer studies. The similar trends of detectability in transverse and longitudinal planes can be appreciated as presented in Fig. 6.
Fig. 8 Sampled (a) transverse and (b) longitudinal plane images with different sphere sizes from 1 mm diameter (left) to 8 mm diameter (right). Unit of the reconstructed values is cm−1 and the display window is [−0.3 0.3]. The image size is 15.2 × 15.2 mm2.
To examine the detection performance of the ideal linear observer for each signal size and slice direction, the task SNR values of the LG CHO as a function of Nc are plotted in Fig. 9. For each signal size and slice direction, the task SNR saturates when Nc is larger than 6, and thus we select Nc = 7 for all experimental conditions.
Figure 10 compares the task SNR and Pc of the LG CHO, DOG CHO without internal noise, DOG CHO with internal noise, and human observers. It is observed that the task SNR of the LG CHO is larger than that of the DOG CHO for all signal sizes. The task SNR of the DOG CHO without internal noise is higher than that of the human observers, and this difference is reduced when internal noise was added. The proportional constant p of Eq. (9) is 4.4 for the transverse plane and 3.5 for the longitudinal plane, which is optimized for the smallest signal size, and then used for all signal sizes. With this internal noise, the SNR of the DOG CHO matches well with that of the human observers. Note that using Pc can be misleading when the values are close to one, especially in the case of LG CHO and DOG CHO without internal noise. In our test images, the ratio of pixel SNR (the ratio of mean to standard deviation) between transverse plane and longitudinal plane is the same for all signal sizes, demonstrating the limitation of pixel SNR for evaluating detectability in CT images.
Fig. 10 SNR and Pc of the LG CHO, DOG CHO, and human observers in the transverse and longitudinal planes.
Table 3 summarizes human efficiency. Human efficiency in the transverse plane decreases as sphere size increases. Except for the 1-mm-diameter sphere, human efficiency is higher in the longitudinal plane, demonstrating that human observers use more information from the longitudinal plane to detect signals. Note that higher human efficiency does not imply better detectability. Table 4 summarizes the relative dose efficiency of human observers for each signal size, indicating that using the longitudinal plane is more beneficial for detecting a small diameter sphere or can achieve the same detectability as with a transverse plane with a reduced dose.
Table 3. Human efficiency
Table 4. Relative dose efficiency of the two planes
4. Discussion and conclusion
We investigated the detection performance of transverse and longitudinal planes from volumetric CBCT images. Detection performance of transverse and longitudinal planes with spheres of different diameters was measured by human observers and compared for each signal size and slice direction. It was shown that longitudinal planes had better detectability for high-contrast smaller spheres (i.e., up to 4 mm in diameter), which was mainly caused by the different noise structures of transverse and longitudinal planes in FDK reconstruction.
The task SNR of transverse and longitudinal planes was also calculated by the DOG CHO, and compared with that of human observer performance. The task SNR of the human observers and DOG CHO showed a similar trend, and the gap between the two observers was reduced by adding internal noise. The level of the internal noise matching between the human observers and DOG CHO was optimized with the smallest signal size, and the same amount of internal noise was used for different signal sizes. Although the task SNR of the DOG CHO with internal noise remains within the error bars of the human observer performance, additional optimization of the DOG CHO can be conducted by tuning channel parameters with different channel numbers, which would be useful to evaluate different tasks in CT images.
As described in Table 3, humans are not good at detecting large signals in correlated backgrounds (e.g., using DOG channels suppresses the DC component to mimic human perception). Perceptually, the longitudinal plane image looks flatter than the transverse plane image; thus, using the longitudinal image is similar to smoothing the texture, which helps improve human detection performance. Conversely, the optimal observer is not limited in perceiving a large signal (i.e., low-frequency contents in the Fourier domain), resulting in an increase in the difference between the optimal (LG CHO) and human performance. However, the difference between the DOG CHO and humans seems to remain constant when the signal size ranges from 4 mm to 8 mm.
The upper bound of detection performance for each signal size and slice direction was calculated by the LG CHO since the LG CHO provided an optimal task SNR of the ideal linear observer for a rotationally-symmetric signal. An alternative is to use the Hotelling observer directly; however, training the Hotelling observer requires a large number of images, which is time-consuming and impractical. For different types of signals, a CHO with partial least square (PLS) channels [51] can be used to approximate Hotelling observer performance. Because PLS channels are estimated directly from images, its performance is not restricted by the shape of the signal, channel parameters, or background statistics. For linear reconstruction, Fourierdomain detectability can be considered for calculation of the optimal SNR, as described in Eq. (5) in [10]. Although it is not presented in this paper, we found that the optimal SNR using Fourier-domain detectability and the LG CHO matched within a 10% error range. Note that Fourier-domain detectability is not effective for calculating optimal SNR when the background is non-stationary or the system is not linear.
In our simulations, we did not include the effect of scatter generated by the object. Scatter degrades detection performance due to reduced signal contrast, but the overall effects are likely to remain the same because scatter affects transverse and longitudinal planes similarly in the case of uniform backgrounds [30]. Detector lag is another factor that can affect detection performance. While the amount of lag is relatively low in current MDCT and flat panel CBCT systems, systems using a direct conversion detector could have more lag effects that introduce additional noise correlation in the reconstructed image [52]. The effect of detector lag on detectability is a subject of future research.
The FDK algorithm causes locally varying noise and resolution properties, especially for the case of a large cone angle [7, 9]. The effect of a large cone angle, known as cone beam artifacts, was not considered in this work because the spherical objects were centered at the iso-center. Because the large cone angle in the FDK algorithm blurs both signal and noise, the detection performance trends in transverse and longitudinal planes can differ, which would be an interesting subject for future research. Although the FDK algorithm produces non-stationary noise properties across the FOV [9], cone-parallel rebinning, which is used in many practical CT scanners, significantly reduces the variation in noise across the FOV [53]. Thus, even with the non-stationary noise statistics caused by the FDK, performance trends similar to those of our detection task are expected. In FDK reconstruction, the choice of the apodization filter and interpolation method in backprojection might change the detection performance. We chose a Hanning-weighted ramp filter with linear interpolation since it produced the highest detection performance in transverse and longitudinal planes [54]. Reconstruction voxel size was selected as half of the intrinsic voxel size at the iso-center with isotropic resolution in order to acquire sufficient sampling and accurate results [55]. The anisotropic resolution produced by detector binning, which is frequently used in flat panel CBCT systems, might change the detection performance in transverse and longitudinal planes. In this study, detector binning was not considered since it produced a suboptimal detection performance [56].
In this work, detection tasks were performed using single slice images; however, our findings can be extended to the detection task of 3-D signals using multislice images. Clearly, using multislice images will increase detection performance in transverse and longitudinal planes [57], but similar results are expected because the noise structure produced by the FDK algorithm is similar along the same slice direction. Our detection task used a uniform background relevant to lesion detection within solid organs such as the liver or abdomen. For a detection task in lung or breast images, it would be necessary to use anatomical backgrounds, which introduce additional noise factors [45, 58, 59]. The extension of the present work to detection tasks with anatomical backgrounds is a subject of future research.
In conclusion, the detection performance of high-contrast small objects (up to a 4-mm-diameter sphere) was higher in the longitudinal planes of a CBCT image. Our results indicate that it is more beneficial to use the longitudinal plane in order to achieve better detection performance of small objects.
Acknowledgments
This research was supported by the MSIP (Ministry of Science, ICT and Future Planning), Korea, under the IT Consilience Creative Programs (IITP-2015-R0346-15-1008) supervised by the IITP (Institute for Information& Communications Technology Promotion), Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2015R1C1A1A01052268) and the framework of international cooperation program managed by National Research Foundation of Korea (NRF-2015K2A1A2067635).
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