^{1,a)}, F. Verhaegen

^{2}and D. A. Jaffray

^{3}

### Abstract

X-ray scatter is a source of significant image quality loss in cone-beam computed tomography (CBCT). The use of Monte Carlo (MC) simulations separating primary and scattered photons has allowed the structure and nature of the scatter distribution in CBCT to become better elucidated. This work seeks to quantify the structure and determine a suitable basis function for the scatter distribution by examining its spectral components using Fourier analysis.

The scatter distribution projection data were simulated using a CBCT MC model based on the EGSnrc code. CBCT projection data, with separated primary and scatter signal, were generated for a 30.6 cm diameter water cylinder [single angle projection with varying axis-to-detector distance (ADD) and bowtie filters] and two anthropomorphic phantoms (head and pelvis, 360 projections sampled every 1°, with and without a compensator). The Fourier transform of the resulting scatter distributions was computed and analyzed both qualitatively and quantitatively. A novel metric called the scatter frequency width (SFW) is introduced to determine the scatter distribution's frequency content. The frequency content results are used to determine a set basis functions, consisting of low-frequency sine and cosine functions, to fit and denoise the scatter distribution generated from MC simulations using a reduced number of photons and projections. The signal recovery is implemented using Fourier filtering (low-pass Butterworth filter) and interpolation. Estimates of the scatter distribution are used to correct and reconstruct simulated projections.

The spatial and angular frequencies are contained within a maximum frequency of 0.1 cm^{−1} and 7/(2π) rad^{−1} for the imaging scenarios examined, with these values varying depending on the object and imaging setup (e.g., ADD and compensator). These data indicate spatial and angular sampling every 5 cm and π/7 rad (∼25°) can be used to properly capture the scatter distribution, with reduced sampling possible depending on the imaging scenario. Using a low-pass Butterworth filter, tuned with the SFW values, to denoise the scatter projection data generated from MC simulations using 10^{6} photons resulted in an error reduction of greater than 85% for the estimating scatter in single and multiple projections. Analysis showed that the use of a compensator helped reduce the error in estimating the scatter distribution from limited photon simulations by more than 37% when compared to the case without a compensator for the head and pelvis phantoms. Reconstructions of simulated head phantom projections corrected by the filtered and interpolated scatter estimates showed improvements in overall image quality.

The spatial frequency content of the scatter distribution in CBCT is found to be contained within the low frequency domain. The frequency content is modulated both by object and imaging parameters (ADD and compensator). The low-frequency nature of the scatter distribution allows for a limited set of sine and cosine basis functions to be used to accurately represent the scatter signal in the presence of noise and reduced data sampling decreasing MC based scatter estimation time. Compensator induced modulation of the scatter distribution reduces the frequency content and improves the fitting results.

I. INTRODUCTION

II. MATERIALS AND METHODS

II.A. Monte Carlo simulation system

II.A.1. X-ray sources and energy

II.A.2. Compensators

II.A.3. Phantoms

II.A.4. Imaging geometry

II.B. Scatter spatial frequency

II.C. Scatter distribution estimation from limited photon and projection simulations

II.C.1. Scatter corrected CBCT reconstruction

III. RESULTS

III.A. Scatter spatial frequency spectrum

III.A.1. Cylinder

III.A.2. Anthropomorphic phantoms

III.B. Scatter distribution estimation from limited photon simulations

III.B.1. Scatter corrected CBCT reconstruction

IV. DISCUSSION

V. CONCLUSIONS

### Key Topics

- X-ray scattering
- 44.0
- Medical imaging
- 35.0
- Photon scattering
- 26.0
- Photons
- 25.0
- Monte Carlo methods
- 22.0

##### A61B6/03

##### G01N33/48

##### G06F19/00

##### G06T

##### G06T5/00

## Figures

(a) Profile of compensator, F1, modeled after the Elekta F1 bowtie (aluminum with a center thickness of 3 mm). (b) Profile for compensator, AL16S, designed to compensate for a 16.4 cm diameter cylinder. AL16S is also composed of aluminum material with a center thickness of 1 mm and a modulation factor of 7.9.

(a) Profile of compensator, F1, modeled after the Elekta F1 bowtie (aluminum with a center thickness of 3 mm). (b) Profile for compensator, AL16S, designed to compensate for a 16.4 cm diameter cylinder. AL16S is also composed of aluminum material with a center thickness of 1 mm and a modulation factor of 7.9.

Axial (a) and sagittal (b) slices showing density values for voxelized head phantom used in the MC simulations.

Axial (a) and sagittal (b) slices showing density values for voxelized head phantom used in the MC simulations.

(a) Axial and (b) coronal slices of the density values for the voxelized pelvis phantom used in the MC simulations.

(a) Axial and (b) coronal slices of the density values for the voxelized pelvis phantom used in the MC simulations.

(a)–(d) The normalized detector scatter distribution, Sn, and (e)–(h) the corresponding logarithm of the spatial frequency, FS, for the 30.6 cm diameter water cylinder at ADD values of 18, 30, 44, and 56 cm.

(a)–(d) The normalized detector scatter distribution, Sn, and (e)–(h) the corresponding logarithm of the spatial frequency, FS, for the 30.6 cm diameter water cylinder at ADD values of 18, 30, 44, and 56 cm.

The normalized scatter distribution (a)–(c) and the corresponding logarithm of the FS (d)–(f) for different bowtie filter implementations: (a) and (d) none, (b) and (e) F1, (c) and (f) AL16S.

The normalized scatter distribution (a)–(c) and the corresponding logarithm of the FS (d)–(f) for different bowtie filter implementations: (a) and (d) none, (b) and (e) F1, (c) and (f) AL16S.

(a) Horizontal profiles along u axis (v = 0) and (b) vertical profiles along v axis (u = 0) for the spatial frequencies of Sn for the 30.6 cm diameter water cylinder with different compensator configurations (none, F1, and AL16S). A strong peak in the horizontal frequency component of FS for the no compensator case is seen around 0.017/cm resulting from the two signal peaks in the horizontal rows of the scatter distribution seen in Fig. 5(a) . Both compensators diminish the strength of these peaks in the horizontal rows resulting in a decrease in FS at the corresponding frequency values.

(a) Horizontal profiles along u axis (v = 0) and (b) vertical profiles along v axis (u = 0) for the spatial frequencies of Sn for the 30.6 cm diameter water cylinder with different compensator configurations (none, F1, and AL16S). A strong peak in the horizontal frequency component of FS for the no compensator case is seen around 0.017/cm resulting from the two signal peaks in the horizontal rows of the scatter distribution seen in Fig. 5(a) . Both compensators diminish the strength of these peaks in the horizontal rows resulting in a decrease in FS at the corresponding frequency values.

Scatter distribution projections, Sn, for frontal views (θ = 0°) of the pelvis (a) and (c) and head (b) and (d) phantom. Images (a) and (b) are without the use of a compensator, whereas images (c) and (d) are with a compensator. An increase in the signal intensity of Sn can clearly be seen at the edges of the pelvis (a) and head (b) phantom when a compensator is not used due to the increased coherent scattering contribution, when a compensator is used [(c)and (d)] these edge effects are significantly diminished.

Scatter distribution projections, Sn, for frontal views (θ = 0°) of the pelvis (a) and (c) and head (b) and (d) phantom. Images (a) and (b) are without the use of a compensator, whereas images (c) and (d) are with a compensator. An increase in the signal intensity of Sn can clearly be seen at the edges of the pelvis (a) and head (b) phantom when a compensator is not used due to the increased coherent scattering contribution, when a compensator is used [(c)and (d)] these edge effects are significantly diminished.

Scatter sinograms for the center row (a) and (c) and center column (b) and (d) of Sn for the pelvis phantom. The first row of images (a) and (b) is without a compensator and the second row (c) and (d) is with the AL16S compensator. Periodic signals can clearly be seen in the angular direction due to the ellipsoidal shape of the pelvis phantom.

Scatter sinograms for the center row (a) and (c) and center column (b) and (d) of Sn for the pelvis phantom. The first row of images (a) and (b) is without a compensator and the second row (c) and (d) is with the AL16S compensator. Periodic signals can clearly be seen in the angular direction due to the ellipsoidal shape of the pelvis phantom.

Scatter sinograms for the center row (a) and (c) and column (b) and (d) of Sn for the head phantom. The first row of images (a) and (b) is without a compensator and the second row (c) and (d) is with the AL16S compensator.

Scatter sinograms for the center row (a) and (c) and column (b) and (d) of Sn for the head phantom. The first row of images (a) and (b) is without a compensator and the second row (c) and (d) is with the AL16S compensator.

(a) Central horizontal profile, (b) central vertical profile, and (c) central angular profile of Sn for both head and pelvis phantoms with (dashed lines) and without (solid lines) the use of the AL16S compensator.

(a) Central horizontal profile, (b) central vertical profile, and (c) central angular profile of Sn for both head and pelvis phantoms with (dashed lines) and without (solid lines) the use of the AL16S compensator.

Logarithm images of FS for the pelvis phantom with (a)–(c) and without (d)–(f) the use of the AL16S compensator for the three central planes (u-v, v-ω, and u-ω). A strong off axis signal with a slope of −1 cm/turn is seen in the image of the u-ω plane shown in (c) and (f), resulting from the rotationally variant shape of the phantom.

Logarithm images of FS for the pelvis phantom with (a)–(c) and without (d)–(f) the use of the AL16S compensator for the three central planes (u-v, v-ω, and u-ω). A strong off axis signal with a slope of −1 cm/turn is seen in the image of the u-ω plane shown in (c) and (f), resulting from the rotationally variant shape of the phantom.

Logarithm images of FS for the head phantom with (a)–(c) and without (d)–(f) the use of the AL16S compensator for the three central planes (u-v, v-ω, and u-ω). Similar to the pelvis phantom an off axis frequency component is seen in the u-ω plane shown in (c) and (f).

Logarithm images of FS for the head phantom with (a)–(c) and without (d)–(f) the use of the AL16S compensator for the three central planes (u-v, v-ω, and u-ω). Similar to the pelvis phantom an off axis frequency component is seen in the u-ω plane shown in (c) and (f).

(a) Contour plot of the resulting RMSE values between the gold standard and the low pass filtered LPS Sn signals for the 30.6 cm diameter water cylinder with no compensator for a range of ucut and vcut values. The optimal cutoff values are found when ucut and vcut are 0.05 and 0.045 cm^{−1}, respectively, resulting in a RMSE value of 6.1. The optimal value is marked with a “+” on the contour plot. (b) The resulting shape of the optimal low pass Butterworth filter in the frequency domain.

(a) Contour plot of the resulting RMSE values between the gold standard and the low pass filtered LPS Sn signals for the 30.6 cm diameter water cylinder with no compensator for a range of ucut and vcut values. The optimal cutoff values are found when ucut and vcut are 0.05 and 0.045 cm^{−1}, respectively, resulting in a RMSE value of 6.1. The optimal value is marked with a “+” on the contour plot. (b) The resulting shape of the optimal low pass Butterworth filter in the frequency domain.

LPS Sn projection for 30.6 cm diameter water cylinder (no compensator) (a) without and (b) with low pass filtering. The low pass filter cutoffs used in (b) are 0.065 and 0.045 cm^{−1} for u and v, respectively. (c) Gold standard scatter simulation Sn result. (d) The percent absolute error between the filtered and gold standard Sn signal. (e) The central horizontal profile of the gold standard, LPS, and filtered LPS Sn signals.

LPS Sn projection for 30.6 cm diameter water cylinder (no compensator) (a) without and (b) with low pass filtering. The low pass filter cutoffs used in (b) are 0.065 and 0.045 cm^{−1} for u and v, respectively. (c) Gold standard scatter simulation Sn result. (d) The percent absolute error between the filtered and gold standard Sn signal. (e) The central horizontal profile of the gold standard, LPS, and filtered LPS Sn signals.

(a)–(c) shows 0° Sn projection for the pelvis phantom for the LPS using 10^{6} photons, low-pass filtered LPS (using optimal cutoff values), and the gold standard (>10^{9} photons) Sn data. The LPS Sn signal uses an angular sampling rate of 1°. (d)–(f) shows the same data but in the form of a sinogram composed of the center horizontal row of Sn signal at each projection angle, θ.

(a)–(c) shows 0° Sn projection for the pelvis phantom for the LPS using 10^{6} photons, low-pass filtered LPS (using optimal cutoff values), and the gold standard (>10^{9} photons) Sn data. The LPS Sn signal uses an angular sampling rate of 1°. (d)–(f) shows the same data but in the form of a sinogram composed of the center horizontal row of Sn signal at each projection angle, θ.

RMSE as a function of the angular sampling rate for each of the four phantom imaging conditions. The results using the SFW and optimal low-pass filter cutoff values are shown as squares (□) and crosses (+), respectively.

RMSE as a function of the angular sampling rate for each of the four phantom imaging conditions. The results using the SFW and optimal low-pass filter cutoff values are shown as squares (□) and crosses (+), respectively.

Optimal low-pass filter cutoff values, (a) ucut, (b) vcut, and (c) ω cut, for the different angular sampling rates used in each of the four phantom imaging configurations. Two outliers at dθ = 72° and 90° were removed from the ω cut data for the pelvis phantom with the AL16S compensator. The optimization for these two points resulted in a selection of the highest value of ω cut searched (35.28 turns^{−1}) indicating that no filtering in the angular direction is optimal for these cases.

Optimal low-pass filter cutoff values, (a) ucut, (b) vcut, and (c) ω cut, for the different angular sampling rates used in each of the four phantom imaging configurations. Two outliers at dθ = 72° and 90° were removed from the ω cut data for the pelvis phantom with the AL16S compensator. The optimization for these two points resulted in a selection of the highest value of ω cut searched (35.28 turns^{−1}) indicating that no filtering in the angular direction is optimal for these cases.

Axial slices for reconstructions of the head phantom with no compensator using (a) uncorrected (primary and scatter), (b) scatter corrected, and (c) primary only projection images. The scatter estimate used in correcting (b) comes from the Fourier filtered and interpolated LPS (10^{6} photons) data with an angular sampling of every 24°.

Axial slices for reconstructions of the head phantom with no compensator using (a) uncorrected (primary and scatter), (b) scatter corrected, and (c) primary only projection images. The scatter estimate used in correcting (b) comes from the Fourier filtered and interpolated LPS (10^{6} photons) data with an angular sampling of every 24°.

Horizontal profile from axial slice of each reconstruction of the head phantom. The location in the axial slice is indicated by the dashed line in Fig. 18(c) .

Horizontal profile from axial slice of each reconstruction of the head phantom. The location in the axial slice is indicated by the dashed line in Fig. 18(c) .

## Tables

Imaging geometry and simulation parameters.

Imaging geometry and simulation parameters.

SFW values (in cm^{−1}) along the horizontal and vertical (u,v) frequency directions for the 30.6 cm diameter cylinder for various ADD, compensator, and detector configurations.

SFW values (in cm^{−1}) along the horizontal and vertical (u,v) frequency directions for the 30.6 cm diameter cylinder for various ADD, compensator, and detector configurations.

SFW values for the pelvis and head phantom with and without the use of the AL16S.

SFW values for the pelvis and head phantom with and without the use of the AL16S.

The optimal ucut and vcut values and corresponding RMSE. The RMSE for using no filter and using a filter with cutoffs selected from the SFW values are also shown for the case with the F1 and AL16S compensators and without the use of a compensator. The error reduction for using the optimal filter cutoffs is also presented.

The optimal ucut and vcut values and corresponding RMSE. The RMSE for using no filter and using a filter with cutoffs selected from the SFW values are also shown for the case with the F1 and AL16S compensators and without the use of a compensator. The error reduction for using the optimal filter cutoffs is also presented.

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