Noise, sampling, and the number of projections in cone-beam CT with a flat-panel detector

Abstract

Purpose: To investigate the effect of the number of projection views on image noise in cone-beam CT (CBCT) with a flat-panel detector.

Methods: This fairly fundamental consideration in CBCT system design and operation was addressed experimentally (using a phantom presenting a uniform medium as well as statistically motivated “clutter”) and theoretically (using a cascaded systems model describing CBCT noise) to elucidate the contributing factors of quantum noise (σ_Q), electronic noise (σ_E), and view aliasing (σ_view). Analysis included investigation of the noise, noise-power spectrum, and modulation transfer function as a function of the number of projections (N_proj), dose (D_tot), and voxel size (b_vox).

Results: The results reveal a nonmonotonic relationship between image noise and N_proj at fixed total dose: for the CBCT system considered, noise decreased with increasing N_proj due to reduction of view sampling effects in the regime N_proj <∼200, above which noise increased with N_proj due to increased electronic noise. View sampling effects were shown to depend on the heterogeneity of the object in a direct analytical relationship to power-law anatomical clutter of the form κ/f ^β—and a general model of individual noise components (σ_Q, σ_E, and σ_view) demonstrated agreement with measurements over a broad range in N_proj, D_tot, and b_vox.

Conclusions: The work elucidates fairly basic elements of CBCT noise in a manner that demonstrates the role of distinct noise components (viz., quantum, electronic, and view sampling noise). For configurations fairly typical of CBCT with a flat-panel detector (FPD), the analysis reveals a “sweet spot” (i.e., minimum noise) in the range N_proj ∼ 250–350, nearly an order of magnitude lower in N_proj than typical of multidetector CT, owing to the relatively high electronic noise in FPDs. The analysis explicitly relates view aliasing and quantum noise in a manner that includes aspects of the object (“clutter”) and imaging chain (including nonidealities of detector blur and electronic noise) to provide a more rigorous basis for commonly held intuition and heurism in CBCT system design and operation.

INTRODUCTION

Among the first questions encountered in developing a new cone-beam CT (CBCT) system is: How many projections should be acquired? The question is inextricably linked to factors of radiation dose, acquisition speed, data handling, and—of course—image quality. With respect to the last, a simple rule of thumb might be invoked: Acquire at least a number of projections (N_proj) such that the angular separation between views (Δθ) amounts to a distance at the edge of the field of view (FOV) equal to the voxel size (b_vox)

$$N_{\mathrm{proj}}\ge 2\pi /\arctan (2b_{\mathrm{vox}}/\mathrm{FOV}),$$ (1)

where for simplicity, we assume a full circular orbit (N_projΔθ = 2π). One could (in fact, should) invoke a measure of spatial resolution or characteristic correlation length that is distinct from the voxel size—for example, the full-width at half maximum of the point-spread function; however, the simple rule of thumb in Eq. 1 essentially assumes voxel size as a coarse surrogate. Along similar lines, one could more strictly invoke the Nyquist theorem in that the sampling distance should be no more than half the desired resolution distance, requiring b_vox → b_vox/2. While Eq. 1 is obviously an oversimplification with respect to numerous factors detailed below, it is not an unreasonable starting point. Why then, does the answer imply something so different from what is typically implemented in CBCT? For example, a CBCT scanner with FOV ∼250 mm and a desired spatial resolution of 0.5 mm would imply N_proj ∼1500 (and Δθ ∼0.23°). The variation with b_vox → b_vox/2 implies N_proj ∼3100. These numbers are not unlike the number of projections acquired in fan-beam multidetector diagnostic CT, but in CBCT—in applications ranging from diagnostic imaging of the breast,^{1, 2} extremities,³ and head⁴ to image-guided surgery^{5, 6} and radiotherapy⁷—acquisition more typically involves N_proj ∼350–450 (with Δθ ∼0.8°–1°). What accounts for the factor of ∼3–8 between such real CBCT embodiments and the answer implied by the (overly simplistic) Eq. 1? And would CBCT image quality improve by increasing N_proj several-fold, assuming other factors such as readout speed and radiation dose could be held fixed?

An experienced practitioner of CBCT would likely answer confidently—No—and a number of reasons for the lower N_proj in CBCT could immediately be offered. For example: (1) Flat-panel detectors (FPDs) have a high level of electronic noise; therefore, increasing N_proj (at fixed dose) would entail a lower exposure per projection and an unacceptably high contribution of electronic noise; (2) Indirect-detection FPDs have blur associated the x-ray converter that exceeds the voxel size; therefore, b_vox in Eq. 1 can be replaced by a larger correlation length, giving a smaller N_proj; and (3) FPDs have relatively slow frame rate, so increasing N_proj would entail unacceptably long scan times and potential for motion artifact. Such empirical rationale—and the conditions under which such rationale fails—are given a more rigorous analytical basis in this paper, including aspects of anatomical clutter, detector nonidealities, and view sampling. The effects of sampling and aliasing in CT have been described in early work by Brooks et al.⁸ and Joseph et al.,^{9, 10} providing insight far beyond that of Eq. 1. Such effects are considered here in the context of CBCT with FPDs (which exhibit less idealized performance with respect to detector blur and electronic noise) and cascaded systems analysis of 3D signal and noise transfer characteristics.

We entertained the question first by way of a simple experiment. A phantom presenting uniform and heterogeneous content was used to measure CBCT image noise as a function of N_proj at fixed total scan dose. The observations were subsequently interpreted using a cascaded systems model^{11, 12} for signal and noise propagation to elucidate the theoretical underpinnings of quantum noise, electronic noise, anatomical clutter, and view sampling effects as a function of N_proj. While previous work has demonstrated the dependence of CBCT imaging performance on factors such as dose, reconstruction parameters, anatomical noise, and the imaging task,^{11, 12, 13, 14, 15, 16, 17, 18, 19} the current work sheds light on the behavior of CBCT image noise with specific regard to view sampling and a finite number of views. It also reveals a new and nontrivial relationship between view sampling and anatomical clutter and demonstrates why the choice of N_proj ∼350–450 for the current generation of CBCT systems is not only a reasonable, practical choice (e.g., in terms of scan time), it is in several respects optimal in minimizing the combined influence of quantum noise, electronics noise, and view aliasing for a typical system configuration.

EXPERIMENTAL METHODS

Imaging bench, techniques, and phantom

All experiments were conducted using the CBCT imaging bench in Fig. 1a. The bench consisted of an x-ray source [DU694 pulsed radiographic/fluoroscopic tube in EA10 housing, Dunlee, Aurora IL], FPD [PaxScan 4343, Varian Medical Systems, Palo Alto, CA], and a computer-controlled motion system: seven linear stages [406XR and HLE60SR linear stages, Parker Hannifin, Cleveland OH] to set overall system geometry and a rotation stage [Dynaserv G3 servo drive, Parker Hannifin, Cleveland, OH)] to rotate the object. The system was operated in a step-and-shoot mode in which the x-ray source delivers pulsed fluoroscopy (typically 10 ms pulses), the FPD is read (typically 3 fps), and the object rotates (e.g., by an angular increment Δθ) in a synchronized manner. The FPD was read at a nominal pixel size of 0.278 mm (referred to as 1 × 1 binning) or 0.556 mm (2 × 2 binning). The system geometry for all experiments entailed source-axis distance (SAD) = 1225 mm and source-detector distance = 1502 mm, calibrated using the two-circle BB phantom method described by Cho et al.²⁰ An antiscatter grid was attached to the FPD (10:1 grid ratio, 103 lpi, and measured transmission factor 0.55, Soyee Products, Korea). A set of 50 dark images and 50 flood images acquired before or after each scan provided correction of pixel offset and gain variations. Image reconstruction was based on 3D filtered backprojection adapted from Feldkamp et al.,²¹ and all scans involved a complete 360° rotation of the object. Reconstructions were performed at isotropic voxel size b_x = b_y = b_z = 0.22 mm (for 1 × 1 binning of the FPD) and 0.44 mm (for 2 × 2 binning).

Figure 1.

Imaging bench and phantom. (a) Illustration of the x-ray imaging bench, components, and system geometry. (b) Illustration of the cylindrical phantom with water (top) and a heterogeneous mixture of plastic spheres (bottom). (c) Example CBCT axial slices of the phantom. The water region included an acrylic annulus as part of the phantom construction, and the small white squares superimposed at constant radius mark the location of ROIs for noise analysis. The spherical clutter region included three electron-density inserts (larger circles at 1, 5, and 9 o'clock positions), and the white square superimposed in the low-density insert shows the location of the ROI for noise analysis.

All scans were performed at 90 kVp (+2 mm Al +0.4 mm Cu added filtration). Beam quality estimated by Spektr²² indicated half-value layer (HVL) = 7.9 mm Al, mean energy = 59.2 keV, and 1.53 mR/mAs at a distance of 1000 mm from the source. Bare-beam exposure was measured (at isocenter and at the surface of the antiscatter grid) using a silicon diode (DDX6-W, Radcal, Monrovia, CA) and exposure meter (Accu-Pro 9096, Radcal, Monrovia, CA). The measured mR/mAs agreed with the Spektr calculation to better than 1%. Dose was measured using a 0.6 cm³ Farmer ionization chamber placed at the center of a 16 cm CTDI phantom at isocenter (with the phantom extended longitudinally above and below by a stack of two additional CTDI phantoms). The scan dose (D_tot) and exposure per projection (X_proj in air at isocenter) can be related by

$$D_{\mathrm{tot}}=N_{\mathrm{proj}}·X_{\mathrm{proj}}·e^{-\mu D/2}·f·f_{\mathrm{BSF}},$$ (2)

where μ was the effective attenuation coefficient of water for the 90 kVp spectrum (0.019 mm⁻¹), D was the diameter of the phantom (15.5 cm as described below), the f factor was taken as 0.9 cGy/R, and the backscatter factor (f_BSF) was empirically determined to be ∼4.5 such that the measured dose and exposure were in close agreement (better than 5%) according to Eq. 2.

As detailed in Table 1, experiments were conducted at two nominal dose levels, D_tot = 2 and 8 mGy. CBCT scans were acquired at fixed total dose by careful variation of the number of projections (N_proj ranging 40–1375) and the mAs/projection (ranging 5–0.126 mAs per projection, giving exposure per projection X_proj = 0.15–5.17 mR). Note that the highest mAs did not saturate the FPD (5 mAs giving ∼75% of sensor saturation) such that all measurements were within a linear regime of detector response. These settings provided total dose as calculated by Eq. 2 within 1% of the nominal dose level for all cases, as shown in Table 1. The experiments therefore spanned more than an order of magnitude in detector exposure, with the intention of varying the influence of electronic noise from a negligible level (high X_proj) to a more appreciable level (low X_proj) all at fixed total dose.

Table 1.

Summary of CBCT acquisition techniques giving constant nominal dose (D_tot = 2 or 8 mGy) for various settings of number of projections (N_proj) and exposure per projection (X_proj) in air at isocenter (SAD = 1225 mm). Careful adjustment of N_proj and X_proj yielded constant D_tot within 1% for all experiments.

Nominal Dtot			Measured Xproj	Measured Xtot	Estimated Dtot
(mGy)	Nproj	mAs/proj	(mR)	(mR)	(mGy)	%err
2	40	5.00	5.17	206.80	2.01	0.72
2	80	2.50	2.59	207.20	2.02	0.91
2	159	1.25	1.30	206.70	2.01	0.67
2	308	0.63	0.67	206.36	2.01	0.50
2	480	0.40	0.43	206.40	2.01	0.52
2	736	0.25	0.28	206.08	2.01	0.37
2	1085	0.16	0.19	206.15	2.01	0.40
2	1375	0.13	0.15	206.25	2.01	0.45
8	160	5.00	5.17	827.20	8.06	0.72
8	213	4.00	3.88	826.44	8.05	0.63
8	318	2.50	2.59	823.62	8.02	0.28
8	488	1.60	1.69	824.72	8.03	0.42
8	634	1.25	1.30	824.20	8.03	0.35
8	959	0.80	0.86	824.74	8.03	0.42
8	1231	0.63	0.67	824.77	8.03	0.42

A phantom was constructed to elucidate the influence of various components of CBCT image noise—viz., quantum noise, electronic noise, anatomical clutter, and view aliasing—as a function of the total scan dose (D_tot), the number of projections (N_proj), and reconstruction voxel size (isotropic b_vox). The phantom is illustrated in Fig. 1, based on a 15 cm diameter plastic cylinder in which two “layers” of material were incorporated (separated by a foam disk). The top layer was water, and the bottom was a mixed collection of plastic spheres of varying diameter (1.6–12.7 mm) and material type (polypropylene, acrylic, and acetal) in water. Such a random assembly of self-similar objects was shown previously¹² to constitute a fractal described by a ∼1/f ³ power-law distribution similar to that measured for anatomical noise-power spectra (“anatomical clutter”) in contexts such as breast^{23, 24} and chest imaging.^{25, 26} The contrast of the spheres (i.e., the amplitude of the simulated anatomical clutter) was chosen to approximate that typical of natural tissues (e.g., polypropylene and acrylic representing soft tissues with ∼100 HU contrast to water and acetal representing bone at several hundred HU contrast). The phantom was thus statistically motivated such that the top layer presented stochastic quantum and electronic noise only, and the bottom presented a combination of stochastic noise and anatomical clutter. Three tissue-simulating, cylindrical electron density inserts [${\rho }_e^w\approx 0.96$ (Breast), ${\rho }_e^w\approx 1.07$ (Liver), and ${\rho }_e^w\approx 1.10$ (B-200, Bone); Gammex RMI, Madison WI] were also incorporated in the bottom layer for analysis of contrast and noise.

Analysis of noise

The experiment permitted analysis of various components of CBCT image noise (i.e., quantum, electronic, anatomical, and view aliasing noise) in terms of the standard deviation in voxel values (σ_vox), and analysis of the noise-power spectrum (NPS) gave additional insight on the spatial frequency content of such noise components. The regions of interest (ROIs) for noise and NPS analysis are illustrated in Fig. 1. In each case, ROIs were within ∼3 cm of the central axial slice (cone angle <1.5° for the geometry in Fig. 1), so the effects of the frequency domain “null cone” associated with violation of Tuy's condition for a circular source-detector orbit were considered negligible. Each ROI comprised a 3D cubic subvolume (not a 2D square implied by the slice images in Fig. 1) of sidelength N_x = N_y = N_z, with the size and number of ROIs varied as follows.

For analysis of (i) voxel noise and (ii) NPS associated with the purely stochastic effects of x-ray quantum noise and readout electronics noise, the ROIs were placed at a fixed radius (∼4 cm) in the water section as in Fig. 1c. For analysis of (i) voxel noise, the standard deviation was analyzed in ROIs of size N_x = 16, giving a total of 512 ROIs in eight contiguous, nonoverlapping slabs within 3 cm of the central axial slice. For analysis of (ii) the NPS, two identical CBCT scans were acquired in succession and subtracted. The NPS was computed from the resulting difference image by discrete Fourier transform in ROIs of size N_x = 64, giving a total of 24 nonoverlapping ROIs, correcting by a factor of 2 to account for the subtraction

$$\mathit{NPS}=\frac{1}{2}\frac{b_x{b}_y{b}_z}{N_x{N}_y{N}_z}⟨{\left|FT\left[\Delta \mathit{ROI}\right(x,y,z\left)\right]\right|}^2⟩,$$ (3)

where the ⟨.⟩ bracket notation represents ensemble average of the NPS estimate from each ROI. The ROIs were not rotated or resampled prior to Fourier transform, so the analysis tended to average out underlying azimuthal angular dependence in the NPS. In each case shown in Table 1, the voxel noise was verified to agree with the volume of the NPS to within ∼2%.

For analysis of (iii) the voxel noise associated with the combined effects of view sampling and stochastic quantum and electronic noise, the standard deviation was analyzed in a single ROI of size (80 × 80 × 80) voxels placed in an otherwise uniform region of the solid water insert in the clutter section as in Fig. 1d. Since the ROI was surrounded by heterogeneous clutter, the standard deviation measured within the otherwise uniform ROI probed a combination of stochastic noise and view aliasing “streaks” from surrounding structures.

EXPERIMENTAL RESULTS

Figure 2 illustrates the central experimental result, showing axial reconstructions of the water layer, the clutter layer, and the associated axial and coronal NPS. Results are shown for the case D_tot = 2 mGy and reconstruction at isotropic voxel size b_vox = b_x = b_y = b_z = 0.22 mm (1 × 1 binning). Other image examples [D_tot = 8 mGy and b_vox = 0.44 mm (i.e., 2 × 2 binning)] are not shown but are discussed below.

Figure 2.

Images and NPS as a function of N_proj. (First row) Axial images in the uniform water region. (Second row) Axial images in the heterogenous clutter region. Zoomed insets illustrate the magnitude and texture of noise as a function of N_proj. (Third row) Axial and (fourth row) coronal NPS analyzed in the uniform water region.

A naïve guess at the behavior of noise versus N_proj might be that since D_tot is fixed, the noise should be constant; of course, that refers only to the quantum noise component, and Fig. 2 reveals a far richer mix of contributing noise factors. Close inspection reveals a change in both the magnitude and frequency content of noise as a function of N_proj. At the lowest N_proj, the noise in water changes only slightly, and there is little or no change in spatial frequency content; however, the noise in the clutter layer increases markedly and demonstrates a dramatic change in spatial frequency content—namely, streaks. The observation points to a conclusion that is clear in retrospect: as the number of projections is reduced, the dominant change in noise characteristic is view sampling that can be well appreciated within a heterogeneous context (clutter) but is not evident in a uniform medium (water). A simple CNR phantom (typically involving a few inserts within uniform plastic) is therefore a poor probe of the effect, whereas a heterogeneous, statistically motivated power-law clutter phantom brings the effect immediately to light.

The observation is quantified in Fig. 3: the noise in water increases monotonically even in the region of low N_proj, whereas the noise in clutter is nonmonotonic and increases at low N_proj due to view sampling. The statistical error in the measurements was such that error bars were approximately the same size as the plot symbols and were therefore not shown for clarity. A different noise component emerges as N_proj increases, also evident in Figs. 23—a monotonic (approximately linear) increase in noise with increasing N_proj. Since D_tot is fixed, and increasing N_proj is associated with lower exposure per projection, X_proj, the culprit for the increase is clear: electronic noise. In both water and clutter, an increase in noise magnitude (with little or no change in noise texture) can be attributed to an increased contribution of electronic noise that adds to the NPS with a similar spatial-frequency characteristic as the quantum noise (i.e., midpass). As shown in Fig. 3, the effect is the same in water and clutter, the latter slightly elevated due to a slightly higher overall attenuation (lower transmission) of the clutter layer. The slope is related to the electronic noise level.

Figure 3.

Voxel noise measured as function of N_proj for 3D image reconstructions at (a) D_tot = 2 and (b) D_tot = 8 mGy.

Figures 23 therefore give two distinct observations: (1) at fixed total dose in a uniform medium (water), noise increases monotonically with N_proj due to increasing contribution of electronic noise, whereas the underlying quantum noise component is constant, and the view aliasing noise component is not observed (since there is no structure in the object to alias) and (2) at fixed dose in a heterogeneous medium (clutter), noise increases at higher N_proj due to the effects just mentioned, and increases at lower N_proj due to view aliasing effects that arise from the clutter. In retrospect, the effects are intuitive, and a theoretical basis for such intuition is provided in Sec. 4.

Finally, there is a subtlety that becomes apparent in the NPS at low N_proj. As seen in the axial NPS of Fig. 4 for N_proj = 40, the NPS exhibits a discrete set of (20) vanes in the Fourier domain associated with the 40 backprojections. In that case, the angular increment is Δθ = 9°, and the second half of the scan is redundant such that backprojected rays overlie the same vanes in the Fourier domain. This effect is also related to the notion of 1/4-pixel offset for optimal sampling.²⁷ The suboptimal sampling becomes apparent at low N_proj. The observation prompted us to consider a separate scan acquired with N_proj = 39 (Δθ = 9.231°) such that the second half of the scan was not redundant. We hypothesized that the NPS would exhibit more complete filling of the Fourier domain but little or no change in the magnitude of the quantum noise (σ_Q), since the total dose was nearly unchanged (to one part in 40). The effect was confirmed as illustrated in Fig. 4, where a reduction by just one projection had a marked effect: the voxel noise in a uniform medium was essentially unchanged (σ_vox = 0.0188 and 0.0186 cm⁻¹ for 39 and 40 projections, respectively, in agreement with the small change in dose of one part in 40); however, as hypothesized, reduction by one projection actually improved the voxel noise in a heterogeneous medium (σ_vox = 0.037 and 0.038 cm⁻¹ for 39 and 40 projections, respectively) and the view aliasing streaks appear visibly reduced. It bears reiteration: the sampling improved despite having one fewer projection due to improved filling of the Fourier domain with nonoverlapping backprojections.

Figure 4.

Images and NPS for N_proj = 39 and 40. The latter case results in redundant views and nearly overlapping backprojections with poorer filling of the Fourier domain and correspondingly increased view sampling effects. The former case reduces such view sampling effects (despite having one fewer projection).

The observation of improved sampling for 39 versus 40 views is a subtlety of academic interest at extremes of angular undersampling and is only evident at very low values of N_proj where differences in Fourier domain filling are more appreciable and is unlikely to affect clinical image quality under conditions of more complete sampling. It is also most evident (or evident at all) near the center of reconstruction (∼4 cm annular radius in the current study) and for systems with an extended SAD, where the discussion above regarding redundant rays is more consistent with a parallel beam approximation. The effect is reduced (or unobservable) at larger values of N_proj, at extended distances from the center of reconstruction, and for shorter SAD (high magnification), although the last two dependencies were not directly investigated in the current work. The effect could also warrant consideration in applications involving low-dose “sparse” CBCT data acquisition.^{28, 29}

AN ANALYTICAL BASIS FOR THE NOISE COMPONENTS

Three components of image noise

The trends evident in the measurements reported above can be understood in terms of three components of image noise—quantum noise, electronic noise, and view sampling—each described below in a simple analytical framework.

Quantum noise

Quantum noise refers to image fluctuations arising from the discrete nature of Poisson-distributed incident x-ray photons, the propagation of which in the image acquisition and reconstruction process is well described by cascaded systems analysis.¹¹ We follow the notation of Tward and Siewerdsen¹¹ and Gang et al.¹² in describing a cascade of gain, spreading, and sampling stages: (1) interaction of x rays in the converter; (2) conversion of x rays to secondary quanta (optical photons); (3) spatial spread of secondary quanta in the converter; (4) coupling of secondary quanta to the pixel aperture; (5) integration of secondary quanta by the pixel aperture; (6) detector readout (2D sampling); (7) addition of electronic noise; (8) normalization and logarithm of processed projection data; (9) application of the ramp filter; (10) application of the smoothing filter; (11) interpolation of detector pixel values; (12) backprojection; and (13) sampling of the 3D voxel grid. The 2D NPS describing quantum noise in a single projection image (i.e., up through stage 7) is³⁰

$$\begin{array}{ccc}\hfill S_7& =& {\overline{q}}_0{a}_{\mathrm{pd}}^4{\overline{g}}_1{\overline{g}}_2{\overline{g}}_4\left(1+{\overline{g}}_4{P}_K{T}_3^2\right)T_5^{2**}{\mathrm{III}}_6+S_{\mathrm{add}}\hfill \\ & =& S_7^Q+S_{\mathrm{add}},\hfill \end{array}$$ (4)

where ${\overline{q}}_0$ is the incident fluence, a_pd is the extent of the integrating aperture, the various ${\overline{g}}_i$ refer to the mean gain of stage i, P_K includes K-fluorescence effects in the x-ray converter,²⁵T_i refers to the transfer function for stage i, III is the comb function representing the sampling distance, and S_add is the NPS associated with the readout electronics. For simplicity in the equations below, we will omit the sampling step—i.e., assume no aliasing from III₆, which is a fair approximation for band-limited indirect-detection FPDs. We will also ignore S_add for the moment (treated separately below) and focus on just the x-ray quantum noise.

The system MTF, T_sys, for backprojection over N_proj views is

$$\begin{array}{ccc}\hfill T_{\mathrm{sys}}& =& \frac{\pi M}{N_{\mathrm{proj}}}{T}_3{T}_5{T}_{10}{T}_{11}{T}_{12}\sum _{i=1}^{N_{\mathrm{proj}}}d\hfill \\ & & \times \mathrm{sinc}\left\{d\left[f_x\cos \left({\theta }_i\right)+f_y\sin \left({\theta }_i\right)\right]\right\},\hfill \end{array}$$ (5a)

where d is an arbitrary distance associated with the reconstruction FOV, and M is the system magnification (SDD/SAD). In the limit of complete sampling (N_proj → ∞),

$$T_{\mathrm{sys}}^{N_{\mathrm{proj}}\to \infty }=T_3{T}_5{T}_{10}{T}_{11}{T}_{12}\frac{M}{f_r}\cong T_3{T}_5{T}_{11}{T}_{12}$$ (5b)

since T₁₀ ($=\frac{f_r}{M}$, where $f_r=\sqrt{f_x^2+f_y^2}$) and the backprojection operator perfectly cancel.

The resulting 3D NPS associated with quantum noise in the filtered backprojection image is

$$\begin{array}{ccc}\hfill S_Q& =& \frac{S_7^Q}{{\left({\overline{q}}_0{a}_{\mathrm{pd}}^2{\overline{g}}_1{\overline{g}}_2{\overline{g}}_4\right)}^2}{T}_{10}^2{T}_{11}^2{T}_{12}^2\left[{\left(\frac{1}{M}\right)}^2{\left(\frac{M\pi }{N_{\mathrm{proj}}}\right)}^2\sum _{i=1}^{N_{\mathrm{proj}}}d\right.\hfill \\ & & \left.\times {\mathrm{sinc}}^2\left\{d\left[f_x\cos \left({\theta }_i\right)+f_y\sin \left({\theta }_i\right)\right]\right\}\right].\hfill \end{array}$$ (6a)

In the limit of infinitely many projections,

$$\begin{array}{ccc}\hfill S_Q^{N_{\mathrm{proj}}\to \infty }& =& \frac{S_7^Q}{{\left({\overline{q}}_0{a}_{\mathrm{pd}}^2{\overline{g}}_1{\overline{g}}_2{\overline{g}}_4\right)}^2}{\left(\frac{f_r}{M}\right)}^2{T}_{11}^2{T}_{12}^2\hfill \\ & & \times \left[{\left(\frac{1}{M}\right)}^2{\left(\frac{M\pi }{N_{\mathrm{proj}}}\right)}^2\frac{N_{\mathrm{proj}}}{\pi }\frac{1}{f_r}\right]\hfill \\ & =& \frac{\pi f_r}{N_{\mathrm{proj}}{M}^2}\frac{S_7^Q}{{\left({\overline{q}}_0{a}_{\mathrm{pd}}^2{\overline{g}}_1{\overline{g}}_2{\overline{g}}_4\right)}^2}{T}_{11}^2{T}_{12}^2.\hfill \end{array}$$ (6b)

The magnitude of quantum noise is

$${\sigma }_Q^2=\int \int \int S_Q(f_x,f_y,f_z)d{f}_{x}d{f}_{y}d{f}_z.$$ (6c)

Equation 6b reflects the midpass characteristic of the 3D NPS described early in the context of transaxial CT by Hanson³¹ and Wagner et al.³² As shown in Tward et al.,^{11, 12} carrying through the triple integral also reproduces the classic relationship of noise, dose, and spatial resolution as described by Barrett et al.³³—i.e., the variance is inversely proportional to dose, slice thickness, etc.—but here accounting for the various nonidealities of the FPD imaging chain.

Electronic noise

Next, we consider the electronic readout noise introduced at stage 7, with NPS assumed to be uncorrelated

$$S_{\mathrm{add}}=a_{\mathrm{pd}}^2{\sigma }_{\mathrm{add}}^2,$$ (7)

where σ_add is the root mean square electronic noise. Propagating S_add through the process of filtered backprojection gives the 3D NPS associated with electronic noise

$$\begin{array}{ccc}\hfill S_E& =& \frac{S_{\mathrm{add}}}{{\left({\overline{q}}_0{a}_{\mathrm{pd}}^2{\overline{g}}_1{\overline{g}}_2{\overline{g}}_4\right)}^2}{T}_{10}^2{T}_{11}^2{T}_{12}^2\left[{\left(\frac{1}{M}\right)}^2{\left(\frac{M\pi }{N_{\mathrm{proj}}}\right)}^2\sum _{i=1}^{N_{\mathrm{proj}}}d\right.\hfill \\ & & \left.\times {\mathrm{sinc}}^2\left\{d\left[f_x\cos \left({\theta }_i\right)+f_y\sin \left({\theta }_i\right)\right]\right\}\right].\hfill \end{array}$$ (8a)

In the limit of infinitely many projections,

$$S_E^{N_{\mathrm{proj}}\to \infty }=\frac{\pi f_r}{N_{\mathrm{proj}}{M}^2}\frac{S_{\mathrm{add}}}{{\left({\overline{q}}_0{a}_{\mathrm{pd}}^2{\overline{g}}_1{\overline{g}}_2{\overline{g}}_4\right)}^2}{T}_{11}^2{T}_{12}^2.$$ (8b)

Note that the 3D electronic NPS has a similar midpass spatial frequency dependence as the quantum noise—a ramp characteristic at low frequencies, peaking at midfrequencies, and rolling off due to apodization and interpolation. The electronic noise component of the CBCT image noise is

$${\sigma }_E^2=\int \int \int S_E(f_x,f_y,f_z)d{f}_{x}d{f}_{y}d{f}_z$$ (8c)

and the noise associated with purely random processes of incident quanta and readout electronics is

$$\begin{array}{c}\hfill {\sigma }_{Q+E}^2=\int \int \int S_Q(f_x,f_y,f_z)+S_E(f_x,f_y,f_z)d{f}_{x}d{f}_{y}d{f}_z.\end{array}$$ (9)

View aliasing

Finally, we consider the effect of view sampling. As evident in Fig. 2, view sampling effects are only manifest in the presence of a heterogeneous object with structure subject to aliasing. A good example is the statistically motivated clutter phantom, designed to exhibit a symmetric 3D power-law NPS

$$S_{\mathrm{obj}}=\frac{\kappa }{{\left(\sqrt{f_x^2+f_y^2+f_z^2}\right)}^{\beta }}=\frac{\kappa }{f^{\beta }}$$ (10a)

the reconstruction of which is subject to smoothing by the system MTF

$$S_B=S_{\mathrm{obj}}{T}_{\mathrm{sys}}^2.$$ (10b)

The NPS associated with view sampling effects is therefore

$$\begin{array}{ccc}\hfill S_{\mathrm{view}}& =& S_B-S_B^{N_{\mathrm{proj}}\to \infty }\hfill \\ & =& \frac{\kappa }{f^{\beta }}\left[T_{\mathrm{sys}}^2-{\left(T_{\mathrm{sys}}^{N\mathrm{proj}\to \infty }\right)}^2\right]\hfill \\ & =& \frac{\kappa }{f^{\beta }}{T}_3^2{T}_5^2{T}_{10}^2{T}_{11}^2{T}_{12}^2\hfill \\ & & \times \left[{\left(\frac{\pi M}{N_{\mathrm{proj}}}\right)}^2\left(\sum _{i=1}^{N_{\mathrm{proj}}}d\mathrm{sinc}\left\{d\right[f_x\cos \left({\theta }_i\right)\right.\right.\hfill \\ & & \left.{\left.+f_y\sin \left({\theta }_i\right)\left]\right\}\right)}^2-\frac{M^2}{f_r^2}\right].\hfill \end{array}$$ (11a)

Note the spatial-frequency content of S_view: the term out front carries the low-frequency characteristic of the object (1/f ^β) apodized by the various transfer functions in the acquisition and reconstruction process; the term in brackets goes as ∼[sinc²(f) − 1/f ²], which itself is low- to midpass; and there is an approximate $1/N_{\mathrm{proj}}^2$ dependence. The resulting S_view is therefore a combination of low-frequency clutter-like noise and midfrequency correlated noise—i.e., streaks—blurred by the system MTF. The view aliasing noise component is simply

$${{\sigma }_{\mathrm{view}}}^2=\int \int \int S_{view}(f_x,f_y,f_z)d{f}_{x}d{f}_{y}df.$$ (11b)

Numerical calculation of T_sys is prone to finite sampling effects where the summation of sinc functions does not nicely cancel out to 1/f even when N_proj is large. For purposes of numerical calculation, therefore, we defined a cutoff frequency as in Gang et al.,¹² below which the object is fully sampled with the transfer function $T_{\mathrm{sys}}^{N_{\mathrm{proj}}\to \infty }$, and above which has transfer function T_sys.

The total CBCT image noise

Considering the components of quantum noise, electronic noise, and view aliasing noise described above, the total image noise in CBCT is simply

$$\begin{array}{ccc}\hfill {\sigma }_{\mathrm{tot}}^2& =& \int \int \int S_E(f_x,f_y,f_z)+S_Q(f_x,f_y,f_z)\hfill \\ & & +S_{\mathrm{view}}(f_x,f_y,f_z)d{f}_{x}d{f}_{y}d{f}_z.\hfill \end{array}$$ (12)

The magnitude and dependencies of each component is analyzed in Sec. 4B, in each case relating back to the observations in Sec. 2. The parameters describing the imaging chain in Eqs. 4, 5, 6, 7, 8, 9, 10, 11, 12 were obtained as in previous work, derived theoretically from a model of the x-ray beam and detector design. The scintillator MTF (T₃) and detector electronic noise (σ_add) were estimated from measurements and taken as empirical input to the model. In the analysis of noise versus N_proj in Sec. 4B, there was no fitting of theory to the measured results, and the level of agreement is simply that suggested from the cascaded systems model in Eqs. 4, 5, 6, 7, 8, 9, 10, 11, 12.

Magnitude and dependence of the noise components

The observed behavior of image noise versus N_proj (at constant D_tot) is described well by the analytical forms in Eqs. 4, 5, 6, 7, 8, 9, 10, 11, 12. Figure 5a, for example, compares the measured voxel noise (in the clutter layer of the cylindrical phantom) in comparison to the prediction of Eq. 12 at D_tot = 2 and 8 mGy and at voxel size b_vox = 0.22 and 0.44 mm. Figure 5b breaks down one of the cases (D_tot = 2 mGy, b_vox = 0.44 mm) into the various noise components, where the behavior of σ_tot (and the optimum at N_proj ∼ 300) is described well by a sum of components that are constant (σ_Q), monontonically increasing (σ_E), and monotonically decreasing (σ_view) as a function of N_proj.

Figure 5.

Analytical model for CBCT noise versus N_proj. (a) Comparison of theory and measurement at various dose (2 and 8 mGy) and voxel size (0.22 and 0.44 mm). (b) Individual contributions of quantum noise (σ_Q), electronic noise (σ_E), and view sampling (σ_view) to the total voxel noise (σ_tot) as a function of N_proj.

The dependence of each noise component is further detailed in Fig. 6. The quantum noise (σ_Q) is seen to be independent of N_proj and varies in inverse proportion to the (square root of) total dose. On the other hand, the electronic noise (σ_E) increases as the (square root of) N_proj when dose per projection is fixed. View aliasing noise (σ_view) decreases approximately as (1/N_proj) and carries an explicit dependence on the value of κ and β in the power-law clutter model. The total noise (σ_tot) exhibits the now familiar nonmonotonic dependence on N_proj, and the optimal N_proj is seen to shift to higher values (or the optimum is lost altogether) as electronic noise is reduced—for example, σ_tot nearly independent of N_proj for N_proj > 400 and σ_add < 2000 e.

Figure 6.

Analysis of each noise component as a function of N_proj. (a) Quantum noise (σ_Q) evaluated at various levels of D_tot. (b) Electronic noise (σ_E) at various levels of D_tot. (c) View aliasing noise (σ_view) at various levels of the power-law magnitude κ. (d) Total voxel noise (σ_tot) at various levels of electronics noise.

DISCUSSION AND CONCLUSIONS

This work revisits fairly fundamental considerations of CBCT image noise in experimental and theoretical analysis of the effect of the number of projections. It elucidates and quantifies trends that are intuitive in retrospect, while calling to light a number of less obvious effects with practical implications for system design and specification of scan protocols.

The measurements in Figs. 234 illustrate the tradeoffs among three noise components—quantum noise, electronics, noise, and view aliasing—under conditions of constant dose. While other work (e.g., Refs. ³⁴ and ³⁵) has investigated the question of noise and number of projections from various perspectives and applications (including, for example, tomosynthesis³⁴), we are unaware of a study that so clearly illustrates the distinct contributions of each noise component at fixed total dose. The (perhaps surprising) appearance of a “sweet spot” (minimum) in the analysis of noise versus N_proj is intriguing, and it certainly defies the naïve logic of the rule of thumb expressed in Eq. 1. That rule of thumb, of course, is based on an oversimplified argument of sampling and spatial resolution, implying simply that more projections is better; however, the noise tells a different story, revealing an “optimal” N_proj that minimizes the contributions of electronic noise and view aliasing relative to the underlying (constant) quantum noise. Perhaps of no surprise to experimentalists at work in CBCT system development over the last decade, the optimum resides in a region about N_proj ∼300, nearly an order of magnitude below that implied by Eq. 1. The existence of the optimum owes directly to the relatively high level of electronic noise in FPDs, and for systems with higher performance detectors (i.e., lower readout noise relative to system gain), the optimum shifts to a higher number of projections and is eventually lost [Fig. 6d] such that a greater number of projections monotonically reduces image noise.

The position of the sweet spot is not a general result, and the dependence associated with any particular CBCT system is expected to depend on system configuration—most notably, the system geometry, choice of FPD, mode of operation, binning, and level of electronics noise. The theoretical analysis detailed in Sec. 4 provides a general framework for understanding such dependencies. The “optimal” N_proj increases as the electronic noise decreases, the total dose increases, and the voxel size increases. For sufficiently low electronic noise (and/or high dose or large voxels), the total noise reduces, the minimum in the plot of noise versus N_proj shifts ever more to the right, and the choice of N_proj falls to practical considerations of frame rate, scan speed, and the amount of projection data to transfer and reconstruct.

The experiments and imaging bench of Fig. 1 involved pulsed exposure “step-and-shoot” CBCT acquisition, thereby minimizing the potential influence of rotation (azimuthal blur) during x-ray exposure. Systems with differing pulse rate and detector readout rate can be expected to demonstrate a different dependence of noise on the number of views. For example, in CBCT systems employing continuous x-ray exposure during rotation, such an effect has been observed to impart a significant component of the MTF.³⁶ In the context of the work reported here, such azimuthal blur is anticipated to reduce view aliasing effects as well as stochastic quantum noise in a manner analogous to other sources of blur and correlation. As evident in both the experimental and theoretical results above, factors that reduce noise {e.g., increased D_tot [Figs. 35a, 6a, 6b], reduced σ_add [Fig. 6d], and increased pixel binning or voxel size [Figs. 35a]} tend to shift the “optimal” number of projections to higher values of N_proj.

Although beyond the scope of the current work, the best choice in N_proj also likely depends on the choice of reconstruction filter (smoother filters providing tolerance in N_proj analogous to increased dose or reduced electronic noise) and—almost certainly—depends on the choice of reconstruction algorithm, with statistical model-based methods providing a greater degree of robustness against sampling effects than filtered backprojection.^{28, 29} The work could certainly also be extended to task-based performance analysis (e.g., calculation of task-based detectability index as a function of N_proj), which is very much within the capacity of the framework detailed in Sec. 4; however, the purpose of the current work was to elucidate the fundamental components underlying image noise, and extension to task-based analysis and optimization is left to future work in the context of specific systems and applications.

The experiments also demonstrate a number of less obvious considerations with respect to sampling. The first is evident upon close inspection of the uniform and heterogeneous regions of the phantom imaged as a function of N_proj as in Fig. 2. The effect of a finite number of projections (i.e., noise associated with view sampling) is not well appreciated in a uniform medium; rather, view sampling is only evident when there is heterogeneous structure present in the image. The effect is most obvious for N_proj < 200 (the left-most portion of the curves in Fig. 3), where the increase in noise is entirely attributable to view sampling of object clutter (whereas the quantum and electronics noise measured in water are immune to such sampling effects). The observation is roughly consistent with findings from Zhao et al.,³⁷ who observed streaks in the NPS when angular separation between projections exceeded ∼2°. Phantoms presenting a uniform medium (or a simple arrangement of contrast inserts as in the ACR phantom, for example) are not well suited to probing the effect of view sampling under such conditions of few projection views. “Clutter phantoms”—popular in the last decade for simulating the anatomical power spectrum in applications such as breast CBCT (Refs. ^{38, 39, 40})—are considerably better suited to such measurement, and a collection of spheres of varying diameter and contrast [providing an approximation to noise of the form 1/f ³ associated with a self-similar object (3D fractal)] was shown to provide a good, statistically motivated phantom for physically probing such sampling effects.¹²

Further consideration involves the effect of redundant rays on the noise and NPS as shown in Fig. 4. Especially under conditions of few projection views (N_proj below the “sweet spot”), N_proj should be selected to avoid redundant views (i.e., views separated by Δθ = 180°). Such redundant rays have little or no effect on quantum and electronics noise (which depend only on the total number of quanta presented to the imaging system), but they result in undersampling of the object (increase in σ_view) that is obvious in the NPS and carries a strong increase in view sampling as evident in the clutter images of Fig. 4 (but not in the corresponding water images). The sparsity of vanes in the NPS (i.e., poor filling of the Fourier domain) is absolved by using a number of projections and angular increment that do not evenly divide with redundant backprojections over the second half of the data. The effect is stronger in scenarios more closely approximating a parallel beam assumption (e.g., near the center of reconstruction and systems with an extended SAD and low magnification) and may be important to consider in applications involving sparse data acquisition as a means to reduce radiation dose and/or reduce computational load or reconstruction time.^{28, 29}

In summary, a common question in CBCT system design and operation was entertained using both experimentation and theory. The experiment elucidated and quantified the tradeoffs among quantum noise, electronics noise, and view aliasing as a function of the number of projections—particularly highlighting the importance of a heterogeneous object (e.g., a statistically motivated “clutter” phantom) in investigating such effects. The theoretical analysis provided an analytical foundation that agreed with the observed effects. Such methods may be potentially useful in designing and operating new CBCT systems for a particular application, can be extended to applications such as limited angle tomosynthesis, hold implications for CBCT image reconstruction from sparse projection data, and could be helpful in understanding image quality effects among scan protocols on clinical CBCT scanners in applications ranging from breast and maxillofacial imaging to image-guided radiotherapy.