Modeling and Control of Nonstationary Noise Characteristics in Filtered-Backprojection and Penalized Likelihood Image Reconstruction

Abstract

Purpose:

Nonstationarity of CT noise presents a major challenge to the assessment of image quality. This work presents models for imaging performance in both filtered backprojection (FBP) and penalized likelihood (PL) reconstruction that describe not only the dependence on the imaging chain but also the dependence on the object as well as the nonstationary characteristics of the signal and noise. The work furthermore demonstrates the ability to impart control over the imaging process by adjusting reconstruction parameters to exploit nonstationarity in a manner advantageous to a particular imaging task.

Methods:

A cascaded systems analysis model was used to model the local noise-power spectrum (NPS) and modulation transfer function (MTF) for FBP reconstruction, with locality achieved by separate calculation of fluence and system gain for each view as a function of detector location. The covariance and impulse response function for PL reconstruction (quadratic penalty) were computed using the implicit function theorem and Taylor expansion. Detectability index was calculated under the assumption of local stationarity to show the variation in task-dependent image quality throughout the image for simple and complex, heterogeneous objects. Control of noise magnitude and correlation was achieved by applying a spatially varying roughness penalty in PL reconstruction in a manner that improved overall detectability.

Results:

The models provide a foundation for task-based imaging performance assessment in FBP and PL image reconstruction. For both FBP and PL, noise is anisotropic and varies in a manner dependent on the path length of each view traversing the object. The anisotropy in turn affects task performance, where detectability is enhanced or diminished depending on the frequency content of the task relative to that of the NPS. Spatial variation of the roughness penalty can be exploited to control noise magnitude and correlation (and hence detectability).

Conclusions:

Nonstationarity of image noise is a significant effect that can be modeled in both FBP and PL image reconstruction. Prevalent spatial-frequency-dependent metrics of spatial resolution and noise can be analyzed under assumptions of local stationarity, providing a means to analyze imaging performance as a function of location throughout the image. Knowledgeable selection of a spatially-varying roughness penalty in PL can potentially improve local noise and spatial resolution in a manner tuned to a particular imaging task.

I. INTRODUCTION

The ability to accurately model image noise is essential in the assessment and optimization of imaging systems. Previous work has successfully modeled the noise characteristics of CT/CBCT systems with a flat-panel detector within the context of filtered backprojection (FBP) reconstruction and validated model predictions of the noise-power spectrum (NPS) and task-based detectability index (d′).^{1, 2} Such work has primarily described the imaging performance at the center of the reconstruction of a circular object, although the ability to describe local, spatially varying aspects of the NPS also falls within the framework.³ Recent advances and proliferation of nonlinear statistical reconstruction algorithms introduces further challenges in noise modeling due to noise and resolution properties that are non-stationary, non-isotropic, object-dependent, and governed by parameters of the iterative reconstruction process (e.g., strength of regularization) that defy - and potentially overcome - conventional noise-resolution tradeoffs. This work extends modeling of CT noise and resolution properties to include description of nonstationary image noise in both FBP and (quadratic penalty) PL image reconstructions and demonstrates the impact of such noise characteristics on task-based detectability. Spatially varying regularization maps in PL reconstruction can be exploited to impose different degrees of smoothing throughout the image, introducing a potential method to fine-tune local noise and resolution in a manner advantageous to a given imaging task.

II. THEORETICAL AND EXPERIMENTAL METHODS

A.). Modeling Nonstationary Noise in FBP

Previous work has established a cascaded systems analysis model which propagates signal and noise transfer characteristics through distinct stages in projection formation and reconstruction, yielding Fourier domain image quality metrics including the modulation transfer function (MTF), noise-power spectrum (NPS), noise-equivalent quanta (NEQ), detective quantum efficiency (DQE), and detectability index (d′).^{1, 4} The 3D NPS for CT/CBCT reconstruction is given by:

$$S=\sum _{i=1}^m\frac{\left({\overline{q}}_0^i{a}_{pd}^4{\overline{g}}_1^i{\overline{g}}_2^i\left(1+{\overline{g}}_4{P}_K{T}_3^2\right)T_5^2**II{I}_6+S_{add}\right)}{{\left({\overline{q}}_0^i{a}_{pd}^2{\overline{g}}_1^i{\overline{g}}_2^i{\overline{g}}_4\right)}^2}{T}_9^2{T}_{10}^2{T}_{11}^2\frac{1}{M^2}{{\Theta }_{12}^i}^2***II{I}_{13}$$ (1)

where the notation for fluence (q_o), aperture size (a_pd), gain (g), transfer function (T), sampling (III), additive NPS (S_add), magnification (M), and backprojection operator (Θ) are defined and described in detail in Ref. 1.

To account for the varying fluence and spectrum transmitted to the detector from different view angles, noise from each view is summed individually. Instead of assuming a continuous 3D 1/f filter for the backprojection operation, each projection NPS was backprojected separately following the transfer function:

$${\Theta }_{12}^i=\frac{M{\theta }_{tot}}{m}\sum _{i=1}^{m}d\text{sinc}\left[d\left(f_x\text{cos\hspace{0.17em}}{\theta }_i+f_y\text{sin\hspace{0.17em}}{\theta }_i\right)\right]$$ (2)

where d is the width of the detector, θ_i is the incident angle of each projection, and the i notation (as either subscript or superscript) marks the i^th projection view. In addition, the incident spectrum ${\overline{q}}_0^i$ and gain factors ${\overline{g}}_1^i$ and ${\overline{g}}_2^i$ are calculated independently for each view as a function of detector location (u, v), allowing calculation of the NPS at any location with an object of arbitrary form. The NPS calculated from the cascaded systems analysis model, including the spatially varying NPS, has been validated with measurements and has been applied to task-based analysis of d′ demonstrating close agreement with human observers for simple imaging tasks across a broad range of imaging conditions.^1,2

B.). Modeling Nonstationary Noise in PL

The penalized likelihood reconstruction, $\widehat{\mu }$, maximizes the objective function, Φ, as a combination of the likelihood term and regularization term:

$$\widehat{\mu }=\text{arg\hspace{0.17em}max\hspace{0.17em}}\Phi (\mu ;y)=\text{arg\hspace{0.17em}max}[\text{log}L(\mu ;y)-\beta R(\mu )]$$ (3)

where L is the conditional probability of measurement/projection data, y, on the true object, μ. The regularization function R penalizes roughness in the data, governed by the regularization strength parameter β. While commonly treated as a constant, β can vary spatially to impart different degrees of smoothing at different locations within the image.

The covariance of an implicitly defined biased estimator, such as PL reconstruction shown in Eq.(3) was derived by Fessler et al.⁵ using the implicit function theorem and Taylor expansion. In the case of transmission tomography with a Poisson noise model, the covariance matrix of voxel j with every other voxel in the image, indicated by the subscript, j ·, is given by:

$${[\text{cov}\{\widehat{\mu }\}]}_j.\approx {\left[{\text{A}}^{\text{T}}\text{D}\{\overline{y}(\mu )\}\text{A}+\beta R\right]}^{-1}{\text{A}}^{\text{T}}\text{cov}\left\{y\right\}\text{A}{\left[{\text{A}}^{\text{T}}\text{D}\{\overline{y}(\mu )\}\text{A}+\beta R\right]}^{-1}{e}_j,$$ (4)

where A is the forward projection operator and its transpose, A^T is the backprojection operator. The weighting matrix, $\text{D}\{\overline{y}(\mu )\}$, for the forward and backprojection operation is diagonal and equal to the mean measurement under the assumption of independent measurement following the Poisson noise model. The regularization operator, R, is the second derivative of R(μ), and is independent of μ for quadratic penalties considered in this study. The terms y and $\text{D}\{\overline{y}(\mu )\}$ introduce object dependence in the noise through the projections of the object. The unit vector, e_j, extracts a column (or equivalently, a row) from the full covariance matrix, $\text{cov}\left\{\widehat{\mu }\right\}$, allowing calculation of the noise at any location within the image.

Assuming the covariance matrix around a small neighborhood, $\mathcal{N}$, to be circulant (i.e., diagonalized by the discrete Fourier transform), the local NPS was calculated by the Fourier transform:

$${\text{NPS}}_{\mathcal{N}}=\text{FT}\left[\text{cov}{\left\{\mu \right\}}_{j,\mathcal{N}}\right].$$ (5)

The point spread function (PSF) of PL reconstruction was derived similarly:⁶

$${[\text{PSF}\{\mu \}]}_j\approx {\left[{\text{A}}^{\text{T}}\text{D}\{\overline{y}(\mu )\}\text{A}+\beta R\right]}^{-1}{\text{A}}^{\text{T}}\text{cov}\left\{y\right\}\text{A}{e}_j.$$ (6)

The local MTF was calculated as the Fourier transform of the PSF, assumed to be circulant within a small neighborhood:

$${\text{MTF}}_{\mathcal{N}}=\text{FT}\left[\text{PSF}{\left\{\mu \right\}}_{j,\mathcal{N}}\right].$$ (7)

In contrast to the MTF of FBP, which is largely independent of the object or spatial location (at least to first-order approximation, ignoring angular sampling effects etc.), the PSF of PL reconstructions carries object and location dependencies that need to be explicitly accounted in analyzing the spatially varying signal and noise performance.

Equations (3) and (6) reveal the possibility to achieve prescribed noise and resolution properties locally by spatial variation of the regularization parameter β. For example, β(x,y) can be arbitrarily shaped to provide more uniform noise throughout the image⁷ or to shape regularization in a manner optimal to a given task. Theoretical predictions of the covariance and PSF of PL reconstructions were validated with simulated data using an ideal detector model, with future work to include validation in realistic detector models (e.g., including effects of correlation (blur) in the detector and electronic noise).

C.). Task-Based Performance

The nonprewhitening observer model was used to relate noise and resolution metrics to task-based detectability index as follows:⁸

$$d^{\text{'}2}=\frac{{\left[\iiint {\left(MTF\cdot W_{\mathit{\text{Task}}}\right)}^{2}d{f}_{x}d{f}_{y}d{f}_z\right]}^2}{\iiint NPS\cdot {\left(MTF\cdot W_{\mathit{\text{Task}}}\right)}^{2}d{f}_{x}d{f}_{y}d{f}_z}$$ (8)

where, instead of decorrelating the noise, a template in the form of the expected signal, MTF · W_Task, is used to match the noise. The task function, W_Task, is defined as the difference in Fourier transforms of the two hypotheses in a binary decision task. Four tasks carrying different spatial frequencies were defined directly in the Fourier domain, as described below.

III. RESULTS AND BREAKTHROUGH WORK

A.). Nonstationarity of the NPS

Figure 1 shows the spatial variation in the axial plane for the theoretical prediction of NPS computed locally for both FBP and PL reconstructions of example phantoms, including a circular water cylinder of 16 cm diameter and an ellipse (16 × 32 cm minor and major axes) with hypodense (−418 HU) and hyperdense (+418 HU) circular inserts. The total dose (to the central point in the axial plane) was 1 mGy in both cases. Each column shows the NPS computed at 4 locations within each phantom - the center and 3 locations of interest. Only at the center of the circular phantom is the NPS roughly isotropic (for both FBP and PL), which is the commonly reported NPS for axial CT. Asymmetry arises at locations off-center in the circular phantom and at all locations throughout the ellipse phantoms. The NPS is lower for PL reconstructions than FBP, due to the strong regularization applied (β = 5×10⁷). There was no attempt to match spatial resolution in these initial results, and the results are not intended as a head-to-head comparison of FBP and PL imaging performance. FBP and PL exhibit distinct levels of asymmetry and nonstationarity in both phantoms, with each governed fundamentally by the fluence (and spectrum) transmitted along each view - for example: longer line integrals (increased noise) along views passing through the hyperdense insert in the heterogeneous ellipse. Comparing the NPS of FBP and PL, the latter appears more uniform and less strongly peaked due to greater smoothing applied to noisier measurements, a property of the PL algorithm not present in FBP.

Figure 1.

Nonstationarity of the NPS in FBP and PL image reconstructions shown at 4 locations (identified in the top row and labeled a-d). The NPS is shown at 4 locations within two phantoms of increasing complexity (a uniform circular cylinder and a non-uniform ellipse). The NPS is notably anisotropic and varies in a manner governed by the path length traversing through the object in each view.

Such a characteristic is also responsible for the anisotropy in MTF shown in Fig.2. For example, noisier measurements along the major axis of the ellipse (f_y axis) are smoothed more than along the minor axis (f_y axis), resulting in a “peanut” shaped MTF that is complementary to the NPS.

Figure 2.

Theoretical predictions of local MTF in PL reconstructions of example phantoms at different spatial locations. Greater smoothing is applied to noisier measurements, resulting in a nonisotropic PSF that is dependent on the obejct and spatial location.

B.). Effect of Nonstationarity on Detectability

Figure 2 shows the detectability index (d′) for FBP and PL computed as a function of location within the heterogeneous ellipse. Four tasks were considered: a delta-function-detection (weighting all frequencies equally); a low-frequency (1.5 mm) Gaussian detetction task; an asymmetric task (elongated object detection) with W_Task oriented along f_x (i.e., object along the minor axis of the ellipse); and the same asymmetric task oriented along f_y (object along the major axis of the ellipse). A fascinating dependence is revealed that demonstrates the effect of nonstationarity on detectability: 1.) for the delta-function task, d′ is slightly elevated in the hypodense insert (higher contrast and lower noise); 2.) for the gaussian task, d′ is considerably elevated in the hypodense insert (and much more so for PL); 3.) for the asymmetric task along the minor axis, d′ is elevated by virtue of a spatial-frequency mismatch between the task and the NPS (i.e., W_Task is highest where NPS is lowest); and 4.) for the asymmetric task along the major axis, d′ is diminished as the noise “masquerades” as the stimulus.

C.). Controlling Nonstationarity

PL reconstruction allows free spatial variation (in the domain of the reconstructed image) of the regularization strength, β(x, y). Such may be employed to improve stationarity of the noise or - interestingly - to purposely impart nonstationarity in the NPS and impulse response in a manner that tunes performance to a given task. The point is demonstrated in Fig. 3, where the covariance (alternatively NPS) is computed at 3 locations in the heterogeneous ellipse of Fig. 1: i.) constant β; ii.) a “certainty-based” β that yields a more stationary covariance; and iii.-iv.) β (x, y) shaped to control the covariance in the region of the hyperdense insert.

Figure 3.

Detectability index (d′) computed as a function of location in the heterogeneous ellipse phantom for FBP and PL (Fig.1, column 4 and 5) and four task functions (W_Task) shown in the top row. Local ROIs corresponding to the d′ values are indicated by the overlay with phantom image in the FBP, all-frequency task (column 1). The spatial dependence of task performance results directly from NPS nonstationarity (Fig.1). Interestingly, this enhancement is more uniform in the low-density insert in FBP, whereas PL presents a stronger peak in d′ at the center. The two asymmetric tasks illustrate best and worst case scenarios in which the spatial frequency of the task can mismatch or align with correlations of the NPS. When the frequency content of the task overlaps with that of the NPS (i.e., noise “masquerading” as signal), detectability is diminished, and vice versa. The effect is more pronounced in PL than FBP due to additional anisotropy in the impulse response function.

IV. DISCUSSIONS AND CONCLUSIONS

This work demonstrates analytical models for computing the non-stationary aspects of signal and noise transfer characteristics in CT imaging by FBP and PL reconstruction (using a quadratic penalty). The degree of nonstationarity in image noise is shown to affect task performance in a manner related to the “mismatch” of NPS and W_Task. The strength of the roughness penalty in PL reconstruction can be varied spatially to achieve nonstationary smoothing within the image, demonstrating the potential to control local image quality to achieve prescribed noise and resolution properties that are advantageous to a particular imaging task. Future work involves regularization design via the spatially varying β map in a manner suitable to a given task to improve detectability. A real detector model will be incorporated into the PL theoretical prediction model for non-ideal data and permit validation of the model in comparison to physical experiments. Modeling will also be investigated for non-quadratic penalties, e.g., the Huber penalty, which has been shown to provide superior performance in comparison to FBP⁹ for tasks benefiting from the edge-preserving noise reduction characteristic of non-quadratic iterative reconstruction techniques.

Figure 4.

Controlling covariance. The variance, covariance (alternatively NPS), and impulse response function in PL reconstruction can be controlled using spatially varying β. Calculations show the covariance matrix computed in the heterogeneous ellipse (Fig. 1) at three locations (center of the low-density circle, center of the ellipse, and center of the high-density circle) spanning a neighborhood of 25 voxels. The first row corresponds to spatially uniform β, resulting in nonuniform noise in PL reconstruction (i.e., greater correlation at the center of the hyperdense insert compared to the hypodense insert). The second row shows a certainty-based penalty³ to achieve more uniform noise throughout the image. By imposing local variation in β (e.g., in the hyperdense insert in rows 3 and 4), one can impart control over noise-resolution tradeoffs in a given region of interest. For example, greater β results in lower noise but lower resolution (row 3), suitable to increasing detectability of a low-contrast, low-frequency task. On the other hand, lower β results in higher noise but higher resolution (row 4), suitable to detection of a high-contrast, high-frequency task.