Impact of the non-negativity constraint in model-based iterative reconstruction from CT data

Abstract

Purpose:

Model-based iterative reconstruction is a promising approach to achieve dose reduction without affecting image quality in diagnostic x-ray computed tomography (CT). In the problem formulation, it is common to enforce non-negative values to accommodate the physical non-negativity of x-ray attenuation. Using this a priori information is believed to be beneficial in terms of image quality and convergence speed. However, enforcing non-negativity imposes limitations on the problem formulation and the choice of optimization algorithm. For these reasons, it is critical to understand the value of the non-negativity constraint. In this work, we present an investigation that sheds light on the impact of this constraint.

Methods:

We primarily focus our investigation on the examination of properties of the converged solution. To avoid any possibly confounding bias, the reconstructions are all performed using a provably converging algorithm started from a zero volume. To keep the computational cost manageable, an axial CT scanning geometry with narrow collimation is employed. The investigation is divided into five experimental studies that challenge the non-negativity constraint in various ways, including noise, beam hardening, parametric choices, truncation, and photon starvation. These studies are complemented by a sixth one that examines the effect of using ordered subsets to obtain a satisfactory approximate result within 50 iterations. All studies are based on real data, which come from three phantom scans and one clinical patient scan. The reconstructions with and without the non-negativity constraint are compared in terms of image similarity and convergence speed. In select cases, the image similarity evaluation is augmented with quantitative image quality metrics such as the noise power spectrum and closeness to a known ground truth.

Results:

For cases with moderate inconsistencies in the data, associated with noise and bone-induced beam hardening, our results show that the non-negativity constraint offers little benefit. By varying the regularization parameters in one of the studies, we observed that sufficient edge-preserving regularization tends to dilute the value of the constraint. For cases with strong data inconsistencies, the results are mixed: the constraint can be both beneficial and deleterious; in either case, however, the difference between using the constraint or not is small relative to the overall level of error in the image. The results with ordered subsets are encouraging in that they show similar observations. In terms of convergence speed, we only observed one major effect, in the study with data truncation; this effect favored the use of the constraint, but had no impact on our ability to obtain the converged solution without constraint.

Conclusions:

Our results did not highlight the non-negativity constraint as being strongly beneficial for diagnostic CT imaging. Altogether, we thus conclude that in some imaging scenarios, the non-negativity constraint could be disregarded to simplify the optimization problem or to adopt other forward projection models that require complex optimization machinery to be used together with non-negativity.

1. INTRODUCTION

A lot of the research in diagnostic computed tomography (CT) is driven by the aim of reducing radiation dose while maintaining image quality. One promising way to achieve this goal is model-based iterative reconstruction (MBIR). Its potential for diagnostic CT imaging has been shown in a number of studies.^1–9 A popular MBIR formulation is penalized weighted least squares (PWLS) reconstruction,¹⁰ which includes two key components: (a) the data fidelity term, which is characterized by the choice of a forward projection model and the option of a statistical weighting of the projections; and (b) the penalty term, which defines a regularization process often involving a potential function applied to voxel differences, together with additional incorporation of a priori knowledge, such as the non-negativity constraint.

To get the most out of the MBIR approach, it is valuable to understand the impact of each component and its subparts. In this context, Thibault et al. gave a thorough discussion on the impact of various parametric expressions for the potential function in the penalty term.¹⁰ This discussion partly relied on knowledge accumulated outside CT.^11–15 Furthermore, Tang et al. provided a comparison with total variation.¹⁶ As they can be designed in many different ways, forward projection models for iterative CT reconstruction have also been given a lot of attention in terms of efficiency and impact on image quality.^17–24 In the same context, the importance of statistical weights in the data fidelity term was examined using both model observers²⁵ and human readers.²⁶

In this paper, we are interested in the non-negativity constraint. Since the attenuation coefficient of x-rays is known to be positive, this constraint appears very natural. However, in the PWLS formulation, non-negativity is not automatically enforced and enforcing it is not straightforward. Applying a non-negativity constraint at the end of the iterations most often will not produce the desired result, although there are special cases where it does.²⁷ Also, applying the constraint at each iteration can violate the process of optimization so that the reconstruction algorithm either diverges or converges to a suboptimal result. For example, well-established general methods like conjugate gradient or nonlinear conjugate gradient²⁸ are known to suffer from this problem. Fortunately, modern proximal splitting methods can correctly accommodate non-negativity.²⁹ Examples for these advanced algorithms include the alternating direction method of multipliers (ADMM), a good review of which is offered by Boyd et al.³⁰; the primal-dual algorithm of Chambolle and Pock³¹; and the iterative soft thresholding algorithm (ISTA)³² or its fast version, FISTA.³³ Note that these algorithms are not only more complex to use; some of them may converge slowly.

The decision to enforce non-negativity or not does not only complicate the algorithmic choices for optimization. It also affects the options for the forward projection model. For example, basis functions like the natural pixels³⁴ are not easily used together with the non-negativity constraint because the coefficients of such basis functions must be allowed to be both positive and negative. An advanced optimization method like the Chambolle-Pock algorithm appears necessary to use natural pixels together with non-negativity, following steps similar to those presented by Rose et al.³⁵ for joint utilization of natural pixels and a TV constraint between voxel values.

From an image quality viewpoint, the non-negativity constraint may not always produce the expected result. Through a theorem that he called the night sky theorem,³⁶ Byrnes showed that the non-negativity constraint can be problematic. Specifically, when the optimization problem amounts to solving a linear underdetermined system of equations with no non-negative solution, enforcing the non-negativity will lead to positive pixels scattered throughout the entire image that resemble bright stars in the dark night sky. Additionally, in PET imaging, it has been observed that enforcing non-negativity can result in undesirable bias for low count scans.^37,38

Nevertheless, enforcing non-negativity is believed to be beneficial for improving image quality in terms of lesion detection in nuclear medicine,³⁹ where it is automatically enforced through the Poisson likelihood model for data fidelity⁴⁰; and non-negativity has been shown to be valuable for rotational x-ray angiography from few views.⁴¹ In CT, the common perception seems to be that the non-negativity constraint is valuable as many important works on MBIR enforce it either exactly or empirically.^10,42–44 Undoubtedly, CT scientists have looked at the issue and forged some opinion in association with their preferred iterative reconstruction scheme. To our knowledge, there is however no publication that reports on an in-depth examination of the impact of the non-negativity constraint on CT image quality. This work aims at filling this gap in the literature. Our motivation for carrying out such an examination is further augmented by the heavily nonlinear nature of MBIR, which makes reliable predictions essentially impossible. For example, consider a CT scan where the only source of data inconsistency is noise in the data. By enforcing non-negativity, we may decrease accuracy within the scanned object because fewer voxels are available to account for the noise in the data, or we may increase accuracy because more information is now provided in the reconstruction process.

In this work, we investigate the impact of the non-negativity constraint in CT through various experiments. For this investigation, we had to decide between carrying out the experiments from the point of view of converged solution to the optimization problem, or from the point of view of an approximate solution obtained through an empirical but efficient algorithm. Both options have merit, which we now briefly discuss. The first option addresses the investigation from a fundamental viewpoint. It is focused on identifying the value of the non-negativity constraint independently of the iterative algorithm employed for reconstruction. If a benefit is observed, the first option does not, however, provide information on how the benefit can be achieved in clinical routine, but it provides a benchmark. The second option addresses the investigation from a practical viewpoint for clinical routine. It may show that the constraint adds robustness, improves image quality, and possibly even allows using fewer iterations. But it cannot show that the maximum benefit of the non-negativity constraint is achieved. Also, if no benefit or negative effects are observed, it does not allow concluding that the constraint is not useful because the root of the observations may lie in the algorithm choice rather than the formulation of the optimization problem.

The investigation consists in five experimental studies that all use real CT data and are designed to challenge the use of the non-negativity constraint in various ways, including noise, beam hardening, parametric choices, truncation, and photon starvation. These five studies all rely on the first option discussed above, implemented by running a provably converging algorithm near convergence. The studies are complemented by a sixth one that relies on the use of an empirical but efficient algorithm based on ordered subsets. To avoid any confounding bias, we systematically start iterations from zero-valued voxels. To accommodate our decisions while keeping the computational load manageable, we focus the investigation on reconstructions of a small number of slices from axial CT scans. Within each study, reconstructions are performed twice: once with the non-negativity constraint, and a second time without. Then, the two settings are compared in terms of image similarity and convergence speed. As needed, we complement the image similarity evaluation with quantitative metrics that provide additional information in terms of noise at matched resolution or closeness to a known ground truth.

The paper is organized as follows. Section 2 provides the mathematical definition of the objective function and specifies the optimization algorithm we use, with an emphasis on how reconstruction with and without the non-negativity constraint is achieved. In Section 3, we describe our experimental setup to evaluate the impact of the non-negativity constraint. This includes the projection data, selected reconstruction studies, and evaluation methods we used. Section 4 presents the results of our evaluation. This section is divided into subsections referring to the six different reconstruction studies. Last, in Section 5, we give our conclusions together with a summary of our contribution and a discussion of various related aspects.

2. BACKGROUND

Model-based iterative reconstruction from CT data is formulated as optimization of a convex objective function. The optimization requires a dedicated algorithm that is consistent with the characteristics of this function. These two aspects are covered in the following subsections.

2.A. Objective function

Let $\overrightarrow{x}\in {ℝ}^N$ be a discrete vector for the three-dimensional reconstruction volume and $\overrightarrow{p}\in {ℝ}^M$ a discrete vector for the measured projection data. The objective function used for our investigations is similar to that introduced by Thibault et al. in 2007.¹⁰ This function, which we denote as F, consists of three parts: the data fidelity term f, the regularization term g with its hyperparameter β ≥ 0, and the indicator function $l_{{ℝ}_{+}}$:

$$F(\overrightarrow{x})=f(\overrightarrow{x})+\beta g(\overrightarrow{x})+l_{{ℝ}_{+}}(\overrightarrow{x}).$$ (1)

The data fidelity is expressed as a weighted squared residual between the forward projected reconstruction and the projection data:

$$f(\overrightarrow{x})=\frac{1}{2}{\Vert W^{-1/2}(A\overrightarrow{x}-\overrightarrow{p})\Vert }_2^2,$$ (2)

where A is the forward projection matrix. The matching back-projection operator is written as its transpose, A^T. The diagonal matrix W is used to assign a statistical weight to each measurement. In this work, note that the expression chosen for W changes from one study to another; a detailed description can be found in Sections 3.A and 3.B.

Let the components of $\overrightarrow{x}$ be written as x_i, i = 1,…,N. The regularization term mitigates noise in the reconstruction and is defined, in its most general form, as

$$g(\overrightarrow{x})=\sum _{i=1}^N\sum _{j=1}^N{c}_{ij}\psi \left(x_i-x_j\right),$$ (3)

where ψ is a potential function used to assign a cost to the difference between the voxel values, and c_ij ∈ [0, 1] is a weight applied to this cost, so that, for example, some differences can be ignored for voxels far away from each other. There is a vast variety of possible definitions for the potential function; the choice has a direct influence on the appearance of the reconstruction.^10,14,15

When we enforce non-negative voxel values for the reconstruction, the indicator function $l_{{ℝ}_{+}}$ is part of the objective function. This non-negativity constraint is defined as

$$l_{{ℝ}_{+}}(\overrightarrow{x})=\{\begin{array}{cc}0& \text{if }{x}_i\in {ℝ}_{+}\text{ for }i=1,\dots ,N\\ +\infty & \text{otherwise}\end{array}.$$ (4)

To simplify the optimization problem, a function f + βg that is strictly convex is often preferred, so that F has a unique minimizer. In diagnostic CT, such a condition is typically met by selecting ψ as a strictly convex function.¹⁰ If ψ is additionally differentiable, then f + βg is also differentiable and many optimization algorithms exist to minimize F. In this paper, we focus on potential functions with these characteristics.

2.B. Optimization algorithm

The first two parts of the presented objective function are convex and differentiable, and thus fairly straightforward to handle. The third part, however, brings challenges to optimize F due to the nonsmooth indicator function. The differential form of the indicator function cannot be described by a single gradient, instead it can only be expressed by a subdifferential which is a set of subgradients.⁴⁵ To handle a subdifferential during the optimization process, the so-called proximity operator can be used.²⁹ This operator maps any vector of ${ℝ}^N$ onto another vector of ${ℝ}^N$. In general, the evaluation of a proximity operator requires solving an optimization problem by itself. In the case of an indicator function of a given set, the proximity operator is the orthogonal projection operator onto the same set. This results in the following closed form solution:

$$\forall i=1,\dots ,N,{\left[{\text{prox}}_{l_{{ℝ}_{+}}}(\overrightarrow{x})\right]}_i=\{\begin{array}{cc}{x}_i& \text{if }{x}_i\in {ℝ}_{+}\\ 0& \text{otherwise}\end{array}.$$ (5)

According to our previous definitions, the optimization algorithm needs to be able to minimize a convex objective function that consists of a smooth and a proximable part. The fast iterative shrinkage-thresholding algorithm, also known as FISTA, meets these demands.³³ It requires only one gradient evaluation per iteration. The pseudocode of FISTA applied to our reconstruction problem is shown in Algorithm 1. We chose the version of FISTA that uses a fixed step size, called λ. The convergence is guaranteed for λ < 1/L, where L is the Lipschitz constant for the gradient of the smooth part, f + βg, of our objective function. The non-negativity constraint is enforced on line 3 of the pseudocode, using the shorter symbol (·)₊ instead of ${\text{prox}}_{l_{{ℝ}_{+}}}(\cdot )$. Note also that we use the conventional ∇ symbol to denote a gradient.

Algorithm 1.

FISTA with constant step size

Input: Parameters β ≥ 0,λ > 0 and initial image ${\overrightarrow{x}}_0$.
1:	${\overrightarrow{y}}_1={\overrightarrow{x}}_0,t_1=1$
2:	for k = 1,2,… do
3:	${\overrightarrow{x}}_k={\left({\overrightarrow{y}}_k-\lambda \left(A^T{W}^{-1}\left(A{\overrightarrow{y}}_k-\overrightarrow{p}\right)+\beta \left(\nabla g\left({\overrightarrow{y}}_k\right)\right)\right)\right)}_{+}$
4:	$t_{k+1}=\left(1+\sqrt{1+4t_k^2}\right)/2$
5:	${\overrightarrow{y}}_{k+1}={\overrightarrow{x}}_k+\left(\frac{t_k-1}{t_{k+1}}\right)\left({\overrightarrow{x}}_k-{\overrightarrow{x}}_{k-1}\right)$
6:	end for

If we do not apply the non-negativity constraint, the indicator function $l_{{ℝ}_{+}}$ is not used and our objective function is purely smooth. In this case, the algorithm simplifies itself to Nesterov’s accelerated gradient descent.⁴⁶ The only change in the presented pseudocode is that the calculation on line 3 is carried out without the operation (·)₊. The convergence condition for the step size remains unchanged.

3. EXPERIMENTAL SETUP

To understand the impact of the non-negativity constraint on the reconstruction result, we performed six experimental studies that challenge the non-negativity constraint in various ways. The first subsection describes the common image reconstruction and data acquisition settings for these studies. The second subsection describes the six studies along with settings that are specific only to them. Afterward, two subsections present the evaluation methods used to compare the reconstructions with and without non-negativity constraint. These methods cover convergence speed and image similarity.

3.A. Common settings

For all six studies, the data acquisition was performed on a state-of-the-art clinical CT system. These studies involve four scanned objects: the American College of Radiology (ACR) phantom, a head phantom, a hip phantom with metal implant, and a real human chest. A multislice axial scan was used that either records 1152 or 2304 projections over 360° depending on whether acquisition is done without or with a lateral flying-focal-spot (FFS). An overview of the scanner geometry can be found in Table I. The FFS option was always activated except for the chest scan.

Table I.

Parameters of scanner geometry.

Source to detector distance	108.56 cm
Source trajectory radius	59.5 cm
Anode angle	7 °
Number of detector channels	736
Angular detector width	0.067864 °
Number of detector rows	8
Detector row height at isocenter	0.06 cm
Number of projections with FFS	2304

To carry out experiments, the parameters in the objective function had to be fully specified. In our implementation, A is modeled using Joseph’s method.⁴⁷ It is a ray-driven approach that provides a good compromise between accuracy and computational cost.²³ Statistical weighting of the data depends on the scanned object. For the ACR, head, and hip phantoms, we simply chose W equal to the identity matrix, that is, we disregarded existing variations in the data noise, which were small in the ACR and head cases. For the human chest, W was related to the noise level in the data as explained later. For the regularization term in Eq. 3, we used c_ij = 1 for the neighbors of each voxel found in the three Cartesian directions and c_ij = 0 otherwise. For the potential function ψ, we chose an edge-preserving potential function that can be fine-tuned with δ > 0:

$$\psi (t,\delta )=\sqrt{t^2+{\delta }^2}-\delta .$$ (6)

Parameter δ controls the importance given to differences between neighboring voxel values.

For all reconstructions, except in the sixth study, 5000 iterations of FISTA were calculated, and after every 25th iteration, the intermediate result was saved. We used ${\overrightarrow{x}}_0=\overrightarrow{0}$ as initial reconstruction volume. The chosen step size λ was based on the Lipschitz constant of the data fidelity term, L(f), computed as the largest eigenvalue of A^TA using the power method.⁴³ To account for the regularization term, we used λ = 0.95/L(f) and the knowledge that L(βg)≪L(f).

3.B. Description of the experimental studies

3.B.1. ACR phantom study

For the first study, we used the ACR CT accreditation phantom (model 464, Gammex-RMI, Middleton, WI, USA). This choice was motivated by the simplicity of the phantom as well as the wide familiarity of CT scientists with its appearance in reconstructions. The ACR phantom has a cylindrical shape with a 20 cm diameter and a length of 16 cm. It is divided into four different modules of which we scanned two for our study, namely module A and module D. Module A has five cylinders representing the attenuation behavior of bone, polyethylene, water, acrylic, and air, respectively. Two ramps are also included that consists of small bars that are visible in 0.5 mm increments in the longitudinal direction. The module can be used to assess positioning and CT number accuracy. Module D contains eight aluminum bar patterns that provide very high contrast relative to the background and are used to assess the spatial resolution for high contrast objects, up to 12 lp/cm. The two modules were scanned separately. During the scan, each module was centered on the rotation axis, and the plane of the source trajectory passed through the middle of the module. The x-ray tube was operated at 80 kV and 500 mAs.

For reconstruction, we used a quadratic potential function in addition to the edge-preserving potential function that was mentioned earlier. It is defined as follows:

$$\psi (t,\delta )=\frac{t^2}{2\delta }.$$ (7)

This additional choice was made to also produce reconstructions that are similar to filtered backprojection (FBP).

The reconstruction was performed using a grid of 512 × 512 × 16 voxels where each voxel had the size of 0.1 × 0.1 × 0.06 cm³. The field-of-view (FOV) radius was 25.0 cm. The step size λ of FISTA was set to 0:000065. The hyper parameter β was 0.1. For the quadratic potential function, δ = 0.005, and for the edge-preserving potential function δ = 0.001. These values for δ were chosen empirically such that the noise is suppressed while maintaining a satisfying spatial resolution as well as a realistic (non cartoon-like) image appearance. A result of the reconstruction of both ACR phantom modules with edge-preserving regularization and the non-negativity constraint can be found in Figs. 1(a) and 1(b).

Fig. 1.

Reconstructions of (a) ACR phantom module A, (b) ACR phantom module D, (c) head phantom, (e) chest scan, and (f) hip phantom, with 5000 iterations, edge-preserving regularization, and non-negativity constraint. In each case, the solid orange curve marks the ROI that is used for the quantitative analysis, and the dashed line marks the image section that is used for display in the following figures. (d) The head phantom with ROIs used for resolution assessment and NPS calculation. The grayscale used for display can be found in the upper corners of each image.

3.B.2. Low-dose study

An important research topic in diagnostic CT is the reduction of x-ray radiation dose for the scanned patient while maintaining a sufficient image quality. This second study is focused on this topic. Specifically, we investigated the effect of the non-negativity constraint on low-dose data acquisitions. The used phantom for this study was an anthropomorphic head phantom (Humanoid Systems, Hawthorne, CA, USA). A water cylinder with a diameter of 4.5 cm was placed next to the head phantom with its long axis along the patient bed. Inside the water cylinder, there is a solid plastic rod of diameter equal to 1.1 cm. The different HU values of water and the plastic rod create a visible contrast between the two materials in the CT image, so that the edge of the plastic rod can be used to compare the resolution in the axial plane for different reconstructions. The selected axial plane for the scan was centered below the level of the eyes. In addition to noise effects, the chosen plane also allows examining the impact of the non-negativity constraint on beam hardening errors. Indeed, the smooth-kernel full-dose FBP reconstruction presented in Fig. 2 shows that the large and complex presence of bones in the scanned plane induces strong beam hardening errors.

Fig. 2.

Filtered backprojection reconstruction of the head phantom using the clinical workstation (smooth B10 filter kernel, 2 mm slice thickness, 320 mAs, 120 kV). Note that a narrow grayscale window is used to demonstrate that the selected slice is subject to prominent beam hardening errors.

Typically, a sequential head scan to rule out bleeding is performed with an x-ray tube setting of 120 kV and an exposure of about 300 mAs. Two approaches were considered to achieve low dose. In the first approach, the x-ray exposure was simply reduced to 40 mAs. In the second approach, the same low exposure was achieved through sparse sampling. That is, the x-ray tube current was set to 320 mAs and only every fourth projection image from one of the two focal spot positions was retained. This results in one-eighth of the original projection data and accordingly in one-eighth of the 320 mAs exposure. To distinguish between the two low-dose acquisitions, we refer to the first one as full view sampling and to the second one as sparse view sampling. Both scans of the head phantom were repeated ten times to enable assessing noise as well as the statistical variability of a quantitative metric of similarity, as discussed later in Section 3.D.

The reconstruction of the head phantom dataset was calculated for a volume of 512 ×512 × 16 voxels with a voxel size of 0.05 × 0.05 × 0.06 cm³. The FOV radius was 25.0 cm. The step size of the optimization algorithm was calculated based on the forward projection matrix and thus was not the same for the two low-dose datasets. For the full view sampling, λ = 0.000466 and for the sparse view sampling, λ = 0.003731. Also, the hyperparameter β was linked to the number of projection images to keep the same balance between the data fidelity and the regularization in the objective function for the different view sampling options.

To decide for a suitable value for the regularization parameters, we examined FBP reconstructions of the full view dataset obtained with two relatively smooth kernels on the workstation of the CT scanner, namely kernels B10 and B30 (Siemens Healthcare GmbH, Forchheim, Germany). Parameters β and δ were selected so that the standard deviation of the gray values in the cerebellum area was between the results of B10 and B30 kernels, and so that the bone sharpness is visually close to that of the B30 kernel. This lead to β = 0.075 and β = 0.009375 for full and sparse view sampling, respectively, and to δ = 0.0025. See Fig. 1(c) for a reconstruction result using the non-negativity constraint.

Although the parameters for the two low-dose approaches are set so as to create comparable images, note that the purpose of our study is to investigate the effect of the non-negativity constraint under two commonly discussed low-dose approaches, not to compare the merits of one approach vs the other.

3.B.3. Parametric space study

The third study was designed with two purposes in mind: (a) to investigate the interplay between the non-negativity constraint and parameters β and δ associated with the regularization, which were both fixed in the previous studies; (b) to examine the effect of the non-negativity constraint on tissues that have very low attenuation such as the lung parenchyma. For this purpose, a chest CT scan was used. This scan was acquired for standard of care at the University of Utah Health (Salt Lake City, UT, USA) and retrospectively saved with IRB approval. It is an axial scan of the thorax at heart level that was taken in the context of a multiphase abdominal CT protocol. Such a scan is used to see if the contrast agent has passed the aorta, to enable a correct timing of data acquisition over the abdomen. The x-ray tube was set to 120 kV and 50 mAs.

Note that the dataset shows a slight truncation of the patient bed. Since this kind of truncation in the data is commonly encountered in clinical practice, we did not attempt to avoid it through the patient selection process. To account for the strong nonuniformity of the scanned object, nonconstant statistical weights were used. Specifically, matrix W from Eq.2 was defined as the square root of the variance of the measured projection data. The square root of the variance rather than the variance was used to mitigate the dynamic range of the statistical weights. Also, to reduce the impact of measurements outside the scanned object, the weights were clipped to a maximum value. Last, the weights were normalized to have in average the value of 1 for the central ray. This normalization simplifies the selection of β. For the central ray, the weight varied between 0.6147 and 1.5516. For all rays together, the weight varied between 0.4293 and 6.0.

The reconstruction was performed on a 640 × 640 × 16 voxel grid with a voxel size of 0.08 × 0.08 × 0.06 cm³. The FOV radius was 25.16 cm and the step size of FISTA was set to λ = 0.000201. The following values were considered for β and δ:

• β = {0.01,0.05,0.1,0.25,0.5}

• δ = {0.01,0.0001}

Combined together, this resulted in ten different reconstruction settings, for each of which a reconstruction was calculated with and without non-negativity.

To display the reconstructions of the chest scan, we use a split grayscale window, as illustrated in Fig. 1(e) for the reconstruction achieved with the non-negativity constraint when β = 0.1 and δ = 0.0001. The narrow window of [−1000, −700] HU on the left side of the image makes image artifacts more visible to the reader. The wide window of [−1100, 100] HU on the right side of the image corresponds to a display setting that is typically used by the radiologist for lung inspection.⁴⁸

3.B.4. Truncated data study

As explained in the introduction, the non-negativity constraint is meant to provide additional information to improve the reconstruction result. When the acquired data are not complete, a priori information can be valuable to avoid or reduce artifacts. An example for incomplete data is truncated projection data. The problem of truncation can appear when the object is too large to be covered by the FOV of the scanner. The fourth study was designed to shed light on impact of the non-negativity constraint for reconstruction from such truncated projection data.

This study employed the same clinical dataset as the one used in the third study. To simulate data truncation, we discarded 184 measurements on each side of the detector, that is, we only used the central 368 of the 736 channels. The amount of discarded data was chosen so that all rays passing through the lungs remain part of the measurements. The reconstruction parameters were identical to the former study, except for the step size of the optimization algorithm that had to be adapted to the change in the scanner geometry and became λ = 0.000118. Hyperparameter β and parameter δ of the edge-preserving potential function were fixed to a preferred setting based on the findings of the previous study. Note that in this truncated data study, the reconstruction from the nonmodified dataset can serve as a ground truth; later, we refer to this reconstruction as that obtained from nontruncated data.

3.B.5. Metal artifacts study

Metal objects in the FOV of the scanner are a well-known source of data inconsistency. As they highly attenuate x-rays, they can lead to strong beam hardening errors as well as photon starvation problems. The fifth study examines how the non-negativity influences the result of a reconstruction that suffers from strong inconsistency in the projection data induced by metal objects.

The dataset for this study was created by using a hip phantom (QRM, Möhrendorf, Germany) that consists in an artificial hip joint made out of metal surrounded by water-equivalent material. The reconstruction was performed on a 640 × 640 × 16 voxel grid with a voxel size of0.08 × 0.08 × 0.24 cm³. The FOV radius of the scanner was 25.16 cm. The step size of the optimization algorithm was 0.000101. Hyperparameter β was set to 0.2 and parameter δ to 0.01. Both values were picked subjectively based on a reasonable image appearance as shown in Fig. 1(f). To mitigate noise, the detector rows were combined so as to create an effective detector pixel height of 0.24 cm at isocenter. Also, the scan was repeated two times and all analyzed reconstructions were averaged over these two scan repetitions.

3.B.6. Empirical algorithm acceleration study

As explained in the introduction, in this work, we decided to use a high number of iterations to examine the effect of the non-negativity constraint on (essentially) converged solutions. Clearly, the high computational effort for 5000 iterations of FISTA is not practical for clinical routine. The sixth and last study looks at a more practical reconstruction approach that uses only 50 iterations but with ordered subsets within the optimization algorithm. The utilization of ordered subsets is known to be very effective to accelerate the reconstruction, particularly in the early iterations. Note that this acceleration gain comes with a cost: convergence of the algorithm is no longer guaranteed, and when convergence occurs, it may not be to the optimum of the objective function. Our implementation is similar to the “OS-mom1” method from Kim et al.⁴⁴ One iteration is counted after all subiterations are performed, meaning that all subsets are used once within one iteration of the algorithm. Note that another interesting way to accelerate FISTA was proposed by Xu et al.⁴⁹

For this study, the clinical chest scan was again used. The settings for the scan and the reconstruction can be found in Section 3.B.3. The parameter values for β and δ were selected based on the results of the third study. The 1152 projections of the dataset were divided into 12 subsets, resulting in 96 projection images for each subset. For simplification of the implementation, a sequential ordering of the subsets was used.

3.C. Assessment of convergence speed

The root-mean-square error (RMSE) was used to assess convergence speed. To compute this figure-of-merit, the result of 5000 iterations was used as a surrogate for the converged solution. Thus, we report

$$\begin{gathered} \text{RMSE}(k)=\sqrt{\frac{1}{\#\Omega }\sum _{i\in \Omega }{\left({\overrightarrow{x}}_k(i)-{\overrightarrow{x}}_{5000}(i)\right)}^2,} \\ \text{for }k=25,50,\dots ,2500 \end{gathered}$$ (8)

where ${\overrightarrow{x}}_k$ refers to the result obtained after k iterations. In this expression, both ${\overrightarrow{x}}_k$ and ${\overrightarrow{x}}_{5000}$ are defined individually for the reconstructions with and without non-negativity; hence, RMSE for k = 5000 is always equal to zero. Note that only k as large as 2500 was used, to mitigate the confounding effect associated with the fact that ${\overrightarrow{x}}_{5000}$ is only a surrogate for the converged solution.

To ignore the irrelevant structures outside of the scanned object, the RMSE was only computed over a region of interest (ROI) called Ω, with #Ω denoting the number of voxels in Ω. These ROIs were all drawn over the central four axial slices where accurate reconstruction is expected for the employed imaging geometry. For the ACR phantom, the head phantom, and the hip phantom, the ROI closely follows the boundary of the object. For the chest scan, the ROI consists of two disjoint regions covering together the entire lung tissue. The ROIs corresponding to each object are depicted with a solid orange curve in Fig. 1.

The purpose of the reported RMSE value is to highlight potential differences in convergence speed between the reconstructions with and without the non-negativity constraint. Differences in image appearance are assessed as discussed hereafter.

3.D. Assessment of image similarity

As stated in the introduction, our primary interest is to assess the impact of the non-negativity constraint on image appearance when the image is defined as the solution of the MBIR optimization problem. The assessment was performed using both qualitative visual impression and quantitative evaluation of differences. For visual impression, difference images that compare the results obtained with and without non-negativity constraint when using 5000 iterations were inspected over each of the four central axial slices where accurate reconstruction is expected. These difference images are restricted to object-specific ROIs highlighted by a solid orange curve in Fig. 1.

For quantitative assessment, the root mean squared difference (RMSD) was evaluated over the previously mentioned ROIs. The formula for the RMSD corresponding to a number k of iterations is

$$\text{RMSD}(k)=\sqrt{\frac{1}{\#\Omega }\sum _{i\in \Omega }{\left({\overrightarrow{x}}_{k,\text{on}}(i)-{\overrightarrow{x}}_{k,\text{off}}(i)\right)}^2}$$ (9)

where ${\overrightarrow{x}}_{k,\text{on}}$ and ${\overrightarrow{x}}_{k,\text{off}}$ are the results with and without non-negativity constraint. To convey the level to which the results are confounded by the use of a finite number of iterations, a mean value, $\overline{\text{RMSD}}$, which corresponds to averaging RMSD(k) over k from 4500 to 5000 by steps of 25, is reported together with the standard deviation observed in RMSD(k) over the same values for k. In addition, the minimum and maximum differences in voxel values are given for k = 5000. Also, for reconstruction from the head phantom dataset, the repeated scans were used to calculate the standard deviation of $\overline{\text{RMSD}}$ due to the CT scan being itself a random variable; this standard deviation is referred to as SV (for Statistical Variability).

For the head phantom dataset, an additional image similarity assessment was performed in terms of image noise at matched resolution. The procedure to match resolution consisted in estimating a normalized edge profile of the plastic rod in the Fourier domain and adjusting hyperparameter β for the reconstruction without non-negativity constraint as needed to match the profiles. In Fig. 1,a dotted line shows the ROI used for the edge profile calculation. The image noise was assessed using the noise power spectrum (NPS), which provides insight on noise magnitude and correlations.^50,51 The NPS computation was carried out in two dimensions as follows. The mean of the ten scan repetitions was subtracted from the reconstructions to create noise images. Afterward, the squared magnitude of a 2D Fourier-transformed rectangular ROI of the noise images was computed, and the NPS was obtained as the average over the ten noise realizations as well as over the four central slices of the reconstruction volume. To further mitigate noise, the two-dimensional (2D) spectrum was last transformed from Cartesian to polar coordinates and then radially averaged to create a 1D NPS curve. This procedure to compute the NPS is consistent with approaches used by others.^50,52,53 As for $\overline{\text{RMSD}}$, the ROI used for computation of the NPS was focused on relevant structures within the head phantom; this ROI is marked with a solid line in Fig. 1(d). It is understood that the analyzed reconstructions likely have shift-variant noise properties, so that the computed NPS does not necessarily carry out the classical meaning of the NPS. We used the NPS in a broad sense to compare the overall noise properties, as discussed by Baek and Pelc.⁵⁴

4. RESULTS

The results are separated into six subsections. Each subsection provides the observations about convergence comparison and image similarity between the approach with and without non-negativity made in one study. The order is the same as in Section 3.B.

4.A. ACR phantom study

The reconstructions of the ACR phantom dataset without the non-negativity constraint can be found in the top row of Fig. 3(a) and (b). The differences between these and the results with non-negativity constraint are presented in the bottom row of Fig. 3(a) and (b). For the images of Fig. 3(a), the edge-preserving regularization was used. For the images of Fig. 3(b), the quadratic regularization was applied. In all cases, the differences in voxel values are very small, so that a narrow grayscale window of [−2.5,2.5] HU is needed for display. Noticeable differences can only be found at the location of high contrast objects.

Fig. 3.

Reconstruction result of the ACR phantom module A and module D using edge-preserving regularization (a) and quadratic regularization (b). The top row shows the results without non-negativity constraint. The bottom row shows the differences between the reconstruction with and without non-negativity constraint. The applied grayscale window is showed in the upper right corner of each image.

The quantitative assessment of differences, reported in Table II, is in agreement with the visual impressions. The values in the difference images stay within a range from −2 to 2 HU. However, there are few voxels where the differences are as large as the observed extremes as confirmed by the fact that $\overline{\text{RMSD}}$ is smaller than 0.3 HU in all cases. Note also that the standard deviation attached with $\overline{\text{RMSD}}$ is very small, showing that the reported $\overline{\text{RMSD}}$ values are not confounded by convergence issues. By comparing the two regularization approaches, we can say that the extreme values have a higher magnitude when an edge-preserving regularization is used, and the $\overline{\text{RMSD}}$ is higher when a quadratic regularization is used. However, the dominant observation is that the differences are similarly small.

Table II.

Quantitative assessment of differences between reconstructions with and without non-negativity for the ACR phantom. The minimum and maximum voxel differences are given, as well as the mean and standard deviation in the root mean square difference over the last 500 iterations. All values are in HU and measured inside the object-specific ROI.

	Min.	Max.	$\overline{\text{RMSD}}$
ACR phantom
Module A, edge-preserving reg.	−1.5	1.8	0.0394 ± 0.0023
Module A, quadratic reg.	−0.2	0.4	0.1828 ± 0.0002
Module D, edge-preserving reg.	−1.6	1.7	0.1295 ± 0.0014
Module D, quadratic reg.	−0.8	0.7	0.2041 ± 0.0001

The RMSE results comparing reconstruction with and without non-negativity constraint in terms of convergence speed are presented in Fig. 4. Similar observations can be made for both ACR modules. For the first 500 iterations, there is no difference in convergence behavior. In the later iterations, differences in RMSE can be observed when using the quadratic regularization, whereas the curves essentially remain the same for the edge-preserving regularization. For the quadratic regularization, we thus see a gain in convergence speed when enforcing non-negativity, but this happens after the RMSE is already below 0.1 HU. Note also that, independently of the non-negativity constraint, convergence is not monotonic and is slower for the edge-preserving regularization. Nonmonotonicity is a known feature of FISTA that can be compensated for at some computational cost.⁵⁵ The slower convergence highlights the known computational challenge that MBIR with edge-preserving regularization faces for clinical translation in CT.

Fig. 4.

Comparison of convergence speed for the reconstruction of the ACR phantom datasets. A solid line and a dotted line are, respectively, used for reconstruction with and without the non-negativity constraint. The potential function is either edge-preserving (blue) or quadratic (orange).

4.B. Low-dose study

Figure 5 presents the reconstruction results for the head phantom dataset. For both sampling settings, the reconstruction without non-negativity constraint is displayed next to the difference image. For the full view sampling, differences between with and without non-negativity are barely visible even in a small grayscale window of [−2.5,2.5] HU. For the reconstruction based on sparse view sampling, the differences are somewhat larger in the uniform regions and particularly much larger near the edges of high-contrast bones. As confirmed later, visibility of the edges does not, however, correspond to a difference in resolution; it rather corresponds to a difference in noise at the edges. Additionally, note that the locations where beam hardening errors were clearly seen in the FBP image of Fig. 2 are not highlighted in the difference images.

Fig. 5.

Reconstructions and difference images of the head phantom. In the top row: results based on full view sampling, in the bottom row: results based on sparse view sampling. Next to the reconstructions without the non-negativity constraint, the difference image between the reconstruction with and without non-negativity constraint is shown. The applied grayscale window is showed in the upper right corner of each image.

Table III provides the results of the quantitative assessment of differences. For the full view sampling, the differences are within −3 and 3 HU and the $\overline{\text{RMSD}}$ is below 0.2 HU. For the sparse view sampling, these values are more than ten times higher. The standard deviation attached to $\overline{\text{RMSD}}$ conveys as before that there is no confounding effect with convergence issues after 5000 iterations. In this table, we also report the statistical variability (SV) over scan repetition for both samplings. This variability is <3% showing little impact of quantum noise in the data over the observed magnitude of differences.

Table III.

Quantitative assessment of differences between reconstructions with and without non-negativity for the head phantom with sparse and full view sampling. The minimum and maximum voxel differences are given, as well as the mean and standard deviation in the root mean square difference over the last 500 iterations. All values are in HU and measured inside the ROI. The last column reports the statistical variability of $\overline{\text{RMSD}}$ over scan repetitions.

	Min.	Max.	$\overline{\text{RMSD}}$	SV
Head phantom
Full view sampling	−2.8	2.6	0.1887 ± 0.0000	0.0044
Sparse view sampling	−48.0	36.1	2.1428 ± 0.0004	0.0289

When looking at Fig. 6, no difference can be observed in the convergence speed between the reconstruction with and without non-negativity constraint. For both sparse and full view sampling, the curves are so similar that they overlap each other.

Fig. 6.

Comparison of convergence speed for reconstruction of the low-dose head phantom datasets. A solid line and a dotted line are, respectively, used for reconstruction with and without the non-negativity constraint. Full view sampling of the projection data is shown in blue and sparse view in orange.

As discussed in Section 3.D, an NPS calculation was performed to further evaluate the differences in the reconstructions with and without non-negativity constraint. To allow for a meaningful comparison of the noise characteristics, the images need to have the same resolution. Figure 7 shows the Fourier transform of the studied edge profiles as obtained when using the same value for β. Based on these curves, the reconstructions with and without non-negativity constraint were deemed to have the same resolution and no adjusting of β for the reconstruction without non-negativity constraint was performed.

Fig. 7.

Resolution test between the result with non-negativity constraint (blue) and without (orange). The curves show the Fourier transform of the edge profile between the plastic rod and the surrounding water; these transforms are normalized to unity at the zero frequency.

The plots in Fig. 8 show the NPS for both reconstructions with and without non-negativity constraint. The range of the abscissa is limited by the Nyquist frequency of 10 cm⁻¹ that arises from the voxel size of 0.05 cm used for reconstruction. For the full view sampling, the NPS curve of the result with non-negativity constraint is identical to the curve of the result without non-negativity constraint. For the sparse view sampling, the NPS curve of the reconstruction with the non-negativity constraint is slightly higher for spatial frequencies above 2 cm⁻¹. However, the maximum discrepancy is only of 1.0% at 3.3 cm⁻¹ frequency. Note that the displayed curves do not tend toward 0 HU² cm² as the frequency increases, which is feasible given that the maximum resolution from the system is about twice larger than 10 cm⁻¹ at field-of-view center thanks to the FFS option. Note also that the NPS at high frequencies is very smooth with no sharp bend, indicating that little if any aliasing occurs.⁵⁶ We attribute the overall appearance of the NPS to the known fact that MBIR induces high noise variance at sharp edges.⁵⁷

Fig. 8.

Noise power spectrum for the low-dose study results with non-negativity constraint (blue) and without (orange).

4.C. Parametric space study

Figure 9 shows the dependence of $\overline{\text{RMSD}}$ on hyperparameter β and parameter δ in the edge-preserving regularizer. The main trend to observe is that, at fixed δ, $\overline{\text{RMSD}}$ decreases with increasing β. This means that the effect of the non-negativity constraint becomes less important as the strength of the edge-preserving constraint is increased, that is, as the reconstruction becomes smoother. The dependence on δ at fixed β is a much smaller effect: most often, decreasing δ results in a slight increase in $\overline{\text{RMSD}}$. The largest $\overline{\text{RMSD}}$ value is achieved for β = 0.01 and δ = 0.01, and the lowest one for β = 0.5 and δ = 0.01. The standard deviation of the $\overline{\text{RMSD}}$ was calculated but is too small to be displayed in the plot of Fig. 9. Its maximal value of 0.0045 can be found for β = 0.01 and δ = 0.0001.

Fig. 9.

Results for the parametric space study, which uses the chest scan dataset. The curves show the dependence of $\overline{\text{RMSD}}$ on hyperparameter β at fixed value for parameter δ of the edge-preserving potential function.

The reconstruction results corresponding to the central three values for β and both values for δ are presented in Fig. 10. For β = 0.25, the images are already overly smooth, with some anatomical details of the lung becoming barely visible. For β = 0.05, the images start showing more noise. Also, the images in the top row of Fig. 10, obtained with δ = 0.01, all show a high-frequency streak pattern that is reduced when δ = 0.0001. Combining these observations with the $\overline{\text{RMSD}}$ values in Fig. 9, we may state that the difference due to using the non-negativity constraint is on the order of 3 HU for visually appealing images.

Fig. 10.

Reconstruction results of the chest scan dataset for different parameter values. The top row shows the results with δ = 0.01. The bottom row shows the results with δ = 0.0001. The applied grayscale window is showed in the upper corners of each image.

The difference images are presented in Fig. 11 using the same arrangement as in Fig. 10. The low-frequency components of the difference images appear largely similar over all shown values for β and δ. The effect of reducing β or increasing δ primarily manifests itself through the introduction of high-frequency patterns of increasing strength and size. Overall, the differences remain however small, particularly when compared with the grayscale window typically used by the radiologist.

Fig. 11.

Difference images between reconstruction with and without non-negativity constraint of the chest scan dataset for different parameter values. The top row shows the results with δ = 0.01. The bottom row shows the results with δ = 0.0001. The applied grayscale window is showed in the upper right corner of each image.

For the sake of brevity, the convergence plots for the different parameter settings are not presented. Instead, the RMSE after 2500 iterations is given in Table IV. The numbers represent the difference between RMSE with and without non-negativity constraint. The presented relative RMSE values are all negative, which means that the reconstruction with the non-negativity constraint is always closer to its converged result than the reconstruction without the non-negativity constraint, but the gain is fairly small with its maximum value falling below 0.4 HU.

Table IV.

Convergence comparison for the parametric space study. The table shows the RMSE for the reconstruction with non-negativity minus the RMSE for the reconstruction without non-negativity constraint. All values are in HU.

	β = 0.01	β = 0.05	β = 0.01	β = 0.25	β = 0.05
δ = 0.01	−0.0883	−0.0124	−0.0115	−0.0042	−0.0014
δ = 0.0001	−0.3665	−0.0603	−0.0221	−0.0038	−0.0060

4.D. Truncated data study

Based on the visual inspection of results in Fig. 10, β = 0.1 and δ = 0.0001 were selected for this fourth study as well as for the sixth study presented later. The reconstruction results for the truncated dataset can be found in Fig. 12. Outside the scanned object, particularly above the patient, there are fewer artifacts when non-negativity is used. However, within the lung tissue, differences between the results with and without the non-negativity are hardly visible.

Fig. 12.

Reconstructions of the truncated chest scan dataset: (left) with non-negativity constraint, (right) without. The grayscale used for display can be found in the upper corners of each image.

Figure 13 shows that, while still being small relative to the grayscale window used by the radiologist, the differences are larger than those observed without truncation. This observation is also supported by the summary values in Table V: the $\overline{\text{RMSD}}$ is more than four times higher for truncated data.

Fig. 13.

Difference between the chest scan reconstructions obtained with and without non-negativity constraint: (left) for truncated projection data; (right) for non-truncated projection data using the same grayscale window, highlighted in the upper right corner of each image.

Table V.

Quantitative assessment of differences between reconstructions with and without non-negativity for both truncated and nontruncated projection data.

	Min.	Max.	$\overline{\text{RMSD}}$
Chest scan
Truncated projection data	−21.9	6.2	7.2014 ± 0.0134
Nontruncated projection data	−19.3	24.6	1.7595 ± 0.0006

To assess which of the two reconstruction options is more accurate, we can use the reconstructions from nontruncated data as ground truth. To avoid any bias, the result with (without) non-negativity constraint from truncated data is compared against the nontruncated result with (without, resp.) non-negativity constraint. Figure 14 shows the described differences in absolute values. The image on the left, where the non-negativity constraint is used, shows less discrepancy, especially on the lower left side of the left lung and the lower right side of the right lung. Expressed quantitatively, the mean absolute difference against the selected ground truths is, respectively, 19.41 and 25.43 HU for the cases with and without constraint.

Fig. 14.

Absolute difference between the reconstruction from truncated data and that from non-truncated data: (left) with non-negativity constraint; (right) without. The applied grayscale window is showed in the upper right corner of each image.

Figure 15 shows the convergence behavior. For the non-truncated data, there is little to no difference between the reconstructions with and without non-negativity. For the truncated data, the use of the non-negativity clearly improves the convergence speed, especially for the first 700 iterations.

Fig. 15.

Comparison of convergence speed for the chest scan dataset. A solid line and a dotted line are, respectively, used for reconstruction with and without the non-negativity constraint. The projection data are either nontruncated (blue) or truncated (orange).

4.E. Metal artifacts study

The reconstruction results for the hip phantom are presented in Fig. 16 using two different grayscale windows to visualize two effects of the non-negativity for this dataset. In the top row, the window is set to [−1100, −900] HU to show that the streak artifacts outside the scanned object caused by the metal implant are strongly reduced when non-negativity is used. In the middle row, the window is set to [−200, 200] HU to show that the image appearance within the object is largely similar. There is, however, one important difference: the non-negativity constraint visibly leads to different HU values near the left boundary of the object. This effect can be further appreciated when examining the difference image displayed in the bottom row of the figure, which highlights that there are also differences near the right boundary and within the implant. Table VI shows that $\overline{\text{RMSD}}$ is about twice the value observed in the truncated data study, and that the minimum and maximum values of the difference are extremely high. We verified that these extreme values occur within the implant.

Fig. 16.

Results for the hip phantom dataset. The first two rows show the reconstruction with (left) and without non-negativity constraint (right) using different gray scale windows. The bottom row shows the difference image. The grayscale used for display can be found in the upper right corner of each image.

Table VI.

Quantitative assessment of differences between reconstructions with and without non-negativity for the hip phantom.

	Min.	Max.	$\overline{\text{RMSD}}$
Hip phantom	−1455.6	1675.6	14.0673 ± 0.0006

To identify which of the two reconstructions is better, Fig. 17 shows a vertical profile (averaged over five contiguous lines and the four central slices) near the left boundary of the phantom and identifies two rectangular ROIs where mean values were computed (also with averaging over the four central slices). For ROI A, which is near the center of the object, the two reconstructions are very similar: the mean value is 58.0 HU with non-negativity and 57.8 HU without. For ROI B, the mean value is, respectively, 101.1 and 79.2 HU for the cases with and without non-negativity, indicating, together with the profile, that the reconstruction without constraint is closer to the true value of 0 HU. It thus appears as though the artifact mitigation offered by the non-negativity constraint outside the object adversely affects image quality near the object boundary. These observed differences are, however, relatively small compared to the overall errors seen inside the object with both reconstructions. Regarding the differences within the implant, we could not assess if one of the reconstructions is better or not; but both appear strongly affected by photon starvation.

Fig. 17.

Comparison of HU values within the water equivalent material of the hip phantom. The orange line in the reconstruction (top) marks the position of the profile plot (bottom). The two ROIs marked with yellow dotted lines are used for quantitative analysis.

Finally, Fig. 18 shows the convergence behavior. As can be observed, the curves for the reconstructions with and without non-negativity are very close to each other. Also, the difference in convergence behavior can only be seen after the first 1000 iterations, at which time the RMSE is below 1 HU.

Fig. 18.

Comparison of convergence speed for the hip phantom dataset. The blue and orange curves, respectively, show the result for reconstruction with and without the non-negativity constraint.

4.F. Empirical algorithm acceleration study

When compared with the reconstruction without ordered subsets and 5000 iterations, the ordered subset implementation of FISTA with (without, resp.) non-negativity constraint has a small RMSE value of 1.72 HU (1.85 HU, resp.) after 50 iterations. Therefore, the ordered subsets approach can be said to provide a satisfactory estimation of the desired image at a computational cost that is 100 times lower.

Figure 19 shows the reconstruction with non-negativity next to the difference image between with and without non-negativity. For reference, the difference image for reconstruction without ordered subsets and 5000 iterations is also displayed. Quantitative assessment of the two difference images is presented in Table VII. Note that the RMSD is not averaged over the last 500 iterations as previously because this feature is not available for the ordered subsets implementation.

Fig. 19.

Results for the study with ordered subsets. The reconstruction with ordered subsets and non-negativity constraint is presented on the left. The difference image between the reconstructions using ordered subsets with and without non-negativity is shown in the middle. As reference, the difference image achieved without ordered subsets is given on the right. The applied grayscale window is showed in the upper corners of each image.

Table VII.

Quantitative assessment of differences between reconstructions with and without non-negativity for the study with ordered subsets.

	Min.	Max.	RMSD
Ordered subsets, 50 iterations	−26.8	27.4	2.2835
No ordered subsets, 5000 iterations	−19.3	24.6	1.7597

The results show that the accelerated algorithm yields additional high-frequency differences between the two reconstructions, but the change in RMSD is about 0.5 HU. Thus, in this particular case, the accelerated algorithm largely reacts to the non-negativity constraint in the same way as the converged solution.

5. DISCUSSION AND CONCLUSIONS

In this work, we reported on an investigation assessing the impact of the non-negativity constraint on the CT reconstruction achieved with MBIR. The investigation involved five experimental studies, all carried out with real data and designed to challenge the non-negativity constraint in various ways, including noise, beam hardening, parametric choices, truncation, and photon starvation. Our primary focus was the examination of the properties of the converged solution, approximated through a suitable number of iterations and initialized from a zero volume to avoid any possibly confounding bias. To make the computational load tractable, circular scans with narrow collimation were used. Additionally, we also examined in a sixth study the effect of replacing the exact iterations by ordered subsets ones that provide within 50 iterations a result that may be more practical for clinical routine. Rather than directly assessing image quality, we evaluated image similarity and complemented this evaluation with quantitative image quality metrics on a case-by-case basis. The choice of the metrics depended on what appeared most appropriate to appreciate the importance of observed differences and the direction in which they were leaning. They included the noise power spectrum at matched resolution or closeness to a ground truth.

Our investigations shed light on the value of the non-negativity constraint, and also indirectly confirmed that making reliable predictions can be challenging, as may be expected due to the heavily nonlinear nature of MBIR. For cases with moderate inconsistencies in the data, associated with noise and bone-induced beam hardening, our results showed that the constraint offers little benefit, even when the attenuation values are low as in the lung study. From the observations based on varying the regularization parameters, it appears that the constraint plays a more important role when the edge-preserving regularization strength is weak; however, in such cases, the image quality is not satisfactory. Hence, sufficient edge-preserving regularization tends to dilute the value of the non-negativity constraint. Differences can be more pronounced when the reconstruction is subject to more aliasing effects, as for the low-dose study with sparse sampling; but these differences are not likely to have much impact, if any, in clinical routine as observed with our NPS measurements.

For cases with strong inconsistencies in the data, the value of the non-negativity constraint was mixed: sometimes, the constraint improved the result, like in the truncation study; sometimes, it did the opposite, like in the metal artifacts study. In both cases, we however observed that the difference between the images with and without the constraint was fairly small within the region of interest relative to the overall level of inaccuracy in the image. Hence, using the constraint or not using it does not play a major role in trying to achieve a good image quality.

In terms of convergence speed, we could only identify one strong effect in using the non-negativity constraint for reconstruction. This effect was observed for the truncated data study, where we could see that the constraint made the convergence process much smoother. The observed effect had however no impact on our ability to obtain the converged solution without the constraint.

The study with ordered subsets was encouraging. It showed that the effect of the constraint was largely similar, except for additional high-frequency errors that only resulted in a 0.5 HU increase in RMSD relative to the converged solution. The outcome may not always be like that. For example, given the complicated convergence process observed for the case without constraint in the truncated data study, it may be in this case that the non-negativity constraint is critical to stabilize the acceleration based on ordered subsets. We have not however investigated this aspect.

Altogether, our study did not highlight the non-negativity constraint as being strongly beneficial for diagnostic CT imaging. Naturally, this observation only holds for the investigated scenarios. From an image quality viewpoint, it could be that the non-negativity constraint plays a useful role for some specific quantification tasks like emphysema scoring. It could also play a useful role for the more classically used helical cone-beam acquisition, due to the long object problem associated with this imaging geometry.⁵⁸ We anticipate that if there was a gain there, it would primarily appear for the outer slices, which are most affected by this problem. From a convergence speed viewpoint, note that our observations are tightly linked with the utilization of one specific (but popular) algorithm, namely FISTA. It is known that enforcing non-negativity can help with other algorithms.^59,42 Last, we would like to reemphasize that our investigation focused on properties of the converged solution. If strong value had been observed for this solution, our viewpoint is that every effort should be made to retain this value when developing an approximate solution that is obtainable within a small number of iterations. Within the context of developing an approximate solution, the non-negativity constraint could, in some circumstances, be useful as a stabilizing force.

To conclude, we presented the results of an investigation on the role of the non-negativity constraint in diagnostic CT. This investigation showed that the value of the constraint is little in cases with moderate data inconsistencies and mixed in cases with strong data inconsistencies. If the non-negativity constraint was highly valuable, it is reasonable to believe that this value would have clearly appeared within our imaging scenarios; however, this was not the case. As a consequence, the non-negativity constraint should at least not be used as a forceful argument to disregard optimization algorithms, or to disregard forward projection models that require a complex optimization strategy.