Harmonization of in-plane resolution in CT using multiple reconstructions from single acquisitions

Abstract

Purpose:

To providea methodology that removes the spatial variability of in-plane resolution using different CT reconstructions. The methodology does not require any training, sinogram, or specific reconstruction method.

Methods:

The methodology is formulated as a reconstruction problem. The desired sharp image is modeled as an unobservable variable to be estimated from an arbitrary number of observations with spatially variant resolution. The methodology comprises three steps: (1) density harmonization, which removes the density variability across reconstructions; (2) point spread function (PSF) estimation, which estimates a spatially variant PSF with arbitrary shape; (3) deconvolution, which is formulated as a regularized least squares problem. The assessment was performed with CT scans of phantoms acquired with three different Siemens scanners (Definition AS, Definition AS+, Drive). Four low-dose acquisitions reconstructed with backprojection and iterative methods were used for the resolution harmonization. A sharp, high-dose (HD) reconstruction was used as a validation reference. The different factors affecting the in-plane resolution (radial, angular, and longitudinal) were studied with regression analysis of the edge decay (between 10% and 90% of the edge spread function (ESF) amplitude).

Results:

Results showed that the in-plane resolution improves remarkably and the spatial variability is substantially reduced without compromising the noise characteristics. The modulated transfer function (MTF) also confirmed a pronounced increase in resolution. The resolution improvement was also tested by measuring the wall thickness of tubes simulating airways. In all scanners, the resolution harmonization obtained better performance than the HD, sharp reconstruction used as a reference (up to 50 percentage points). The methodology was also evaluated in clinical scans achieving a noise reduction and a clear improvement in thin-layered structures. The estimated ESF and MTF confirmed the resolution improvement.

Conclusion:

We propose a versatile methodology to reduce the spatial variability of in-plane resolution in CT scans by leveraging different reconstructions available in clinical studies. The methodology does not require any sinogram, training, or specific reconstruction, and it is not limited to a fixed number of input images. Therefore, it can be easily adopted in multicenter studies and clinical practice. The results obtained with our resolution harmonization methodology evidence its suitability to reduce the spatially variant in-plane resolution in clinical CT scans without compromising the reconstruction’s noise characteristics. We believe that the resolution increase achieved by our methodology may contribute in more accurate and reliable measurements of small structures such as vasculature, airways, and wall thickness.

1 |. INTRODUCTION

The in-plane resolution is a key parameter in the definition of an acquisition protocol for clinical purposes. The effective resolution depends on the X-ray focal spot size and shape, detector aperture, scanner geometry, and reconstruction algorithm.¹ To provide a varied set of reconstruction algorithms, manufacturers have developed their own sets of reconstruction kernels with specific characteristics depending on the clinical purpose. For instance, sharp kernels are specially designed to provide higher spatial resolution at the expense of image noise, whereas soft kernels are used for routine whole-body scans because they produce reasonably low-noise images and lower resolution. Therefore, the reconstruction method is carefully chosen, considering the trade-off between signal-to-noise ratio (SNR) and spatial resolution.

An additional inconvenience is that the in-plane resolution not only varies across reconstruction methods, it is also spatially variant across the image. For this reason, the recommendation for clinical practice is to place the targeted structure to be analyzed near the isocenter—where the resolution is higher—and, whenever this is not possible, choose the reconstruction method carefully to reach the desired resolution with reasonable image noise.^2–4 The variability of spatial resolution mainly relies on two factors affecting different directions with respect to the isocenter¹:

• Radial. The shape change of the projected focal spot as a function of the angle between the source-detector ray and the source-isocenter ray. This results in an effective increase in the point spread function (PSF) in the radial direction.

• Tangential. The difference in arc length swept out by the ray increases with the distance from the isocenter. The effect is a reduction of resolution in the tangential direction.

Although these effects are well known and their clinical implications have been reported in the literature,⁵ their nature stems from the device design intrinsically and there is no straightforward solution. The scientific community is aware of this problem and has focused efforts on designing new reconstruction techniques to increase the overall resolution of the CT scans.

A common approach is to increase the overall resolution by modifying the acquisition process. For instance, the number of samples per rotation can be increased by deflecting the focal spot on the X-ray tube⁶ and the effective resolution at the isocenter improves by inserting tantalum grids in front of the CT detectors at the expense of reducing the dose efficiency.^7,8 Note, however, that this approach aims at increasing the overall resolution of CT scans by reducing the tangential factor instead of removing the spatial variability.

La Riviere and Vargas proposed reducing the radial effect by directly modeling the sinogram’s local blurring effects and compensating them using a penalized-likelihood sinogram restoration model within a single-slice circular fan-beam geometry.^9,10 Although direct modeling of the local blurring allows for a spatially variant correction of the in-plane resolution, it requires modeling the source, detectors, and anode angulation, which is not usually provided by manufacturers. These limitations, the need for the sinograms, and complications of extending to other geometries make the implementation of this method impractical for clinical use.

A pragmatic approach that clinical researchers have adopted to deal with the trade-off between SNR and resolution is to include several reconstruction kernels in their acquisition protocols. This way, the soft kernels are used for density-related measures and the sharp kernels for structural analysis. Unfortunately, kernels are vendor-dependent and there is no explicit correspondence among them.¹¹ For this reason, multicenter studies with different CT scanners may exhibit significant differences in measures that are attributable to vendors.^12,13 The application of iterative reconstruction methods does not alleviate this problem because they exhibit a nonlinear behavior that may introduce biases in densities.¹⁴ To minimize variability across manufacturers, researchers suggest using kernels that affect similarly to resolution, SNR, density measures, and texture.¹¹ This is the approach adopted by multicenter studies such as SPIROMICS¹⁵ and COPDGene,¹⁶ which include several reconstruction kernels per manufacturer in their protocols to ensure the best conditions for density analysis or structural measurements. These approaches, however, do not avoid the trade-off between noise and resolution because the detection and analysis of structures remain limited by the spatial resolution in the case of softer kernels (where the SNR is higher) or by the noise in sharper kernels (with lower SNR).

Missert et al. used the increasing availability of reconstructions in clinical studies to synthesize an image from multiple kernels using deep convolutional neural networks.¹⁷ Although promising, the method depends on the specific kernels used for the training, and it cannot be directly used when the number or kind of reconstructions used vary. Furthermore, because the training is performed with spatially variant resolutions, the resulting resolution remains spatially variant.

In this work, we propose a multikernel in-plane resolution harmonization methodology that removes the spatial variability of resolution and provides a sharp image with high SNR from a set of reconstructions and without the need for sinograms or a specific training dataset. The method takes advantage of the different reconstructions obtained from the same acquisition, which have different noise and resolution characteristics. The method formulates those reconstructions as observations of the same object that differ in resolution and noise characteristics. This is a reasonable assumption because all of them come from the same information (same acquisition). The principal contributions of our methodology are as follows:

1. It is formulated as a reconstruction problem where we model the desired sharp image as an unobservable variable to be estimated from observations with different spatially variant resolutions. This has the advantage that we can employ any arbitrary combination of reconstructions to estimate the sharp image.

2. For each reconstruction, the blurring operator is modeled as a general shift-variant filter that operates across the image to be estimated. Therefore, both iterative and backprojection-based reconstructions can be employed as observations indistinctly.

3. The noise in the estimated image is controlled using a density harmonization technique that reduces the CT numbers’ variability across reconstructions. This way, the increase of resolution does not increase the noise in the resulting image.

2 |. MATERIALS AND METHODS

2.1 |. Harmonization of in-plane resolution

Without loss of generality, we denote the images as column vectors of dimension N to formulate linear transformations as matrix products over the image.

Our resolution harmonization assumes that an unobservable high-resolution image, $x\in {ℝ}^N$, generates the M available observations (reconstructions), $x_1,\dots ,x_M\in {ℝ}^N$. This is a reasonable supposition considering that all the reconstruction are originated from the same sinogram acquired. Therefore, the main differences between reconstructions are the in-plane resolution and the noise variance. In Figure 1, we show a scheme of the methodology proposed, where a set of M observable reconstructions coming from the same sinogram are employed. Those reconstructions may be generated by backprojection kernels or iterative methods indistinctly. The reconstruction process is modeled as a certain linear transformation ($A_m\in {ℝ}^{N\times N}$ for m = 1, …, M) applied over the unobservable high-resolution image. The reconstruction also introduces spatially variant noise, ε_m = [ε_m,1, …, ε_m,N]^T, and bias, $b_m\in {ℝ}^N$, that account for the different noise characteristics and biases reported in the literature.^13,18,19 Therefore, the mth reconstruction becomes x_m = A_mx + b_m + ε_m.

FIGURE 1.

Schematic description of the in-plane resolution harmonization proposed. The methodology assumes that an unobservable high-resolution, low-noise image x generates the available observations, x₁, …, x_M. The methodology comprises three steps. First, a density harmonization removes the biases due to the reconstruction methods and reduce the noise. Then, the spatially variant point spread function (PSF) is estimated in each reconstruction. Finally, the unobservable image x is estimated by solving a regularized least squares (LS) problem. The regularization term leverages the PSF estimates and the density harmonization to improve the resolution while keeping the noise low

As a first step of the methodology, we introduce a density harmonization stage to remove the spatially variant biases and reduce the noise variance per reconstruction. This is a key step because it will remove the discrepancies across reconstructions that are not due to resolution differences. The harmonization effect over the unobserved image is also assumed to be linear, being B_m the mth transformation matrix. Therefore, after harmonization, the mth reconstruction x_m turns into y_m = B_mA_mx + η_m, where η_m = [η_m,1, …, η_m,N]^T is the noise after harmonization with local variance Var{η_m,n} < Var{ε_m,n} for n = 1, …, N.

After density harmonization, the PSF is estimated from each harmonized reconstruction. The estimation should allow for a spatially variant and anisotropic PSF to account for the in-plane resolution variability. Therefore, the PSF in the mth reconstruction, hm(r|s), is a 2D function that depends on the location $s\in {ℝ}^2$. The shift-variant PSF is a linear mapping over the image x and, therefore, it can be represented as a matrix product H_mx, where H_m contains all the shift-variant PSFs across the image conveniently arranged. Note that this PSF estimate accounts for the combination of the reconstruction and the density harmonization step (i.e., $H_m=\widehat{A_m{B}_m}$).

Finally, the multiple reconstructions and PSF estimations allow us to formulate the estimation as a multi-image deconvolution problem.^20,21 A deconvolution step estimates the unobserved image by leveraging the multiple resolution information provided by the reconstructions. Note that the inclusion of the density harmonization step allows us to precondition the deconvolution problem without biases appropriately. Additionally, it also allows us to introduce a regularization term that prevents the increase of noise due to deconvolution.

The following step-by-step explanation details an implementation of the proposed methodology illustrated in Figure 1.

2.1.1 |. Density harmonization

The goal of this step is to reduce density variations caused by the reconstruction method.

A homogenization of CT numbers across reconstructions can be achieved using the standardization method proposed by Chen-Mayer et al.¹² This method treats the CT number by modeling the incoherent, coherent, and photoelectric processes that contribute to the attenuation following empirically determined energy dependence proposed by Schneider et al.²² and simplified a single parameter, α(E), by Martinez et al.²³ The homogenization requires the estimation of a reconstruction-dependent parameter $\alpha (\overline{E})$ that also depends on the average energy of the emission spectrum $\overline{E}$. This technique, however, does not account for the spatially variant noise and bias.

On the other hand, the density harmonization approach introduced by Vegas-Sánchez-Ferrero et al.¹³ proposes a noise stabilization and autocalibration methodology that accounts for spatially variant noise and reduces the noise-dependent bias across the image. This technique employs a statistical characterization of noise in polychromatic CT scans after the post-log transformation²⁴ and has been validated for multiple vendors, reconstruction methods, and doses.²⁵ The statistical characterization provides signal and noise estimates that are used to remove the spatially variant bias due to noise.¹⁹ The resulting harmonized image shows homogeneous noise characteristics across the image with an effective reduction of biases due to noise and reconstruction algorithms and with minimal effects in the resolution. We employ this method because it reduces the spatially variant noise in each reconstruction (Var{η_m} < Var{ε_m}) and also removes the biases for each reconstruction (b_m ≈ 0).

2.1.2 |. Point spread function estimation

The PSF is usually defined as the system output response to an input point object. The PSF estimation is a problem deeply studied in the literature,^20,21 and several approaches have been proposed for computed tomography.^26–29 Any of the state-of-the-art methods for PSF estimation can be used in this step.

An alternative approach is to estimate the line spread function (LSF) for a set of orientations. This approach is easier to implement because the transitions between densities—the edge spread function (ESF)—can be better defined by phantoms with subpixel precision.³⁰ Then, the LSF is calculated by numerical differentiation for each direction.

In our specific implementation, we follow the methodology proposed by the American Association of Physicists in Medicine (AAPM) in their TG233 report.¹⁴ In that report, the ESF estimation is performed in the boundaries of high-contrast materials with known shapes applying the method proposed by Richard et al.³¹ and refined by Chen et al.³² for circular structures. The known shape characteristics allow calculating the ESF in the normal direction to the phantom’s boundary and the pixel-edge distance with subpixel accuracy. Furthermore, if the PSF is assumed to be isotropic and the distance of the edge to the isocenter is approximately constant, one can aggregate all the samples around the edge and obtain an ESF estimate with subpixel resolution. This aggregation of samples around the edge is precisely the methodology used in the TG233 report to calculate the LSF and the modulated transfer function (MTF).

The LSF estimation is also preferable in the absence of phantoms because the ESF can be estimated in the external contour of the scanned subject as proposed by Sanders et al.²⁹

A detailed analysis of the specific shape of the PSF exceeds the scope of our methodology. Therefore, for our implementation, we consider the results reported by Wang et al.³³ and Chen and Ning³⁴ who measured the ESF in spheres and concluded that the PSF is efficiently approximated by a 3D Gaussian function. We will assume the PSF follows an anisotropic spatially variant function to account for the resolution variability across the image and to consider the effects of radial and tangential effects in the resolution.

It is worth mentioning that our methodology and its formulation is not limited to a Gaussian PSF. Alternative shapes of anisotropic and spatially variant PSF functions such as the ones proposed by Schwarzband and Kiryati²⁶ can be directly adopted. Nevertheless, the Gaussian assumption will suffice to show the resolution improvement achieved by our methodology.

The Gaussian standard deviation estimation, $\widehat{\sigma }$, is performed for each location and orientation by measuring the edge decay as the length between 10% and 90% of the ESF amplitude, denoted as ${\Delta }_{10\text{\%}}^{90\text{\%}}$. This parameter is related to σ and provides the following estimate:

$$\widehat{\sigma }={\Delta }_{10\%}^{90\%}/\mathrm{2.564.}$$ (1)

The derivation of this estimator is given in Appendix A.

2.1.3 |. Deconvolution

After the density harmonization step, the resulting images ${\left\{y_m\right\}}_{m=1}^M$ are related to the target image, x, through the following linear forward model:

$$\left[\begin{array}{c}{\mathit{y}}_1\\ {\mathit{y}}_2\\ ⋮\\ {\mathit{y}}_M\end{array}\right]=\left[\begin{array}{c}{H}_1\\ H_2\\ ⋮\\ H_M\end{array}\right]\mathit{x}+\mathit{\eta },$$ (2)

where $H_m\in {ℝ}^{N\times N}$ represents the linear blurring operator of the mth harmonized reconstruction, and $\eta ={\left[{\eta }_1^T,{\eta }_2^T,\dots ,{\eta }_M^T\right]}^T\in {ℝ}^{MN}$ models the remaining noise after the density harmonization step.

Note that MN ≥ N, so the equation system in Equation (2) is overdetermined and the estimate of x can be calculated by the LS method. In this specific case, the LS solution can be improved by recalling that the noise term in Equation (2) is highly correlated between kernels. This implies that the main discrepancies lay in the edges of the images. This property allows us to include a regularization term that imposes that the unobserved image’s intensity is similar to the weighted pixel-wise average of the reconstruction images ${\left\{y_m\right\}}_{m=1}^M$ in those areas where there is no edge discrepancy. This regularizing mechanism acts as a noise removal technique and does not compromise the resolution enhancement. With these considerations, the estimated high-resolution image, $\widehat{x}$, can be obtained as the solution to the following regularized LS problem:

$$\widehat{x}=\underset{x}{\text{arg\hspace{0.17em}min}}{\Vert Y-Hx\Vert }_2^2+\lambda {\Vert W(x-\overline{Y})\Vert }_2^2,$$ (3)

where λ is the regularization parameter, W is a diagonal weighting matrix, and $\overline{Y}$ is the inverse-variance weighting:

$$\overline{Y}=\frac{{\sum }_{m=1}^M{y}_m/\text{Var}\left\{y_m\right\}}{{\sum }_{m=1}^{M}1/\text{Var}\left\{y_m\right\}},$$ (4)

where Var{y_m} is the local variance of image y_m (i.e., the variance of each element of η_m). We chose this weighted average because it minimizes the variance among all weighted averages. Although these variances are unknown a priori, we can find an appropriate surrogate noting that Var{y_m} has an inverse relationship with the PSF spread parameter, σ_m. Therefore, we can use the variance estimate for each reconstruction, ${\widehat{\sigma }}_m^2$, calculated in Equation (1) instead:

$$\overline{Y}=\frac{{\sum }_{m=1}^M{\widehat{\sigma }}_m\odot y_m}{{\sum }_{m=1}^N{\widehat{\sigma }}_m},$$ (5)

where ⊙ is the element-wise product.

The weighting matrix W = diag(w₁, w₂, …, w_N) is defined to apply the regularization in homogeneous regions with no structural details. This can be achieved by noting that the noise is correlated across reconstructions because they come from the same sinogram.

Therefore, the structural details affected by the different PSFs can be detected by analyzing the variability across reconstructions. A simple way to detect the pixels with structural details, D = [d₁, d₂, …, d_N], is by direct thresholding the discrepancies among reconstructions:

$$d_k=\{\begin{array}{ll}1,\hfill & \text{if }{\text{max}}_m\left\{Y_m(k)\right\}-{\text{min}}_m\left\{Y_m(k)\right\}\ge t_d\hfill \\ 0,\hfill & \text{if }{\text{max}}_m\left\{Y_m(k)\right\}-{\text{min}}_m\left\{Y_m(k)\right\}<t_d,\hfill \end{array}$$ (6)

where the maximum and minimum operators are calculated pixel-wise across the M reconstructions. Note that D defined through Equation (6) is a mask containing the structural details affected by PSFs. To extend the effect of those structural details, one can apply a smoothing operator such as the PSFs estimated in the previous step. Finally, the weights are calculated as the complementary of the details:

$$\left[\begin{array}{c}{w}_1\\ w_2\\ ⋮\\ w_N\end{array}\right]=\left[\begin{array}{c}1\\ 1\\ ⋮\\ 1\end{array}\right]-C\left[\begin{array}{c}{d}_1\\ d_2\\ ⋮\\ d_N\end{array}\right],$$ (7)

where C is a spatial smoothing operator. Note that the threshold t_d in Equation (6) is flexible and can be defined as a spatially variant function.

Finally, solving the optimization problem of Equation (3) is equivalent to solve³⁵:

$$\left(H^{T}H+\lambda W\right)x=H^{T}y+\lambda W\overline{Y}.$$ (8)

We employ the conjugate gradient method to find this solution iteratively. Note that H^TH + λW is symmetric and positive-semidefinite, thereby convergence conditions hold.³⁵ In general, the construction of H and H^T required by the conjugate gradient algorithm is not trivial. In Appendix B, we detail how to construct H and H^T for the PSFs considered. Convergence for a single slice takes about 30 s in a 2.9 GHz Quad-Core Intel Core i7, 16 GB with Python (SciPy 1.5.0, NumPy 1.18.5 libraries).

2.2 |. Experimental validation

The validation of the proposed methodology is performed in a physical phantom with different clinical scanners. Additionally, we show the performance in clinical images reconstructed with several reconstruction methods.

2.2.1 |. Physical phantom

The CTP698 Lung Phantom II (The Phantom Laboratory, www.phantomlab.com) was scanned in an ultra low-dose (LD) clinical scenario (CTDIVol = 0.65 mGy) with three different Siemens scanners (Definition AS, Definition AS+, and Drive), slice thickness 0.75 mm, tube voltage 120 kVp, pitch 1, and pixel size 0.72 mm. Four reconstruction methods were employed for the resolution harmonization in each scanner, two filtered backprojection and two iterative reconstruction methods. An additional high-dose (HD) acquisition (CTDIVol = 15.20 mGy) was performed with identical acquisition parameters and reconstructed with a sharp kernel for the assessment of in-plane resolution. This reconstruction is not employed in the kernel harmonization and will serve as a low-noise and high-resolution reference in the quality metrics considered throughout the paper. A summary of the acquired images is shown in Table 1.

TABLE 1.

Summary of reconstruction methods employed in the CTP698 Lung Phantom II acquisition

Scanner	Reconstruction method	kVp	mA (mGy)
Definition AS	HD B45Fa	120	400 (15.20)
	LD B31F	120	84 (0.65)
	LD B45F	120	84 (0.65)
	LD I31F 2	120	84 (0.65)
	LD I31F 5	120	84 (0.65)
Definition AS+	HD B45a	120	400 (15.20)
	LD B31	120	84 (0.65)
	LD B45	120	84 (0.65)
	LD I31 2	120	84 (0.65)
	LD I31 5	120	84 (0.65)
Drive	HD Bf39a	120	400 (15.20)
	LD Bf39	120	84 (0.65)
	LD Bf39 Qr43	120	84 (0.65)
	LD Bf39 2 ADMIRE	120	84 (0.65)
	LD Bf39 5 ADMIRE	120	84 (0.65)

Abbreviations:HD, high dose; LD, low dose.

^aNot employed during the resolution harmonization.

The resolution harmonization methodology was applied for the LD reconstruction methods exclusively (two filtered backprojection and two iterative). Figure 2 shows the phantom employed. This phantom comprises eight regions with different densities and eight tubes with different orientations simulating airways with several wall thicknesses. We employ the eight regions to assess noise after resolution harmonization and the four tubes that are normal to the short-axis plane to assess the accuracy in wall-thickness estimation after resolution harmonization (tube numbers follow the phantom datasheet).

FIGURE 2.

Physical phantom employed comprising a variety of internal holes and structures. The eight regions will be used for assessment of noise, whereas the four tubes depicted are used to assess the accuracy in wall-thickness estimation after resolution harmonization

The anisotropy and spatial variation of the resolution were studied by measuring the ESF estimation in the edges of high-contrast regions (acrylic insert and exterior ring). This way, the noise effect in the ESF estimate due to the LD acquisition is minimized. Additionally, the phantom’s elliptical shape allows us to measure the ESF at different distances and angles from the isocenter.

We use the 90%–10% edge decay, ${\Delta }_{10\text{\%}}^{90\text{\%}}$, as the in-plane resolution metric. The analysis of the anisotropy and spatial variation was performed by considering the following potential factors that may affect the in-plane resolution:

• Quantization (Q). Interpolation error due to the estimation of ${\Delta }_{10\text{\%}}^{90\text{\%}}$ in the normal direction of the edge. It is proportional to the distance between the sampling location and the closest integer location in the lattice.

• Radial (R). Dependence on the distance to the isocenter.

• Angular (A). Dependence on the angle with respect to the isocenter.

• Longitudinal (L). Dependence on the long-axis location.

The model selection is performed by considering regressions with an increasing number of factors and comparing them through the Akaike information criterion (AIC) and the R² parameter. The AIC is a useful exploratory tool that estimates the relative distance between the fitted model and the actual unknown mechanism that generated the observed data.³⁶ Computing AIC in regression statistics usually results in arbitrary large negative values, where the minimum AIC value, AIC_min, is the model that best fits the data. The model selection is performed comparing the differences with the minimum, ΔAIC_i = AIC_i − AIC_min, for the ith model, and considering that the relative likelihood of the ith model is proportional to exp(−ΔAIC_i/2). Note that the exponential function makes the relative likelihood of differences beyond 10, ΔAIC_i > 10, negligible. The R² parameter is also employed to assess the explained variance for each model.

The performance of the statistical models in the regression analysis showed the minimum AIC for the radial component (R) in all the scanners considered (see Table C1). The difference in AIC for the rest of models was always beyond 10. Therefore, there is strong evidence that the factor that best fits the observed edge decay is the radial factor (R).³⁶ The R² parameter confirms that adding more predictive variables does not increase the explained variance. Consequently, the spatial dependence of resolution can be effectively modeled by an isotropic PSF that changes radially.

The radial dependence is modeled by regression using the predicted ${\Delta }_{10\text{\%}}^{90\text{\%}}$ to define the parameter of the Gaussian kernel through Equation (1).

The effect of the resolution harmonization in the phantom is assessed in five different ways:

1. Measuring the reduction of spatially variant resolution: We assess the reduction of radial dependence after resolution harmonization with the same regression analysis employed to estimate the ${\Delta }_{10\text{\%}}^{90\text{\%}}$ parameter.

2. Measuring the wall thickness of the phantom tubes: The wall thickness was measured following the full-width half-maximum technique and compared to the reference values provided in the phantom datasheet.^37,38

3. Measuring noise characteristics: The standard deviation of CT numbers was measured for the eight different densities enumerated in Figure 2.

4. Comparing the MTF calculated in the phantom’s external ring: The MTF was computed following the AAPM methodology presented in their TG233 report¹⁴ detailed previously.

5. Measuring the detectability in low-contrast objects: The detectability was evaluated for the insert that shows the lowest contrast with the surrounding material (insert 3 in Figure 2)using a channelized Hotelling observer with Laguerre–Gauss (LG) channels^39,40 as described by Racine et al.⁴¹ We employed the detectability index, $d^{\prime }=\sqrt{2\frac{{\left({\mu }_s-{\mu }_a\right)}^2}{{\sigma }_s^2+{\sigma }_a^2}}$, as the figure of merit, where μ_s, μ_a, σ_s, and σ_a are the means and variances of the model’s decision variable under the signal-present and signal-absent conditions, respectively.^42,43 The assessment was performed with a Monte Carlo cross-validation, where two disjoint sets of regions (training and test) with dimensions 45 × 45 pixels were randomly chosen. The training set comprised 60 signal-present and 60 signal-absent regions and was used to train the optimal observer, whereas the test dataset included 20 signal-present and 20 signal-absent regions. Each region was channelized in 10 LG channels with a scale factor of 24 (see Reference 44 for a detailed discussion on the selection of the scale parameter). The Monte Carlo validation was repeated 500 times.

2.2.2 |. Clinical CT

The clinical CT images evaluation was performed in a set of reconstructed images acquired in a clinical Siemens Definition scanner with 120 kVp tube voltage and tube current set to 100 mA (CTDIVol ~ 2.87 mGy), pixel size 0.61 × 0.61 mm, and slice thickness 0.5 mm. The reconstructions were performed with four methods, one backprojection kernel (B31f) and three iterative reconstruction methods (I31F2, I31F5, I44F2).

The same subject was also scanned with a higher dose (tube current 400 mA, CTDIVol ~ 13.47 mGy) and identical acquisition parameters and reconstructed with a sharp kernel (B45f). The resolution harmonization was performed with the LD reconstructions, whereas the HD scan was used as a reference for comparison purposes (not in the resolution harmonization). Figure 3 shows the clinical CT scan used as a reference.

FIGURE 3.

Clinical image used for visual assessment. The image is divided in two halves to show tissues in two different window densities (left for soft tissues [−1024, 500] HU; right for lung parenchyma [−1024, −700] HU). The four cylinders located at the gantry are used to assess the noise characteristics after resolution harmonization

We evaluate the changes in resolution by estimating the ESF and MTF in the subject’s external contour by following the method proposed by Sanders et al.²⁹ and adapted to axial planes.¹³ We are aware that this MTF estimation method is not as precise as the one used for the phantoms. However, it has been effective in distinguishing resolution differences between reconstruction methods.²⁹ The ESF estimates were also employed to calculate the PSF per reconstruction as detailed in Appendix A.

Finally, the noise response assessment in the resolution harmonization is performed in the four gantry cylinders (see Figure 3) by measuring each cylinder’s standard deviation. Those cylinders are made of a homogeneous foam with defined density that is usually employed as a CT number reference or, as in this case, for noise assessment.

2.3 |. Parameter selection

2.3.1 |. Density harmonization parameters

The density harmonization adopted proposed in Reference 13 is performed within a two-step methodology that involves a noise stabilization²⁵ and an autocalibration stages.¹⁹ The first stage performs a statistical characterization using a finite mixture model with a prefixed number of components with predetermined CT values.²⁵ Those components are references that serve to distinguish different tissues locally and calculate the expected CT value per pixel more accurately by discarding those samples that are statistically different from the pixel under study. As shown in the original paper, the specific values are not critical as long as they allow us to distinguish between different tissues locally. Therefore, we adopt the same parameter configuration in Reference 25: nine tissues ranging from −1000 HU to 400 HU to ensure the whole range of CT densities within a general CT scanner were represented. The specific values we chose were −990, −825, −555, −380, −185, 0, 100, 237, 340. The implementation is performed in the 3D volume considering a local neighborhood of 3 × 3 × 3. The result of this stage is a denoised 3D volume, an estimate of the local variance of noise, and a stabilized noise (no spatially variant).

During the autocalibration stage, the spatially variant biases are identified and removed by establishing the functional relationship between CT values observed in the denoised 3D volume and the observed variance of noise.¹⁹ Finally the systematic biases are removed by normalizing the CT values using anatomical references of identified tissues such as air and blood. We employ the trachea and the descending aorta to perform the normalization to the nominal values of air (−1000 HU) and blood (50 HU). The trachea and the descending aorta can be manually identified or employing any threshold-based methodology. The resulting image is a 3D volume with reduced noise and harmonized CT numbers across the image.

2.3.2 |. Deconvolution parameters

The regularizer is defined considering three parameters:

• Detail threshold (t_d defined in Equation (6)). This threshold is employed to detect pixels with structural details across the acquisitions and depends on the discrepancies between images and their noise characteristics. As a rule of thumb, this parameter is similar to the noise standard deviation observed in the image with sharpest kernel.

• Smoothing operator (defined in Equation (7)). This operator is performed as a local average filter. Because the purpose of this filter is to extend the effect of structural details affected by the PSF, a reasonable choice is to select an operator that covers the whole effect of the PSF in an edge. Experiments showed that the 90% to 10% ESF decay is slightly above 2 pixels so we adopt a local average filter of twice the ${\Delta }_{10\text{\%}}^{90\text{\%}}$ parameter to ensure that the 100% of the edge decay is covered. That is, a local average filter with a window size 5 × 5.

• Regularization parameter (λ defined in Equation (3)). This parameter balances the effect of the regularizer and the deconvolution. As the λ parameter increases, the deconvolution becomes less pronounced. The selection of this parameter depends on the desired characteristics we pursue in the edges. A conventional approach is to minimize the mean squared error (MSE).^45,46 However, the minimization of the MSE does not guarantee the elimination of potential Gibbs effects that may cause undesired edge enhancements.

The optimal selection of parameters involved in the regularization is a problem that depends on the images, regularizer definition, and balance between the data fidelity and the regularizer. In Figure 4, we show how the regularization parameter and detail threshold affect the MTFs calculated in the Definition AS scanner with the physical phantom (the MTFs for the Definition AS+ and Drive scanners were similar and we omitted them for brevity). We also include a reference MTF simulated for an elliptical phantom with the same geometrical characteristics as the one used (see details in Appendix D).

FIGURE 4.

MTFs for different values of the regularization parameter λ and detail threshold d_t

Note that the MTFs exhibit an overshoot for low values of λ that is mitigated as the regularization increases its contribution. We selected the regularization parameters experimentally by ensuring no overshoot in the MTFs for each scan. The parameters selected are: Definition AS: d_t = 49, λ = 150; Definition AS+: d_t = 54, λ = 100; Drive: d_t = 25, λ = 18. The differences in λ and d_t parameters between the Definition scanners and the Drive scanners are due to the different reconstruction kernels. In the Drive scanner, all the reconstruction kernels employed in the resolution harmonization perform a pronounced noise reduction at the expense of lower resolution. Therefore, the structural details are detected with a lower threshold, and the data fidelity versus regularization balance is achieved with a lower value of λ. For the clinical image, we employed the same configuration as for the Definition AS: d_t = 49, λ = 150.

3 |. RESULTS

3.1 |. Physical phantom

3.1.1 |. Correction of resolution spatial dependence

In Table 2 we gather the regression equations modeling the dependence on the distance to the isocenter, D, of the ESF decay parameter, ${\Delta }_{10\text{\%}}^{90\text{\%}}$, for all scanners and reconstructions.

TABLE 2.

Linear regression and R² parameters for the edge spread function decay, ${\Delta }_{10\text{\%}}^{90\text{\%}}$, depending on the distance to the isocenter (D)

		Regression equation	R 2
Definition AS	HD B45Fa	1.20 + 1.65×10−5 D2	0.62
	LD B31F	2.08 + 7.69×10−6 D2	0.14
	LD B45F	1.26 + 9.14×10−6 D2	0.26
	LD I31F2	2.01 + 7.04×10−6 D2	0.13
	LD I31F5	1.90 + 5.97×10−6 D2	0.11
	Resolution harmonization	1.30 + 7.35×10−6D2	0.10
Definition AS+	HD B45Fa	1.27 + 1.71×10−5 D2	0.54
	LD B31F	2.25 + 6.35×10−6 D2	0.10
	LD B45F	1.33 + 9.30×10−6 D2	0.19
	LD I31F2	2.17 + 5.83×10−6 D2	0.10
	LD I31F5	2.07 + 4.79×10−6 D2	0.08
	Resolution harmonization	1.37 + 4.74×10−6 D2	0.03
Drive	HD Bf39 Qr43a	1.62 + 1.58×10−5 D2	0.46
	LD Bf39	1.90 + 9.03×10−6 D2	0.19
	LD Bf39 Qr43	1.62 + 1.02×10−5 D2	0.20
	LD Bf39 2 ADMIRE	1.76 + 1.06×10−5 D2	0.23
	LD Bf39 5 ADMIRE	1.57 + 1.17×10−5 D2	0.26
	Resolution harmonization	1.27 + 1.02×10−5 D2	0.23

Abbreviations:HD, high dose; LD, low dose.

^aNot employed during the resolution harmonization.

Figure 5 shows the edge sampling scheme used to estimate ${\Delta }_{10\text{\%}}^{90\text{\%}}$ for different distances to the isocenter and the regression curves for the Definition AS scanner. The edge sampling was performed in the normal directions of the external ring (in red) and acrylic (in blue) boundaries of the phantom. Note that a quadratic dependence on the distance, D, properly fits the loss of resolution (increase in ${\Delta }_{10\text{\%}}^{90\text{\%}}$). The regression plots for the rest of scanners were similar and, for brevity, we omit them.

FIGURE 5.

Edge spread function sampling normal to the external ring (in red) and acrylic inset (in blue) boundaries and regression models fit to the radial dependence of the edge spread function decay parameter, ${\Delta }_{10\text{\%}}^{90\text{\%}}$

A careful inspection of the regression equations in Table 2 shows that the sharp kernels (HD B45F and LD B45F) provide the lowest values of ${\Delta }_{10\text{\%}}^{90\text{\%}}$ with similar regression curves. This result was expected because the dosage is not directly related to the resolution. In contrast, using a soft kernel (LD B31F, LD Bf39) produces a significant increase of ${\Delta }_{10\text{\%}}^{90\text{\%}}$, though the dependence on the distance to the isocenter is slightly reduced. This reduction is due to a loss of resolution near the isocenter. Therefore, although the soft kernels achieve a relative homogeneous resolution across the image, it is reached at the expense of losing resolution in the whole image. The iterative methods (I31F2, I31F5) exhibit similar behavior to the soft kernel, indicating that noise reduction still implies a loss of resolution.

The results obtained with our resolution harmonization show, on average, an excellent performance for the whole range of distances. It is homogeneous and achieves a high resolution (low ${\Delta }_{10\text{\%}}^{90\text{\%}}$) for all distances. For instance, in the Definition AS scanner, the regression curve after resolution harmonization drops the distance factor (7.35 · 10⁻⁶), which is a 55% reduction compared to the HD and high-resolution reference (HD B45F). The Definition AS+ and Drive scanners show a similar reduction: 72% for the Definition AS+ (HD B45F: 1.71 × 10⁻⁵, Resolution harmonization: 4.74 × 10⁻⁶), and 35% for the Drive scanner (HD Bf39 Qr43: 1.58 × 10⁻⁵, Resolution harmonization: 1.02 × 10⁻⁵). The lower resolution dependence on the distance is also reflected in the decrease of the R² parameter when compared to the HD and high-resolution reference (Definition AS: 0.62–0.10; Definition AS+: 0.54–0.03; Drive: 0.46–0.23).

In Figure 6, we show the four reconstructions employed in the in-plane resolution harmonization for the Definition AS scanner, the harmonized image, and the HD, sharp kernel reconstruction used as a reference. The LD, sharp kernel reconstruction (LD B45F) shows a good contrast between regions with clearly defined edges, although it also exhibits high noise. This noise is partially removed in the LD B31F at the expense of a significant loss in edges’ definition. Both iterative reconstructions (LD I31F2 and LD I31F5) exhibit a more pronounced noise reduction with similar behavior in the edge contrast than the soft kernel. The resolution harmonization achieves a good balance between noise and edge definition. The noise reduction is comparable to the iterative reconstruction methods with the advantage of preserving the edge contrast. The images for the rest of the scanners studied show similar characteristics and are omitted for brevity.

FIGURE 6.

Phantom CT scans employed in the in-plane resolution harmonization for the Definition AS scanner, reference high-dose and sharp reconstruction (HD B45F), and resolution harmonization image for the Definition AS scanner

3.1.2 |. Wall thickness measurements

Wall thickness measurements are detailed in Table 3. The results show that the harmonized image improves accuracy in all cases. The reduction of absolute error for thicknesses of 1.2 and 1.5 mm is especially remarkable, where the nonharmonized images exhibit a relative error or around 25%. In those cases, the resolution harmonization reduces the absolute error to less than 0.05 mm (< 5% of relative error). For smaller wall thicknesses, the measurements become less accurate in all scanners. This was an expected consequence of the full-width half-maximum technique because it is known to have a positive bias for low measurements.^47,48 However, even in those cases, the resolution harmonization error remains the lowest and reduces the relative error from 10 to 50 percentage points.

TABLE 3.

Wall thickness measurements from airway tubes of the phantom

		Tube 2			Tube 3			Tube 5			Tube 6
		Wall thickness (mm)	Absolute error (mm)	Relative error (%)	Wall thickness (mm)	Absolute error (mm)	Relative error (%)	Wall thickness (mm)	Absolute error (mm)	Relative error (%)	Wall thickness (mm)	Absolute error (mm)	Relative error (%)
Reference value		0.6	–	–	0.9	–	–	1.2	–	–	1.5	–	–
Definition AS	HD B45Fa	1.47	0.87	144.70	1.43	0.53	58.44	1.47	0.27	22.54	1.63	0.13	8.81
	LD B31F	1.60	1.00	166.10	1.66	0.76	84.09	1.75	0.55	45.63	1.90	0.40	26.71
	LD B45F	1.43	0.83	138.18	1.41	0.51	56.72	1.45	0.25	20.52	1.61	0.11	7.66
	LD I31F2	1.57	0.97	161.51	1.65	0.75	82.87	1.74	0.54	44.68	1.89	0.39	26.30
	LD I31F5	1.52	0.92	153.29	1.63	0.73	81.10	1.72	0.52	43.34	1.89	0.39	25.78
	Resolution harmonization	1.30	0.70	116.34	1.26	0.36	39.85	1.23	0.03	2.21	1.47	0.03	1.76
Definition AS+	HD B45Fa	1.50	0.90	149.49	1.43	0.53	58.87	1.49	0.29	24.53	1.68	0.18	11.77
	LD B31F	1.61	1.01	168.74	1.70	0.80	88.97	1.78	0.58	48.20	1.94	0.44	29.60
	LD B45F	1.49	0.89	148.52	1.43	0.53	59.26	1.48	0.28	22.94	1.67	0.17	11.06
	LD I31F2	1.60	1.00	165.88	1.69	0.79	87.50	1.77	0.57	47.23	1.94	0.44	29.07
	LD I31F5	1.60	1.00	166.49	1.67	0.77	85.56	1.75	0.55	45.89	1.92	0.42	28.27
	Resolution harmonization	1.35	0.75	124.88	1.27	0.37	40.93	1.17	0.03	2.23	1.47	0.03	2.31
Drive	HD Bf39 Qr43a	1.56	0.96	160.64	1.51	0.61	67.64	1.65	0.45	37.25	1.82	0.32	21.47
	LD Bf39	1.61	1.01	167.69	1.59	0.69	76.18	1.68	0.48	40.40	1.88	0.38	25.07
	LD Bf39 Qr43	1.53	0.93	155.72	1.51	0.61	68.04	1.59	0.39	32.75	1.79	0.29	19.33
	LD Bf39 2 ADMIRE	1.57	0.97	161.90	1.58	0.68	75.23	1.68	0.48	39.85	1.87	0.37	24.34
	LD Bf39 5 ADMIRE	1.57	0.97	161.61	1.56	0.66	73.66	1.66	0.46	38.73	1.85	0.35	23.52
	Resolution harmonization	1.26	0.66	109.37	1.42	0.52	58.08	1.27	0.07	5.93	1.48	0.02	1.36

Note. Bold letters represent the most accurate measurement. The resolution harmonization reduces the measurement error in all the cases.

Abbreviations:HD, high dose; LD, low dose.

^aNot employed during the resolution harmonization.

3.1.3 |. Noise assessment

Table 4 shows the results for each scanner, reconstruction, and region. In the Definition AS and AS+ scanners, the resolution harmonization has a lower noise standard deviation than the LD reconstructed images employed except for LD I31F5, where similar values were found. However, it is worth noting that the noise reduction in LD I31F5 comes at the expense of a resolution loss, as shown in Table 2, where the intercept for ${\Delta }_{10\text{\%}}^{90\text{\%}}$ was 1.90 and 2.07 for the Definition AS and AS+, respectively. The resolution harmonization maintains a low-noise standard deviation while preserving a low edge decay as shown before in Table 2. The same conclusion can be extracted for the Drive scanner.

TABLE 4.

Noise standard deviation calculated in the eight regions described in Figure 2

Region		1	2	3	4	5	6	7	8
Definition AS	HD B45Fa	14.07	15.11	11.41	13.96	13.53	10.21	10.75	21.79
	LD B31F	20.98	21.55	16.48	20.53	19.68	13.37	15.30	32.62
	LD B45F	35.78	35.63	28.11	35.15	32.36	20.04	26.15	55.58
	LD I31F2	16.12	17.00	12.60	15.75	15.03	10.94	11.68	25.52
	LD I31F5	9.09	10.76	6.97	8.77	8.15	6.61	6.38	15.34
	Resolution harmonization	9.30	10.52	7.13	8.27	8.64	5.76	6.57	17.11
Definition AS+	HD B45Fa	13.83	18.31	11.74	14.53	13.30	9.84	10.74	21.81
	LD B31F	19.65	23.11	16.15	20.18	18.79	12.50	14.71	30.83
	LD B45F	33.39	36.27	27.25	33.86	30.60	18.60	24.74	51.97
	LD I31F2	15.16	19.01	12.54	15.96	14.81	9.89	11.48	24.47
	LD I31F5	8.91	13.56	7.51	10.36	9.83	5.69	7.04	15.74
	Resolution harmonization	8.87	15.33	7.50	9.60	9.70	5.55	7.06	17.45
Drive	HD Bf39 Qr43a	10.62	12.53	9.64	11.53	9.88	8.64	8.93	16.57
	LD Bf39	23.55	25.65	20.18	24.37	23.53	15.37	19.11	33.79
	LD Bf39 Qr43	27.54	29.69	23.64	28.49	26.75	17.27	22.36	39.36
	LD Bf39 2 ADMIRE	18.11	20.14	15.63	19.01	19.29	12.75	14.90	26.61
	LD Bf39 5 ADMIRE	9.49	11.63	8.94	10.75	14.02	8.16	8.75	15.21
	Resolution harmonization	10.55	12.88	9.00	11.18	13.84	6.85	8.85	21.26

Abbreviations:HD, high dose; LD, low dose.

^aNot employed during the resolution harmonization.

3.1.4 |. Modulated transfer function

Figure 7 shows the MTF estimates and the 50% cutoff frequency, f₅₀. We also include a reference MTF simulated for an elliptical phantom with the same geometrical characteristics as the one used. The simulated MTF was calculated with a noise-free synthetic image whose densities in the contours were set considering the pixel’s proportion intersected by the ellipse (see details in Appendix D). The resolution harmonization frequency response notably increases for the three scanners, showing a 50% cutoff frequency beyond 7.5 line pairs per centimeter. This result confirms that the proposed methodology effectively improves the resolution beyond the levels found in the reconstructed images with given kernels.

FIGURE 7.

Modulated transfer function for the three scanners. The 50% cutoff frequency obtained with the resolution harmonization methodology surpasses the sharpest kernel in all cases

3.1.5 |. Low-contrast detectability

Table 5 shows the results obtained for the d′ metric. The resolution harmonization increases the detectability of the LD images for all scanner models. This result evidences that the noise after resolution harmonization is effectively controlled and the detectability improved.

TABLE 5.

The detectability index d′ increases after the resolution harmonization is compared to all the reconstructions employed in the resolution harmonization (LD reconstructions)

Scanner	Reconstruction method	d′ (95% CI)
Definition AS	HD B45Fa	52.29 (51.81–52.76)
	LD B31F	33.35 (32.98–33.73)
	LD B45F	29.72 (29.36–30.09)
	LD I31F 2	35.78 (35.38–36.17)
	LD I31F 5	41.92 (41.50–42.33)
	Resolution harmonization	46.72 (46.22–47.21)
Definition AS+	HD B45a	52.30 (51.82–52.78)
	LD B31	31.36 (30.83–31.90)
	LD B45	26.56 (26.10–27.02)
	LD I31 2	35.43 (34.88–35.97)
	LD I31 5	40.47 (39.94–41.01)
	Resolution harmonization	41.60 (41.10–42.10)
Drive	HD Bf39a	44.07 (43.36–44.78)
	LD Bf39	31.02 (30.67–31.36)
	LD Bf39 Qr43	30.99 (30.69–31.29)
	LD Bf39 2 ADMIRE	32.45 (32.06–32.83)
	LD Bf39 5 ADMIRE	33.68 (33.29–34.07)
	Resolution harmonization	38.46 (38.10–38.82)

Abbreviations: HD, high dose; LD, low dose.

^aNot employed during the resolution harmonization.

3.2 |. Clinical scans

Figure 8 shows the harmonized image, the LD kernels used, and the HD sharp kernel acquired for visual comparison. We divided the images into two halves to show tissues in two different window densities (left for soft tissues [−1024, 500] HU; right for lung parenchyma [−1024, −700] HU). A region of emphysema is zoomed to ease the visualization of small structures such as vessels and septal walls.

FIGURE 8.

Reconstructed images of the same subject for high-dose and sharp kernel (HD B45f), and low-dose with different reconstruction methods (backprojection: B31F, iterative: I31F2, I31F5, I44F2). The proposed methodology increases the contrast and resolution, and vascular structures and pulmonary septa are enhanced

The zoomed regions clearly show the differences in noise and resolution between reconstructions. The LD B31F shows more noise patterns than the iterative reconstructions and a lower resolution than the HD B45F. The LD I31F5 reconstruction achieves a good reduction of noise but at the expense of a loss of resolution, which is visually comparable to the LD B31F. The LD I44F2 reconstruction shows the best resolution among the LD reconstructions. However, the noise increases remarkably and there is an appreciable bias towards higher CT numbers in the emphysema regions that is not observable in the HD reference image. The resolution harmonization image achieves a good compromise between noise reduction and resolution. The emphysema regions are distinctly visible and the vasculature and septal walls are clearly defined.

Figure 9 shows the ESF and MTF estimates for all the reconstructions. The densities acquired in the normal direction to the contour are represented as dots. The ESF estimates were calculated after rebinning those samples in with a bin size equal to 0.1 times the pixel size as recommended in the AAPM TG233 report.¹⁴ The presence of samples with a high slope close to location 0 evidence the increase in resolution. The ESF estimates also show a higher slope after resolution harmonization. This effect is reflected in the MTF plot because the 50% cutoff frequency increases beyond all other reconstructions.

FIGURE 9.

Edge spread function and modulated spread function estimated in the subject’s chest contour. Note that the resolution harmonization achieves a sharper transition between the densities of the subject and the external air. The sharp estimate of the modulated transfer function confirms the sharp transition by showing an overall higher frequency response

The noise standard deviation measurements for the four regions are shown in Table 6. The noise achieved by the resolution harmonization is around the average of the LD kernels employed and in the same range as the sharpest reconstruction obtained in HD. It is important to note that, although the LD I31F5 reconstruction shows even lower noise than the HD counterpart, its frequency response is lower and, therefore, the resolution is compromised. The reasonably low noise obtained by the resolution harmonization combined with the increase in resolution shows that the proposed methodology finds a good compromise between noise and resolution.

TABLE 6.

Noise standard deviation (HU) estimated in the gantry cylinders shown in Figure 3 across the volume for all the reconstructions

Region	1	2	3	4
HD B45Fa	31.39	30.59	28.91	29.20
LD B31F	56.88	57.03	55.72	55.49
LD I31F2	44.34	44.68	43.44	43.54
LD I31F5	26.35	26.95	25.81	26.42
LD I44F2	58.76	58.17	55.61	55.30
Resolution harmonization	35.46	39.77	34.66	39.80

Abbreviations: HD, high dose; LD, low dose.

^aNot employed during the resolution harmonization.

4 |. DISCUSSION AND CONCLUSIONS

We propose a versatile methodology that removes the spatial variability of in-plane resolution in CT scans. The methodology takes advantage of the set of reconstructions usually available in imaging protocols used in routine care and clinical studies. The proposed approach provides a high-resolution and low-noise CT reconstruction with harmonized in-plane resolution. The methodology does not require any training or specific reconstruction method.

The methodology is formulated as a reconstruction problem where the desired sharp image is modeled as an unobservable variable to be estimated from an arbitrary number of observations with spatially variant resolution. The methodology includes a density harmonization step that removes the density variability across reconstruction methods and reduces the noise per reconstruction. Therefore, both backprojection and iterative reconstruction methods can be employed indistinctly. Once the density discrepancies across reconstructions are removed, the spatial resolution is characterized for each reconstruction by estimating the PSF. Our formulation contemplates a spatially variant PSF of an arbitrary shape with no limitations in the anisotropic or spatially variant behavior of the in-plane resolution. Finally, the deconvolution step is formulated as a regularized LS problem.

The regularization term was established taking advantage of the specific characteristics of the problem: the observable images are always instances of the same information that were reconstructed in different ways (filter backprojection or iterative methods). This implies that the noise is highly correlated between images, they are perfectly aligned, and the only discrepancies between them are the biases originated by the reconstruction and noise variance. Therefore, if the spatially variant noise and biases are properly treated, the discrepancies across images can be reduced to resolution. We mitigate those discrepancies with the density harmonization stage and identify the resolution discrepancies across reconstructions pixel-wise. Therefore, the regularization term accurately acts in regions where no structural information is compromised.

Besides, after the density harmonization, the discrepancies across reconstructions are limited to differences in resolution. These characteristics lead to a solution that shows low-noise and sharp resolution.

Along with the proposed methodology, we suggested an implementation that includes the density harmonization presented in Reference 24, because it is based on a statistical characterization of the signal and is not limited to the reconstruction technique. Therefore, backprojection and iterative reconstruction methods can be used. The PSF estimation was performed considering different oriented LSFs and was calculated according to the AAPM TG233 report’s recommendations.¹⁴ Following the results of Wang³³ and Chen,³⁴ we assumed a Gaussian shape for the PSF. The PSF spread factor was calculated using the 90%–10% decay of the ESF estimated for each orientation and location. We employed this methodology because it can be easily extended to clinical scans utilizing the subject’s contour.^13,29 Finally, the LS problem was solved using the conjugate gradient method with an inverse-variance weighting as the regularization term. This term ensures that the sharp estimate will have low noise.

It is worth noting that the methodology we proposed is versatile and its formulation allows for more general assumptions than the one we made in the suggested implementation. For instance, although we considered oriented LSFs in the PSF estimation, alternative methods can be equally employed. Obviously, the suitability of the estimation methodology depends on the phantom used. Additionally, the methodology’s formulation is not limited to a Gaussian PSF and any other shape can be employed because our formulation does not include any restriction in shape, spatial variability, or anisotropy.

The experimental validation was performed with CT scans acquired with different scanners in phantoms. Four LD acquisitions reconstructed with backprojection and iterative methods were used for the resolution harmonization, whereas a HD, sharp reconstruction was used as a reference. The different factors affecting the in-plane resolution were carefully considered and analyzed with statistical models measuring the ESF decay for different orientations and distances from the isocenter. The statistical model was selected with the AIC, showing that the radial component was the factor that best fits the ESF decay, ${\Delta }_{10\text{\%}}^{90\text{\%}}$ (Table C1).

Results of the ESF decay with our resolution harmonization methodology (Table 2) showed that the in-plane resolution improves remarkably (small intercepts), and the spatial variability observed in the phantoms is substantially reduced (decrease in the radial factor). The MTFs (Figure 7) estimated in the phantoms also confirmed a pronounced increase in resolution.

Results also indicated that, although iterative reconstructions significantly reduce the noise, it is achieved at the expense of a loss in resolution, similar to the one observed for soft kernels. Our methodology can contribute to overcoming the loss of resolution in LD scenarios, where iterative reconstructions are commonly recommended.⁴⁹

The resolution improvement was also tested by measuring the wall thickness of tubes simulating airways. The resolution harmonization showed that a relative error was remarkably reduced in all cases (up to 50% points) and obtained better performance than the HD, sharp reconstruction used as a reference (Table 3).

Finally, the noise was assessed by measuring the noise’s standard deviation in eight regions with homogeneous densities (Table 4). Results showed that the resolution harmonization methodology achieves a good balance between resolution and noise, obtaining comparable noise levels than the reconstruction with lower noise. Interestingly, the resolution harmonization exhibited less noise than the HD sharp kernel employed. This result confirms the excellent balance between resolution and noise reduction achieved by the proposed methodology.

The resolution harmonization has been presented as a methodology that leverages different reconstructions within the same acquisition. This is a realistic scenario because the sinogram is not usually accessible in clinical studies. Researchers are aware of this limitation and, therefore, clinical studies’imaging protocols include multiple reconstructions to reduce the variability across manufacturers and choose reconstructions depending on the purpose of measurements. However, when the sinogram is available, our methodology can incorporate that information as an additional regularization term that constrains the solution, x, to the loss of resolution within the sinogram:

$$\widehat{x}=\underset{x}{\text{arg\hspace{0.17em}min}}{\Vert Y-Hx\Vert }_2^2+\lambda {\Vert W(x-\overline{Y})\Vert }_2^2+\delta {\Vert (P\mathit{x}-s)\Vert }_2^2,$$ (9)

where δ is a regularization parameter, s is the sinogram, and P is the projection operator that projects the high-resolution image into the sinogram space. This operator can be performed considering a spatially variant PSF in the sinogram domain as proposed by La Riviere and Vargas.¹⁰

An essential characteristic of our methodology for multicenter studies (where there is high variability in scanners and reconstruction methods) is that it does not require any specific number of reconstructions or training. This property also distinguishes our methodology from deep learning (DL) approaches limited to a fixed number of inputs, whose specific training may compromise its generalizability. However, there is an undeniable potential in applying DL techniques. For instance, the DL method proposed by Missert et al.¹⁷ showed that a synthetic image can be generated from a set of inputs that offers a resolution comparable to the sharpest input image with the noise characteristics of the softest reconstruction employed. Although promising, this result does not consider the spatial variability of resolution and can only achieve the same resolution characteristics of the sharpest kernel used. As we showed in our work, modeling the resolution’s spatial variability allows us to achieve a homogeneous and even higher resolution than the input images. Additionally, our methodology is compatible with DL techniques in all of its steps. Therefore, nonlinear and spatially variant operators can be derived without compromising the methodology’s generalizability. In future work, we will focus our efforts on the application of DL techniques in each stage.

This work has some limitations that are worth noting. First, the presented evaluation is confined to common reconstruction techniques for Siemens scanners. Although the spatially variant resolution is implicit to the CT acquisition physics, we do not have data from other vendors that could lead to more general conclusions related to other manufacturers. Second, although our validation was performed with implemented acquisitions protocols similar to COPDGene¹⁶, it does not consider several fields of view or pixel sizes. We plan to extend this validation to a wider set of acquisition configurations and applications in future work.

The resolution’s improvement and spatial stability is a crucial factor that contributes toward more accurate and reliable measures of small structures such as nodules, plaques, airway walls, and vascular structures. The resolution radial effect might highly impact quantitative metrics extracted from measurements defined at different locations from the isocenter, such as tree structures formed by the pulmonary vascular and bronchial systems. Changes in small lung airways and vessels have been strongly associated with the pathophysiological development of COPD, asthma, and pulmonary hypertension.^50–53 Therefore, the ability to detect small changes in the morphology of those structures is critical to the early diagnosis of the disease as well as the definition of its progression. We believe that the results presented in this paper may contribute to a more homogeneous comparison across the morphological properties of those structures as they go from proximal central locations near the isocenter to more distal subpleural locations in the periphery of the reconstruction field of view. This could impact the quantitative assessment of progression as small effects can be confounded by the variable resolution.⁵⁴ The harmonization of the resolution could also prove crucial in the definition of more robust biomarkers to track therapeutic response, for example, in the definition of the local effect of vasodilators in the pulmonary vasculature.⁵⁵

APPENDIX A: RELATIONSHIP BETWEEN ${\Delta }_{10\text{\%}}^{90\text{\%}}$ AND σ FOR GAUSSIAN PSFs

Given an ideal ESF, u(x), and a Gaussian PSF, ϕ(x|σ²), defined as

$$u(x)=\{\begin{array}{cc}{u}_{\text{max}},& \text{if }x\ge 0\\ u_{\text{min}},& \text{if }x<0\end{array}\text{ and }\varphi \left(x\mid {\sigma }^2\right)=\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{x^2}{2{\sigma }^2}}.$$ (A1)

Their convolution results in the following smoothed ESF:

$$\begin{gathered} u(x)⊛\varphi \left(x\mid {\sigma }^2\right)={\int }_{-\infty }^{\infty }u\left(x-x^{\prime }\right)\varphi \left(x^{\prime }\mid {\sigma }^2\right)d{x}^{\prime } \\ ={\int }_{-\infty }^x\left(u_{\text{max}}-u_{\text{min}}\right)\varphi \left(x^{\prime }\mid {\sigma }^2\right)d{x}^{\prime }+u_{\text{min}} \\ =\left(u_{\text{max}}-u_{\text{min}}\right)\Phi \left(x\mid {\sigma }^2\right)+u_{\text{min}}, \end{gathered}$$ (A2)

where Φ(x|σ²) is the cumulative distribution function (CDF) of a Gaussian distribution with zero mean and variance σ² and ⊛ is the convolution operator.

Therefore, the ESF decay between 10% and 90% of its amplitude is the difference between 10th and 90th percentiles of the Gaussian distribution (P_10%,σ and P_90%,σ, respectively):

$${\Delta }_{10\text{\%}}^{90\text{\%}}=\left|P_{90\text{\%},\sigma }-P_{10\text{\%},\sigma }\right|=\left|P_{90\text{\%},1}-P_{10\text{\%},1}\right|\sigma .$$ (A3)

The normalized 10th and 90th percentiles of a Gaussian distribution are −1.282 and 1.282, respectively. Therefore, |P_90%,1 − P_10%,1| = 2.564, and the estimate becomes $\widehat{\sigma }={\Delta }_{10\text{\%}}^{90\text{\%}}/2.564$.

APPENDIX B: CONSTRUCTION OF H AND H^T FOR A SPATIALLY VARIANT KERNELS

The experimental observations showed that the PSF can be effectively approximated by isotropic and spatially variant Gaussian kernels (Section 3.1). The symmetric shape of the Gaussian function and the separability of the 2D Gaussian function allows us to construct the matrix H efficiently as follows.

First, we consider a 2D spatially variant Gaussian kernel, $h\left(m,n\mid {\sigma }_{i,j}^2{I}_2\right)$, defined in the image domain (m, n) ∈ Ω = Ω_rows × Ω_columns, at location (i, j) ∈ Ω and diagonal covariance, ${\sigma }_{i,j}^2{I}_2$. For simplicity, we assume the dimensions of both columns and rows are equal to D. Let us denote $x\in {ℝ}^N$ the image to be filtered arranged as a column (the columns of the 2D image are stacked as a column vector and N = D²).

The isotropic covariance makes the Gaussian kernel separable,

$$h\left(m,n\mid \left[\begin{array}{cc}{\sigma }_{i,j}^2,& 0\\ 0,& {\sigma }_{i,j}^2\end{array}\right]\right)=\varphi \left(m\mid {\sigma }_{i,j}^2\right)\varphi \left(n\mid {\sigma }_{i,j}^2\right),$$ (B1)

where ϕ(·) is a 1D Gaussian kernel defined in Equation (A1). Consequently, the filtering can be performed sequentially in each dimension as a matrix product, Hx, with a filtering matrix H associated to the spatially variant variance $\Sigma ={\left\{{\sigma }_{i,j}^2\right\}}_{i,j}=0,\dots ,N-1$:

$$H=P_{r\to c}{G}_{{\Sigma }^T}{P}_{c\to r}{G}_{\Sigma },$$ (B2)

where the matrices P_r→c and P_c→r are permutation matrices that transform the image arrangement from row to column and from column to row, respectively. G_Σ and $G_{{\Sigma }^T}$ are block diagonal matrices:

$$G_{\Sigma }=\left[\begin{array}{cccc}{G}_{\Sigma }^0& \mathbf{0}& \cdots & \mathbf{0}\\ \mathbf{0}& G_{\Sigma }^1& \ddots & ⋮\\ ⋮& \ddots & \ddots & \mathbf{0}\\ \mathbf{0}& \cdots & \mathbf{0}& G_{\Sigma }^{N-1}\end{array}\right],$$

$$G_{{\Sigma }^T}=\left[\begin{array}{cccc}{G}_{{\Sigma }^T}^0& \mathbf{0}& \cdots & \mathbf{0}\\ \mathbf{0}& G_{{\Sigma }^T}^1& \ddots & ⋮\\ ⋮& \ddots & \ddots & \mathbf{0}\\ \mathbf{0}& \cdots & \mathbf{0}& G_{{\Sigma }^T}^{N-1}\end{array}\right],$$

where 0 is a zero matrix of dimension N × N and $G_{\Sigma }^j\in {ℝ}^{N\times N}$ filters the jth column and is defined as

$$G_{\Sigma }^j=\left[\begin{array}{ccc}\varphi \left(0\mid {\sigma }_{0,j}^2\right),\varphi \left(1\mid {\sigma }_{0,j}^2\right),& \dots ,& \varphi \left(N-1,{\sigma }_{0,j}^2\right)\\ \varphi \left(-1\mid {\sigma }_{1,j}^2\right),\varphi \left(0\mid {\sigma }_{1,j}^2\right),& \dots ,& \varphi \left(N-2,{\sigma }_{1,j}^2\right)\\ ⋮& & \\ \varphi \left(-(N-1)\mid {\sigma }_{N-1,j}^2\right),& \dots ,& \varphi \left(0\mid {\sigma }_{N-1,j}^2\right)\end{array}\right].$$ (B3)

Similarly, $G_{{\Sigma }^T}$ is a block matrix where the ith block in the diagonal, $G_{{\Sigma }^T}^i$, filters the ith row and is defined as

$$G_{{\Sigma }^T}^i=\left[\begin{array}{ccc}\varphi \left(0\mid {\sigma }_{i,0}^2\right),\varphi \left(1\mid {\sigma }_{i,0}^2\right),& \dots ,& \varphi \left(N-1,{\sigma }_{i,0}^2\right)\\ \varphi \left(-1\mid {\sigma }_{i,1}^2\right),\varphi \left(0\mid {\sigma }_{i,1}^2\right),& \dots ,& \varphi \left(N-2,{\sigma }_{i,1}^2\right)\\ ⋮& & \\ \varphi \left(-(N-1)\mid {\sigma }_{i,N-1}^2\right),& \dots ,& \varphi \left(0\mid {\sigma }_{i,N-1}^2\right)\end{array}\right].$$ (B4)

Note that the sequential filtering is firstly performed over the columns. Then, the transformation P_c→r arranges the filtered columns into a column vector whose elements are the rows stacked vertically. Finally, the rows are filtered with their corresponding spatially variant kernels and rearranged into column format through P_r→_c.

The transpose matrix required by the conjugate gradient algorithm becomes

$$H^T=G_{\Sigma }^T{P}_{c\to r}^T{G}_{{\Sigma }^T}^T{P}_{r\to c}^T=G_{\Sigma }^T{P}_{r\to c}{G}_{{\Sigma }^T}^T{P}_{c\to r},$$ (B5)

where we applied the property of permutation matrices $P_{r\to c}^T=P_{r\to c}^{-1}=P_{c\to r}$.

Finally, from Equations (B2) and (B5), the term H^T H in Equation (8), becomes

$$H^{T}H={\left(P_{c\to r}{G}_{\Sigma }\right)}^T{G}_{{\Sigma }^T}^T{G}_{{\Sigma }^T}{P}_{c\to r}{G}_{\Sigma }.$$ (B6)

Note that the filtering is trivially generalized for M different PSFs by stacking the set of matrices ${\left\{H_m\right\}}_{m=1}^M$ associated to their corresponding filters with spatially variant variances ${\left\{{\Sigma }_m\right\}}_{m=1}^M$ as it is described in Equation (2).

APPENDIX C: MODEL SELECTION ANALYSIS

Table C1 shows the results for the model selection considering quantization (Q), radial (R), angular (A), and longitudinal (L) factors. The analysis considered an increasing number of factors and showed that the radial factor was one with the least AIC value. The R² parameter confirmed that increasing the number of factors does not increase the variance explained by de model. These results suggest that the angular (A) and longitudinal (L) factors do not improve the fitting of the statistical model considered. Therefore, in agreement with Wang et al.³³ and Chen and Ning,³⁴ our results suggest that the in-plane resolution is isotropic and nondependent on the longitudinal location. However, there is a strong dependence on the distance to the isocenter.

TABLE C1.

Goodness of fit of the statistical models for the edge spread function decay, ${\Delta }_{10\text{\%}}^{90\text{\%}}$, considering the following factors: quantization (Q), radial (R), angular (A), and longitudinal (L)

		${\Delta }_{10\text{\%}}^{90\text{\%}}~Q$			${\Delta }_{10\text{\%}}^{90\text{\%}}~Q+R$			${\Delta }_{10\text{\%}}^{90\text{\%}}~Q+R+A$			${\Delta }_{10\text{\%}}^{90\text{\%}}~Q+R+A+L$
		AIC	ΔAIC	R 2	AIC	ΔAIC	R 2	AIC	ΔAIC	R 2	AIC	ΔAIC	R 2
Definition AS	HD B45F	−77 386.16	15 392	0.603	−92 778.21	0	0.686	−92 498.24	279	0.686	−92 504.11	274	0.686
	LD B31F	−22 681.99	17 624	0.051	−40 306.79	0	0.275	−39 689.79	617	0.272	−39 702.04	604	0.273
	LD B45F	−54 373.33	8390	0.214	−62 763.43	0	0.309	−61 425.99	1337	0.307	−61 425.00	1338	0.307
	LD I31F 2	−31 982.99	19 168	0.065	−51 151.03	0	0.302	−50 402.02	749	0.300	−50 456.82	694	0.301
	LD I31F 5	−43 321.90	21 002	0.089	−64 324.14	0	0.339	−63 484.38	839	0.337	−63 682.49	641	0.339
Definition AS+	HD B45	−48 969.91	27 807	0.499	−76 777.76	0	0.678	−73 991.39	2786	0.673	−73 989.68	2788	0.673
	LD B31	−19 661.10	15 696	0.012	−35 357.78	0	0.231	−34 207.01	1150	0.225	−34 220.79	1136	0.225
	LD B45	−23 042.12	10 284	0.170	−33 326.56	0	0.296	−32 203.68	1122	0.293	−32 271.09	1055	0.294
	LD I31 2	−28 961.15	16 821	0.022	−45 783.12	0	0.252	−44 436.25	1346	0.246	−44 610.00	1173	0.248
	LD I31 5	−40 897.81	17 285	0.046	−58 183.42	0	0.276	−56 600.56	1582	0.270	−57 442.78	740	0.280
Drive	HD Bf39	−41 920.67	64 649	0.155	−106 570.54	0	0.696	−102 332.81	4237	0.693	−102 477.00	4093	0.694
	LD Bf39	−10 490.74	21 461	0.000	−31 952.13	0	0.288	−31 312.02	640	0.288	−31 357.45	594	0.289
	LD Bf39 Qr43	−1879.37	20 621	0.036	−22 501.32	0	0.304	−22 015.03	486	0.304	−22 058.19	443	0.305
	LD Bf39 2 ADMIRE	−12 796.97	31 014	0.000	−43 811.23	0	0.387	−43 132.18	679	0.388	−43 285.21	526	0.389
	LD Bf39 5 ADMIRE	−15 577.71	44 142	0.011	−59 719.85	0	0.507	−58 969.99	749	0.508	−59 395.78	324	0.511

APPENDIX D: MTF SIMULATION IN ELLIPTICAL PHANTOMS

An illustrative reference of a good MTF in a phantom with known geometry acquired with a certain pixel size can be estimated by simulating a synthetic image of the phantom whose resolution is limited by the sampling imposed by the pixel size.

In our specific case, the synthetic image of an elliptical phantom with known dimensions and materials’ CT number can be calculated for a specific reconstructed pixel size as follows.

Let us consider an elliptical phantom, aligned to the isocenter (x_iso, y_iso), with major radius R and minor radius r. Now, assuming that a pixel with coordinates (x_pixel, y_pixel) can be seen as a 1 × 1 square, the intersection between the ellipse and the pixel boundaries are given by the following equation system:

$$\frac{{\left(x-x_{\text{iso }}\right)}^2}{R^2}+\frac{{\left(y-y_{\text{iso }}\right)}^2}{r^2}=1$$

$$x=x_{\text{pixel }}\pm 0.5$$

$$y=y_{\text{pixel }}\pm \mathrm{0.5.}$$

For each (x_pixel, y_pixel) in the image, this set of equations give two or more intersection points if the ellipse intersects the pixel, one if the ellipse is tangent to the pixel’s border, and zero if the pixel is internal or external to the ellipse.

The assignment of the CT numbers for each pixel is then performed considering the percentage of the internal area defined by the ellipse that falls within the pixel. To do so, one can easily identified the internal pixels as those with 0 or 1 intersection points that meet the condition $\frac{{\left(x-x_{\text{iso }}\right)}^2}{R^2}+\frac{{\left(y-y_{\text{iso }}\right)}^2}{r^2}<1$. These pixels can be assigned to the CT number because 100% of the pixel falls within the ellipse. Likewise, those pixels with 0 or 1 intersections satisfying $\frac{{\left(x-x_{\text{iso }}\right)}^2}{R^2}+\frac{{\left(y-y_{\text{iso }}\right)}^2}{r^2}\ge 1$ can be assigned to −1000 HU because they fall outside the ellipse.

For the intersected pixels, the ellipse area that falls within the pixel can be approximated by the polygon’s area defined by the intersection points and the vertices of the pixel that are internal to the ellipse. Note that this approximation is reasonable because the ellipse’s dimensions are much bigger than the pixel size. Therefore, the curvature of the ellipse segment within each pixel is extremely low.

We can calculate the polygon’s area defined by a set of ordered vertices with the shoelace method. Finally, we assign the phantom CT number times the pixel’s internal area as the intensity value at that location. The resulting image is a free-noise elliptical phantom whose edges exhibit the highest resolution possible for the pixel size considered.

The MTF can be calculated from this image in the same way as calculated in the real phantoms acquired.

DATA AVAILABILITY STATEMENT

The data that support the findings are not publicly available.