Generalized Penalized Weighted Least-Squares

Reconstruction for Deblurred Flat-Panel CBCT Steven Tilley II, Jeffrey H. Siewerdsen, J. Webster Stayman

Department of Biomedical Engineering, Johns Hopkins University, Baltimore, MD

Abstract—An increase in achievable spatial resolution would

enable application of flat-panel detector cone-beam CT (FP-

CBCT) in high-resolution applications, such as 3D breast and bone

imaging. Improved reconstruction techniques exploiting more

sophisticated models of noise and spatial resolution characteristics

may provide such improvement. In this work, we propose a

statistical model of measurement mean and covariance, with

particular attention given to correlations in measurement data. An

iterative reconstruction algorithm based on this model was applied

to simulated and real experimental data to evaluate its

performance compared to an uncorrelated model under various

imaging conditions. Focal spot and detector blur of a CBCT

testbench were measured and used to inform the model-based

image reconstruction. Simulation studies demonstrate that

incorporation of the correlated noise model yields improved

reconstructions, especially in the presence of focal spot blur.

Improved noise-resolution tradeoffs were quantified, and

application to high-resolution imaging of trabecular bone was

demonstrated.

Keywords— Spatial resolution, Cone-beam CT, Correlated noise,

Model-based Reconstruction, Generalized Least-Squares Estimation

I. INTRODUCTION

Flat-panel cone-beam computed tomography (FP-CBCT)

has found many applications in high spatial resolution imaging

due to the large volumetric coverage and fine detector pixels.

Despite this advantage, the spatial resolution required for some

applications remains just out of reach. For example,

visualization of microcalcifications in CBCT mammography or

highly detailed trabecular structure in quantitative CT of

musculoskeletal extremities both stand to benefit from

improvements beyond conventional spatial resolution limits.

Current breast imaging solutions include projection

mammography, tomosythesis, and CBCT, with the requirement

to detect microcalcifications at a scale of ~0.1 mm. Similarly in

trabecular bone imaging, the gold standard is microCT which

tends to have a very small field of view, making it difficult to

apply to in vivo applications. In this paper, we seek to extend

the high spatial resolution capabilities of FP-CBCT through

careful modeling of system blur and correlated noise within a

statistical reconstruction framework.

Typical model-based iterative reconstruction algorithms

balance accuracy and complexity of the imaging system model

based on desired resolution, noise properties, algorithm

complexity, and execution time. The standard approach is to

adopt a projector model with little or no blur and independent

measurement noise. Such assumptions have been supported in

part by work done by Hofmann et al. [1], which shows that

modeling source blur in clinical CT systems does not

significantly improve image quality for typical focal spot sizes

(~0.5 mm) and detector pixels sizes (~0.6 mm). While this may

be true for current diagnostic CT systems and current spatial

resolution goals, there are a number of factors that suggest an

advantage by way of improved system models for FP-CBCT.

Such factors include 1) finer detector pixels (70-200 m) in flat

panels; 2) larger measurement blur due to light spread in the

scintillator; 3) increased focal spot blur due to widespread use

of inexpensive fixed anode x-ray tubes and compact geometries

in dedicated FP-CBCT systems; and 4) the presence of

significant noise correlation in FP-CBCT data due to light

spread after the conversion of primary x-ray quanta to

secondary light photons.

Model-based reconstruction approaches that account for

system blur but without a correlated noise model have been

previously developed. [2], [3] Similarly, sinogram restoration

methods have also been proposed to correct for system blur [4]

and noise correlation [5]. We propose a combination of

sinogram restoration (deblurring) followed by iterative

reconstruction, tracking the correlations through the deblur

processing and accounting for them in the reconstruction step.

Flat-panel CBCT systems can be characterized by multiple

sources of blur and noise. We consider blur due to an extended

x-ray source and due to the detector scintillator, the latter of

which adds correlations to the noise. Because noise correlation

is a result of only one of these two sources of modeled blur, we

explore the effects of varying the magnitude of these blurs in

simulation. Understanding the scenarios that benefit from blur

and correlation models will not only improve reconstructions

on current FP-CBCT systems, but will help to guide new system

design (e.g., geometry, focal spot size, and scintillator

thickness).

In the following sections we present an overview of the

reconstruction methodology. This approach is a refined version

of the methods reported in [6] with a modified strategy for

deblurring. A simulation investigation for various system

designs with different magnitudes of source and detector blur

follows. Subsequently, a brief description of the physical

measurement of source and detector blur for a FP-CBCT

testbench is presented. Finally, the measured blur models are

incorporated within the proposed methodology and applied to

FP-CBCT extremity data for trabecular bone imaging.

II. METHODS

A. Imaging System Model and Reconstruction

We consider a system model based on the first and second

order statistics. Thus, the measurements are specified by the

mean vector (�̅�) and the covariance matrix (𝐊𝑌 ) which are

functions of the attenuation (), related by a forward projection

operator (A), a gain operator (G), and the source and detector

scintillator blurs (Bs and Bd respectively). These models can be

represented compactly in vector form as

�̅� = 𝐁𝑑𝐁𝑠𝐆 exp(−𝐀μ̅) (1)

𝐊𝑌 = 𝐁𝑑𝐷{𝐁𝑠𝐆 exp(−𝐀μ̅)}𝐁𝑑𝑇 + 𝐊𝑟𝑜 (2)

where Kro denotes additive readout noise in the detector. (Note

that source blur is approximated as a projection domain effect –

implying that depth dependent magnification effects are small.)

Transforming measurements into line integral estimates and

presuming Gaussian noise permits accommodation of a non-

diagonal covariance matrix within a linear least-squares

framework. However, such linearization requires an initial

deblur operation. We introduce a “modified” deblur operator

𝐂−1 ≈ 𝐁𝑑−1𝐁𝑠

−1 (3)

which is approximately equal to the inverse of the source and

detector blurs. Exact inversion may not be possible or desirable

due to null spaces and noise magnification.

The line integral estimate and its covariance then become

𝑙 = − log(𝐆−1𝐂−1𝑦) (4)

𝐊𝐿 ≈ 𝐷 {

1

𝐂−1𝑦} 𝐂−1𝐊𝑌[𝐂−1]𝑇𝐷{

1

𝐂−1𝑦}

(5)

yielding the penalized weighted least-squares equation:

�̂� = arg min𝜇

||𝑙 − 𝐀𝜇||𝐊𝐿

−1+ 𝛽𝑅(𝜇) (6)

B. Practical Implementation

In this work, we use a quadratic penalty function, allowing

equation (6) to be written in a closed form.

�̂� = [𝐀𝑇𝐊𝐿−1𝐀 + 𝛽𝐑]−1𝐀𝑇𝐊𝐿

−1𝑙 (7)

We assume all blurs are shift invariant, allowing blur operations

to be done in the Fourier domain. We choose the deblur

operation in (3) so that it applies a binary mask (M) zeroing any

nulled frequencies, approximated as the frequencies where the

blur MTF is less than 10-2 (simulation) or 10-4 (testbench data).

𝑪−1𝑥 = 𝐹−1 (𝑀𝐹(𝑥)

𝐹(𝑩𝑠𝑩𝑑))

(8)

where F denotes the Fourier transform. Consequently:

[𝑪−1]−1 ≈ 𝑪∗𝑥 = 𝐹−1(𝐹(𝑩𝑠𝑩𝑑)𝑀 𝐹(𝑥)) (9)

which allows the inverse covariance matrix to be written as

𝐊𝐿−1 ≈ 𝐷{𝐂−1𝑦}[𝐂∗]𝑇 𝐊𝑌

−1[𝐂∗]𝐷{𝐂−1𝑦} (10)

where KY is approximated as

𝐊𝑌 ≈ 𝐁𝑑𝐷{𝐂−1𝑦}𝐁𝑑𝑇 + 𝐊𝑟𝑜 (11)

The null space in C* forbids an inversion of 𝐊𝐿 , but using

iterative methods we may compute a generalized inverse and

solve the likelihood equation within the unmasked subspace.[7]

A similar approach was used for null spaces in Bd when

inverting 𝐊𝑌 in simulation studies.

Solving of (7) can be divided into two parts, preprocessing

and iterative steps. In the preprocessing step line integral

estimates are computed from the measurements using the

modified deblur, the inverse covariance matrix is applied, and

the result is backprojected. The inverse in (7) is solved

iteratively using the conjugate gradient method (100 iterations

for simulated data, 300 for real data). Each iteration requires a

convolution with the penalty kernel, a forward projection, an

application of the inverse covariance, and a backprojection. The

application of the covariance matrix requires an application of

the inverse of 𝐊𝑌, also via the conjugate gradient method (1000

iterations in the preprocessing step and 100 in each iteration of

the iterative step, or until the residual decreased to 0).

C. Simulation Studies

Figure 1 shows the simulation phantom. Projection data were

generated using a high resolution version of the phantom

(25 m voxels) onto a high resolution detector (1x7000 line of

35 m pixels at 720 angles), which were then downsampled to

1x1750 at 720 angles. The data were reconstructed into a

1000x1000 grid of 0.1 mm voxels. Gaussian noise was added

to the data to simulate correlated quantum noise and white

readout noise. Noiseless data generation from line integrals is

shown in (12), and noise injection is show in (13).

�̅� = 𝐁𝑑𝐁𝑠𝐆exp (−𝑙) (12)

𝑦 = �̅� + 𝑩𝑑𝑁 (0, √𝐁𝑠𝐆 exp(−𝑙)) + 𝑁(0, 𝜎𝑟𝑜) (13)

The variance of the readout noise was 3.5 photons2. Gain was a

constant 106 photons.

Two reconstruction methods were studied: 1) using the full

covariance matrix in equation (10); and 2) an uncorrelated

covariance matrix composed of only diagonal elements:

𝐊𝐿−1 ≔ 𝐷{𝐂−1𝑦} + 𝐷{𝜎𝑟𝑜

2 } (14)

which represents an independent noise assumption. When this

noise model is used, (7) is a traditional penalized least-squares

reconstruction with deblurred measurements. The noise models

were compared by plotting two resolution-variance curves from

reconstructions with varying penalty strength.

Resolution was measured as an edge response to the disc in

the phantom, and variance was approximated as the spatial

variance of the data inside the blue ring shown in Figure 1.

Additional FDK reconstructions were conducted with the cutoff

frequency at Nyquist and no apodization.

To evaluate the importance of the two types of blurs, various

blur scenarios were tested. There were two subsets of

experiments where: 1) the total blur was kept constant, and the

distribution of blur was altered from source dominated to

detector dominated; and 2) the ratio of the blurs was kept

constant, and the value of the total blur was varied. All blurs in

the simulation study were Gaussian.

D. Testbench Characterization

In order to apply this algorithm to physical FP-CBCT data,

we estimated the source and detector blurs on our x-ray

testbench. This was done by imaging a tungsten edge,

calculating the edge spread function (ESF), and then converting

this to a modulation transfer function (MTF).[8] The scintillator

Figure 1: Simulation

phantom with fat (a),

muscle (b),

trabecular bone (c),

line pairs (d), and a

uniform disc (e) for

resolution (edge).

and noise

measurements

(a)

(c)

(d) (e)

(b)

blur was assumed to be radially symmetric. The source blur is

known to not be radially symmetric, so MTFs were taken in the

(approximate) vertical, horizontal, and 45 degree directions.

These results were compared with MTFs taken from a pinhole

image of the source, acquired by placing a lead plate with a

pinhole on the filter and using a geometry with a high

magnification.

E. Physical FP-CBCT testbench data studies

Our testbench includes a Varian 4030CB detector and a

Varian Rad-94 x-ray tube. A custom wrist phantom (The

Phantom Laboratory, Greenwich, NY) was imaged with large

(~1.1 mm) and small (~0.5 mm) focal spots. We used the large

focal spot for testing our method, and the small focal spot to

create a high resolution reference image using FDK. The data

were reconstructed into a 210x600x600 volume of .15 mm

voxels using the correlated and uncorrelated noise models with

readout noise modeled as a constant 3.5 photons2. These two

iterative reconstruction methods were noise matched.

Deblurred and non-deblurred data were also reconstructed

using FDK. FDK reconstructions were done using a cutoff at

Nyquist and no apodization.

III. RESULTS

Figure 2 shows resolution variance curves for simulated

systems with different combinations of source and detector

blur. Points positioned on the curve in the diagram all have the

same total blur, while points on the straight line have the same

ratio of source to detector blur. The correlated noise model

equals or outperforms the uncorrelated noise model in all cases.

When source blur is large there is a significant advantage to the

correlated noise model whereas when source blur is minimal,

the two methods are nearly equivalent. This is consistent with

the fact that only the detector blur adds correlation and when

source blur is absent, the deblurring operation whitens the

noise, making the uncorrelated noise assumption accurate.

When detector blur is absent, the data is uncorrelated and

deblurring source blur adds off-diagonal values to the

covariance matrix making the correlated noise model important.

When the ratio of the blurs is kept constant and total blur

Figure 4: Testbench MTF Measurements. A: Detector MTF, with gauss*sinc model. B: Source pinhole image. C: Source MTF from pinhole

image. D: Profiles from pinhole MTF. E: Source MTF from tungsten edge response.

Figure 3: Simulated data reconstructions.

Top half is difference with truth. Bottom

half is reconstruction with a zoomed

image of the line pair target. Values are in

unites of mm-1. The uncorrelated model

yields higher noise in resolution matched

reconstructions and coarser resolution in

noise matched reconstructions as

compared with the correlated noise model

images.

Figure 2: Resolution-Variance

plots. Subplots A-G correspond to

different systems with varying

source/detector blurs (top left).

Uncorrelated noise model:

triangles, correlated noise model:

squares, FDK: star. Note that

everything to left of FDK

represents an improved spatial

resolution and that the correlated

noise model has the greatest

advantage in the case of systems

dominated by source blur (E).

Smaller total blur (F) permits a

finer resolution limit while larger

blur (G) increases this fine limit.

increases, the relative performance of the methods is similar.

The fine resolution limit increases as blur increases.

Figure 3 shows reconstructions from a simulated system with

approximately twice as much source blur as detector blur (point

D in Figure 2). When the images are resolution matched, the

correlated noise model produces a reconstruction with less

noise. (See background in the difference images.) When noise

matched, using the correlated noise model produces a sharper

reconstruction (note line pairs) than the uncorrelated noise

model. Also note, while variances are matched the correlations

in the reconstruction differ substantially.

Figure 4 summarizes the testbench characterization studies.

Figure 4A shows the detector MTF (scintillator blur and pixel

sampling) measured using the tungsten edge. The solid line

shows a fit to Gaussian times a sinc function (representing the

detector aperture). The FWHM of this Gaussian (in the spatial

domain) is 0.85 pixels, or 0.33 mm. Figure 4E shows the source

MTF measured using the tungsten edge placed at the center of

rotation. The MTF is asymmetric. The 45 degree MTF has its

first zero at a higher frequency than the vertical or horizontal

MTFs, which is consistent with a source resembling a 2D rect

function. The pinhole image of the source (Figure 4B) supports

this. The “horns” at the top and bottom of the image may

explain the differences in the first side lobes of the vertical and

horizontal MTFs. The MTF was also calculated (although at a

different magnification) by taking the Fourier transform of the

pinhole image (Figure 4C). The vertical, horizontal, and 45

degree line profiles of this 2D MTF are shown in Figure 4D,

and approximately match the shape of the MTF calculated using

the edge spread. The solid line in Figure 4E is the Gaussian blur

used in the reconstruction of our wrist phantom. The blur

focuses on the main lobe and is slightly larger than reality,

which decreases instability due to inaccuracies in the model.

Figure 5 shows real data reconstructions. Figure 5A is a high

resolution reference image, with a zoom region (Figure 5B) and

patch for variance estimation identified. A coarser resolution

FDK reconstruction where most of the high resolution

trabecular structure has been lost is shown in Figure 5C.

Attempting to deblur (using the Gaussian approximations

discussed previously) prior to applying FDK results in

significant noise amplification (Figure 5D). In comparison,

noise matched uncorrelated (Figure 5E) and correlated (Figure

5F) noise model reconstructions are also shown. Again, the

correlated noise model achieves higher resolution - showing

finer structures than the uncorrelated case at matched noise.

Comparisons with the high resolution reference indicate that

these fine structures are representative of true trabecular details.

IV. DISCUSSION

In this work we have presented a general framework for

modeling both source and detector blur including correlated

noise effects. Investigations in both simulations and in physical

FP-CBCT data show that proper noise modeling can improve

the noise-resolution trade-off. This advantage is most dramatic

for systems with significant source blur, though deblurring and

an accurate noise model outperforms conventional FBP in all

cases considered.

This methodology has potential application in a variety of

applications requiring high spatial resolution, including

trabecular bone imaging and breast CBCT. Moreover, the

studies presented here begin to characterize the improvements

possible for specific system designs (e.g., varying amounts of

source and detector blur). This may have potential impact in

system design – for example, permitting use of larger focal spot

x-ray sources (improved power requirements) and thicker

scintillators (improved detection efficiency) – if the proposed

reconstruction methodology is integrated into the system.

V. ACKNOWLEDGEMENTS

This work was supported in part by NIH grants R21-EB-014964 and T32-EB010021, and an academic-industry partnership with Varian Medical Systems

(Palo Alto, CA.)

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(a) (b)

(d)

(c)

Figure 5: Testbench data reconstructions. In split images, top half

is difference with the high resolution reference image. Units are mm-1.

(a): soft tissue. (b) cartilage. (c) trabecular bone. (d) cortical bone.