TOMOSYNTHESIS IMPLEMENTATION OF MULTIPLE IMAGE RADIOGRAPHY

Keivan Majidi1, Jovan G. Brankov1, and Miles N. Wernick1, 2

1ECE Dept., Illinois Institute of Technology, Chicago, IL 60616, USA 2Dept. of Biomedical Engineering, Illinois Institute of Technology, Chicago, IL 60616, USA

ABSTRACT

Multiple-Image Radiography (MIR) is an analyzer-based phase sensitive X-ray imaging method, which simultaneously generates three planar images containing information about absorption, refraction and scattering properties of the object. These three planar images are linear projections of the corresponding properties inside the object. The linearity of the projections allows reconstruction of volumetric images by computed tomography (CT) methods. In this work, we explore the use of tomosynthesis, a limited-angle tomography method, to reconstruct volumetric images of the object. We investigated the accuracy of reconstructions as a function of the number of angular views when the total object radiation exposure is fixed.

Index Terms— Multiple-Image Radiography,

Tomosynthesis, Diffraction-Enhanced Imaging, Phase Sensitive Imaging

1. INTRODUCTION

Multiple-Image Radiography [1] is a planar imaging

method which is an improvement of the diffraction-enhanced imaging (DEI) technique [2]. MIR is able to extract simultaneously three different images from a set of measurements made with a system schematically shown in Figure 1. These images represent three properties of an object: absorption, refraction and ultra-small-angle x-ray scatter (USAXS). The refraction image shows the integrated effect of refractive index variations along the beam path and is suitable for visualizing soft tissues, which have small absorption coefficients, e.g. tendons and cartilages. The USAXS image represents the sub-pixel textural structure of the object and is suitable for observing textural soft tissues such as breast tumors or calcaneal fat pad.

MIR images are virtually immune to degradation due to scatter at higher angles. For this reason, these images have good contrast and are a potential alternative for conventional radiography.

Planar images made by MIR are linear projections of the object properties [3]. Therefore, it is feasible to reconstruct 3D volumetric images of the object properties from a set of projections acquired at different angular views. In [4] we

demonstrated a CT version of MIR called CT-MIR. CT-MIR uses a full range of angular projections to reconstruct the entire 3D volumetric images. In cases where a full range of angles cannot be obtained or where scan time is limited, limited-angle tomography could be useful. For example, we envision MIR tomosynthesis being used for imaging of cartilage in the knee [5] and in mammography [6].

Figure 1. Schematic view of MIR imaging system.

Tomosynthesis [7] is a limited-angle tomography method which allows reconstruction of an arbitrary ROI from the object from a limited number of angular views. The basic idea of tomosynthesis was first presented in [8] which used a simple "shift and add" principle. In 1972, Grant [7] showed a proof of concept and coined the term "tomosynthesis". In the late 1990s, with the invention of high-resolution digital flat-panel detectors and improvement in computing cost, interest in tomosynthesis was renewed, with applications such as mammography, chest imaging, and orthopedic imaging [9].

In this work, we evaluate the applicability of tomosynthesis to MIR imaging. We explore the case where the total radiation exposure of the object remains fixed regardless of the number of collected angular views. We evaluate the accuracy of the reconstructed images as a function of the number of angular views.

Presented results suggest that an optimal number of angular views exists and it is different for each MIR image.

2. MIR IMAGING MODEL

The schematic of MIR imaging system is shown in

Figure 1. ( , , )x y z and ( , , )x y z denote the coordinates in the object and the imaging system domain, respectively. The object is illuminated with a collimated, monochromated X-ray beam. While the X-ray beam passes through the object,

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it will be affected by the object properties, which will change the beam angular intensity profile (AIP). Next, an analyzer crystal, which is a narrow angular filter, is used to analyze the beam AIP. This angular filter passes only beam components that are traveling at or near its Bragg angle, b , to the surface of the measuring detector. By measuring the intensity at different angular positions of the analyzer, , we can effectively measure and analyze the AIP of the beam. One way to analyze the beam AIP is to calculate the MIR parametric images, which will be described later. First, we will describe an analytical model for the measured AIP.

Using radiation transport theory [10] we derived and experimentally tested the image formation model in MIR [3]. Using the image formation model in [1] and [3], one can derive a simplified approximation of beam AIP at the detector surface as following:

( ; , ) ( ) ( ; , )g x y R f x y , (1)

where ( ; , )g x y is the AIP at location ( , )x y on the two dimensional detector surface, ( )R is AIP which would be seen in absence of the object and is a characteristic of the imaging system, ( ; , )f x y represents object function, and

denotes convolution with respect to . The object function is the AIP one would observe on the analyzer surface if the illuminating beam is perfectly collimated, i.e., the beam AIP is a Dirac delta function. We showed in [3] that ( ; , )f x y can be approximated by following Gaussian curve:

2

0 22

( , )1( , )( ; , ) exp2 ( , )2 ( , )

x yx yf x y I ex yx y

(2)

where I0 is the initial intensity of the beam,

( , )( , ) ( , , )

l x yx y A x y z dl is the integral of absorption

coefficient, ( , , )A x y z , along the beam path , l

( , )( , ) ( , , )

l x yyx y n x y z dl is the angular shift of the

beam centroid which is the integral of refractive-index gradient, ( , , )y n x y z , and 2

( , )( , ) ( , , )

l x yx y usaxs x y z dl

is the angular beam divergence (like variance) which is the integral of USAXS parameter, . ( , , )usaxs x y z

Now using equation (2) the measured discrete version of ( ; , )g x y can be modeled as:

, [ ] ( ; , )m n l m ng l g x y

)

, (3)

where m and n are the indices of detector pixels at location ( ,m nx y , l=1,2,…,L represents the index of the analyzer angle l and denotes the measurement noise model.

As showed in [4], the three MIR parametric images could be directly estimated from the sampled , [ ]m ng l as follows:

First, for each detector pixel, we define the normalized AIP as:

, , ,1

[ ] [ ] [ ]L

m n m n m nl

G l g l g l , (4)

and the AIP shift of the imaging system:

0

1 12

1

( ) [ ]L

LI

l

R l R l , (5)

where is the angular spacing between the measurements. Now we can estimate the three MIR parametric images

as follows:

,

1

0

[ ]( , ) ln

L

m nl

m n

g lx y

I, (6)

,1

1( , ) ( ) [ ]2

L

m n m nl

Lx y l G l R (7)

22

,1

2

10

1( , ) ( ) ( , ) [2

1 1( ) [ ]2

L

m n m n m nl

L

l

L ]x y l x y G

Ll R R l

I

l (8)

The parametric images described above are planar images. In order to enable tomography or tomosynthesis, one needs to acquire planar images from different angular views, . This can be done in two ways. One is to rotate the imaging system around the stationery object or to rotate the object inside the stationary imaging system. The later approach is adopted in our system.

3. TOMOSYNTHESIS

The simplest method for tomosynthesis approach

involves only shift and add steps [9] to reconstruct volumetric ROI data. However, more complex algorithms provide a better accuracy. In this work we adopted Simultaneous Iterative Reconstruction Technique (SIRT) [12] which is a modification of Algebraic Reconstruction Technique (ART) [13]. SIRT uses all of the projections simultaneously for update at each iteration step and it is more robust to noise than ART. For an overview of these methods application in tomosynthesis see [11].

In the proposed approach an isocentric object motion was used which is consistent with imaging model presented in CT-MIR [4]. In this section, we will describe the reconstruction procedure using absorption data but the same approach was applied on the other two parametric data.

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Let ( , , )i n kA x y z denote the voxelized volumetric absorption data where i, n and k are the indices of each voxel. Now the forward imaging model can be described as: ( , ; ) ( , , , ) ( , , )m n i n k

i kpx y W i k m p A x y z (9)

where p is the tomographic projection angle and

is a weighting function that describes the influence of

( , , , )W i k m p( , , )i n kA x y z voxel on the ( , ; )m n px y

measurement. Note that because the beam in the imaging system can be approximated by parallel beam and its direction is perpendicular to y-axis weightings,

, are independent of n. ( , , , )W i k m pIn the reconstruction, our goal is to reconstruct

( , , )i n kA x y z using the measured ( , ; )m n px y values. The SIRT reconstruction method is an iterative

algorithm that starts with an initial 1

( , , )i n kA x y z estimate. This estimate is iteratively updated until desired stopping criterion is satisfied, described next.

At qth iteration, we can calculate:

( , ; ) ( , , , ) ( , , )q

m n p i n ki k

qx y W i k m p A x y z (10)

Next, we can calculate estimation error in projection domain as:

( , ; ) ( , ; ) ( , ; )qq

m n p m n p m n perr x y x y x y , (11) The estimation error is used to update the pervious

volumetric data estimate as:

1( , , ) ( , , )

( , , , ) ( , ; )1 ( , , , )

q q

i n k i n k

qm n p

p mi k

A x y z A x y z

W i k m p err x yP W i k m p

, (12)

where P is the number of projection angles. Next, repeat Eqs. (10) - (12) until the following iteration

stopping criterion, , is satisfied: 1

2 2

2

( , ; ) ( , ; )

( , ; )

q qm n p m n p

qm n p

err x y err x y

err x y,

where 2

. denotes L2-norm. 4. SIMULATION

To test the proposed method we created a 125x125x125

phantom made of ellipsoids (Figure 2). To calculate projections of the absorption, refraction and USAXS, we assigned a uniform value of the corresponding object parameters to each ellipsoid (see the paragraph after eq. (2) for corresponding parameters). The parameters values are chosen to match the data obtained from a human thumb study.

Figure 2. Slices from experimental phantom.

The data acquisition was simulated using a Poisson noise model. This was followed by parametric images estimate using Eqs. (4) - (9).

The procedure above was repeated for every projection angle. This was followed by tomosynthesis to estimate parametric volume data.

To evaluate the proposed method performance we vary the range of the tomographic angle, , from [ 5,5] to

[ 70,70] with one sample per degree resolution. This will result in number of projection angles, , to vary from 11 to 144. Let us denote the maximum tomographic angle by

P

. Therefore, we will have that 5,70 . Next, the radiation dose delivered to the object was

fixed. This is to say that the total number of photons striking the object was constant regardless of the number of projection angles P. If there were no constraint on delivered object dose, the reconstruction would improve as the number of projection angles increases.

The reconstructed images using (o38 77P ) are shown in Figure 3. It could be seen that reconstructed images near the focal plane ( 0z z ) are more fateful to the phantom, shown in Figure 2, than images which are out of focal plane.

Next, to quantify the algorithm accuracy we defined the following ROI error function:

21( ) [ ( , , ) ( , , )]ROI

ROI

K

i n k i n ki j k K

MSE A x y z A x y zC

(13)

where C is the number of voxels in the ROI. The mean square error, for each reconstructed parameter,

as a function of is shown in Figure 4. It can be seen that the optimal range for each parametric image is different. For refraction parameter, this range is from o30 to , for USAXS parameter it is from to and

o40o50 o60

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for absorption coefficient, the range is o55 and above. Images using four different are shown in Figure 5 for visual comparison. The image columns are marked by letters A, B, C, and D. These letters correspond to the same letters in the plot (Figure 4). ( )MSE

Figure 3 Reconstructed images for projection angular range from -38 :38 .

Figure 4. MSE for each parameter vs. .

5. CONCLUSION

We have proposed and tested a limited angle version of CT-MIR [3]. This method is capable of accurately reconstruct data confined in a region of interest. We also investigated the optimal projection angular range for the data acquisition with fixed radiation exposure. In future studies, we plan to verify these results by applying the

algorithm to experimental MIR data and investigate clinical applicability of MIR-tomosynthesis.

Figure 5 Reconstructed images ( 5z z ) for at points show in

Figure 4

ACKNOWLEDGMENTS This research was supported by NIH/NIAMS

Grant No. AR48292 and NIH/NCI Grant No. CA111976.

6. REFERENCES

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[8] Ziedses des Plantes, “Eine neue methode zur differenzierung in der roentgenographie,” Acta Radiol. , vol.13, pp. 182–192, 1932.

[9] J. T. Dobbins III, and D.J. Godfery, “Digital x-ray tomosynthesis: current state of the art and clinical potential,” Phys. Med. Biol., vol. 48, pp. 65-106, 2003.

[10] A. Ishimaru, Wave Propagation and Scattering in Random Media, Piscataway, New Jersey, 1997.

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[12] P. Gilbert, “Iterative methods for the 3D reconstruction of an object from projections ” J. Theor. Biol., vol 76, pp. 105–117, 1972.

[13] M. S. Kaczmarz, “Angenaherte Auflosung systemen linear Gleichungen,” Acad. Polon. Sci. Lett. Bull. A., vol 3, pp. 355-357, 1937.

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