Optimized algebraic local tomogragphy

OPTIMIZED ALGEBRAIC LOCAL

TOMOGRAPHY

I. Frosio, N. A. Borghese Applied Intelligent System (AIS) Lab., Computer Science Dept., University of Milan

Abstract: In this paper, we describe how to obtain optimized tomographic images by

acquiring a limited number of projection images from a limited angle of view.

A classical algebraic method has been modified to eliminate typical artifacts

introduced at the image boundaries. Experimental results are reported in the

field of maxillo-facial radiology; they demonstrate that the proposed method

produces images of high quality valuable for planning and following-up the

results of implantology operations.

Key words: Dental local tomography – artifact suppression – algebraic reconstruction

techniques – ordered subset – equalization.

1. INTRODUCTION

Computerized tomography (CT) is nowadays widely used for imaging

the patient’s body in many clinical fields; however, the cost of the machinery

and the high dose delivered to the patient is not justified for many

applications. Accordingly to the ALARA (As Low As Reasonably

Achievable) principle, many efforts have been done to develop techniques

and machines aimed at reconstructing the local anatomical districts of

interest, reducing the potential damage that can be produced inside the

irradiated volume [1-3].

In this regard, an alternative to CT is represented by conventional

tomography, whose basis was proposed during the 30s of the last century

[1]. In particular, a set of different radiographic images are taken from

different points of view and stored on the same film sensor. With this

modality, only one planar section of the irradiated volume can be obtained,

whereas the out-of-plane structures are blurred out. Thanks to its simplicity,

conventional tomography is nowadays largely diffused especially in

2 I. Frosio, N. A. Borghese

maxillofacial radiology, where many ortopantomographs are capable of

acquiring such images [4-6].

However, the limitation of conventional tomography of producing the 3D

reconstruction of only a single plane largely limits its use. For instance,

when the doctor wants to analyze the state of an old implant, or plan the

introduction of a new one, the need of a 3D volume larger than a single

plane is evident. Besides this, an extremely accurate positioning of the

patient is required to obtain the desired 3D section, but there are no means to

guarantee it.

A more versatile technique, which has recently become feasible thanks to

the introduction of digital X-ray sensors, is represented by tomosynthesis

[1]. In this domain, the different radiographic images taken from different

points of view, are acquired and stored on file independently. A volumetric

reconstruction of the irradiated area can be obtained a-posteriori though

back-projection, which is obtained shifting one image with respect to the

other and averaging the absorption coefficients in the different images. Since

an arbitrary number of planes can be reconstructed, tomosynthesis allows

navigating inside the reconstrcuted volume.

Both in conventional tomography and in tomosynthesis the reconstructed

slices are blurred because of the limited number of images and different

strategies have been proposed to limit the blur impact. These are based on

linear filtering [1, 7, 8], inverse 3D filtering [9], iterative deblurring [10] or

approximate matrix inversion [11]. All these approaches divide the

reconstruction problem into two steps: first, back-projection is performed to

reconstruct the volume through tomosynthesis; then, a deblurring strategy is

used to improve its resolution [1].

A more general approach to volume reconstruction is represented by

algebraic reconstruction techniques, where the reconstruction and the

deblurring phases are mixed together [2]. Besides blur, a second problem in

local tomography with limited angle is represented by band and truncation

artifacts [13, 14] that may significantly degrade the quality of the

reconstructed volume: correction algorithms are required to get satisfactory

results.

In this paper, we describe how algebraic methods can be optimized to

reconstruct local districts starting from a limited set of projections, removing

band and truncation artifacts. The method has been applied to maxillo-facial

radiology in which, projection images can be obtained adapting an

ortopantomograph that is traditionally used for panoramic and cephalometric

imaging [4-6]. The results reported here demonstrate that the proposed

approach can provide highly detailed 3D images valuable for planning of

implantology operations, or post-interventional follow-up.

OPTIMIZED ALGEBRAIC LOCAL TOMOGRAPHY 3

2. METHOD

2.1 Acquisition instrumentation and trajectory

An ortopantomograph (Hyperion by MyRay, [5]), equipped with a 1536x64

Time Delay Integration (TDI) sensor, was used to acquire a set of 11

projection images, partially overlapped, over an angle of 40° [17] (Fig. 1).

Each projection is constituted of 1536x602 pixels, for an image size of

147.5mm x 57.8mm, with a grey level resolution of 12 bpp. The total time

required for acquiring the set of 11 projections is about 1 min; the X-ray tube

is active for 26s, the rest of the acquisition time is devoted to positioning the

acquisition apparatus.

Figure 1. Schematic representation of the acquisition trajectory. It was designed to acquire

images almost parallel to the tooth cross-section; only the first and the last projections are

depicted here.

The number of projections and the acquisition angle are results of a

compromise among acquisition time, dose and resolution. It should be

remarked that, when a limited acquisition angle is used, the resolution of the

reconstructed volume perpendicular to the acquisition cones is low (Fig. 1)

and structures parallel to this direction are hardly distinguishable [15-16].

For this reason, the images are acquired perpendicular to the dental arch to

obtain the maximum visibility of the cross-section of the tooth and of the

nerve channel in particular (Fig. 2, 3). Moreover, anisotropic voxels are

used: 50 slices perpendicular to the dental arch (X direction in Fig. 1),

spaced by 1mm, are reconstructed; each slice measures 700x267 pixels, each

of 0.15mm x 0.15mm. The 6th projection is centered in the middle of the

reconstructed volume; the first and the last ones are slanted by +/- 20°.


Figure 2. The same slice, reconstructed after 5 iterations of the ART, SART and SIRT

algorithms, is shown here. The ART reconstruction is characterized by high contrast; it is also

very noisy. The SIRT reconstruction is smooth (low noise) ant it has a low resolution. SART

offers a good compromise between ART and SIRT. The nerve channel is indicated by arrows.

2.2 Reconstruction algorithm

Under the general hypothesis of negligible refraction and monochromatic

radiation, the absorption p measured along a ray path can be expressed by

the De Beer’s law [2, 3] as:

p = -log(i/i0) = (x)dx (1),

where i0 is the radiation intensity exiting from the tube, i is the measured

intensity, (x) is the linear absorption coefficient in position x and the

integral is computed over the ray path. It follows that, descretizing equation

(1) for all the measured pixels, each tomographic problem can be written as:

p = Wx (2),

where p is the vector containing the measured absorptions (Eq. 1), x is the

unknown vector of the absorption coefficients of the volume and W is the

projection matrix, which describes how each voxel contributes to the

measured data. In particular, the element Wij describes the length of the i-th

X-ray accross the j-th voxel. For instance, Wij=0 means that the i-th X-ray

does not cross at all the j-th voxel.

Solving the linear system (2) through least squares is not feasible because

of the large size of the system (9.345,000 unknowns and 10,171,392 data in

the present case), and iterative optimization algorithms have been adopted to

this scope. In general, given an initial guess x0, the absorption coefficients of

the volume are iteratively updated as [2]:


xk+1

= xk – W

T][D] ([W]x

k – [p]) (3),

where xk is the estimate of the volume at the k-th iteration, D is a

diagonal matrix with Dii = 1/(WiWiT), Wi is the i-th row of the matrix W,

is a damping factor, T is the transpose operator and [] indicates a subset

of the rows of a matrix / vector. In practice, starting from the current

estimate of the volume absorption coefficients xk, an estimate Wx

k of the

projection data is obtained; this estimate is compared with the measured data

p and the difference, (Wxk - p), is weighted by the matrix D and back-

projected into the volume through the matrix WT. An iteration is completed

when all the equations in (3) have been considered [2].

Depending on the size of the subset considered, we obtain ART

(Algebraic Reconstruction Technique, one equation per subset – historically,

this is the first method proposed [19]), SART (Simultaneous ART, each

subset is composed by the equations associated to one projection [20]) or

SIRT (Simultaneous Iterative RT, one subset comprises all the system

equations [2]) methods. In the original ART method, the volume is updated

after considering one pixel in one image. This leads to a quick convergence

that generally produces a high resolution but very noisy volume [21] (Fig.

2). With the SIRT method, the volume update occurs only at the end of the

iteration when all the pixels in all the images have been examined; in this

case, the noise is largely reduced, but convergence of the algorithm is too

slow and only poorly contrasted volumes can be obtained in a reasonable

amount of time (fig. 2). For these reasons, SART approach has been widely

adopted, which represents a good compromise between convergence speed

and quality of the reconstructed volume.

In SART, updating is carried out after considering all the pixels in one

image. An aspect that affects the convergence speed of the algorithm, is the

order in which the projections are considered: maximizing the orthogonality

between the equations of two consecutive subsets produces faster

convergence. As a general rule for tomography, two consecutive subsets

should be associated to largely spaced projections and an optimal access

scheme could be derived from this rule (ordered subset scheme) [2].

However, similar results can also be obtained also through a randomized

access order [2, 22], which is the solution adopted in our algorithm.

Notice that Eq. (3) can be interpreted as the updating equation of an

iterative minimization scheme through ordered subset gradient descent of the

following cost function [18]:

E = ½ (Wx - p)T D (Wx - p) (4),


which represents a weighted least squares system where D plays the role

of the weighting matrix. We remark here that the shorter is the path of an X-

ray inside the object, the higher is the weight (a lot of elements of Wi are

zero and as a consequence, Dii = 1/(WiWiT) is large). This observation is

fundamental to develop efficient solutions for suppressing the truncation

artifacts.

2.3 Band and truncation artifacts suppressions

Fig. 3a shows 5 slices of the anterior molar district of an

anthropomorphic phantom, reconstructed with the original SART algorithm.

Two types of artifacts can be observed in these slices: first, a set of vertical

bands appear at the margins of the slices; these are especially visible for the

slices far from the central one: the rightmost and leftmost ones in Fig. 3a.

We will refer to this kind of artifact as “band artifact” [14].

The second kind of artifact (truncation artifact, [15]) appears as a narrow,

white stripe at the right margin of the central slice in Fig. 3a-b; it also

appears, somehow attenuated, as a larger, bright stripe in the second and

fourth slices of Fig. 3a-b.

Band artifacts introduce spurious high frequencies in the reconstructed

volume and they can partially hide small anatomical features occurring at the

border of the reconstructed volume. Their effect can be even more

impressive when image treatment, like unsharp masking, is applied to the

observed section to enhance the visibility of small anatomical details. On the

other hand, truncation artifacts produce light areas in the reconstructed

volume and they may significantly reduce the grey level dynamics of the

image, thus also reducing the overall contrast.

It is known that band artifacts are generated by SART during the iterative

reconstruction process [14] as the projection cone associated to each

projection only partially crosses the volume. As a consequence, a

discontinuity is generated at the margins of the projection cone, as the voxels

inside the cone are updated, whereas the external ones are not (Fig. 4a).

To avoid the introduction of the band artifacts, a local equalization

strategy inspired by [14] has been adopted. In particular, after each volume

update trough (3) and for each slice of the volume, the margins of the

projection cone are individuated and the difference (k) of the absorption

coefficients between the last voxel inside the cone and the first one outside is

computed. The voxels outside the projection cone are then updated as:

xijk xijk + kdijk/Dk) (5),


where xijk is the voxel in position (i, j, k), Dk is the distance of the

projection cone margin from the border of the reconstructed volume for the

k-th slice, dijk is the distance of the voxel (i, j, k) from the margin of the

projection cone in the same slice. In practice, a value k is added to the

absorption coefficients of the voxels outside the projection cone, such that

the discontinuity at the cone margin is canceled; furthermore, the value of k

is modulated such that the local equalization has the larger effect close to the

margin of the projection cone, but it does not modify the voxels far from it

(Fig. 4b). Comparison of Fig. 3a and Fig. 3b shows that the proposed

algorithm is effective in removing the band artifacts.

Figure 3. In panel (a), five slices of the anterior molar district of an antropomethric phantom

are shown (SART, 5 iterations). The distance between two slices is 12mm. Panels (b) shows

the same slices when the band artifact suppression algorithm is adopted. In panel (c) the

truncation artifact algorithms is also adopted: contrast and resolution are enhanced.


Figure 4. In panel (a), the introduction of the band artifacts is illustrated. Panel (b) shows the

correction strategy.

Figure 5. A short and a long ray crossing the reconstructed volume are highlighted

To explain how truncation artifacts are introduced in local tomography,

we must consider that, out of the reconstructed volume, other structures can

absorb the X-ray radiation; as a consequence, the absorption coefficients of

the voxels are overestimated. However, the overestimate is not uniform

inside the volume, but it is enhanced by the weighting system introduced by

D for the rays that only partially cross the reconstructed volume (Fig. 5). As

a matter of fact, short rays are associated with high weights (Eq. (4)), their

update is consequently large (Eq. (3)) and their convergence towards a

(biased) solution is fast. In practice, the absorption coefficients of the voxels

crossed by short rays are overestimated, and truncation artifacts like those in

Fig. 3a-b appear.

In these cases, the proposed solutions are based on modeling the

absorption of the external structures [12-14], taking into account with

lumped values the true absorption along the rays. If this cannot be done,

more rough solutions are required.

In particular, to avoid the introduction of truncation artifacts, and starting

from the observation above, we have chosen to modify the weighting system

by modifying Eq. (3) as:


xk+1

= xk – W

T][D

-1] ([W]x

k – [p]) (6).

This guarantees that the shortest rays are associated to small weights and

therefore do not significantly bias the reconstructed volume. Fig. 3c shows

that, using (6) instead of (3), the truncation artifacts are strongly reduced; as

a consequence, the grey level dynamics is less compressed and the overall

contrast of the reconstructed slice is also improved.

3. RESULTS

The proposed method for local volume reconstruction was widely tested

on anthropometric phantoms and human patients.

In Fig. 3, the reconstruction of an anthropometric phantom shows that the

quality of the reconstruction is similar to that obtained by other methods on

the market [4], but it is faster as the total reconstruction time is 118s (for a

Mobile DELL Precision M4400, Intel Core Duo @3.06GHz, RAM 3.5G).

Furthermore, it has to be noticed that 24s are sufficient to complete the first

of the five iterations performed by SART; already after the first iteration, the

partially reconstructed volume can be shown to the doctor for its evaluation.

Most of the computational time (>70%) is spent for forward and back

projection of the data. Suppression of the band and truncation artifacts does

not appreciably affect the reconstruction tome, as it requires less than 3s.

Figure 6. In panel (a), 3 slices from a volume without band and truncation artifacts

suppression are shown. Panel (b) shows the same slices after artifact suppression. The arrows

indicate the nerve channel; the insertion of a new implant is also simulated. In both cases,

unsharp masking (7x7 mask, gain 3) and gamma correction (gamma value 0.8) were applied.

Fig. 6 shows 3 slices of a volume reconstructed from a real patient.

Because of imprecise patient positioning, the jaw is partially hidden by the

band artifacts in the first slice of Fig. 6a. When the artifact suppression

strategies are adopted, the entire jaw profile becomes visible. This allows


accurate measurements, for instance of the bone thickness or of the position

of the nerve cannel, aimed for instance at planning the installation of a new

implant, as shown in Fig. 6b.

4. DISCUSSION AND CONCLUSION

Using the mechanical capabilities and the small area sensor of an

ortopantomograph, we have created a method for local tomography, that can

be applied in maxillo-facial radiology and in particular, in implantology.

With respect to CT, two advantages are introduced: first, a limited dose is

delivered to the patient; this is particularly important as the implant has often

a limited extension, and a small part of the dental arch has to be analyzed to

plan its installation. Another advantage is that reconstruction is performed

using a traditional ortopantomograph, which is widely diffused, thus making

possible a tomographic examination without upgrading the hardware.

The limited size of the ortopantomograph sensor forces to acquire the set

of projections in TDI mode; a minute is necessary to complete the

acquisition, during which the patient should be still to avoid the introduction

of movement artifacts. This could appear as a strong limitation of the

proposed method, but one has to consider that even a panoramic radiography

requires about 20s; moreover, the patient has to bite a blocking device solid

with the machine to avoid undesired movements. Finally, the experimental

results on real patients demonstrate that the patients can comfortably remain

still, without introducing significant artifacts.

Band artifacts represent a more critical aspect, which can dramatically

reduce the quality of the reconstructed volume (Fig. 3, 6). This kind of

artifact has received some attention in the field of breast tomosynthesis,

where the reconstructed tissue is generally homogeneous [12-14]; in this

case, the local equalization strategy is aimed at equalizing the mean

absorption coefficient inside and outside the projection cone. The same

approach cannot be used in the dental field, where air, soft tissue and bone

often appear close each other in the same slice. With respect to [14], the

algorithm proposed in our paper operates at a smaller scale: it perfectly

equalizes the absorption coefficients of the voxels inside and outside the

projection cone at the margins of the cone itself, but it also decreases the

effect of the equalization as the distance from the margin of the cone

increases. This approach produces experimentally better result than those

produced by the approach proposed in [14].

Correction of the truncation artifacts [12-14] was obtained though a

careful analysis of the SART updating formula (Eq. (3)). This provides

reliable results when the object of interest is entirely contained in the


reconstructed volume. In this case, short rays cross a small number of voxels

producing reliable measurement data, to which a high weight can be

assigned; moreover, in this case short rays are also associated with short

paths along the patient volume therefore the measured absorption is low. On

the other hand, in case of local tomography, short rays suffers from over-

estimation of the absorption coefficients; moreover, because of the higher

weight, their update is large if compared with the update along the other X-

rays, and convergence is faster; in this case, assigning a higher weight to

these rays produces biased solution, as shown in Fig. 3a-b and 6a. Modifying

the weighting system of the SART algorithms allows this phenomenon to be

limited, producing the reliable results illustrated in Fig. 3c and 6.

The formulation given in (3) and (4) for the updating formula of the

ART, SART and SIRT algorithms allows to interpret these methods as

different versions of the ordered subset scheme applied to the steepest

descent method for the minimization of (4). In particular, the gradient

[WT][D]([W]x

k–[p]) can be computed quickly, but in an approximate

manner, considering only a subset of the equations. The smaller the number

of equations considered, the most speed-up is achieved; however, in this case

the update is not guaranteed to be a descendent direction for (4); as a result,

the reconstructed volume suffers from an excess of noise; this is the case of

the ART method. On the other hand, the updates produced by SIRT are

necessarily descent directions for (4); the method produces a smooth

reconstruction of the volume, but the updates are short and convergence

requires a high number of iterations, incompatible with the clinical practice.

The subset dimension of SART is large enough to guarantee that the

approximation of the gradient in (4) is reliable and, at the same time, the

convergence of the algorithm is fast.

Overall, the proposed method for local tomographic analysis has

demonstrated reliable in producing volumes of adequate resolution and low

noise level, that can be used for planning and following-up of operations in

maxillofacial radiology. Accurate measurements of the local anatomical

characteristics are obtained delivering a small radiation dose into the patient

tissue and at a limited cost.

In the future, speed-up of the reconstruction process through an

unmatched projector / backprojector pair [23] and multithreading will be

investigated. Moreover, the advantage of incorporating Bayesian filtering

[14-15] will also be assessed.

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Optimized algebraic local tomogragphy

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