A review of geometric calibration for different 3-D X-ray imaging systems

LOW ENERGY ACCELERATOR, RAY AND APPLICATIONS

A review of geometric calibration for different 3-D X-ray imaging systems

Yao Yang，

Liang Li，

Zhi-Qiang Chen

Nuclear Science and Techniques

Vol.27, No.3

Article number 76

Published in print 20 Jun 2016

Available online 18 May 2016

DOI：10.1007/s41365-016-0073-y

64902

A precise knowledge of geometry is always pivotal to a 3-D X-ray imaging system, such as computed tomography (CT), digital X-ray tomosynthesis, and computed laminography (CL). To get an accurate and reliable reconstruction image, exact knowledge of geometry is indispensable. Nowadays, geometric calibration has become a necessary step after completing CT system installation. Various geometric calibration methods have been reported with the fast development of 3-D X-ray imaging techniques. In these methods, different measuring methods, calibration phantoms or markers, and calculation algorithms were involved with their respective advantages and disadvantages. This paper reviews the history and current state of geometric calibration methods for different 3-D X-ray imaging systems. Various calibration algorithms are presented and summarized, followed by our discussion and outlook.

3-D X-ray imaging systemComputed tomographyGeometric calibrationReconstruction

1 Introduction

The importance of geometric calibration cannot be underestimated. In a typical 3-D X-ray imaging system, computed tomography (CT) for example, the X-ray tube assumes different discrete positions along a trajectory. For each source position, a projection image (a radiograph) is acquired [1]. Reconstruction of the imaged object is accomplished by the imaging system. In the last few decades, a large number of reconstruction algorithms have been published devote to improving the image quality, which includes some famous algorithms like FDK, Grangeat’s algorithm, Katsevich’s algorithm, back-projection filtration (BPF) algorithm, etc. However, the image quality promotions are limited by the lack of a precise knowledge of the system geometry. 3-D X-ray system geometry refers to the relative positions of the x-ray focal spots and the location of the digital detector. A precise calibration of the system geometry can overcome problems, such as the blurring effect and artifacts, and becomes the most important factor for a better reconstructed image.

The development of geometric calibration was a substantial improvement to computer imaging. We can easily obtain a high image quality and a high level of detail resolution of a reconstructed structure in the imaged object with calibrated system geometry. Various calibration methods have been employed in medical or industrial imaging, which have already achieved good results in practical applications. This paper will summarize the current state of geometry calibration in different 3-D X-ray imaging systems. Also, a brief history of geometric calibration will be given, followed by a description of various methods. As these methods are suitable for different 3-D X-ray imaging systems, we will introduce them with a classification according to the systems where they are employed.

2 Background

In order to get a relatively accurate system geometry, various calibration methods have been reported. Due to some technical limitations, most of them were carried out within the past two decades.

Early methods determined the system geometry by physically measuring the spatial location of the different components. Although these methods can get a relatively accurate geometry, they require a number of precise measurements and are typically complex. After early methods, researchers proposed a series of mathematical methods which employ some point-like markers and calibrate the system geometry with the markers’ trajectory on the detector plane. These methods can calibrate the system geometry with just several simple materials. However, they cannot calibrate the entire geometrical parameters and they are only suitable to some general scanning trajectory, such as cone-beam CT or spiral-CT. Besides, some calibration methods were also developed for the imaging system, which has a specific scanning trajectory, such as tomosynthesis, computed laminography and limited-angle CT [2-4]. While these systems may not have a conventional scanning mode, some markers and their locations in the projected image were used to determine the relative positions between the components of the system. Last but not least, methods with a well-designed phantom also existed. Although these methods can acquire an accurate geometry, the phantoms used were always very expensive and were not convenient for practical applications. Nowadays, more and more practical methods have been introduced for geometric calibration. These methods may determine the system geometry with only some simple point-like markers located at any position within the imaged space and can obtain relatively accurate system geometry when dealing with practical applications.

In conclusion, we cannot get a reconstructed image of a high quality without an exact knowledge of the system geometry. These existing methods have achieved great success, however, geometry calibration leave us much room for future improvement.

3 Geometric calibration for different 3-D X-ray imaging systems

The existing methods are suitable for different 3-D X-ray imaging systems, which have a specific scanning trajectory. We will describe the typical methods according to the system where they are employed.

3.1 Calibration for CT

Computed tomography was first introduced in 1971. Since then, it has been widely used in clinical applications. With the development of flat-plane detector and computer hardware, it is possible to get a high resolution reconstructed image. In a typical CT system, the image resolution depends on many factors, such as the resolution of the detector, the focal spot size of the X-ray source, the reconstruction method, and the accuracy of the system geometry. In a practical system, the geometry parameters are always inaccurate. This inaccuracy severely impacts the quality of the reconstructed image and makes geometric calibration very necessary [5].

In a typical CT system, such as circular orbit CT or spiral CT, some common parameters should be taken into account [6-8], as shown in Fig.1:

Fig. 1.

Geometry parameters of a typical 3-D x-ray imaging system.(a) Definition of the parameters

D, R, (u_{0}, v_{0})

.(b) Definition of the parameters

η, σ, φ

$D$ ：The distance between the X-ray source and the detector

$R$ ：The distance between the X-ray source and the rotation center

$u_{0}$ : The horizontal coordinate of the detector center

$v_{0}$ : The horizontal coordinate of the detector center

$φ$ : The oblique angle of the detector in the horizontal direction

$σ$ : The pitch angle of the detector

$η$ : The oblique angle of the detector in the detector plane

$Δ β$ : The angle sampling interval

In order to calculate all the parameters above, some methods, which use well-designed phantoms, were proposed [9-11]. For example, in Cho’s method, a specially designed phantom which consists of 24 steel ball bearings in a known structure, is used. 12 ball bearings are spaced evenly at 30 degree in two plane-parallel circles separated by a given distance along the tube axis. Using the ellipse trajectory of the ball bearings in the detector, he can calibrate all the geometric parameters of a CT system. Although this method can calculate all the parameters, it needs a specially designed phantom, which may be very expensive. Besides, the design deviation will influence the accuracy of the calibration, which makes the method hard to be employed in practical applications.

Most mathematical methods do not calculate all the parameters. They use some point-like markers or feature points without specific arrangement in the space. The only prior knowledge may be the distance between each two markers. We will introduce Yang’s method as an example in the following section [12].

Yang’s method is based on a number of feature points and can estimate the following five system parameters, $D$ , $R$ , $u_{0}$ , $v_{0}$ , and $η$ . For a cone beam CT system, if we assume that only the object rotates and the X-ray tube and detector remain stationary, the orbit of a point in the object during the scan is a circle in a plane parallel to x-y plane. The projection of this circle on the detector plane will be an ellipse. Individual points on this ellipse correspond to the marker’s angular positions on the circle. Yang referred to two points on the circular orbit that are exactly 180 degree out of phase as a radial pair, as shown in Fig. 2(a). Thus, the distance, $ρ$ , between a radial pair of points on the detector plane can be calculated. It can be proved that $ρ$ will have the maximum and minimum values when the feature point is on the x axis or y axis for a cone beam CT system with a fan angle less than 60 degree and a cone angle less than 30 degree, as shown in Fig. 2(b). We can define these feature points as $A_{i k} (u_{i k}, v_{i k})$ , here $i$ refers to the index number of each individual marker, and $j$ is the index number for the four benchmark points on each marker orbit.

Fig.2.

Calibration geometry of Yang’s method. (a) Projection orbit and radial pair. (b) Four benchmark points on each marker orbit [12].

Thus, we can calculate the five parameters. For each individual marker, defined

X_{i} = \frac{v_{i 1} - v_{i 2}}{| A_{i 3} A_{i 4} |} Y_{i} = \frac{v_{i 1} + v_{i 2}}{2} .

(1)

A linear function $X = a + b Y$ can fit all $(Y_{i}, X_{i})$ , and we can determine $D$ and $v_{0}$ ,

D = b v_{0} = a .

(2)

To calculate $η$ and $u_{0}$ , when we define the intersection of $A_{i 1} A_{i 2}$ and $A_{i 3} A_{i 4}$ as $(u_{i 0}, v_{i 0})$ , we have

(3)

We found that all $(u_{i 0}, v_{i 0})$ can fit the linear function $u_{0} = a + b v_{0}$ . After we calculate a and b, we have

η = \tan^{- 1} b .

(4)

To calculate $R$ , we need the distance $l$ between two markers. Define $r_{1}$ and $r_{2}$ the two radii of each marker’s orbit, as shown in Fig. 3.

Fig. 3.

Distance between two markers in Yang’s method

We have

l^{2} = h^{2} + r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} \cos α .

(5)

Here

h = \frac{| A_{10} A_{20} |}{d} R, r_{1} = \frac{| A_{13} A_{14} |}{2 d} R, r_{2} = \frac{| A_{23} A_{24} |}{2 d} R .

(6)

If we define the original angular position of the two markers as $α_{10}$ and $α_{20}$ , then

\begin{array}{l} α_{1} = α_{10} - α_{14} α_{2} = α_{20} - α_{24} . \\ α = α_{1} - α_{2} \end{array}

(7)

Then, we can calculate R

R = \frac{l \cdot d}{\sqrt{{| A_{10} A_{20} |}^{2} + {(\frac{| A_{13} A_{14} |}{2})}^{2} + {(\frac{| A_{23} A_{24} |}{2})}^{2} - \frac{| A_{13} A_{14} | \cdot | A_{23} A_{24} | \cdot \cos α}{2}}} .

(8)

Then, we have all the five parameters with Yang’s method. There are many similar methods, which also calculate part of the system parameters. For example, Noo proposed his analytical method in 2000 and von Smekal proposed a method based on Fourier transform in 2004 [13-14]. Beque calibrated the geometry using an object with three spheres [15]. The common point of these methods is that they use more than one projection to determine the geometric parameters for each source position. Other methods calculate parameters for each projection separately and independently of all other projections [16-19]. Besides, there are also methods which calibrate the system geometry by estimating a general projection matrix [20]. Due to the space limitations, we’ll not discuss these methods in detail in this paper.

In conclusion, all these methods are designed for a typical CT system with a circle-plus-arc trajectory and may not be suitable for other imaging systems. Each of them has their own limitations and advantages. Some can give an accurate system geometry and calculate all the geometric parameters with well-designed phantoms. However, the phantoms may be very complicated, precise, and expensive, which make these methods only applicable to experimental research. Other methods are more suitable for practical applications due to their low requirement to the calibration phantoms. However, they can only calculate parts of the system parameters. All these methods can help us to improve the quality of the reconstructed image, and we can choose an appropriate method according to the needs of different applications.

3.2 Calibration for digital X-ray tomosynthesis

Tomosynthesis was first introduced at almost the same time as CT. Due to some technical limitations, it was not widely used until the appearance of flat panel detector [21]. Nowadays, digital X-ray tomosynthesis has become more and more popular in practical applications, especially in clinical applications, such as breast tomosynthesis and dental tomosynthesis [22].There are three typical scanning modes in a tomosynthesis system, as shown in Fig. 4 [23]. A completely isocentric motion is shown in Fig. 4(a). Both the x-ray source and detector rotate around a common axis within the object. Figure 4(b) is the most classic tomosynthesis scanning method, a partial isocentric motion. The detector moves in a line or a circle within one plane. And, the X-ray source rotates around a certain rotation center. In some applications, the detector has to be stationary due to the limited space, as shown in Fig. 4(c). Both the object and detector stay still while the X-ray source moves along a trajectory around them, for example, the breast tomosynthesis system.

Fig. 4.

Three typical digital tomosynthesis scanning mode.

In a typical tomosynthesis system, the imaged object is fixed in a certain position and the X-ray source assumes different discrete positions along a trajectory in space. For each source position, a projection radiography image is acquired and then sent to the tomosynthesis system. Accurate image reconstruction also requires a precise knowledge of the system geometry. There have been many methods that calibrate the 3-D X-ray tomosynthesis system geometry. However, the tomosynthesis system includes a different geometry for each acquired image, typically because of the change in the X-ray source location for each acquired image. This means that methods may not acquire a satisfying result [24-26].

Many tomosynthesis geometric calibration methods are based on some well-designed phantoms. For example, Wang and Godfrey proposed similar methods, which employ some point- like markers to estimate possible deviation in tomosynthesis geometry and reduce these deviations to get a better reconstructed image [27-28]. Hui et al. proposed a method using a phantom with ten fiducial markers. With this phantom, the projection matrices of an experimental digital tomosynthesis prototype is acquired from each projection view under a series of misalignment conditions [29]. Although these methods can acquire a relatively accurate system geometry, the phantom used is relatively complex and is not convenient for clinical applications. In 2005, GE (General Electric Company) proposed a method using some fiducial markers with non-determined positions. In GE’s method, the geometry is determined by arbitrarily locating at least two markers within the imaged volume and locating the projections of the markers within at least two images corresponding to different positions of a focal spot of the X-ray source. For every X-ray source position one image acquired must be included in at least two images, as shown in Fig. 5 [30].

Fig. 5.

Principle diagram of GE’s calibration method

$P$ and $Q$ denote two different focal spot positions of the X-ray source, $A$ and $A^{'}$ denote two different fiducial marker positions. $B$ and $B^{'}$ and $C$ and $C^{'}$ denote the respective locations of the projections of the markers $A$ and $A^{'}$ , generated by acquiring a projection image with $P$ and $Q$ . Then, the method calibrates the geometry by first selecting an arbitrary focal spot position, $P$ , and selecting the first marker position, $A$ , as an arbitrary point between $P$ and the corresponding projection, $C$ , located on the line through $P$ and $C$ . Then, the method determines the additional marker, $A^{'}$ , as the intersection of the line through $A$ and $O$ with a line through the focal spot, $P$ , and the corresponding projection, $C$ of $A^{'}$ . Then, we can determine the remaining focal spot position, $Q$ , as the intersection of the family of lines. This method can calibrate tomosynthesis geometry using only some fiducial markers with a non-determined position, which is very suitable for practical applications. However, the stability of the method needs to be tested in more experiments.

Generally speaking, these calibration methods for a tomosynthesis system are more tailored, typically because the system has some specific characteristics. The limitations and advantages of each method are similar to those for a CT system. As the development of existing calibration methods is not very consummate, geometry calibration in tomosynthesis leaves us much room for future improvement.

3.3 Calibration for computed laminography (CL)

Laminography techniques are widely used to produce cross-section images of selected planes within objects. In some cases, it provides a viable alternative to CT. There are always three scanning types for the CL system: linear, planar, and rotational. Only the rotary bearing movement has a simple geometrical structure and the space requirement is not as high as the other modes. Furthermore, the beam angle does not need to be as wide as the conventional laminography [31-32]. We will discuss the rotational scanning mode in the following section. As shown in Fig. 6.

Fig. 6.

Principle diagram of cone beam CL scanning

The scanning geometry consists of three parts, an X-ray source, a two-dimensional digital flat plane detector, and a rotation gantry. For a practical CL scanning system, the projection of the X-ray focuses on the imaging plane, namely point $O$ , which is not known and cannot be measured by direct means. However the CL reconstruction algorithm needs the position of the projection of the X-ray focus, so the calibration of the projection coordinate system is the essential step. Barry Eppler introduced a method based on the empirical data gathered during physical calibration and data analytically derived from the empirical data. Other methods may also use some point like markers to calibrate the system geometry [33]. In 2012, Yang proposed a method using several spherical objects at two geometrical magnification ratio positions in the cone X-ray beam to calibrate the system geometry. We’ll give a brief introduction of this method in the following section [34].

As shown in Fig. 7, we need to determine the projection of the X-ray focus (point O).

Fig. 7.

Principle diagram of calibration method

The projection of the point $p_{1} (x_{1}, y_{1}, z_{1})$ in the cone-beam is point ${p^{'}}_{1}$ . When the point $p_{1}$ moves to $p_{2} (x_{1}, y_{1}, z_{1} + Δ z)$ with the projection of ${p^{'}}_{2}$ on the imaging plane along the z-axis by a step of $Δ z$ ,the equation of the lines $F P 1$ and $F P 2$ are:

\begin{array}{l} F P 1 : \frac{x}{x_{1}} = \frac{y}{y_{1}} = \frac{z - D}{z_{1} - D} . \\ F P 2 : \frac{x}{x_{1}} = \frac{y}{y_{1}} = \frac{z - D}{z_{1} + Δ z - D} \end{array}

(9)

Then, we can derive the coordinates of ${p^{'}}_{1}$ and ${p^{'}}_{2}$ :

\begin{array}{l} {p^{'}}_{1} \to (\frac{- D x_{1}}{z_{1} - D}, \frac{- D y_{1}}{z_{1} - D}) . \\ {p^{'}}_{2} \to (\frac{- D x_{1}}{z_{1} + Δ z - D}, \frac{- D y_{1}}{z_{1} + Δ z - D}) \end{array}

(10)

Then, the equation of line ${p^{'}}_{1}$ ${p^{'}}_{2}$ is

{p^{'}}_{1} {p^{'}}_{2} : y = \frac{y_{1}}{x_{1}} x .

(11)

Obviously, line ${p^{'}}_{1}$ ${p^{'}}_{2}$ passes through $O$ . Thus, for any point in the cone-beam, after moving a distance along the central X-ray, the line connecting its new projection point with its original projection point always passes through the projection point of the X-ray focus. So we can set any two points at position $p_{1}$ in the cone-beam and get their projections, then move them to position $p_{2}$ along the central X-ray and get their new projections. The intersection point of the two lines, which connects the projections of the same point, is the projection of the X-ray focus. Then, we have the calibrated geometry.

As we can see, the geometric calibration for the CL system is different from that of other systems, such as CT and tomosynthesis. However, the essence of the calibration is the same. Thus, we can also learn from these calibration methods and this may help us better design calibration methods for other systems.

3.4 Other calibration methods

All the calibration methods discussed above are suitable to some specific imaging modalities. There are some other methods fit for some special imaging modalities, such as linear CT, saddle-line CT, and tuned-aperture CT (TACT). As these imaging systems are not widely used in practical applications, calibration methods for these systems are not as well developed as those discussed above.

Due to the relatively special scanning trajectory, calibration methods for those systems always employ several simple markers and determine the system geometry with their locations and projection locations [35]. In 1997, a machine named TACT was introduced for clinical dental diagnosis. TACT is a quais-3D X-ray imaging technique that may reconstruct the image slices of the region-of-interest at any depth based on a series of intra-oral radiographs taken from different directions [36]. The principle of TACT is similar to tomosynthesis and its scanning trajectory is always arbitrary. In order to get an accurate positional relationship between the source and detector, we put some steel or ceramic beads behind the irradiated teeth and took several photos in any source location. Then we calibrated the geometry with the projection locations of the spheres. However, it is very inconvenient to put markers into a patient’s mouth and this makes TACT not used clinically nowadays [37-42].

Although these imaging systems are not widely used in practical applications, the calibration methods for them are worth considering, and we can get valuable reference from them when studying a new calibration method.

4 Numerical simulations

In order to explain the effect of geometric calibration, we chose some typical methods, and some numerical simulations were carried out. The results are shown as follows.

Firstly, we did some simulation with Yang’s method in a CT system. The geometry of the system is shown in Fig.8.

Fig.8.

Geometry of the CT system

We set the phantom between the source and detector. The size of the sensor is $128 \times 128 m m^{2}$ with $512 \times 512$ pixels. The distance, $D$ , between source and detector is 300 mm. The distance between the $R$ source and the coordinate origin is 180 mm. The source, coordinate origin, and the center of the detector plane are located in a straight line. Then, we add some random errors to both $D$ and $R$ . Also we give the detector a micro offset. After that, we get 360 projections from 0~360 degrees. And the images are reconstructed using the FDK algorithm with three kinds of system geometry. They are the accurate geometry, the calibrated geometry, and the geometry with random errors. The center slices of different results are shown in Fig.9.

Fig.9.

Center slice of different reconstructed results.

As shown in Fig.9, Fig9(a) is the reconstructed image with accurate system geometry, while Fig9(c) is the one that is reconstructed with calibrated geometry. Compared to Fig(b), which is reconstructed without geometric calibration, we find that Fig9(c) has a higher contrast in details and can be the one that reconstructed the image with accurate geometry. This explains the effects of the geometric calibration methods.

Also, we carried out a simulation using a tomosynthesis system with GE’s method. The simulation geometry is shown in Fig.10.

Fig.10.

Simulation using tomosythesis system

We set the distance of the X-ray focal spot to the center of the sensor at 300 mm. The origin of the coordinate was located at the center of the sensor. The detector remained stationary during IDT scanning. The size of the sensor was $20 \times 20 m m^{2}$ with $1000 \times 1000$ pixels. The phantom we used were consisted of 27 balls distributed in three layers. The size of the phantom was $10 \times 10 \times 10 m m^{3}$ with $512 \times 512 \times 512$ pixels, as shown in Fig.10 (b). The X-ray source moved with angular ranges of $\pm 30$ degrees, and an angular interval of $3$ deg. The images were reconstructed on $512 \times 512 \times 512$ grids. The distance between the trajectory and the detector plane is $300 m m$ . Then, we added some random errors to the 3-D coordinates of each source position. Using these new source positions, we got 21 projections. After that, we reconstructed the image with three kinds of system geometry. The center slices of the different results are shown in Fig.11.

Fig.11.

Center slice of different reconstructed results

As shown in Fig.11, Fig.11(a) is the reconstructed image without geometric calibration, Fig11(b) is reconstructed with calibrated geometry, and Fig.11(c) is the one that was reconstructed with accurate geometry. As the offsets of our original source positions are limited, the differences between the reconstructed images are not so significant. However, we also found that the reconstructed slice with a calibrated geometry is better than that without geometric calibration.

In a conclusion, we can see from the above simulations that geometric calibration can help us get better reconstructed images. We can easily obtain a high image quality and a high level of detail resolution of the reconstructed structure of the imaged object with calibrated system geometry.

5 Discussion and conclusion

A precise knowledge of geometry is always pivotal to a 3-D X-ray imaging system. We can easily obtain a high image quality and a high level of detail resolution of the reconstructed structure in the imaged object with an accurate system geometry. Various calibration methods have been proposed in order to get a relatively reliable geometry. Most of them use some markers and calculate the system geometry with a mathematical method, such as the analytical method, transform domain method, or iterative approach. These methods are easy to operate when dealing with practical applications. However, they are only suitable for some specific imaging modalities and can only calibrate part of the geometric parameters. Other methods may get a better calibrated geometry, as they use a custom phantom designed according to the imaging system. However, these phantoms are always very complex and expensive. Besides, phantoms need a special design and the design deviation will influence the accuracy of the calibration. All these limitations make them hard to be employed in practical applications. Furthermore, calibration methods, which can be applied to any kind of imaging modalities, also exist, but the accuracy and stability remain a problem to be solved.

Although these existing methods have acquired great achievements, there is much room for improvement in geometric calibration. Future work may focus on methods which can apply to different imaging modalities with fewer markers. Furthermore, we may use some iterative methods to optimize the system geometry when doing image reconstruction, thus we can eliminate the calibration step.

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