**2. Performance of tomosynthesis**

### **2.1 General tomosynthesis reconstruction methods**

Existing tomosynthesis algorithms can be divided into three categories: (1) backprojection algorithms, (2) filtered backprojection (FBP) algorithms, and (3) iterative algorithms. The Fig.1 view shown here highlights the difficulty in visualizing three-dimensional (3D) information in X-ray radiography. In the Fig.2 view it turns an acquisition direction to avoid superimposition of an object. The three resulting projection images may be shifted and added (SAA) so as to bring either the circles or triangles to coincide (i.e., focus), with the complementary object smeared out. The basis for FBP is the backprojection of data acquired in projections acquired over all angles. This procedure is performed for each pixel in a projection, and for all possible angles of the projected data, then one has created a simple backprojection image of the object (Fig.3). The backprojection algorithm is referred to as "SAA", whereby projection images taken at different angles are electronically SAA to generate an image plane focused at a certain depth below the surface. The projection shift is adjusted so that the visibility of features in the selected plane is enhanced while that in other planes is blurred. Using a digital detector, image planes at all depths can be retrospectively

X-Ray Digital Linear Tomosynthesis Imaging of Arthoroplasty 97

object in Fourier space is related to the Radon transform of the object. The relationship between the Radon transform and cone-beam projections has been well studied and solutions to the cone-beam reconstruction have been provided (Smith 1985).The FBP algorithm generally provides a high degree of precision for 3D reconstruction images when an exact type algorithm is employed (Feldkamp et al 1984) (see Appendix). Therefore, this method has been adopted for image reconstruction of 3D tomography and multi-detector

An iterative algorithm performs the reconstruction in a recursive fashion (Ruttimann et al 1984 , Bleuet et al 2002), unlike the one-step operation in backprojection and FBP algorithms. During iterative reconstruction, a 3D object model is repeatedly updated until the model converges to the solution that optimizes an objective function. The objective function defines the criteria of the reconstruction solution. The objective function in the maximum likelihood (ML) algorithm is the likelihood function, which is the probability of getting the measured projections in a given object model. The solution of the ML algorithm is an object model that

Anteroposterior (AP) radiograph of these joints demonstrate the excellent visualization of the prostheses in this view. AP radiograph is difficult to visualize 3D information in an AP radiograph as shown (Fig.4). An alternative approach to tomosynthesis imaging is to determine the number of views that can be acquired given imaging constraints (e.g., time restrictions from patient motion, dose restrictions of the detector). The tomographic angle can be selected to yield images with an acceptable level of artifacts. This is certainly the case with FBP algorithm and is suspected to be the case with simple backprojection (or the SAA algorithm). It is possible, however, that this restriction could be reduced by the use of an

Fig. 3. Overview of the filtered backprojection (FBP) algorithm.

maximizes the probability of getting the measured projections.

**2.2 Radiography vs. tomosynthesis** 

alternative reconstruction scheme.

cone-beam CT.

reconstructed from one set of projections. The SAA algorithm is valid only if the motion of the X-ray focal spot is parallel to the detector.

Fig. 1. Overview of the conventional radiography method.

Fig. 2. Overview of the shifted and added (SAA) algorithm.

FBP algorithms are widely used in CT in which many projections acquired at greater than 360 degrees are used to reconstruct cross-sectional images. The number of projections typically ranges between a few hundred to about one thousand. The Fourier central slice theorem is fundamental to the FBP theory. In two-dimensional (2D) CT imaging, a projection of an object corresponds to sampling the object along the direction perpendicular to the X-ray beam in the Fourier space (Kak et al 1988). For many projections, information about the object is well sampled and the object can be restored by combining the information from all projections. In 3D cone-beam imaging, the information about the

reconstructed from one set of projections. The SAA algorithm is valid only if the motion of

the X-ray focal spot is parallel to the detector.

Fig. 1. Overview of the conventional radiography method.

Fig. 2. Overview of the shifted and added (SAA) algorithm.

FBP algorithms are widely used in CT in which many projections acquired at greater than 360 degrees are used to reconstruct cross-sectional images. The number of projections typically ranges between a few hundred to about one thousand. The Fourier central slice theorem is fundamental to the FBP theory. In two-dimensional (2D) CT imaging, a projection of an object corresponds to sampling the object along the direction perpendicular to the X-ray beam in the Fourier space (Kak et al 1988). For many projections, information about the object is well sampled and the object can be restored by combining the information from all projections. In 3D cone-beam imaging, the information about the

Fig. 3. Overview of the filtered backprojection (FBP) algorithm.

object in Fourier space is related to the Radon transform of the object. The relationship between the Radon transform and cone-beam projections has been well studied and solutions to the cone-beam reconstruction have been provided (Smith 1985).The FBP algorithm generally provides a high degree of precision for 3D reconstruction images when an exact type algorithm is employed (Feldkamp et al 1984) (see Appendix). Therefore, this method has been adopted for image reconstruction of 3D tomography and multi-detector cone-beam CT.

An iterative algorithm performs the reconstruction in a recursive fashion (Ruttimann et al 1984 , Bleuet et al 2002), unlike the one-step operation in backprojection and FBP algorithms. During iterative reconstruction, a 3D object model is repeatedly updated until the model converges to the solution that optimizes an objective function. The objective function defines the criteria of the reconstruction solution. The objective function in the maximum likelihood (ML) algorithm is the likelihood function, which is the probability of getting the measured projections in a given object model. The solution of the ML algorithm is an object model that maximizes the probability of getting the measured projections.
