Siddon algorithm developments

Siddon [1985] was the first to provide an exact, efficient and reliable algorithm for calculating the radiological path through a 3D CT array. The developed algorithm is able to calculate the intersection lengths by following the path of each ray through a volume, using a paremetric representation and considering the intersection points with the grid edges rather than the individual pixels (Figure 2.13). This is named ray-tracing. Siddon’s algorithm is summarised in six action steps described in Figure 2.14. A more detailed description can be found in Siddon [1985].

Figure 2.13: The pixels of the CT array (left) are consdered as the intersection areas of an orthogonal grid. The intersections of the ray with the grid edges are

calculated, rather than the individual pixels. Reproduced from Siddon [1985].

Figure 2.14: The flow of Siddon’s radiological path algorithm. For the typical problems he studies in 1985, the relative amount of computation time required in each part of the algorithm is given by the respective percentages to the right of each

part description. Adapted from Siddon [1985].

The main disadvantage of this approach is that the calculation of the indices of the intersected pixels is very complex and time-consuming. In addressing these, notable improvements were published by Jacobs et al. and Christiaens et al. in 1998 which significantly decreases the required computational time, reaching a speedup of a factor of 5.0 and 13.0 respectively for the total re-construction time. The main advancement in both methods is that the new algorithms restrict the calculation of the intersected pixel coordinates to once per ray instead of once per pixel. It also surpasses the need to allocate memory for the different arrays.

2.6.2 Rigid registration

Image registration is a challenging image processing problem in many applica-tions which is required when comparison or integration of multimodal imaging such as images resulting from different times, devices, and/or perspectives is needed [Haber and Modersitzki, 2006]. Imperative for reliable results in im-age registration is the incorporation of prior knowledge [Modersitzki, 2007].

One refers to rigid registration when the imaged object is rigid itself. Exten-sive overviews of image registration techniques can be found in Maintz and Viergever [1998], Brown [1992], Maurer and Fitzpatrick [1993], Van den Elsen et al. [1993].

In simple terms image registration is the process of estimating a mapping between a pair of images. One image is assumed to remain stationary (the reference image), whereas the other (the source or template image) is spatially transformed to match it [Ashburner and Friston, 2004]. The registration task

is illustrated by an example described in Fischer and Modersitzki [2004], shown in Figure 2.15. Starting with the reference R (Figure 2.15:a) and template T (Figure 2.15:b) images, the aim is to find a reasonable spatial transformation such that the resulting transformed template T’ matches R, or is similar to R subject to a suitable distance measure. In this example, Figure 2.15:c is the result of a linear matching strategy which reduces its difference to R by

∼ 35%. In Figure 2.15:d, non-linear matching is applied which further reduces the difference by another 30%.

There are various methods which spatial transformation is based on for a particular image, such as the use of markers, landmarks, distinct greyvalues, anatomical features, etc. These are used as constraints in the spatial trans-formation.

Figure 2.15: Registration results for X-rays of a human hand depicting (a):

reference R, (b): template T, (c): transformed template T’ after (affine) linear registration which reduces its difference to R by ∼35%, and (d): transformed template T’ after non-linear (curvature) registration which further reduces the

difference by another 30%. Adapted from Amit [1994].

2.6.3 ‘Shift-and-add’ reconstruction

Tomosynthesis (section 2.5) is a method of generating slice images from a finite set of projection images. There are various different reconstruction methods used in tomosynthesis, the simplest and most common being the ‘shift-and-add’ (SAA) technique. SAA involves calculation of the differential shifts of the images which are shifted back to a common centre and added together. It can be used for a number of difference tomosynthesis system set-ups.

The SAA method implements the fact that objects at different height distances z above the detector will experience different degrees of parallax (apparent dis-placement) as the x-ray tube moves. Hence, the objects will be projected onto the detector at respective positions, relative to z.

A basic representation of the principles of SAA are given in Figure 2.16. This figure shows the three locations of an x-ray tube around an object containing a circular and a triangular object in two adjacent planes. If the x-ray tube and detector are moving synchronously in parallel planes, as is the case in the depicted example, the objects’ magnification is purely dependent on z and not on the locations of the x-ray system components. Then, by shifting and adding the acquired image, the structures in certain planes (regions or objects of inter-est) are made to line up and thus come in focus. Structures in other planes are smeared out as they are distributed over the image and appear blurred.This method requires prior knowledge of the dimensions of the imaged volume and the respective rotation angle at which each projection was acquired. By im-plementing the SAA image reconstruction method in tomosynthesis, full 3D images of the screened volume can be created.

Figure 2.16: Principles of tomosynthesis image acquisition and SAA reconstruction. (a) Tomosynthesis imaging. The circle in Plane A and triangle in Plane B are projected in different locations when imaged from different angles. (b) The acquired projection images are appropriately shifted and added to bring either

the circle or triangle into focus; structures outside the plane of focus are subsequently spread across the image (i.e. blurred). Reproduced from Dobbins III

and Godfrey [2003].