Centreline based method were the first applied to prone to supine CT colonography (CTC) registration [Acar et al., 2001, Nain et al., 2002, Li et al., 2004, de Vries et al., 2006, Wang et al., 2009]. Here, linear shrinking and stretching operations based on relative centreline path geometries are used to find corresponding path positions.
Coupling of extracted centreline features, such as local extrema [Acar et al., 2001, Li et al., 2004] in prone and supine acquisitions have been used. Furthermore, extracted features of the endoluminal surface such as average radial distance, circumference and surface curvatures [Nain et al., 2002] have been used to match each pair of points on the centreline. Salient anatomical points such as the flexures and junctions tend to be identifiable between patient positions and can be used to better align the cen- treline paths [Wang et al., 2009]. Polyp matching can be performed by comparing registered centreline positions, and with some methods [Li et al., 2004], polyp registration may be performed using further
information about the polyp geometry and centreline offset.
These methods can only provide correspondence information in a single dimension corresponding to the colon centreline path. They do not account for the twisting of the colon around the centreline are therefore inherently limited in their potential registration accuracy. Areas of clinical interest, such as polyps, are found on the endoluminal surface and so the correspondence of these locations cannot be found directly using these methods.
2.1.1.1
Extraction of the Centreline
Figure 2.1: A centreline extraction algorithm [Nain et al., 2002] using a steady-state distribution of temperature across an extracted isosurface. Each level set of this function consists of a loop around the colon surface, shown here as individual segments of colour (left). The centre of mass of each segment can be used to fit points along the centreline (right).
A centreline is defined by Blum et al. as the locus of centres of sphere contained in the shape [Blum, 1967]. Desired properties of a centreline have been defined by Wan et al. [Wan et al., 2002]:
• Connectivity - requires the centreline to be a sequence of directly connected voxels. • Centricity - the centreline should be at maximal distance from the colon wall. • Singularity - the centreline should be a single path.
• Detectability - centreline extraction should detect and tolerate branching or looping due to topo- logical changes in colon dataset.
• Robustness - algorithms should perform consistently regardless of defined start and end point. • Automation - end points should be determined automatically.
The segmentation and topological thinning of the colon has been established as an accurate method for centreline extraction of the colon [Ge et al., 1999, Paik et al., 1998, Sadleir and Whelan, 2005]. Topological thinning of the binary colon volume however, usually results in more than a single path from caecum to rectum, producing a graph structure. Therefore a subsequent graph search algorithm must be performed to remove any loops and branched created by the thinning.
Other centreline extraction algorithms are based on level set methods such as ‘fast marching’ [Sethian, 1996, Cohen and Kimmel, 1997]. Here an image-based measure is built, defining a speed function with which a wave front is propagated through the image, using the Eikonal equation (which physically models wave-light propagation). These techniques have been extended [Deschamps and Co- hen, 2001] to give an approach that is relevant in 3D, and adapted to the problem of tubular anatomical structure extraction; as well as improvements made to the front propagation technique and reduction of user interaction.
Nain et al. [2002] create a triangulated surface model of the colon with a ‘marching cubes’ [Lorensen and Cline, 1987] isosurface extraction algorithm. Centreline extraction is performed based on a physical model. Colon start and end points are held at a constant temperature of 0 and 1. A steady- state distribution of temperature across the surface is used to give a smooth temperature distribution from end to end. Each level set of this function consists of a loop around the colon surface, the centre of mass of which can be used to fit points along the centreline (figure 2.1).
Other methods are based on graph theory. Wan et al. [2001] use a distance from boundary (DFB) field, containing the Euclidean distance from each voxel inside the volume to the nearest boundary point, to create a directed weighted graph. This weighted graph is then used to build a minimum-cost span- ning tree using a dynamic programming, shortest path algorithm [Dijkstra, 1959]. A fast heap-sorting technique is used to detect the node with the minimum DFB cost. [Jiang and Gu, 2005] have improved on these methods by employing a boundary voxels cutting (BVC) method to the Dijkstra shortest path calculation. This speeds up the algorithm by removing colon boundary voxels which contribute nothing to the centreline.
Subvoxel accuracy is achieved in [Van Uitert and Bitter, 2007]; rather than using distance fields based on a binary mask, a subvoxel precise Euclidean distance field is used based on the level set time crossing of the object. Instead of computing the centreline on the voxel grid and then applying a smooth- ing technique, a smooth centreline at subvoxel accuracy is directly extracted from the subvoxel precise distance field. This work has been extended [Van Uitert and Summers, 2007] to automatically and accurately determine the centreline when the colon is over- or under- distended due to insufflation. Pre- viously, this would give erroneous centrelines, with the insufficiently distended colon resulting in the centreline not being fully extended throughout the entire colon, and the over-distended colon resulting in the centreline crossing through the colon wall.