Fig. 11.8 A skull surface extracted from a CT image showing typical staircase artifacts. These can be reduced by smoothing the surface or by blurring the 3D image before applying marching cubes. This latter approach is often better

(the corners of a cube). This could provide 256 potential cases, but by adjusting for rotational and other symmetries these can be reduced to 15 cases. Ambiguities in 3D, if not dealt with correctly, can lead to holes in the resulting surface. There are a number of possible solutions to this, for example using tetrahedra rather than cubes.

Volumetric medical images usually do not have resolutions of comparable quality to what is achieved with range data. Partly as a consequence of this, surface extraction algorithms such as marching cubes [47] may produce surfaces with typical ‘staircase’ effects. Further processing may smooth the surface or decimate it. It is often better to smooth the 3D volume before running marching cubes as this reduces the noise in the dataset. As mentioned above, the triangulation can contain singularities (zero angled triangles), holes, duplicate triangles, folds and so on, if the implementation is not robust.

Example: Curvatures and Geometric Tools Once the surface has been extracted, different operations are possible for medical imaging applications. We will illustrate briefly with the example of curvatures. Curvatures are an intuitive quantity but, since they involve second derivatives, they tend to be very sensitive to surface irregularities [6, 7] They can be used as a quantification of the surface or as a tool to evolve the surface [5]. Figure 11.9 shows some illustrative examples.

11.3.2 Volume Rendering

It is also possible to produce a rendering of the anatomy from a model without extracting a surface first. This method is known as volume rendering. The idea originated in computer graphics before its application to medical imaging, but this method is ideally suited to rendering of volumetric data such as CT or MRI scans. A volume rendering of a CT image is shown in Fig. 11.10. The idea is very similar to

Fig. 11.9 (a) Colon image colored by curvature. The colon is deformed by a curvature (‘Ricci’) flow into a cylinder, from which comparisons can be made [69]. (b) Mean curvature of the cortical surface. These are used to quantify the folding of the cortex, as a biomarker of brain shape [6]

raytracing, in that a ray is cast from a given 2D image pixel through the volume. The process aims to simulate the passage of light through a semi-transparent medium. If the ray has a current color Crn, as it passes through a voxel v with color Cv and opacity αv , the color of the ray is changed using the over operator:

The scalar values in the image need to be mapped to color and opacity by the transfer function, which determines the color and opacity of a voxel depending on the greyscale values in the image (i.e. Cv (I ) and αv (I ) are functions of the image intensity, I ). There are some simple transfer functions, such as the maximum intensity projection, which just calculates the maximum value along the ray. This has proven useful in viewing vessels in angiography images.

It is also possible to composite the rays back to front rather than front to back, in which case the slightly different under operator is used:

volume rendering and many tricks to speed up the process. Even on modern graphics cards and CPUs, however, the interactivity of volume rendering is not the same as surface rendering. A common option is to display the volume rendered scene at a lower resolution during interaction and display the full image only when interaction stops. This will become less of a problem as graphics performance increases.

In general, to achieve a good result, it is necessary to roughly segment the organ of interest first before performing surface or volume rendering. In the next section

Fig. 11.10 A volume rendering of the CT image from Fig. 11.8. There is smoothing of the skull surface, and we are able to view this through the transparent skin

we will look at the problem of segmentation. A further issue is how to incorporate a lighting model into volume rendering. In the volumetric lighting approach, a normal is calculated at each voxel from finite difference approximation of the scalar gradient:

The normal is calculated as n = | II | and the interaction of the light with each voxel is calculated during composition.

11.3.3 Summary

This section illustrated how some of the techniques described in this book can be applied to medical imaging data. Bear in mind the message from the previous section about limitations of the original images. If there is one message to take from surfaces from medical images, it is that they can be rather low quality, with discretization (‘staircase’ artifacts), holes, missing parts, foldovers and so on. On the other hand, the number of triangles in a triangulated surface can be huge and feel unnecessary with regards to graphics requirements, but one has to be very careful when smoothing or performing other manipulations. In medical imaging, the clinician may not be interested in final image quality, but might want to focus on small imperfections or other nearly indistinguishable features that could be indicative of disease. Medical diagnosis is inherently different from computer graphics in this respect. Volume rendering has the nice property that the voxel data is rendered without