Experimental results

Figures 1 and 2 show the projections of 20 3D objects that depict the first segment of the colon of 10 subjects that suffer from chronic non-ulcerous colitis and 10 who do not. Each volume image is in size. The side of a voxel corresponds to 1.14 mm in all directions. For each one of these objects the chord-angle histogram was constructed with N =30, M =12 and . These histograms may be thought of as characterisations of the whole surface. For the description of roughness we use only the first four bins in d , corresponding to maximum d value of .

Ideally, we would like to have a very large population of healthy subjects, from which the mean ``healthy'' histogram can be constructed reliably. Then all test histograms should be compared with that and their distance from the ``average normal'' could be used as a diagnostic aid. However, the collection of data from healthy patients is neither easy nor morally acceptable: Clinicians do not easily subject healthy people to unnecessary examinations just to collect data for research. Because of that, the most compact class of images we have is the class ``colitis''. The 10 images which form the non-colitis class are too diverse to form a compact class with a well defined mean. As a result, we shall measure the distance of each histogram, from this mean, and expect to observe a trend showing the distance from class colitis increasing for non-colitis patients. Indeed, such a trend can be observed in the results shown in figure 3, where we plot along the vertical axis the distance of each image from the mean of class colitis and along the horizontal axis the identity of each image counting sequentially from top left to bottom right the images shown in figures 2 and 1.

In figure 4 we plot the energy of each histogram and the value of . It can be seen that characterises the class colitis in a much more compact way than the energy.

The roughness of the surface of the brain is something that characterises the human brain. We present an example here of quantifying the smoothness of a human brain and the brain of a monkey. The values of the two human brain hemispheres are for the left and for the right hemisphere. The corresponding values for the monkey brain are 60.6 and 79.9 respectively. The images used and the histograms computed from them are shown in figure 5. Care was taken in creating these histograms so that the difference in size of the two brains was removed by appropriate scaling.