Boundary model

The methods presented in this thesis make the task of volume rendering surfaces more intuitive and less labor intensive than previous methods. The algorithm is based on the computation of boundary characteristics in a volume and using this information to guide the user towards opacity function settings which readily display the boundaries. Towards that end, a mathematical model has been chosen for what constitutes an ideal boundary. We now describe the boundary model upon which the techniques of this thesis are based.

We assume that at their boundary, objects have a sharp, discontinuous change in the physical property measured by the values in the dataset. In MRI this property might be density of hydrogen nuclei, in CT and EM it is radio-opacity of the tissue. Also, we assume that the data is band-limited prior to sampling, but not with the usual Nyquist criterion. Rather, the measurement process is assumed to have a Gaussian frequency response, which effectively blurs the sharp edges with a Gaussian filter.

Figure 3.1: The ideal physical boundary (a) is convolved with a Gaussian filter (b) representing the frequency response of the measurement process, producing the measured boundary prior to sampling (c) which exhibits the shape of the $\operatorname {erf}()$ function.

Figure 3.1 shows a step function representing an ideal boundary prior to measurement, the Gaussian filter which performs the band-limiting by blurring, and the resulting measured boundary (prior to sampling). The resulting curve (Figure 3.1(c)) happens to be the integral of the Gaussian kernel, which is called the error function, notated `` $\operatorname {erf}()$''[KK68]. Actual measurement devices necessarily band-limit, so they always blur boundaries somewhat. Their frequency response is never exactly a Gaussian function, however, since this requires infinite support, and infinitely high frequencies cannot be measured. Although certain mathematical properties of the Gaussian function are exploited later, the inexact match of real-world sampling to the Gaussian ideal has not been found to severely limit application of the techniques presented here. A final assumption made for the purposes of this analysis is that the blurring is isotropic, that is, uniform in all directions. Again, the algorithms presented in this thesis will often work even if a given dataset doesn't have this characteristic, but results may be improved if it is pre-processed to approximate isotropic blurring. Having previously assumed that objects are made of material which attains the same data value everywhere in their interior, this is effectively equivalent to assuming that the boundaries of objects will be uniform over their entire surface. The consequences of these various assumptions will be evident when histogram volume calculation is discussed in Chapter 4.