Histogram volume creation

Having decided on a structure with which to represent the probabilistic relationship between $f$, $f'$, and $f''$, we now must find a way to measure that relationship as it occurs in a given dataset.

In Chapter 3, we found a position-independent relationship between $f$, $f'$, and $f''$ that characterized an ideal boundary. However, to find that relationship, we afforded ourself the luxury of first knowing where the boundary was in the idealized dataset. For example, when plotting the data value and its derivatives as a function of position in Figure 3.4, we knew exactly where to place a path which crossed through the boundary. In the case of real volume data, we do not know where the boundaries are, and as discussed previously, we should not need to know. Yet we need some way to measure $f$, $f'$, and $f''$ so as to reveal the same relationship.

This thesis has taken the simple approach of measuring $f$, $f'$, and $f''$ at each point of a uniform lattice. Figure 4.2 helps illustrate why this can work.

Figure 4.2: Sampling the boundary: from continuous to discrete. Even when the measurements are taken over a grid which has no particular spatial relationship to the boundary, the relevant curves in the scatterplot are recorded, indicating the presence of a boundary.

In 4.2(a), the thickness of the boundary is sampled continuously to produce smooth graphs of the data value versus derivative relationships, as was done in Figure 3.4. In 4.2(b), the boundary is sampled at a high frequency along the same path. Instead of producing a smooth plot of first or second derivative versus data value, we now have a scatterplot, but the sequence of measurements traces out the same relationships as before⁵. Finally, in 4.2(c) the boundary is sampled everywhere on a grid which uniformly covers the dataset, and which has no particular spatial relationship to the boundary. Nonetheless, the scatterplots of first and second derivative versus data value have the exact same shape as before, although now the points are distributed differently. Specifically, a much greater number of sample points have accumulated near the $f(x)$ axis, where the derivative values are nearly zero, because there are many more samples within the homogeneous regions inside and outside the cylinder.

Figure 4.2(c) encapsulates the ramifications of our previous assumptions. The reason why curves appear in the scatterplots at all is our initial assumption of band-limited data acquisition, which causes all sharp physical boundaries to be measured as smooth transitions. We also made an assumption about material uniformity by stating that when materials are measured, they attain the same data value everywhere. This implies that the scatterplots will be coherent along the data value axis. Variations in the data value associated with the exterior or interior of the object would ``smear'' the scatterplots horizontally. In addition, we had made the assumption that the band-limiting was isotropic, which causes the measured boundaries to be uniform regardless of position or orientation. Variations in the thickness of the boundary would change the values of the derivatives, which would smear the scatterplots vertically.

The sole purpose of the histogram volume is to capture and represent the scatterplots of Figure 4.2(c) in a single structure. Thus, to the extent that our various assumptions about boundaries and band-limiting are valid, for every boundary that occurs in a dataset, we should be able to discern in the histogram volume the relationship between the data value and its derivatives. These relationships were first seen in Figure 3.8 and are evident in the scatterplots of Figure 4.2(c).

Footnotes

... before⁵

For visual clarity in the depiction of samples along the grayscale boundary (Figure 4.2(b), left), fewer samples are indicated than are plotted in the two scatterplots.