The transformation from position domain to value domain

Figure 3.5: Relationships between $f$, $f'$, $f''$

Remember that our ultimate goal is to arrive at a transfer function that makes the values at the boundary opaque. Observe that in Figure 3.5(a), the data value increases monotonically⁴. This means there is a one-to-one relationship between the data value and position. Therefore, it is possible to transform the first and second derivatives from functions of position to functions of data value. Five data values, $v_1$ through $v_5$, have been indicated on the vertical axis. The pattern of horizontal and vertical lines shows that for each of these five data values, we have an associated first and second derivative values. This association is formed through the relationship between data value and position which is governed by the graph of $f$. If we were to plot the derivatives associated with data value as we moved through the range of data values, we would get the curves seen in Figure 3.5(b). Data value, not position, is on the horizontal axis, on which $v_1$ through $v_5$ are shown. The crucial change is that information about position has been entirely removed from the picture, and we are left with the relationship between the data value and its derivatives.

There is a more elaborate conception of the graphs portrayed in Figure 3.5(b) that is well worth describing, since it can give us a more intuitive feel for why those graphs have their particular shape. For the time being, consider just the relationship between data value and the first derivative. Both of these quantities are functions of position, so we could draw a three dimensional graph showing this, as in Figure 3.6.

Figure 3.6: The three dimensional plot of $f$ and $f'$ as functions of position $x$ is projected along the three different axes so as to demonstrate the relationship between each pair of variables.

Below the three dimensional curve we see data value versus position, and to the right, first derivative versus position. Both of these graphs are projections of the three dimensional curve. But there is a third way to project this curve: parallel to the position axis, so as to cast away all position information. We are then left with the relationship between data value and first derivative, exactly the curve seen in Figure 3.5(b).

The same can be done for the relationship between data value and its second derivative, as seen in Figure 3.7.

Figure 3.7: The three dimensional plot of $f$ and $f''$ as functions of position $x$ is projected along the three different axes so as to demonstrate the relationship between each pair of variables.

In the projections which are below and to the right of the three dimensional curve, we recognize the graphs of data value versus position and second derivative versus position. But again, we can project the three dimensional parametric curve along the position axis, distilling out the relationship between data value and its second derivative, to get the second curve in Figure 3.5(b)

Now that Figure 3.5(b) is more thoroughly motivated, we can represent its content in a different way. Since we have ``projected away'' position information and reformulated the first and second derivatives as functions of data value, we can create a three dimensional plot of the first and second derivatives as functions of data value, as seen in Figure 3.8.

Figure 3.8: The three dimensional parametric plot of $f$, $f'$, and $f''$ is projected along the three different axes so as to reveal the position-independent relationships between these three variables. The cross-hairs indicate the positions along the curves corresponding to the middle of the boundary.

We recognize that two of the projections of this curve are the same as in Figure 3.5(b). The full three dimensional curve has an approximately helical shape, as evidenced by the fact that the third projection of the curve (upper right) closes back on itself.

The significance of these curves, the three dimensional one as well as its projections, is that they provide a basis for automatically generating opacity functions. By analyzing an ideal boundary we have arrived at a position-independent relationship between the data value and its derivatives. Therefore, if a three dimensional record of the relationship between $f$, $f'$ and $f''$ for a given dataset contains curves of the type shown in Figure 3.8, we can assume that they are manifestations of boundaries in the volume. With a tool to detect those curves and their position, one could generate an opacity function which makes the data values corresponding to the middle of the boundary (indicated with cross-hairs) the most opaque, and the resulting rendering should show the detected boundaries. Short of that, one could use a measure which responds to some specific feature of the curve (the zero crossing in $f''$, or the maximum in $f'$) and base an opacity function on that. This is the approach taken in this thesis.

Footnotes

... monotonically⁴

This is, of course, the ideal situation. If the segment along which $f$ and its derivatives has been measured does not lie directly between large regions of two distinct materials, the monotonicity condition may not hold. Again, we rely on the statistical properties of the histogram to provide the overall picture of the boundary characteristics.