Spiny Dendrite (hippocampus): Experimenting with $g_{thresh}$ and $b(x)$ peak width for one dimensional opacity functions

We now analyze the neuron dataset first seen in the comparison of surface versus direct volume rendering (Figure 1.4). The dataset is a tomographic reconstruction of a spiny dendrite on a hippocampal pyramidal neuron from a rat. The specimen is courtesy of Prof. K. Hama of the National Institute for Physiological Sciences, Okazaki, Japan. The National Center for Microscopy and Imaging Research acquired a series of images of the two micron thick specimen with an intermediate high voltage electron microscope. The images were processed with single tilt axis tomography to create a volume of floating point values. This was quantized to eight bits by histogramming the raw data to determine the floating point range that best contained the important values. The histogram volume generated for this dataset has a resolution of $256^3$, and was calculated using the Hessian second derivative measure.

Figure 6.14: Analysis of hippocampal neuron dataset

As was mentioned in Section 4.4, the scatterplots for this dataset (Figures 6.14(b) and 6.14(c)) do not show clear evidence of any boundaries, as compared to the scatterplots for the other CT datasets analyzed in previous sections. This is also evident from the graphs of $g(v)$ and $h(v)$ (Figures 6.14(d) and 6.14(e)). Instead of having a noticeable maximum at the data value associated with the boundary, and being low elsewhere, the predominant feature of $g(v)$ is its minimum at the data value associated with the background value (around 80). The graph of $h(v)$ is also atypical-- instead of having one distinct zero-crossing, it lies very close to the $v$ axis along the range of values approximately between 75 to 140.

We show next the results of two $p(v)$ calculations, based on different choices for the value of $g_{thresh}$. Figure 6.14(d) illustrates how $g_{{thresh}_\mathbf{1}}$ was chosen to coincide with the average gradient magnitude within the background value. The resulting $p(v)$ (Figure 6.14(f)) has three zero-crossings in the data value range from 75 to 140, theoretically indicating the presence of three boundaries in the volume. Yet the cross-section shows that there are basically just two types of material in the dataset, background and dendrite, and we are interested in the (single) boundary between the two. The problem is that the histogram volume did not succeed in properly measuring the soft boundary that exists between the dendrite and the background. As a result, straight-forward application of Equation 5.10 with what appears to be the correct $g_{thresh}$ did not create a position function $p(v)$ which is suitable for further analysis.

However, we can choose a $g_{thresh}$ so as to create a position function $p(v)$ with a single zero-crossing. Raising $g_{thresh}$ tends to stretch the graph of $p(v)$ away from the $v$ axis, eliminating multiple zero-crossings. Using the $g_{{thresh}_\mathbf{2}}$ indicated in Figure 6.14(d) leads to the position function $p(v)$ seen in Figure 6.14(g). The new $p(v)$ has only one zero-crossing, which is circled for clarity.

This kind of experimentation with $g_{thresh}$ should be explained, since it may appear to be somewhat ad hoc. As one can see from the scatterplot of data value and second derivative (Figure 6.14(c)), there is a general tendency for the second derivative to decrease as data value increases. This property is better represented by the fainter pattern of voxels with very high and very low second derivatives (located in the upper left and lower right portions of the scatterplot, respectively), than by the much larger number of voxels clustered along the region of near-zero second derivative values. Looking at the second derivative scatterplot, one can visually interpolate between the faint distribution of voxels with very high and very low second derivative values in order to find the approximate zero-crossing of this faint pattern. In our experience, the single zero-crossing in $p(v)$ that is revealed by increasing $g_{thresh}$ is well-correlated with the zero-crossing in the scatterplot. We now proceed with the second stage of opacity function generation, by experimenting with different boundary emphasis functions.

Figure 6.15: Renderings of hippocampal neuron dataset

For an initial $b(x)$ we start with a tent function centered at zero, spanning 0.3 voxels on either side, seen in Figure 6.15(a). The reason for the small scale in the domain of $b(x)$ is that the non-ideal shape of the $g(v)$ and $h(v)$ graphs led to a very inaccurate estimation of the boundary blurring parameter $\sigma$, calculated to be 0.45 with Equation 5.8. This implies a boundary thickness of $2\sigma = 0.9$, but inspection of dataset cross-sections would suggest the thickness is closer to 2.5. Still, the initial $b(x)$ is too wide, because the resulting opacity function (Figure 6.15(b)) is non-zero for too many different values, so the rendering (Figure 6.15(c)) shows no clear structure. This can be improved by narrowing the peak in $b(x)$ (Figure 6.15(d)), which similarly narrows the peak in $\alpha (v)$ (Figure 6.15(e)), and the new rendering (Figure 6.15(f)) is somewhat clearer.

Figure 6.16: Renderings of hippocampal neuron dataset, continued

Continuing to make the $b(x)$ peak narrower (Figure 6.16(a)) leads to an even narrower peak in $\alpha (v)$ (Figure 6.16(b)), and finally the rendering (Figure 6.16(c)) shows the surface structure of the neuron. Again, this process was not as automatic as would be ideal, but to arrive at this rendering (starting with the $p(v)$ calculated previously) required empirically decreasing only one variable, the width of the triangular peak in $b(x)$. From here, we can further decrease the amount of ``haze'' surrounding the neuron by moving the peak in $b(x)$ slightly to the right, to slightly higher position values (Figure 6.16(d))¹⁷. The peak in the new opacity function moves in the same direction, putting greater emphasis on higher data values, so as to emphasize the boundary region closer to the interior of the neuron. The change in the resulting opacity function is slight (Figure 6.16(e)), but there is less semi-transparent material surrounding the neuron in the rendering (Figure 6.16(f)).

Footnotes

...fig:pyra-bemph3)¹⁷

The peak in $b(x)$ is centered at 0.0025 instead of 0.0.