Analysis of Uncertain Scalar Data with Hixels - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 39

a

b

f

Fig. 3.7 Slicing a histogram. For an isovalue κ , we can use the histogram to count the number of
voxels with function values above ( a ) and below ( b ) i . ©IEEE reprinted, with permission, from
Thompson et al. [ 10 ]

3.2.3 Fuzzy Isosurfacing

When down-sampling larger datasets, hixels enable preserving the presence of an
isosurface within the data. In particular, because hixels store the counts of all function
values present within a block, we can compute the likelihood of the presence of
an isosurface within that block. Given a hixel h i and an isovalue κ ,we slice the

κ κ

histogram at . These two

counts, a and b for the count above and below, respectively, provide an indication
as to how frequently the isosurface

and compute the number of voxels above and below

may exist within the block. Alternatively these
values can approximate the surface area of the isosurface within the block, Fig. 3.7
visualizes this slicing process.
Using the values a and b , we can then compute a likelihood field. We let g

κ

a b

=

b − a .

For hixels that have a = b , g takes on the value 0, while g > 0 for hixels that are

strongly above κ and g < 0 for hixels that are strongly below. If a = 0, we set

g a . By volume rendering the g field we can get a

“fuzzy” depiction of where the isosurface exists in a hixelated field. By comparison,
naive down-sampling of the scalar field could either move or destroy isovalues. By
visualizing the field g we get a more honest depiction about where that isovalue was
originally in the dataset, and can thus preserve that information.
Figure 3.8 shows visualizations of the stag dataset for

= b , and when b = 0weset g =

κ = 580, down-sampled

from its original size of 832 × 832 × 494 to 208 × 208 × 123, 104 × 104 × 61,

7. Hixels of block size b 3 used 2 b 2
bins. By tracking a histogram of values, at lower resolutions we can preserve the
fidelity of the isosurface and display a more expressive view of the data. Using only
a single value, it is challenging to preserve the thin features of the isosurface, as the
legs, antenna, and mandibles are hard to preserve. Figure 3.2 shows a side-by-side
comparison of the isosurfaces produced at

52 × 52 × 30, 26 × 26 × 15, and 13 × 13 ×

580 for the mean and lower-left fields
as compared to the volume rendering of the g field when the hixel block size is 16 3 .

κ =