Chapter summary and solutions to end-of-chapter exercises for "Data Visualization: Principles and Practice" book by Alexandru C. Telea We presented a number of fundamental methods for visualizing scalar data: color mapping, contouring, slicing, and height plots. Color mapping assigns a color as a function of the scalar value at each point of a given domain. Contouring displays all points within a given two- or three-dimensional domain that have a given scalar value. Height plots deform the scalar dataset domain in a given direction as a function of the scalar data. The main advantages of these techniques are that they produce intuitive results, easily understood by users, and they are simple to implement. However, such techniques also have s number of restrictions.

1. DS-620 Data Visualization
Chapter 5. Summary
Valerii Klymchuk
June 26, 2015
0. EXERCISE 0
5 Scalar Visualization
Scalar datasets, or scalar fields, represent functions f : D → R, where D is usually a subset of R2
or R3
.
This chapter presents most popular scalar visualization techniques: color mapping, contouring, and height
plots.
5.1 Color Mapping
Color mapping associates a color with every scalar value. For every point of the domain if interest D, color
mapping applies a function c : R → Colors that assigns to that point a color c(s) ∈ Colors which depends
on the scalar value s at that point. There are two of the most common forms to define such a scalar-to-color
function c: color look-up tables and transfer functions. Color look-up table C, also called a colormap,
is a uniform sampling of the color-mapping function c:
C = {ci}i=1..N , where ci = c
(N − i)fmin + ifmax
N
.
In practice equation above is implemented as a table of N colors c1, ..., cN , which are associated with
the scalar dataset values f, assumed to be in the range [fmin, fmax]. The colors ci with low indices i in the
colormap represent low scalar values close to fmin, whereas colors with indices close to N in the colormap
represent high scalar values close to fmax.
This works well, when we can determine this range in advance of the visualization. Imagine an application
where we want to visualize a time-dependent scalar field f(t) with t ∈ [tmin, tmax]. If we do not know f(t)
for all values t before we start the visualization, we cannot compute the absolute scalar range [fmin, fmax].
In such situations, a better solution might be to normalize the scalar range separately for every time frame
f(t). This implies drawing different color legends for every time frame as well.
Besides using a sampled scalar-to-color function to a discrete look-up table, one can also define the
function c analytically, if desired. This is done by defining three scalar functions cR : R → R, cG : R →
R, and cB : R → R, whereby c = (cR, cG, cB). The functions cR, cG and cB are also called transfer
functions. In practice, one uses predefined look-up tables when there is no need to change the individual
colors at runtime and transfer functions when the investigation goals require dynamically changing the
color-mapping functions.
5.2 Designing Effective Colormaps
A color-mapping visualization is effective if, by looking at the generated colors we can easily and accurately
make statements about the original scalar dataset that was color-mapped. Different types of statements and
analysis goals require different types of colormaps. Such goals include the following:
1. Absolute values: Tell the absolute data values at all points in the displayed dataset.
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2. 2. Value ordering: Given two points in the displayed dataset, tell which of the corresponding two data
values is greater.
3. Value difference: Given two points in the displayed dataset, tell what is the difference of data values
at these points.
4. Selected values: Given a particular data value finterest (or a compact interval of data values), tell
which points in the displayed data take the respective value finterest.
5. Value change: Tell the speed of change, or first derivative of the data values at given points in the
displayed dataset.
Designing a colormap that achieves all above goals in equal measure is a challenging task.
Color legends are required for any application of color mapping where we require to map a color to a
data-related quantity. We must be able to mentally invert the color-mapping function c; that is, look at a
color of some point in the visual domain DV and tell its scalar value f - we must know the color mapping
function. In practice this is achieved by drawing a so-called color legend. This mechanism has some
conditions succeed. First, the color-mapping function c must be invertible, this requires the function to be
injective, meaning that every scalar value in the range [fmin, fmax] is associated with a unique color. It is
not sufficient that the colors have different numerical values. We must also be able to easily perceive them
visually as being different. Second, the spatial resolution of the visualized dataset must be high enough as
compared to the speed of variation of the scalar data f.
Rainbow colormap is a blue-to-red colormap, where blue suggests low values, whereas red suggests
high values. These are one of the widest used colormap types, however the rainbow colormap has also several
important limitations:
• Focus: Perceptually, warm colors attract attention more than cold colors (attention attracted more
towards the high data values.) This can be desirable, however if these may not be the values where we
want to focus on.
• Luminance: Luminances of the rainbow colormap entries vary non-monotonically. This leads to users
being potentially attracted more to certain colors than to others, and/or perceiving certain color
contrasts as being more important that others.
• Context: Hues can have application-dependent semantics. Interpretation of the rainbow colormap
as a heat might be a good encoding for temperature dataset, but this association is much harder or
unnatural to do in case of a medical dataset.
• Ordering: The rainbow colormap assumes that users can easily order hues from blue to green to yellow
to red, however we cannot assume that every user will order hues in this particular manner. When
this assumption does not hold, goal 2 is challenged.
• Linearity: Besides the colormap invertibility requirement, visualization applications often also require
a linearity constraint to be satisfied. Some users perceive colors to change “faster” per spatial unit
in the higher yellow-to-red range than in the lower blue-to-cyan range. If a linear signal would be
perceived by the user as nonlinear, with possibly undesirable consequences for goals 3 and 5.
Other colormap designs.
Grayscale: here we map data values f linearly to luminance, or gray value, with fmin corresponding to
black and fmax corresponding to white. The grayscale colormap has several advantages. First, it directly
encodes data into luminance, and thus it has no issues regarding the discrimination of different hues. Second,
color ordering is natural (from dark to bright), which helps goal 2. Rendering grayscale images is less sensitive
to color reproduction variability. However, telling differences between two gray values, or addressing goal 3,
is harder than when using hue-based colormaps.
Two-hue: can be seen as a generalization of the grayscale colormap, where we interpolate between two
colors, rather than between black and white. If the two colors are perceptually quite different (differ both
in hue and in perceived luminance), the resulting colormap allows an easy color ordering, and also produces
a perceptually more linear result than the rainbow colormap. The two-hue colormap uses both hue and
luminance differences of its two defining colors to create a higher dynamic range. However, used on a 3D
shaded surface, the luminance perceived in the final visualization is ambiguous (doe to the colors or due to
the shading of the scene.) A simple way to correct this issue is to use a corrected isoluminant two-hue
2

3. colormap, where all entries have the same luminance, similar to the luminance-corrected rainbow colormap.
The disadvantage of this design is that less colors can be individually perceived, leading to challenges for
goals 1, 4, and 5.
Heat map: The intuition behind colors of heated body colormap is that they represent the color of an
object heated at increasing temperature values, with black corresponding to low data values, red-orange hues
for intermediate data ranges, and yellow-white hues for the high data values respectively. The heat map uses
a smaller set of hues that a rainbow colormap, but adds luminance as a way to order colors in an intuitive
manner. However, its strong dependence on luminance makes the heat map less suitable for color coding
data on 3D shaded surfaces.
Diverging: (double-ended scale) colormaps are constructed starting from two typically isoluminant hues.
However, we now add a third color cmid for the data value fmid = (fmin + fmax)/2 located in the middle of
the considered data-range, and use two piecewise-linear interpolations between cmin and cmid, and between
cmid and cmax. Additionally, cmid is chosen so that it has a considerably higher luminance than cmin and
cmax. Diverging colormaps are good for situations where we want to emphasize the deviation of data values
from the average value fmid, and effectively support the task of assessing value differences.
Zebra colormap: In some applications, we want to emphasize the variations of the data rather than
absolute data values (goal 5). This is useful when we are interested in detecting the dataset regions where
data changes the most quickly or, stays constant. A solution is to use a colormap on the scalar dataset f
containing two or more alternating colors that are perceptually very different. If we use a colormap that
maps the value range to an alternating pattern of black and white colors, we obtain the zebra-like result
shown in Figure 5.3(b). Thin, dense stripes indicate regions of high variation speed of the function, whereas
the flat areas in the center and at the periphery of the image indicate regions of slower variation. WE can
now also see better which are the regions where the data has similar rates of variation across our domain.
Interpolation issues.
Results of texture-based color mapping are better, since the texture-based method interpolates the scalar
value s, and then applies the color mapping c separately at every rendered pixel.
c(p) = c
i
siφi(p) .
In contrast, the vertex-based color mapping c applies color mapping at vertices only, and then interpolates
the mapped colors themselves.
c(p) =
i
c(si)φi(p).
A better solution is to store for each vertex pi the value si−smin
smax−smin
,, i.e., our scalar values normalized
in a range of [0, 1], as a texture coordinate. Next, we render our surface cells textured with 1D texture
containing the colormap. 1D textures are described by a single texture coordinate and are stored as a 1D
image, i.e., a vector of colored pixels. Texture-based color mapping produces reasonable results even for
a sparsely sampled dataset. However, this technique requires rendering texture-based polygons, an option
that may not be available on low-end graphics hardware or in situations when we want to use the available
texture for other purposes.
Color banding.
Choosing a small number of colors N would inevitably lead to the color banding effect. It is usually
not desirable, as it creates discrete visual artifacts in the rendered image that do not reflect any of the input
data. Color banding can be avoided by increasing the number of colors in the colormap. The more quickly
the data varies spatially, the smother the colormap has to be to avoid color banding.
Additional issues. Selecting an optimal colormap is further influenced by a number of additional
considerations, as follows:
• Geometry: Not all colors are equally strongly perceived when displayed on surfaces that have the same
area. Perceiving a color accurately is strongly influenced by the colors displayed in neighboring areas.
Densely packed colors are not perceived separately under a certain spatial scale, but blended together
by the human eye. This becomes an issue when the areas in question are very small, such as when
color mapping a point cloud.
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4. • User group: 6 to 10 percent of all men would not be able to correctly separate red from green (or larger
variety of hues). This and other forms of colorblindness should be taken into account when designing
colormaps in critical applications intended to be used by a large public.
• Medium: It is hard to design rich colormaps, containing more than a dozen colors that look the same
on all the devices. A common mistake in practice is to use hue-based colormaps, such as rainbow
colormap, and display the resulting visualization on luminance-based devices, such as black-and-white
printed material.
• Contrast: Some techniques work on the colors of an existing image: modifying the gray values, or
colors in order to emphasize details of interest. Such techniques are discussed in the context of image
processing.
Designing effective colormap involves knowledge of the application domain conventions, typical data
distribution, visualization goals, general perception theory, intended output devices, and the user preferences.
There is no universally effective colormap.
5.3 Contouring
Color banding is related to fundamental and widely used visualization techniques called contouring.
Points located on such a color border, drawn in black in Figure 5.7(a), are called a contour line, or
isoline. Formally, a contour line C is defined as all points p in a dataset D that have the same scalar value,
or isovalue s(p) = x, or
C(x) = {p ∈ D|s(p) = x}.
We can actually combine the advantages of contours and color mapping by simply drawing contours for
all values of interest on top of a color-mapped image that uses a rich colormap. Besides indicating points
where the data has specific values, contours can also tell us something about the data variation itself. Areas
where contours are closer to each other, indicate higher variations of the scalar data. The scalar distance
between consecutive contours (which is constant) divided by the spatial distance between the same contours
(which is not constant) is exactly the derivative of the scalar signal.
Contour properties. First, isolines can be either closed curves, they never stop inside the dataset
itself - they either close upon themselves, or stop when reaching the dataset border. Second, isoline never
intersects (crosses) itself, nor does it intersect an isoline for another scalar value.
Contours are perpendicular to the gradient of the contoured function. The gradient is the direction of
the function’s maximal variation, whereas contours are points of equal function value, so the tangent to a
contour is the direction of the function’s minimal (zero) variation.
Computing contours. Since dataset mathcalD is defined as a set of cells carrying node or cell scalar
data, plus additional basis functions, being an intersection of the function graph with a horizontal plane, the
isoline of such dataset is piecewise linear and has topological dimension 1, this is a polyline.
For every edge e = (pi, pj) of the cell c in dataset we test whether the isoline value v is between the scalar
vertex attributes vi and vj corresponding to the edge end points pi and pj. If the test succeeds, the isoline
intersects e at a point
q =
pi(vj − v) + pj(v − vi)
vj − vi
.
We repeat the previous procedure for all edges of our current cell and finally obtain a set of intersection
point S = {qi} of the isoline with the cell. Next, we must connect these points together to obtain the actual
isoline. Since we know that the isoline is piecewise linear over a cell, we can use line segments to connect
these points. If S contains more than 2 points, as illustrated in Figure 5.11 for the quad cell which has
4 intersection points, there are two possibilities for connecting the four intersection points: creating two
separate loops, or creating a single contour loop. When we use triangular mesh, such ambiguities do not
exist, because it is shifted now to the quad-splitting process: splitting diagonally from the lower-left to the
upper-right, or from the upper-left to the lower-right vertex.
Contouring needs to have at least piecewise linear, C0
dataset as input. This means that we cannot
directly contour image data, for example, which is actually a uniform grid with piecewise constant interpo-
lation, and thus represents a discontinuous function. Resampling image datasets to piecewise linear datasets
4

5. can be done, however this has the hidden effect of changing the continuity assumptions on the dataset
from piecewise constant to piecewise linear. While this should not pose problems in many cases, there are
situations when such changes can lead to highly incorrect visualizations.
The most popular methods that reduce the number of operations done per cell, is the marching squares
method, which works on 2D datasets with quad cells, and its marching cubes variant which works on 3D
datasets with hexahedral cells.
5.3.1 Marching Squares
The method begins by determining the topological state of the current cell with respect to the isovalue. A
quad cell has, thus, 24
= 16 different topological states. The state of a quad cell can be represented by a
4-bit integer index, where each bit stores the inside/outside state of a vertex. This integer can be used to
index a case table (see figure 5.12.
The marching squares algorithm constructs independent line segments for each cell, which are stored in
an unstructured dataset type, given that isolines have no regular structure with respect to the grid they are
computed on. A useful post processing step is to merge the coincident end points of line segments originating
from neighbor grid cells that share an edge. Besides decreasing the isoline dataset size, this also creates a
dataset on which operations such as computing vertex data from cell data via averaging is possible.
5.3.2 Marching Cubes
The algorithm accepts 3D instead of 2D scalar datasets and generates 2D isosurfaces instead of 1D isolines.
Since a hex cell has 28
= 256 different topological cases, in practice, this number is reduced to only 15 by
using symmetry considerations. Marching cubes generates a set of polygons for each contoured cell, which
includes triangles, quads, pentagons, and hexagons. Resulting 3D isosurface is saved as an unstructured
dataset. An additional step needed for marching cubes is the computation of isosurface normals. These are
needed for smooth shading. Normals can be computed by normal averaging. Alternatively, since isosurfaces
are orthogonal to the scalar data gradient at each point, normals can be directly computed as being the
normalized gradients of the scalar data.
As a general rule, most isosurface details that are under or around the size of the resolution of the
isosurfaced dataset can be either actual data or artifacts, and should be interpreted with great care.
We can say, that the slicing and contouring operations are commutative.
Marching algorithm variations. There exist similar algorithms to marching cubes for all cell types,
such as lines, triangles, and tetrahedra. These algorithms can treat all grid types, as long as all encountered
cell types supported. All variants produce unstructured grids.
Isosurfaces can be also generated and rendered using point-based techniques.
Dividing cubes algorithm. A classical algorithm for generating and rendering isosurfaces using clouds
is dividing cubes. It works for 3D uniform and rectilinear grids (box shaped cells). Dividing cubes approx-
imates the isosurface using constant basis functions, whereas marching cubes uses linear basis functions.
For dividing cubes, the ideal minimal cell size is that of a screen pixel, in which case every point primitive
is of pixel size and there are no visual gaps or other rendering artifacts in the rendered isosurface.
5.4 Height Plots
Height plots, also called elevation or carpet plots can be described by the mapping operation, given a
two-dimensional surface Ds ∈ D, part of a scalar dataset D:
m : D∫ → D, m(x) = x + s(x)n(x), ∀x ∈ D∫ ,
where s(x) is the scalar value of D at the point x and n(x) is the normal to the surface D∫ at x. The hight
plot mapping operation “warps” a given surface D∫ included in the dataset along the surface normal, with
a factor proportional to the scalar value.
Most common variant of height plots warps a planar surface D∫ . However, we can produce height plots
starting from different basis surfaces Ds, such as torus, shown in Figure 5.18. In this image, both the height
and the color encode the scalar value: the two visual cues straighten each other to convey the scalar value
5

6. information. If desired, one can encode two different scalar values with a height plot, one into the plot’s
height and the other into the plot’s color.
5.4.1 Enridged Plots
The height plots, color mapping, and contouring techniques have their own strengths and limitations:
• Height plots are easy, to learn, intuitive, generate continuous images, and show the local gradient of
data. However, quantitative information can be hard to extract from such plots, it is not easy to tell
which peak is the highest. Also, 3D occlusion effects can occur.
• Color mapping share advantages of height plots and do not suffer from 3D occlusion problems. How-
ever, making quantitative judgments based on color data can be hard, and requires carefully designed
colormaps.
• Contour plots are effective in communication precise quantitative values. However, such plots are less
intuitive to use. Also, they do not create a dense, continuous, image.
Combining height plots, contour plots, and color mapping alleviates some but not all, of the problems of
each plot type.
An alternative solution is proposed by ** enridged contour maps.** The idea is to use a non-linear
mapping given by
z(x, y) = sf(x, y) + sh g
f(x, y)mod h
h
,
instead of the linear mapping z(x, y) = sf(x, y), where s > 0 is the plot’s scaling factor, g(u) = au(1 − u) is
a parabolic function. Effect is to add parabolic bumps of height a to consecutive intervals in the range of f
of size h.
Enridged plots combine the appearance of contour plots and height plots. The nested cushion-like shapes
that emerge in this type of plot convey a sensation of height which is much stronger than in classical height
plots.
Enridged plots can be also extended to use a hierarchy of contour levels, by using two instances of
parameter-pairs (h1, a1) (larger intervals) and (h2, a2) (denser contours) such that a2 < a1 and h1 = kh2.
Enridged plots can be further tuned to emphasize the nesting relationship between different regions. For
this we simply replace the symmetric parabolic profile g(u) = u(1−u) by an asymmetric one g(u) = u(1−u)n
,
which increases sharply close to u = 0 and is relatively flat close to u = 1. The image now resembles the
structure of Venn-Euler diagram. These structures can be useful for visualizing data hierarchies in different
contexts, beyond scalar fields.
5.5 Conclusion
We presented a number of fundamental methods for visualizing scalar data: color mapping, contouring,
slicing, and height plots. Color mapping assigns a color as a function of the scalar value at each point of
a given domain. Contouring displays all points within a given two- or three-dimensional domain that have
a given scalar value. Height plots deform the scalar dataset domain in a given direction as a function of
the scalar data. The main advantages of these techniques are that they produce intuitive results, easily
understood by users, and they are simple to implement. However, such techniques also have s number of
restrictions.
What have you learned in this chapter?
This chapter presents most popular scalar visualization techniques: color mapping, contouring, and height
plots. It talks about design of effective colormaps: important design decisions pertaining to the construction
of colormaps, contouring in two and three dimensions. Marching squares and marching cubes algorithms are
well explained; strengths and limitations of standard techniques are outlined.
What surprised you the most?
Specific colormaps for categorical datasets have been designed. An excellent online resource that allows
choosing a categorical colormap based on data properties and task types is ColorBrewer. It is also surprising
to know how we can draw more than a single isosurface of the same dataset in one visualization. The process
6

7. of rendering several nested semitransparent isosurfaces that correspond to a discrete sequence of isovalues
can be generalized to the continuous case, as we shall see with the volume rendering technique in Chapter
10.
What applications not mentioned in the book you could imagine for the techniques ex-
plained in this chapter?
Perhaps, we can visualize probabilities as height plots or scalar fields with isolines and colors mapped to
the scalar value. Besides a visual clue, those images (datasets) can also suggest shortest paths from, i.e.,
arbitrary points A and B situated on top of the surface of the same plot.
1. EXERCISE 1
Consider the simple scalar visualization of a 2D slice from a human head scan shown in the figure below.
Here, six different colormaps are used. Explain, for each of the sub-figures, what are the advantages (if any)
and disadvantages (if any) of the respective colormap.
Figure 1: Scalar visualization of a 2D slice dataset using six differentcolormaps (see chapter 5).
Hints: Consider what types of structures you can easily see in one visualization as compared to another
visualization.
Table 1: Advantages and disadvantages of different colormaps.
Colormap
type
Advantages Disadvantages
rainbow (a) We can relatively easy distinguish the
harder tissues (color mapped to yellow, or-
ange, and red), and the air (color mapped
to dark blue)
Focus: warm colors attract more attention
that the cold colors. Depending on applica-
tion, these may not be the values where we
want to focus on. Linearity: Colors change
faster per spatial unit in the higher yellow-
to-red range than in the lower blue-to-cyan
range. Linear signal would be perceived as
nonlinear.
7

8. Continuation of Table 1
Colormap
type
Advantages Disadvantages
grayscale (b) Grayscale directly encodes data into lumi-
nance, and thus it has no issues regarding
the discrimination of different hues. Col-
ormaping is natural (from dark to bright),
which helps goal 2. Rendering grayscale
images is less sensitive to color reproduc-
tion variability issues on a range of devices.
Focus: warm colors attract more attention
that the cold colors. Depending on applica-
tion, these ma not be the values where we
want to focus on. Linearity: Colors change
faster per spatial unit in the higher yellow-
to-red range than in the lower blue-to-cyan
range. Linear signal would be perceived as
nonlinear.
two-color (c) If two colors are perceptually quite differ-
ent in hue but also in luminance, the result-
ing colormap allows an easy color order-
ing, and produces more linear result than
the rainbow colormap. In case none of two
colors is too bright, it is suitable for non-
compact data displayed on a white back-
ground.
It arguably offers less dynamic range.
While using this colormap on 3D shaded
surface, the luminance perceived in the fi-
nal visualization is ambiguous.
heat map (d) Compared to rainbow colormap it uses a
smaller set of hues, but adds luminance as
a way to order colors in an intuitive man-
ner. Allows one to discriminate between
more data values, than the two-hue col-
ormap, since it uses more hues.
Its strong dependence on luminance makes
it less suitable for color coding data on 3D
shaded surfaces, just as the two-hue non-
isoluminant colormaps.
diverging (e) This can be interpreted as a temperature
colormap, where the average value (white)
is considered neutral, and we want to em-
phasize points which are colder, respec-
tively warmer, than this value.
*
diverging (f) Good for situations where we want to em-
phasize the deviation of data values from
the average value fmid, and also effectively
support the task of assessing value differ-
ences. Since the maximum luminance of
this colormap is lower than that of pure
white, we can use this colormap also on
non-compact data domains. More hues
than in two-hue colormaps allow one to dis-
criminate between more data values.
*
End of Table
2. EXERCISE 2
Consider an application where we use color-mapping to visualize a scalar field defined on a simple 2D
uniform grid consisting of quad cells. The scalar field values, recorded at the cell vertices, are in the range
[s {min},s {max}]. Next to the color-mapped grid, we show, as usual, a color legend, where the two extreme
colors (at the endpoints of the color legend) correspond to the values smin and smax, respectively. Given
this set-up, can we say for sure that any color shown in the color legend will appear in at least one point
(pixel) of the color-mapped grid? If so, argue why. If not, detail at least one situation when this will not
occur.
Yes, it may happen in case in isoluminant clormaps or in case of texture-based colormaps, which produce
pixel-accurate renderings - reconstructions of our given sampled signal. Scalar fields produce such legends
too. It also depends on color interpolations done by graphics hardware: most widespread vertex-based
8

9. color-mappings produce suboptimal results for dataset regions where the data varies too quickly.
No, in case if resolution of the output image is low which affects our visualization in a way, that non-linear
hues on color legend start being perceived as misleading. Images that exhibit color banding can have fewer
hues than the color legend, displayed as continuous strip.
3. EXERCISE 3
A simple, though not perfect, way to create 2D contours or isolines, is to use a so-called delta colormap.
Given a scalar contour-value (or isovalue) a, and a dataset with the scalars in the range [m, M], a delta
colormap maps all scalar values in the ranges [m, a − d] and [a + d, M] to one color, and the values in the
range [a − d, a + d] to another color. Here, d is typically very small as compared to the scalar range M − m.
Consider now the application of this delta colormap to a scalar field stored on a simple uniform 2D grid
having quad cells, as compared to drawing the equivalent isoline on the same grid. Detail at least two
drawbacks of the delta colormap as compared to the isoline visualization. Next, explain which parameters
we could fine-tune (e.g., value of d, sampling rate of the grid, color-interpolation techniques used on the grid
cells, or other parameters) in order to improve the quality of the delta-colormap- based visualization.
Drawbacks of applying delta colormap to the scalar field on uniform 2D grid with quad cells:
• Delta colormap on a scalar field does not draw the isoline itself, it highlights a sub-cloud of points,
which are close in terms of their scalar values. This does not tell us much about topology.
• Some bright hues can appear invisible on while backgrounds, causing a loss of visual information
depicted.
It is better to use isoluminant colors for delta colormap. It is also good to keep parameter d adjusted to
sampling resolution of the grid to help marching algorithms.
Contouring needs to have at least piecewise linear datasets as input. We can not directly contour the image
data, when scalar field represents piecewise constant interpolation of a discontinuous function. Resampling
to piecewise-linear dataset and changing the continuity assumptions on the dataset from piecewise-constant
to piecewise-linear can lead to incorrect visualizations.
4. EXERCISE 4
Contouring can be implemented by various algorithms. In this context, the marching cubes algorithm
(and its variations such as marching squares or marching tetrahedra) propose several optimizations. Com-
pared to a ‘naive’ implementation of contouring, that does not use the marching technique, what are the
main computational advantages of the marching technique?
In naive implementation for each cell we must test whether the isovalue intersects every cell edge, and,
if so, compute the exact intersection location. * marching algorithms reduces number of operations done
per cell * computes edge-isovalue intersections only on those edges that are known to be intersected for
a specific topological state. * besides decreasing the isoline dataset size, unstructured dataset produced
supports operations such as computing vertex data from cell data via averaging.
5. EXERCISE 5
Consider a 2D constant scalar field, f(x, y) = k, for all values of (x, y). What will be the result of
applying the marching squares technique on this field for an isovalue equal to k? Does the result depend
on the cell type or dimension – for instance, do we obtain a different result if we used marching triangles
on a 2D unstructured triangle-grid rather than marching squares on a 2D quad grid? Hints: Consider the
vertex-coding performed by the marching algorithms.
Marching squares will produce a contour line, which crosses each edges of the cells in our dataset.
Marching triangles have no ambiguous cases, but they will also produce the same result as marching squares.
Isolines have no structure with respect to the grid they are computed on.
6. EXERCISE 6
9

10. An isoline, as computed by marching squares, is represented by a 2D polyline, or set of line-segments.
Can two such line segments from the same isoline intersect (we do not consider segments that share endpoints
to intersect). Can two such line segments from different isolines of the same scalar field intersect? If your
answer is yes, sketch the situation by showing the quad grid, vertex values, isoline segments, and their
intersection. If your answer is no, argue it based on the implementation details of marching squares.
An isoline never intersects (crosses) itself, nor does it intersect an isoline for another scalar value.
7. EXERCISE 7
Consider a two-variable scalar function z = f(x, y) which is differentiable everywhere. Figure 5.9 (also
displayed below) shows that the gradient of the function is a vector which is locally orthogonal to the contours
(or isolines) of the function. Give a mathematical proof of this.
Hints: Consider the tangent vector to the contour at a point (x, y), and express the variation (increase
or decrease) of the function’s value in a small neighborhood around the respective point.
Whereas contours are points of equal function value, so the tangent to a contour is the direction of the
function’s minimal (zero) variation: ∆f = 0. The gradient of the function is the direction of the maximal
variation.
8. EXERCISE 8
Contouring and slicing are commutative operation, in the sense that slicing a 3D isosurface with a plane
yields the same result as contouring the 2D data-slice obtained by slicing the input 3D scalar volume with
the same plane. Consider now the set of 2D contours obtained from a 3D scalar volume by applying the
2D marching squares algorithm on a set of closely and uniformly spaced 2D data-slices extracted from that
volume. Describe (in as much detail as possible) how you would use this set of 2D contours to reconstruct
the corresponding 3D isosurface.
If the set is dense, i.e., the slice panels are close to each other, and the dataset does not exhibit to-sharp
variations, we can connect points on the isolines from the previous and next slice and construct the 3D
isosurface.
9. EXERCISE 9
Consider an application in which the user extracts an isosurface, and would like to visualize how thick the
resulting shape is, using e.g. color mapping. That is, each vertex of isosurface’s unstructured mesh should
be assigned a (positive) scalar value indicating how thick the shape is at that location. For this problem:
• Propose a definition of shape thickness that would match our intuitive understanding (you can support
your explanation by drawings, if needed)
• Propose an algorithm to compute the above definition, given an isosurface stored as an unstructured
mesh
• Discuss the computational complexity of your proposed algorithm
• Discuss the possible challenges of your proposal (e.g., configurations or shapes for which your definition
would not deliver the expected result, and/or the algorithm would not work correctly with respect to
your definition).
Hints:
• Consider first the related problem of computing the thickness of a 2D contour, before moving to the
3D case.
• Consider first that the contour is closed, before treating the case when it is open because it reaches
the boundary of the dataset.
Thickness of a contour generally differs depending on the direction of measurement.
I propose to define the thickness in the direction of inverted normal. At each sample point pi we compute
a distance d between a sample point itself and the point e where inverted normal n exits a closed contour.
10

11. The length of a vector between pi and e can be derived from pi − d × n = e. Once we detected the point e,
we can compute d = ||pi − e|| and store the value at the sample point pi.
Vertex normal values can be computed as the area-weighted average of the polygon normals for all the
cells that use a given vertex using formula:
nj =
A(Cj)n(Ci)
A(Ci)
,
where A(Ci) is the area of the cell Ci, n is a cell’s normal.
Implementing an unstructured grid implies storing coordinates of a collection of all sample points {pj}
and then storing all vertex indices for all cells {ci}. These indices are usually stored as integer offsets in the
grid sample point list {pj}.
In order to calculate coordinate-vector e we use the following steps:
• ∀ cells ci where pj is a vertex we compute normals and compute vertex normal nj by averaging
• for each vertex and its normal we cast a ray in the direction of inverse normal, starting a loop with a
small number d >= 0 and increasing d with each iteration, until point-vector e = pj − d × nj appears
to belong to at least one of the cells in our unstructured contour mesh.
• once we have obtained such a scalar d, we store vector-point e in the vertex pj. We can always easily
compute and use ||e|| as a measure of contour’s thickness in the direction of the normal.
10. EXERCISE 10
One of the challenges of practical use of isosurfaces in 3D is selecting an ‘interesting’ scalar value, for which
one obtains an ‘interesting’ isosurface. We cannot solve this challenge in general, since different applications
may have different definition of what an interesting isosurface is. However, consider an application where you
want to automatically select a small number of isovalues (say, 5) and display these to give a quick overview
of the data changes. How would you automatically select these isovalues? Hints: Define, first, what you
consider to be the interesting structures in the data, e.g., areas of rapid scalar variation, or areas where an
isosurface changes topology).
I would pick 5 numbers automatically by dividing scalar range = [min, max] with isovalues as follows:
i1 = min+1/3 range. Then, I would pick second isovalue inside the largest interval [i1, max] as i2 = i1 +1/3·
(max−i1). Third isovalue I would pick as i3 = i2+1/3 (max−i2)], forth and fifth as i4 = max−2/3 (max−i3)
and i5 = min + 2/3 (i1 − min) accordigly.
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