PV251 Visualization
Autumn 2024
Study material
Lecture 5: Spatial data visualization
Spatial data visualization, which includes the entire field of scientific visualization,
assumes that the input datasets implicitly contain spatial or spatiotemporal attributes.
This fact significantly helps to create and interpret visualizations of such data, because
the mapping of data attributes to the graphic ones is intuitive and mostly
straightforward. Our visual system constantly receives and interprets images of the
physical phenomena that surround us. Therefore, it is relatively natural for humans to
process images on the screen in a similar way.
The main differences between the perception of the surrounding world and the spatial
data on the screen are:
- When observing the real world, we are not limited to two-dimensional, discrete, lowresolution
projection.
- Using the screen, we can visually examine various real and simulated phenomena at
different scales.
- On the screen we can dynamically change the contrast, lighting, resolution, density,
and other parameters of the displayed data.
- On the screen, we can interactively explore places that are very difficult to access in
the real world.
- We can interactively add and delete pieces of data on the screen to improve data
context information or remove meaningless data.
Mapping the attributes
When visualizing spatial data, we must first decide which spatial data attributes to map
to the spatial attributes (locations) on the screen. This phase includes various types of
transformations, such as scale, rotation, translation, skew, and projection. Then it is
necessary to map the remaining attributes of the input data to other visualization
components. These attributes include, for example, color or texture, but also the size
and shape of graphic entities.
1D data
One-dimensional data is often obtained by sampling a physical phenomenon as it moves
along a curve in space.
Let us have a one-dimensional sequence of data with one variable. Then we can map the
spatial data to one of the dimensions (axes) of the screen and plot the data values
themselves either to the second dimension of the screen (creating a bar graph) or to the
color of the marker or region along the first axis (color bar). The data must be scaled so
that we can display it over a range of screen dimensions (number of pixels) or within a
range of screen attributes (such as the number of colors supported). Parts of the display
can be reserved for supporting visualization elements, such as the display of axes, labels,
etc.
Algorithm for drawing 1D data
Suppose we have calculated values (datamin, datamax) for the input dataset, which define
the minimum and maximum value in this data, and the variable datacount determines the
number of points (data) to be displayed (it can be all data or just a certain subset) . Next,
assume that the part of the screen on which the data will be displayed is a rectangular
area defined by (xmin, ymin, xmax, ymax). The following code can be used to draw a line graph
from the points of a field marked as "data".
DRAW-LINE-GRAPH(data, dataCount, xMin, xMax, yMin, yMax)
1. dataMin <- computeMin(data, dataCount)
2. dataMax <- computeMax(data, dataCount)
3. xFrom <- xMin
4. yFrom <- worldToScreenY(data[0], dataMin, datamax, yMin, yMax)
5. for i<- 1 to dataCount
6. do xTo <- worldToScreenX(i, dataCount, xMin, xMax)
7. yTo <- worldToScreenY(data[i], dataMin, dataMax,
yMin, yMax)
8. drawLine(xFrom, yFrom, xTo, yTo)
9. xFrom <- xTo
10. yFrom <- yTo
worldToScreenX(index, dataCount, xMin, xMax)
return (xMin + index * (xMax - xMin)/dataCount)
worldToScreenY(value, dataMin, dataMax, yMin, yMax)
return (yMin + (value – dataMin) * (yMax yMin)/(dataMax
/ dataMin))
The worldToScreenX function is simpler than the worldToScreenY function because we
assume that the array of values is indexed from 0. If we wanted to remove this
assumption and generalize the function, it would look like this:
worldToScreenX(index) {
return (xMin + (index - indexMin) * (xMax - xMin) / (indexMax -
indexMin));
}
The whole graph drawing function can be improved in various ways. For example, if the
number of entry points exceeds the number of available pixels, we can preprocess the
entry points (e.g., sample, average…). We may also scale the displayed data to better
convey that information.
A similar algorithm can be applied to plot data in the form of a color bar, simply replacing
line drawing by drawing rectangles (most often of uniform size), the color of which is
proportional to the value of the displayed data and the result is transformed into the
range of available colors provided by the output device.
1D multivariate data
If the input data is marked as multivariate (it contains more variables or more values for
one data input), it is possible to extend the previous technique to be able to process such
data. This can be achieved by juxtapositioning or superimpositioning. For line graphs,
this means that the visualization will contain a set of isolated graphs (especially in cases
where the variables have different scales) or a single graph plotting data on two or more
variables. In this case, it's a good idea to draw individual lines in a different style and / or
color so that we can distinguish the individual data.
A bar graph for color can be composed similarly. Here is a more intuitive use of the
approach, where we layer individual columns on top of each other.
2D data
Data containing two spatial dimensions is displayed mainly by mapping the spatial
attributes of the data to the spatial attributes of the screen. The result can be one of the
following types of visualizations:
1. Image
2. Rubber sheet
3. Cityscape
4. Scatterplot
5. Map
6. Contours
Image
An image is created as a result of 2D data visualization if one data value in each position
is mapped to a color and all intermediate pixels are colored by interpolation. An example
is an image obtained from a tomography.
Rubber sheet
Let us have input data that can be regularly or even irregularly distributed in space. If we
map this data to the height of the respective point in three-dimensional space and, in
addition, the points are triangulated in such a way that a surface is created, we speak of
a visualization of a relief, a surface, or a rubber sheet.
An example is the figure, which shows the sea level and its transition to the mainland in
Florida.
Cityscape
This type of visualization is created by drawing 3D objects (generally blocks) in various
places in the plane. The individual data then controls the attributes of these graphic
objects (e.g., height, color). The figure shows an example of such a visualization, where
we see the density of air traffic over the United States at a certain point in time.
Scatterplot
A classic representation, where each data point is placed in the graph, individual data
values affect the color, shape, or size of individual marks. It is worth to realize that,
unlike the image, there is no interpolation.
Map
The result of the visualization is a map if the input data contain linear and planar
properties, but also point objects.
A linear property can be considered a path or river and is represented as a sequence of
connected coordinates, which are drawn as line segments.
Area properties, such as a lake or the area of a state defined by its boundary, are
represented by a closed curve - a set of coordinates, where the coordinates of the first
and last point are identical. They are usually displayed as a polygon, which can be filled
with a color, texture, or repeating symbol.
Point objects (for example, high voltage pylons, schools) are generally represented by a
certain symbol placed in a given location. Due to the possibility of overlapping some
symbols, the variant of the technique is often chosen, where the symbol does not lie
exactly at the corresponding place on the map but is slightly shifted.
Individual objects usually have their own labels, which again complicates the whole
process of displaying the map. We will learn more about the so-called geovisualization in
one of the following lectures.
Contours, isobars
This type of visualization displays boundary information that is derived from an image
and represents a certain continuous phenomenon, such as altitude or temperature. It is
also referred to as isovalue (e.g., isobar), which means "one value", so the contour in the
map determines the boundary between points above and below this value. Each
"isovalue" can generate several closed contours. Multiple isobars can be drawn at once,
using color, line weight, or markings to distinguish them.
The figure shows the distribution of several contours obtained from a 2D image of a
water molecule. Isobar values are distinguished by color.
2D multivariate data
As with 1D data, we can extend these techniques to multivariate data using
juxtapositioning and superimpositioning.
Juxtapositioning - simple stacking of several 2D visualizations of one variable into a 3D
visualization. This approach allows you to preview all data at once, however, due to
overlays, it can be difficult to detect relationships between data. In addition, this
technique is limited by the number of variables that can still be clearly displayed,
especially for techniques that already use the third dimension (for example, relief or
cityscape).
Superimpositioning - similarly limited by the number of variables that can be displayed
in this way. For example, for reliefs, we draw individual data using translucent surfaces.
Excessive overlap is also not very suitable for maps, as it makes it difficult to extract
individual components.
An alternative approach to displaying multivariate 2D data is to use "non-spatial
multivariate" visualization, which we will discuss in later lectures.
In addition to visualization techniques that display entire data sets, we can only visualize
one-dimensional subsets, projections, or various summaries of data. Then we can use
any of the previous techniques. So, let's focus on some of the projection options:
- Frequency histograms
- Merging rows and columns
- Linear "probes"
Frequency histograms
Summarization of data sets of an arbitrary dimensionality can be performed by calculating
the frequency at which the given values or intervals of values appear in the data. The result
is then displayed in the form of a bar chart. If we work with subsets of input values, we must
choose the number of individual subsets very carefully and how they are distinguished from
each other. If the wrong choice is possible, the loss of important data properties is possible.
In this case, the best distribution of the data will be suggested by a person who is familiar
with the content of the data. On the other hand, the often-used simple division into a fixed
number of equally large subsets is not very effective in this case.
Merging rows and columns
A useful mechanism for locating the boundaries of regions of interest and regions with low
or high variability is a technique that summarizes the rows and / or columns of an image.
Various techniques are used for this purpose, such as sum, mean, median, standard
deviation, maximum or minimum of values. The resulting 1D visualization can then be
displayed separately or more often displayed in combination with 2D visualization as
additional information.
Color bars, line charts, and bar charts are most commonly used for this technique.
Linear probes
A one-dimensional probe for 2D data sets can be compared to a drilled hole in a mineral. A
line passes through the input data, and the data that intersects this line is displayed using
one of the previous techniques for displaying 1D data. Like a drill bit drilling a hole in a
mineral. To achieve this, we use two mathematical tools: parametric equations and bilinear
interpolation.
We start by defining the probe using a parametric equation. The definition is based on user
input (either a pair of points or a point and direction vector). If we assume that the line is
defined by two points P_1 and P_2 (their coordinates), then the parametric equation of the
line is defined as
P(t) = P1 + t(P2 – P1), where 0.0 <= t <= 1.0
We can now obtain the coordinates of any point on this line by choosing the value of t. In
general, these points on the line are evenly distributed and the number of points is
determined in proportion to the length of the line. If we already have the coordinates of the
sampled points, we can use interpolation to calculate their values, which can then be
visualized as a 1D data set.
3D data
As with 2D data, 3D spatial data can be described by vertices, edges, and polygons in the
form of discrete continuous patterns or as a structure. In fact, most scientific and technical
visualizations involve a combination of these two representations, such as the flow of air
around an aircraft wing. We will first focus on exploring the basic techniques for visualizing
these types of data, and then discuss the methods that combine these techniques.
The basic techniques for 3D data visualization include:
- Visualization of explicit surfaces
- Visualization of volume data
- Implicit surfaces
Visualization of explicit surfaces
An explicit surface is a surface that is defined in one of the following ways:
1. List of 3D vertices, list of edges between them (specified as a pair of indexes to the
list of vertices), list of planar polygons (usually specified as a fixed or variable list of
indexes to the list of edges).
2. A set of parametric equations defining x, y, z coordinates of points on a surface
together with a strategy of their connection (for example triangulation), which
creates edges and polygons. By determining the step of the parameter in the
equations, we can influence the smoothness of the surface.
Example 1:
Let's give an example of a unit cube, which can be represented as follows. Note that each
edge is shared by exactly two edges, and each vertex is part of three edges. In addition, each
wall has a defined front and rear wall by edge orientation (clockwise or counterclockwise).
This condition is especially important for the correct calculation of surface normals.
vertex[0] = (0., 0., 0.)
vertex[1] = (0., 0., 1.)
vertex[2] = (0., 1., 1.)
vertex[3] = (0., 1., 0.)
vertex[4] = (1., 0., 0.)
vertex[5] = (1., 0., 1.)
vertex[6] = (1., 1., 1.)
vertex[7] = (1., 1., 0.)
edge[0] = (0, 1)
edge[1] = (1, 2)
edge[2] = (2, 3)
edge[3] = (3, 0)
edge[4] = (0, 4)
edge[5] = (1, 5)
edge[6] = (2, 6)
edge[7] = (3, 7)
edge[8] = (4, 5)
edge[9] = (5, 6)
edge[10] = (6, 7)
edge[11] = (7, 4)
face[0] = (0, 1, 2, 3)
face[1] = (8, 9, 10, 11)
face[2] = (0, 5, 8, 4)
face[3] = (1, 6, 9, 5)
face[4] = (2, 7, 10, 6)
face[5] = (3, 4, 11, 7)
Example 2:
Another example is the parametric definition of a unit cylinder located in the y-axis.
y = 1.0, x = cos Θ, z = sin Θ,
0.0 ≤ Θ ≤ 2π (top base)
y = 0.0, x = cos Θ, z = sin Θ,
0.0 ≤ Θ ≤ 2π (bottom base)
y = h, x = cos Θ, z = sin Θ,
0.0 ≤ Θ ≤ 2π, 0.0 ≤ h ≤ 1.0 (middle part)
By manipulating the parameter Θ we can change the smoothness of the cylinder surface. If
we set the value of h to 1 and the value of Θ to π / 2, then we obtain a cube of height 1.
Other examples of parametric surfaces commonly encountered in computer graphics are
Bézier curves and B-splines.
In general, the visualization of spatial data using explicit surfaces depends on whether the
input data is associated with vertices, edges, or walls. Examples for each of these cases are:
- Temperature or weight of the node (information about peaks)
- Chemical bond strength (edge information)
- Area coverage by map (wall information)
The displayed information can be mapped to one of the non-spatial graphic attributes color,
transparency, texture, etc.
The image comes from the article Explicit Mesh Surfaces of Particle Based Fluids published at
the Eurographics 2012 conference.
Volumetric data visualization
Just as pixels are used for 2D visualization, we use voxels (volume elements) for 3D
visualizations. Volumetric data is generally generated by sampling a continuous
phenomenon and can be captured by sensors (e.g., tomography) or generated by various
simulations (e.g., CFD - computational fluid dynamics). In both cases, we receive
multidimensional data with regularly or irregularly distributed positions, and the goal is to
provide the user with information about the structure, repeating patterns and anomalies in
this data.
Most approaches to volume data visualization fall into one of the following categories:
- Slicing - the use of a cutting plane, which can be axially aligned or can have any
orientation. The result is a 2D sheet cut from data, which can then be displayed using
one of the 1D or 2D visualization techniques.
- Isosurfaces - the user specifies the input parameters according to which the
description of the data set surface is generated and is visualized using one of the
explicit surface visualization techniques.
- Direct volume rendering - two methods: the first is based on throwing a beam into a
scene containing a solid and calculating the values in individual pixels based on the
voxels that the beam intersected. The second method projects each voxel onto the
projection plane using one of the methods of accumulating voxel effects on pixels.
Resampling
For all the above-mentioned volume data visualization techniques, initial resampling
is an essential element.
For example, for isosurfaces we must find places where the data correspond to the
selected "isovalue", which in most cases is somewhere between the points of the
original data set - a process very similar to creating contours in 2D.
When plating, especially if the clipping plane is not axially aligned with the coordinate
system, it is necessary to resample the input data to obtain an evenly distributed set
of pixels. In addition, if the input data is unevenly distributed, interpolation must also
be used.
In the technique of direct volume rendering, we have to sample data along the rays,
which again implies resampling. Resampling is not required only when using parallel
projection along the major axes.
Thus, it is clear that the resampling process plays a very important role in the vast
majority of volume data visualization techniques.
Plating of volumetric data by cutting planes
As with 2D data visualization, we can use the probe technique to create a subset of
lower-dimensional input data. A popular technique for volumetric data based on this
principle is to use clipping planes, where the data block is "cut" by the plane of the
position and the orientation and data that the plane intersects are mapped to the
screen.
The simplest implementation of this technique limits the orientation of the cutting plane in
such a way that the normal of the plane coincides with one of the major axes. The user then
specifies the row, column or depth in the data block and the corresponding plate is then
displayed using one of the 2D spatial data visualization techniques we have become familiar
with earlier.
If we remove the restriction on the orientation of the crop plane, then any voxel that is
intersected by the crop plane can affect the value in one or more pixels. In this case, we can
decide to resample the input dataset at the points where it intersects the clipping plane by
placing a regular grid on the clipping plane. Alternatively, we can select the voxel closest to
the pixel of the clipping plane, which then represents the volume at this point in the plane.
Another approach is to combine the values in the closest voxels to a given pixel on the
clipping plane, their contribution being calculated as the weight assigned based on the
distance of the center of the voxel from the clipping plane.
To specify the cutting plane, we must set six parameters - three positional and three to
define the plane normal.
There are many variations of this technique, each focusing on specific details within the
volumetric data. An example might be:
- Non-planar plates
- Sequence of plates with different orientations
- Plates stacked "on top of each other", displayed in parallel
- Plates placed orthogonally, displayed in parallel
Obtaining an Isosurface Using Marching Cubes
The Marching Cubes algorithm was created in 1987 by Lorensen and Cline and focuses on
rendering the surfaces of volumetric data. A similar technique was developed one year
earlier (Wyvill et al.), however, marching dice have become the most popular algorithm for
this problem.
The basis is the definition of a voxel as a cube with values contained in its eight corners. If
one or more corners of the cube contain values lower than the user-specified "iso value" and
one or more corners also contain values higher than the iso value, it is clear that such a voxel
will in some way contribute to the iso-surface. By determining which edges of the cube are
intersected by the isosurface, we create triangular patterns that separate the region lying
"inside" the isosurface from the region lying outside. The resulting combination of samples
from all cubes at the boundary of the iso-surface gives a final surface representation.
We will now describe this algorithm in more detail. The algorithm consists of two main
components. The first is responsible for defining the section or sections of the surface that
"chop" the individual cubes. If we classify each corner of the cube as falling below or above
the defined isovalue, we obtain 256 possible configurations. Two of them are trivial - all the
corners of the cube fall in or out. In this case, the cube does not contribute to the isosurface.
For all other configurations, we must determine which edges of the cube intersect
the isosurface, and these intersections are then used to create one or more triangular
patterns forming the isosurface.
If we consider symmetry, then there are only 14 unique configurations in the remaining 254
formulas. If only one corner contains a value less than the isovalue, a single triangle is
created whose vertices lie on the edges adjacent to that corner. The normal of the triangle
points from the corner. Thanks to the symmetries, this case generates a total of 8
configurations. One such configuration is shown in the figure on the left.
If the two corners have a value less than the iso value, 3 unique configurations are generated
(see figure second from the left). These depend on whether the corners belong to the same
edge or to the same wall or are "hit" diagonally.
For three corners with a value lower than the iso value, we again have three unique
configurations (picture second from the right). These depend on whether they have 0, 1, or 2
shared edges.
If we have "hit" four corners, we have 7 unique configurations (picture on the right). The
result depends on whether the corners share 0, 2, 3 or 4 edges.
Each of the non-trivial configurations adds 1 to 4 new triangles to the iso-surface. The
vertices of the triangles can be calculated by interpolation along the edges, or the center of
the edge is taken for simplicity. Obviously, the interpolated peaks give us a smoother
surface.
The following figure shows a hydrogen atom rendered using a simple version of Marching
Cubes, in which the centers of the corresponding edges of the cubes are taken as the
vertices of the triangles. The result is also unsatisfactory due to the small input data set.
Now that we can construct a part of the isosurface for one voxel, we can apply this
procedure to the entire volume. We can either process each cube separately or we can
proceed by taking the cubes that share the edges one by one.
Problems of Marching Cubes
One of the obvious problems is the memory requirements when saving the resulting surface.
Each boundary cube can generate up to 4 faces. The size of the stored data can be reduced
by sharing vertices and edges or even merging coplanar samples into larger areas. Another
solution may be to "fit" parametric surfaces to groups of boundary points. However, this
approach is difficult for complex surfaces.
Another problem occurs if we do not have a volume completely filled with voxels. This can
happen if the data is poorly scanned. The result is holes in the data to which some values
must be assigned, most often by interpolation. It is obvious that the newly added values may
reduce the validity of the resulting surface.
Direct volume rendering
Direct volume rendering techniques do not use any 3D polygons for rendering. Instead, the
pixels of the resulting image are calculated individually, either by throwing a beam into the
scene for each pixel or by projecting the voxels onto the projection plane.
The basic process of rendering volumetric data begins with the transformation of the
positions of individual voxels into a visual coordinate system, using a standard 3D graphics
pipeline. The user must specify the view reference point and view direction vector, the width
and height of the image to be projected on the projection plane, and for perspective
projection, the distance of the camera from the projection plane must also be specified.
Then we have the option to choose one of the following methods:
- Forward mapping - projecting each voxel onto the projection plane and
determining which pixels will be affected and how.
- Reverse mapping - also called "ray casting". It is the transmission of a ray from
each pixel of the projection plane through the volume data, where the values of
the data that the ray intersects are sampled and thus determines the resulting
value for each pixel.
Forward mapping - problems:
F1. How to handle pixels that are affected by multiple voxels
F2. How to handle pixels that have no voxel mapped to
F3. How to deal with the fact that voxels are usually projected into positions between pixels.
Inverse mapping - problems:
I1. How to choose the number of points to be sampled along the ray.
I2. How to calculate the value at these points, which often fall between the voxels.
I3. How to combine the points that the ray crossed in its path.
Solution:
Problems F2 and F3 can be solved by mapping each voxel to a region of the projection plane,
allowing it to partially affect the value of several pixels adjacent to the position at which the
voxel was projected. The method using this principle is based on setting the weights of
individual voxels for a given pixel, the weight being derived from the distance between the
pixel and the projected location of the given voxel. In general, a maximum of four pixels are
affected in this way.
Another method using this principle is the so-called "splatting", where we associate a small
region of texture with each voxel and this region is projected onto the projection plane.
Problem I1 can be easily solved by determining the spacing between the voxels and setting
the sampling frequency to a value less than these spacings. Thus, it is not possible to omit
some properties and characteristics of the displayed volume. The sampling itself may be
accompanied by resampling or interpolation discussed earlier.
Problems F1 and I3 are commonly solved using a technique known as "compositing". The
simplest forms of this technique, such as calculating a maximum volume value or averaging
all volume values associated with a pixel / beam. More common approaches assume that
each voxel has an associated transparency value and use this when integrating all voxels
mapped to a given pixel. If we assume that voxel i has color ci and transparency oi, then its
contribution to the resulting pixel value is
In other words, we must determine the accumulated transparency between the projection
plane and the voxel and use it to assign the intensity (ci * oi) of the voxel. The resulting pixel
value is then given by the calculation
Implicit surfaces
A typical method of modeling surfaces in computer graphics is the use of parametric
equations to define points on the surface. These points are then connected into polygonal
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networks (mesh). This representation is particularly advantageous for the application of
transformations and the calculation of surface normals. An alternative method is to use the
so-called implicit representation, where the surface is defined as a so-called zero contour
(initial contour) function with two or three variables.
Implicit representations have their strengths when applying operations such as blending or
metamorphosis.
In the figure, we see the difference between the polygonal representation of the model and
the same model displayed using an implicit surface, which is defined by 800 vertices and
their normals.
Combined techniques
Most effective visualizations are in fact a combination of two or more of the techniques
described above. Each of the techniques has its strengths and weaknesses in terms of
information that it is or is not able to display effectively. The strengths of the individual
techniques can be highlighted by their combination, while of course we must ensure that the
overlap is minimized. At the same time, there is increasing pressure to display data
simultaneously in several different ways to better understand the information being
communicated. For example, the weather forecast includes the display of temperature, wind
speed, relative humidity, and several other factors that affect the accuracy of the forecast.
In this section, we will focus on visualizations that were created by combining previous
techniques.
Plates combined with isosurface
The figure illustrates the technique where the surface of medical data is combined with
orthogonal slicing of the same data set. The surface is mapped to one color (red) and the
resulting values after plating are displayed in blue. The surface can reveal a surface structure
that would be difficult to identify by plating the volume itself. The plate provides detailed 2D
information, especially if the color assignment is well chosen. It can provide the user with
information about relatively uniform regions as well as those in which dynamic changes are
taking place. Another advantage is that the plate also displays the inner (nested) regions in
the appropriate range of values, while the general isosurface shows only the outer surface.
When designing this type of visualization, it is important to consider the following:
- It is not appropriate to support rapid changes in isosurface values
- The position and orientation (along the three axes) of the plate should be easy to
control by the user, including the possibility of animating the position of the plate
- To mediate the view of data from all sides, it is necessary to allow the user to change
the position and orientation of the camera
- Color assignment is essential - some areas of interest inside the plate can only be
discovered by carefully choosing a color scheme. At the same time, it is important to
use a different color to display the iso-surface that does not appear in the color map
of the plate to avoid misinterpretation.
- The user should be able to easily hide either of these two visualization components,
or at least change the transparency.
Combination of iso-surface and pictograms
As already mentioned, iso-surfaces are suitable for conveying details of 3D surfaces, but
generally do not contain any other aspects of the input data. Pictograms (glyphs), such as a
regular arrow, are used to display the magnitude or direction of change in a dataset — either
as a gradient in static data or as a flow in dynamic data. Glyphs can be placed near the iso-
surface.
The image shows a storm cloud, where the surface shows the density of the water inside the
cloud and the arrows show the direction and strength of the wind. In addition, a cutting
plane is used to display water density details. By moving the starting positions of the arrows
(they are connected to the moving plane) we can reveal "quiet" regions and, conversely,
regions with turbulence.
When designing this type of combined visualization, it is advisable to follow similar rules as in
the previous case. Interactive control of visualization parameters - isosurface values, basic
glyph position, angle of view - contribute to streamlining the visualization. Other aspects are
needed to improve the information content of glyphs, such as:
- Changing glyph density
- Glyph size scaling
- Assign color to glyphs
Calculation of the basic position of glyphs based on areas of interest in a vector field
Another potential improvement of this technique may be the use of a cutting plane or a
rubber sheet.
Summary
We got acquainted with several commonly used techniques for visualization of
spatiotemporal data. Thanks to the fact that we went through the techniques according to
the dimensionality of the data, which we gradually increased, we learned the basics on which
the individual techniques are based. Since several different approaches can be used to
visualize a particular dataset, it is important to understand the pros and cons of each
technique - especially which data properties clearly present the technique. It is often
advantageous to combine individual techniques.