5. Spatial data visualization
www.hypergridbusiness.com
visthis.blogspot.com
blogs.library.duke.edu
www.automation-drive.com
Spatial data visualization
• Input data contains spatial or spatio-temporal
attributes
Real world vs. screen
• In real world, we are not limited by 2D space,
discrete representation, low resolution
• On screen:
– Exploring data in different scales
– Dynamic changes of contrast, lighting, resolution
– Interactive exploration of space inaccessible in
real world
– Interactive adding and removing parts of the data
Mapping of attributes
• Phase no. 1:
– Mapping of spatial attributes of data to spatial
attributes of the screen (transformation)
• Phase no. 2:
– Mapping of the remaining attributes – color,
texture, size, shape of graphical entities, …
1D data
• Sequence of 1D data with one variable
– Graph
– Color bar
http://www.opendx.org
1D graph drawing - algorithm
• Input:
– datamin, datamax – minimal and maximal value in
data
– datacount – number of data items to be drawn
– Screen for data visualization – rectangle (xmin, ymin,
xmax, ymax)
1D multivariate data
• More variables or more values for one data
input
• Extension of the previous technique
– Juxtapositioning
– Superimpositioning
www.usenix.org
en.wikipedia.org
2D data
• Two spatial dimensions – mapping of spatial
data attributes to screen space attributes
• Typical visualizations of 2D data:
– Scatterplot
– Map
– Image
– Cityscape
– Contours, isobars
Scatterplot
• Each data item influences color, shape, and
size of the selected glyph
• No interpolation
www.geoafrica.co.za
Map
• Linear objects – continuous line segments
(rivers, roads)
• Planar objects – closed polygons with color,
texture, … (lakes, countries)
• Point objects – specific symbols (school,
church)
• Labels
Image
• Data value mapped onto color in given
position, color between pixels has to be
interpolated
Interactive Data Visualization - Foundations, Techniques and Applications. Matthew Ward
Cityscape
• Drawing 3D blocks in plane, data mapped onto
their attributes (height, color, …)
Interactive Data Visualization - Foundations, Techniques and Applications. Matthew Ward
Contours, isobars
• Border information representing a continuous
phenomenon (elevation, temperature)
• Determines the boundary between points
with higher and lower values
http://www.opendx.org/
2D multivariate data
• Juxtapositioning
– Stacking of 2D visualizations to 3D
• Superimpositioning
– Overlapping 2D visualizations
• Both limited by the number of variables
en.wikipedia.org
boscoh.com
Studying 2D data
• Simplification of input data – visualization of
subsets of the data, projections,
summarizations
• Then using previous techniques
• Projection techniques:
– Frequency histograms
– Merging rows and columns
– Linear „probes“
Frequency histograms
• Calculating the frequency in which given
values are appearing in the data
• Result is displayed as bar chart
www.microbiologybytes.com
Merging rows and columns
• Localization of regions of interest with high or
low variability
• Merging by adding, averaging, calculating
median, standard deviation, maximum,
minimum
• Color bars, line charts, bar charts
Merging rows and columns
Linear „probes“
• Line (ray probe) passing through the input
data
• Using parametric equations and bilinear
interpolation
• Defined by two points P1 and P2 or by one
point and direction vector
• Parametric equation for line:
P(t) = P1 + t(P2 – P1), where 0 ≤ t ≤ 1.0
3D data
• Discrete samples of a continuous
phenomenon or set of vertices, edges, and
polygons
• Mostly combination of both
• Basic techniques:
– Visualization of explicit surfaces
– Volumetric visualization
– Implicit surfaces
Visualization of explicit surfaces
• Explicit surface defined as:
– List of 3D vertices, edges, planar polygons
– Set of parametric equations defining x, y, z
coordinates of points, along with strategy for their
connection (edges, polygons)
www.shapeways.com
Example
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 – unit cylinder in 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)
Examples
• Input data associated with:
– vertices – temperature, weight of vertex
– edges – strength of chemical bond
– polygons – map coverage of area
http://pub.ist.ac.at/group_wojtan/projects/meshSPH/index.html
Volumetric visualization
• Using voxels
• Categories:
– Slicing – using clipping plane
– Isosurfaces – generating surface
– Direct volume rendering –
ray casting or
projecting of voxels
to projection plane
vidi.cs.ucdavis.edu
http://www.docstoc.com/docs/92958007/Direct-Volume-Rendering-_DVR_-Ray-casting
Voxel
Resampling
• Important for most of the volumetric
visualization techniques, as we need to obtain
a uniformly distributed samples for their
displaying
– Slicing
– Isosurfaces
– Direct volume rendering
Slicing of volumetric data using clip
planes
• Creates a subset of input data in lower
dimension
http://doc.instantreality.org/tutorial/volume-rendering/
Slicing of volumetric data using clip
planes
• Orientation of clip plane
– Normal vector of the plane corresponding to one
of the main axes
– Arbitrary orientation
amath.colorado.eduwww.mathworks.com
Generating isosurface using Marching
Cubes
• Lorensen, Cline (1987)
• Voxel = cube with vertices
• Algorithm creates triangles based on the
correspondence between vertices and
isosurface
en.wikipedia.org
Marching Cubes – details
• 256 configurations, thanks to symmetry only
16 unique (1 = whole cube inside, 1 = whole
cube outside)
• Generating corresponding triangles
http://www.opendx.org
Marching Cubes - details
(http://www.opendx.org
Marching Cubes - problems
• High memory requirements
• Holes in data – poor quality of input data
www.cescg.org
williamaadams.wordpress.com
Direct volume visualization
• Pixels of the resulting image computed
individually – using ray casting or voxel
projection
• Methods:
– Forward mapping
– Inverse mapping
(ray casting)
www.volviz.com
Forward mapping - problems
• F1: How to deal with pixels which are
influenced by more voxels?
• F2: How to deal with pixels without any voxels
mapped onto them?
• F3: How to deal with situation when voxels are
projected to positions between pixels?
Inverse mapping - problems
• I1: How to choose correct number of points
along ray which will be sampled?
• I2: How to calculate the value in these points
if they hit the space between voxels?
• I3: How to combine points hit by the ray?
Solution
• F2: How to deal with pixels without any voxels
mapped onto them?
• F3: How to deal with situation when voxels are
projected to positions between pixels?
• Solution: Mapping of each voxel to a region of
the projection plane. Each voxel then partially
influences values of several neighboring pixels
Solution
• I1: How to choose correct number of points
along ray which will be sampled?
• Solution: Determining the spacing between
pixels and setting the sampling frequency to
the smaller value than this spacing
Solution
• F1: How to deal with pixels which are influenced by more
voxels?
• I3: How to combine points hit by the ray?
• Solution: Compositing
– Each voxel has associated the transparency value
– Voxel i has color ci and transparency oi , then its contribution to
the resulting pixel value is:
– Resulting pixel value is then determined as:

−
=
−
1
0
** )1(
i
j
jii ooc

−
==
−=
1
0
**
0
)1(),(
i
j
ji
n
i
i oocyxI
Classification
• Important aspect is to determine the
transparency and color associated with data =
classification
• Setting the regions of interest – they can have
different color and transparency
• Transfer function
vidi.cs.ucdavis.edu
Lighting and shading
• No information about normals – we need to
calculate gradient:
– e.g., for each voxel (vx, vy, vz) we calculate the
gradient value in x axis as (vx-1, vy, vz) and (vx+1, vy, vz)
– Let be gx the x component of gradient. Then:
• Central difference operator
vx – vx-1
• Central difference estimate of gradient
vx+1 – vx-1
Examples
http://www.opendx.org http://old.vrvis.at/via/resources/PR-CBerger-2/index.html
Implicit surfaces
• Surface is defined as zero contour for function
with two or three variables
http://www.cs.umd.edu/class/spring2005/cmsc828v/papers/vimp_tog.pdf
Implicit surfaces
• Defining the surfaces that
correspond to given
values
Combined techniques
• Combination of techniques enables to
highlight their strong points
www.srh.noaa.gov
Slices combined with isosurfaces
• Isosurface of medical data in combination
with orthogonal slicing
• Video
http://www.opendx.org
Slices combined with isosurfaces
• Design of this type of visualization should take
into account:
– Do not support fast changes of isosurface values
– User-driven position and orientation of slices
– User-driven position and and orientation of
camera
– Color assignment
– Hiding individual visualization components or
making them transparent
Combining isosurfaces and pictograms
• Isosurfaces for showing details of 3D surface,
pictograms for showing size or direction of
change in the dataset
http://www.opendx.org
Combining isosurfaces and pictograms
• The following rules should be kept:
– Interactive control of parameters
– Changing the density of pictograms
– Changing the size of pictograms
– Different colors of pictograms
– Calculating basic position of pictograms
Summary
• Different techniques for data in different
dimensions
• We need to understand pros and cons of the
techniques
• Their combination is beneficial
profs.etsmtl.ca
www.ii.uib.no
Examples
• Ivan Viola – Importance-Driven
Volume Rendering
• http://www.cg.tuwien.ac.at/
research/publications/2004/Viola-2004-
ImpX2/
Examples
• Åsmund Birkeland - View-Dependent Peel-Away
Visualization for Volumetric Data
• https://vis.uib.no/wp-content/papercite-
data/pdfs/birkeland09peeling.pdf
Examples
• Visible human project
• https://www.nlm.nih.gov/research/visible/visi
ble_gallery.html