Visualisation : Lecture 5
Discrete Data Structure and
Colour Mapping
Computer Animation and Visualisation –
Lecture 10
Taku Komura
Institute for Perception, Action & Behaviour
School of Informatics
Taku Komura
Colour Mapping 1
Visualisation : Lecture 5
Overview
Discrete data structure
Topology and interpolation
Topological dimension
Data Representation
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Structure
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Topology - Cells
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Geometry - points
Attributes
Colour Mapping
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Discrete Vs. Continuous
Real World is continuous
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Data is discrete
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finite resolution representation of a real world (of abstract) concept
Difficult to visualise continuous shape from raw discrete
sampling
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eyes designed towards the perception of continuous shape
we need topology and interpolation
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Topology
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Topology : relationships within the data invariant
under geometric transformation
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Which vertex is connected with which vertex by an edge?
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Which area is surrounded by which edges?
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Interpolation & Topology
If we introduce topology our visualisation of
discrete data improves
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why ? : because then we can interpolate based on the
topology
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Interpolation & Topology
Use interpolation to shade whole cube:
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Interpolation: producing intermediate samples from a
discrete representation
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Taku Komura
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Importance of representation
What happens if we change the representation?
Discrete data samples remain the same
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topology has changed ⇒ effects interpolation ⇒ effects visualisation
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Taku Komura
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Topological Dimension
Topological Dimension: number of independent continuous variables
specifying a position within the topology of the data
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different from geometric dimension (position within general space)
point
Topological
0D
Geometric
2D/3D
e.g. 2D (x,y)
e.g. 3D point on curve
curve
1D
2D/3D
surface
2D
3D (in general)
volume
3D
3D
Temporal volume
4D
3D (+ t)
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e.g. MRI or CT scan
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Visualisation : Lecture 5
Structure of Data
Structure has 2 main parts
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topology : “set of properties invariant under certain
geometric transformations”
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determines interpolation required for visualisation
“shape” of data
OR
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geometry
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instantiation of the topology
specific position of points in geometric space
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Data Representation
Data objects :
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structure + value
referred to as datasets
Abstract
Concrete
Structure
- Topology
- Geometry
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Data Attributes
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Consists of
Cells, points (grid)
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Scalars, vectors
Normals, texture
coordinates, tensors etc
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Dataset Representation of the
Structure
Points specify where the data is known
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specify geometry in ℝN
Cells allow us to interpolate between points
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specify topology of points
Point with known
attribute data
(i.e. colour ≈ temperature)
Cell (i.e. the triangle) over
which we can interpolate
data.
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Cells
Fundamental building blocks of visualisation
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our gateway from discrete to interpolated data
Various Cell Types
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defined by topological dimension
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specified as an ordered point list
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primary or composite cells
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(connectivity list)
composite : consists of one or more primary cells
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Zero-dimensional cell types
Vertex
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Primary zero-dimensional cell
Definition: single point
Polyvertex
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Composite zero-dimensional cell
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composite : comprises of several vertex cells
Definition: arbitrarily ordered set of points
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One-dimensional cell types
Line
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Polyline
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Primary one-dimensional cell type
Definition: 2 points, direction is from first to
second point.
Composite one-dimensional cell type
Definition: an ordered set of n+1 points, where
n is the number of lines in the polyline
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Two-dimensional cell types - 1
Triangle
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Primary 2D cell type
Definition: counter-clockwise ordering of 3 points
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Triangle strip
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Composite 2D cell consisting of a strip of triangles
Definition: ordered list of n+2 points
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n is the number of triangles
Quadrilateral
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Primary 2D cell type
Definition: ordered list of four points lying in a plane
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order of the points specifies the direction of the surface normal
constraints: convex + edges must not intersect
ordered counter-clockwise defining surface normal
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Two-dimensional cell types - 2
Pixel
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Primary 2D cell, consisting of 4 points
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numbering is in increasing axis coordinates
Polygon
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Primary 2D cell type
Definition: ordered list of 3 or more points
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topologically equivalent to a quadrilateral
constraints: perpendicular edges; axis aligned
counter-clockwise ordering defines surface normal
constraint: may not self-intersect
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Three-dimensional cell types
Tetrahedron
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Definition: list of 4 non-planar points
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Hexahedron
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Six edges, four faces
Definition: ordered list of 8 points
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six quadrilateral faces, 12 edges, 8 vertices
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constraint: edges and faces must not intersect
Voxel
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Topologically equivalent to Hexahedron
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constraint: each face is perpendicular to a coordinate axis
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“3D pixels”
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Attribute Data
Information associated with data topology
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usually associated to points or cells
Examples :
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temperature, wind speed, humidity, rain fall (meteorology)
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heat flux, stress, vibration (engineering)
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texture co-ordinates, surface normal, colour (computer graphics)
Usually categorised into specific types:
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scalar (1D)
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vector (commonly 2D or 3D; ND in ℝN)
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tensor (N dimensional array)
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Attribute Data : Scalar
Single valued data at each location
ozone levels
elevation from reference plane
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volume density (MRI)
simplest and most common form of visualisation data
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Attribute Data : Vector Data
Magnitude and direction at each location
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3D triplet of values (i, j, k)
Wind Speed
Force / Displacement
Magnetic Field
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Also Normals – vectors of unit length (magnitude = 1)
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Attribute Data : Tensor Data
K-dimensional array at each location
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Generalisation of vectors and matrices
Tensor of rank k can be considered a k-dimensional table
 Rank 0 is a scalar
 Rank 1 is a vector
 Rank 2 is a matrix
 Rank 3 is a regular 3D array
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Example : Tensor Stress
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Visualisation Algorithms
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Generally, classified by attribute type
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scalar algorithms
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vector algorithms
(e.g. glyphs)
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tensor algorithms
(e.g. tensor ellipses)
(e.g. colour mapping)
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Scalar Algorithms
Scalar data : single value at each location
Structure of data set may be 1D, 2D or 3D+
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we want to visualise the scalar within this structure
Two fundamental algorithms
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colour mapping
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contouring
(transformation : value → colour)
(transformation : value transition → contour)
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Colour Mapping
Map scalar values to colour range for display
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e.g.
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scalar value = height / max elevation
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colour range = blue →red
Colour Look-up Tables (LUT)
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provide scalar to colour conversion
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scalar values = indices into LUT
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Colour LUT
Assume
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scalar values Si in range {min→max}
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n unique colours, {colour0... colourn-1} in LUT
Define mapped colour C:
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if Si < min then C = colourmin
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if Si > max then C = colourmax
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LUT
colour0
Si
colour1
C
colour2
else
C = colour
S i − min
colourn-1
max− min
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Colour Transfer Function
More general form of colour LUT
A continuous function instead of a discrete LUT
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scalar value S; colour value C
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colour transfer function : f(S) = C
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e.g. define f() to convert densities to realistic
skin/bone/tissue colours
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Colour Spaces - RGB
Colours represented as R,G,B intensities
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3D colour space (cube) with axes R, G and B
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each axis 0 → 1
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Black = (0,0,0) (origin); White = (1,1,1) (opposite corner)
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Problem : difficult to map continuous scalar range to 3D space
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(below scaled to 0-255 for 1 byte per colour channel)
can use subset (e.g. a diagonal axis) but imperfect
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Colour Spaces - Greyscale
Linear combination of R, G, B
Defined as linear range
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easy to map linear scalar range to grayscale intensity
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can enhance structural detail in visualisation
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The shading effect is emphasized
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as distraction of colour is removed
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not really using full graphics capability
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lose colour associations :
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e.g. red=bad/hot, green=safe, blue=cold
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Example
Winding numbers :
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say we have a curve on a 2D plane
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How much a point is surrounded by the curve
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We can calculate the winding numbers at every location
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Winding numbers
1/4
1
0
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Winding numbers visualized by
grey scale
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Cup with an open top
Star with an open top
A bar
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Color Spaces – RGB
Combining different channels to define a mapping function
B(x)
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R(x)
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Winding numbers visualized by
red and blue
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Cup with an open top
Star with an open top
A bar
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Colour spaces - HSV
Colour represented in H,S,V parametrised space
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H (Hue) = dominant wavelength of colour
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“vibrancy” or purity of colour
V (Value) = brightness of colour
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colour type {e.g. red, blue, green...}
S (Saturation) = amount of Hue present
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commonly modelled as a cone
brightness of the colour
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Colour spaces - HSV
HSV Component Ranges
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Hue = 0 → 360o
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Saturation = 0 → 1
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e.g. for Hue≈blue
0.5 = sky colour; 1.0 = primary blue
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Value = 0 → 1 (amount of light)
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e.g. 0 = black, 1 = bright
All can be scaled to 0→100% (i.e. min→max)
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use hue range for colour gradients
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very useful for scalar visualisation with colour maps
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Example : HSV image components
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Winding numbers by HSV (only
changing hue)
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Cup with an open top
Star with an open top
A bar
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Different Colour Spaces
Visualising gas density in a combustion chamber
− Scalar = gas density
− Colour Map =
A: grayscale
B: hue range blue to red
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B
C: hue range red to blue
D: specifically designed
transfer function
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C
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D
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Color maps from Matlab
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Colour Table/Transfer Function
Design
“More of an art than a science”
Key focus of colour table design
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emphasize important features / distinctions
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minimise extraneous detail
Often task specific
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consider application (e.g. temperature change, use hue red to blue)
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Rainbow colour maps – rapid change in colour hue
representing a ‘rainbow’ of colours.
shows small gradients well as colours change quickly.
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Examples of Manually Designed LUT
3D Height Data
HSV based colour transfer function
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continuous transition of height
represented
8 colour limited lookup table
− discrete height transitions
− Based on human knowledge
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Visualisation : Lecture 5
Summary
Discrete data structure
Topology and interpolation
Topological dimension
Data Representation
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Taku Komura
Structure
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Topology - Cells
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Geometry - points
Attributes
Colour Mapping
Colour Mapping 43