discrete data structure and colour mappinghomepages.inf.ed.ac.uk/tkomura/cav/presentation10.pdf ·...

Visualisation : Lecture 5

Discrete Data Structure and Colour MappingColour Mapping

Computer Animation and Visualisation –Lecture 10

Taku Komura

Taku Komura Colour Mapping 1

Institute for Perception, Action & BehaviourSchool of Informatics


Overview

� Discrete data structure

� Topology and interpolation

Topological dimension� Topological dimension

� Data Representation

− Structure � Topology - Cells

� Geometry - points

− Attributes


− Attributes

� Colour Mapping


Discrete Vs. Continuous

� Real World is continuous− eyes designed towards the perception of continuous shape

� Data is discrete� Data is discrete− finite resolution representation of a real world (of abstract) concept

� Difficult to visualise continuous shape from raw discrete sampling

we need topology and interpolation


− we need topology and interpolation


Topology� Topology : relationships within the data invariant

under geometric transformation

− Which vertex is connected with which vertex by an edge?− Which vertex is connected with which vertex by an edge?

− Which area is surrounded by which edges?



Interpolation & Topology

� If we introduce topology our visualisation of discrete data improves


− why ? : because then we can interpolate based on the topology


Interpolation & Topology

� Use interpolation to shade whole cube:


− Interpolation: producing intermediate samples from a discrete representation


Importance of representation

� What happens if we change the representation?


� Discrete data samples remain the same− topology has changed ⇒ effects interpolation ⇒ effects visualisation


Topological Dimension� Topological Dimension: number of independent continuous variables

specifying a position within the topology of the data

− different from geometric dimension (position within general space)

Topological GeometricTopological Geometricpoint 0D 2D/3D e.g. 2D (x,y)

curve 1D 2D/3D e.g. 3D point on curve

surface 2D 3D (in general)

volume 3D 3D e.g. MRI or CT scan

Temporal volume 4D 3D (+ t)



Structure of Data

� Structure has 2 main parts

− topology : “set of properties invariant under certain geometric transformations”geometric transformations”

� determines interpolation required for visualisation

� “shape” of data

− geometry

� instantiation of the topology

OR


� specific position of points in geometric space


Data Representation

� Data objects : structure + value

− referred to as datasets

Structure- Topology- Geometry

—————————Cells, points (grid)

—————————

Consists of

Abstract

Concrete


—————————Data Attributes

—————————Scalars, vectorsNormals, texture

coordinates, tensors etc


Dataset Representation of the Structure

� Points specify where the data is known

− specify geometry in ℝN

Cells allow us to interpolate between points

ℝ

� Cells allow us to interpolate between points

− specify topology of pointsPoint with known attribute data (i.e. colour ≈ temperature)


Cell (i.e. the triangle) over which we can interpolate data.


Cells

� Fundamental building blocks of visualisation

− our gateway from discrete to interpolated data

� Various Cell Types

− defined by topological dimension

− specified as an ordered point list (connectivity list)

− primary or composite cells

composite : consists of one or more primary cells


� composite : consists of one or more primary cells


Zero-dimensional cell types� Vertex

− Primary zero-dimensional cell− Definition: single point

� Polyvertex− Composite zero-dimensional cell

� composite : comprises of several vertex cells

− Definition: arbitrarily ordered set of points



One-dimensional cell types� Line

− Primary one-dimensional cell type− Definition: 2 points, direction is from first to

second point.second point.

� Polyline− Composite one-dimensional cell type− Definition: an ordered set of n+1 points, where

n is the number of lines in the polyline


n is the number of lines in the polyline


Two-dimensional cell types - 1� Triangle

− Primary 2D cell type

− Definition: counter-clockwise ordering of 3 points� order of the points specifies the direction of the surface normal

� Triangle strip

− Composite 2D cell consisting of a strip of triangles

− Definition: ordered list of n+2 points� n is the number of triangles

� Quadrilateral

− Primary 2D cell type


− Definition: ordered list of four points lying in a plane� constraints: convex + edges must not intersect� ordered counter-clockwise defining surface normal


Two-dimensional cell types - 2� Pixel

− Primary 2D cell, consisting of 4 points� topologically equivalent to a quadrilateral

constraints: perpendicular edges; axis aligned� constraints: perpendicular edges; axis aligned

− numbering is in increasing axis coordinates

� Polygon− Primary 2D cell type− Definition: ordered list of 3 or more points

� counter-clockwise ordering defines surface normal� constraint: may not self-intersect


� constraint: may not self-intersect


Three-dimensional cell types� Tetrahedron

− Definition: list of 4 non-planar points

� Six edges, four faces

� Hexahedron

− Definition: ordered list of 8 points

� six quadrilateral faces, 12 edges, 8 vertices

� constraint: edges and faces must not intersect

� Voxel

− Topologically equivalent to Hexahedron


− constraint: each face is perpendicular to a coordinate axis

− “3D pixels”


Attribute Data� Information associated with data topology

− usually associated to points or cells

� Examples :

− temperature, wind speed, humidity, rain fall (meteorology)

− heat flux, stress, vibration (engineering)

− texture co-ordinates, surface normal, colour (computer graphics)

� Usually categorised into specific types:

− scalar (1D)

− vector (commonly 2D or 3D; ND in ℝN)


− vector (commonly 2D or 3D; ND in ℝN)

− tensor (N dimensional array)


Attribute Data : Scalar

� Single valued data at each location


− simplest and most common form of visualisation data

volume density (MRI)elevation from reference plane

ozone levels


Attribute Data : Vector Data

� Magnitude and direction at each location

− 3D triplet of values (i, j, k)− 3D triplet of values (i, j, k)

Wind Speed


− Also Normals – vectors of unit length (magnitude = 1)

Magnetic Field

Force / DisplacementWind Speed


Attribute Data : Tensor Data

� K-dimensional array at each location

− Generalisation of vectors and matrices− Tensor of rank k can be considered a k-dimensional table− 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


� Example : Tensor Stress


Visualisation Algorithms� Generally, classified by attribute type

− scalar algorithms (e.g. colour mapping)

− vector algorithms (e.g. glyphs)− vector algorithms (e.g. glyphs)

− tensor algorithms (e.g. tensor ellipses)



Scalar Algorithms

� Scalar data : single value at each location

� Structure of data set may be 1D, 2D or 3D+� Structure of data set may be 1D, 2D or 3D+

− we want to visualise the scalar within this structure


� Two fundamental algorithms

− colour mapping (transformation : value → colour)

− contouring (transformation : value transition → contour)


Colour Mapping

� Map scalar values to colour range for display− e.g.

� scalar value = height / max elevation� scalar value = height / max elevation

� colour range = blue →red

� Colour Look-up Tables (LUT)

− provide scalar to colour conversion

− scalar values = indices into LUT



Colour LUT� Assume

− scalar values Si in range {min→max}

− n unique colours, {colour0... colourn-1} in LUT0 n-1

� Define mapped colour C :

− if Si < min then C = colourmin

− if Si > max then C = colourmax

− else

colour0

colour1colour2

SiC

LUT


− else

colourn-1

C= colour�S

i− min

max− min�


Colour Transfer Function� More general form of colour LUT

� A continuous function instead of a discrete LUT

− scalar value S; colour value C− scalar value S; colour value C

− colour transfer function : f(S) = C

− e.g. define f() to convert densities to realistic skin/bone/tissue colours


skin/bone/tissue colours


Colour Spaces - RGB� Colours represented as R,G,B intensities

− 3D colour space (cube) with axes R, G and B

− each axis 0 → 1 (below scaled to 0-255 for 1 byte per colour channel)

− Black = (0,0,0) (origin); White = (1,1,1) (opposite corner)

− Problem : difficult to map continuous scalar range to 3D space


− Problem : difficult to map continuous scalar range to 3D space

� can use subset (e.g. a diagonal axis) but imperfect


Colour Spaces - Greyscale� Linear combination of R, G, B

� Defined as linear range

− easy to map linear scalar range to grayscale intensity

− can enhance structural detail in visualisation� The shading effect is emphasized

� as distraction of colour is removed

− not really using full graphics capability

− lose colour associations :


− e.g. red=bad/hot, green=safe, blue=cold


Example� Winding numbers :

− say we have a curve on a 2D plane

− How much a point is surrounded by the curve− How much a point is surrounded by the curve� Winding numbers

− We can calculate the winding numbers at every location


� 1/4 1 0


Winding numbers visualized by grey scale

� Cup with an open top Star with an open top

� A bar


� A bar


Color Spaces – RGB Combining different channels to define a mapping function

B(x) R(x)



Winding numbers visualized by red and blue


� A bar


� A bar


Colour spaces - HSV � Colour represented in H,S,V parametrised space

− commonly modelled as a cone

� H (Hue) = dominant wavelength of colour

− colour type {e.g. red, blue, green...}

� S (Saturation) = amount of Hue present

− “vibrancy” or purity of colour

V (Value) = brightness of colour


� V (Value) = brightness of colour

− brightness of the colour


Colour spaces - HSV � HSV Component Ranges

− Hue = 0 → 360o

− Saturation = 0 → 1

� e.g. for Hue≈blue

0.5 = sky colour; 1.0 = primary blue

− Value = 0 → 1 (amount of light)

� e.g. 0 = black, 1 = bright

� All can be scaled to 0 →100% (i.e. min→max)

use hue range for colour gradients


− use hue range for colour gradients

− very useful for scalar visualisation with colour ma ps


Example : HSV image components



Winding numbers by HSV (only changing hue)


� A bar



Different Colour Spaces� Visualising gas density in a combustion chamber

− Scalar = gas density

− Colour Map =

A BA: grayscale

B: hue range blue to red

C: hue range red to blue

D: specifically designedtransfer function

� highlights contrast


C D

� highlights contrast


Color maps from Matlab



Colour Table/Transfer Function Design

� “More of an art than a science”

� Key focus of colour table design

− emphasize important features / distinctions

− minimise extraneous detail

� Often task specific

− consider application (e.g. temperature change, use hue red to blue)


− consider application (e.g. temperature change, use hue red to blue)

− Rainbow colour maps – rapid change in colour hue

representing a ‘rainbow’ of colours.

shows small gradients well as colours change quickly.


Examples of Manually Designed LUT3D Height Data


HSV based colour transfer function

− continuous transition of height represented

8 colour limited lookup table

− discrete height transitions

− Based on human knowledge


Summary

� Discrete data structure

� Topology and interpolation

Topological dimension� Topological dimension

� Data Representation

− Structure � Topology - Cells

� Geometry - points

− Attributes


− Attributes

� Colour Mapping

discrete data structure and colour mappinghomepages.inf.ed.ac.uk/tkomura/cav/presentation10.pdf ·...

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