Visualisation : Lecture 4

Data Representation in Visualisation

Visualisation – Lecture 4 Taku Komura

Institute for Perception, Action & Behaviour School of Informatics

Taku Komura Data Representation 1 Visualisation : Lecture 4 Data Representation

● We have a system architecture for visualisation – the visualisation pipeline – rendering techniques ● Data is ......

– discrete (in representation) – structured / unstructured (e.g. grid / cloud) – of a specific D (e.g. 2D / 3D)

Taku Komura Data Representation 2 Visualisation : Lecture 4 Discrete vs. Continuous

● Real World is continuous ● Data is discrete – Computers are good in handling discrete data – discrete representation of a real world (of abstract) concept

● Difficult to visualise continuous shape from raw discrete sampling?

– we need topology and interpolation

Taku Komura Data Representation 3 Visualisation : Lecture 4 Interpolation & Topology

● If we introduce topology our visualisation of discrete data improves

– The topology helps the visual system to do the interpolation

Taku Komura Data Representation 4 Visualisation : Lecture 4 Topology

● Topology : relationships within the data invariant under geometric transformation (i.e. Orientation, translation)

i.e. the information which vertex is connected to which edge, which face is composed of which edges , etc.

Taku Komura Data Representation 5 Visualisation : Lecture 4 Interpolation & Topology

● What if these colour represented spatial temperature at these 8 discrete points?

– e.g. colour scale blue(=cold)→green→red( = hot)

– How easy is it to visualise the temperature field over the whole cube?

Taku Komura Data Representation 6 Visualisation : Lecture 4 Interpolation & Topology

● Use interpolation to shade whole cube:

– Interpolation: increasing resolution of a discrete representation by producing intermediate samples

— in example: producing intermediate colour pixels over cube topology from discrete vertex samples

Taku Komura Data Representation 7 Visualisation : Lecture 4 How to interpolate over a ?

● Will the red color dot on the right lower corner affect the color of the near the left top?

Taku Komura Data Representation 8 Visualisation : Lecture 4 Importance of representation

● What happens if we change the representation?

● Discrete data samples remain the same – topology has changed ⇒ effects interpolation ⇒ effects visualisation

Taku Komura Data Representation 9 Visualisation : Lecture 4 How to interpolate in this case?

● Will the red color dot on the right lower corner affect the color of point near the left top?

Taku Komura Data Representation 10 Visualisation : Lecture 4 Interpolation in digital images ● Some digital cameras use interpolation to produce a larger image than the sensor captured or to create digital zoom. ● Also using the topology data

No interpolation

Taku Komura With interpolation Data Representation 11 Visualisation : Lecture 4 Topological Dimension

• Data has an inherent topological dimension – I.e minimum number of independent continuous variables needed to specify the location inside the data – Points : 0D, – curves : 1D, – surfaces : 2D, – volumes : 3D – Time dependent volumes : 4D

Taku Komura Data Representation 12 Visualisation : Lecture 4 Data Representation

● Data objects : structure + value – referred to as datasets

Abstract

Concrete (e.g. VTK) Structure - Topology Consists of - Cells, points (grid) ————————— ————————— Data Attributes Scalars, vectors Normals, texture coordinates, tensors etc

Taku Komura Data Representation 13 Visualisation : Lecture 4 What is a dataset?

● Dataset consists of 2 main components – structure of the data – value – attributes associated to particular parts of the structure – structure gives spatial meaning to the attributes

values = { blue, green, green, green, green, green, turquoise, red}

–

– – attributes alone are meaningless without structure

Taku Komura Data Representation 14 Visualisation : Lecture 4 Structure of Data

● Structure has 2 main parts – topology :

— determines interpolation required for visualisation — “shape” of data

OR – geometry

— instantiation of the topology — specific position of points in geometric space

Taku Komura Data Representation 15 Visualisation : Lecture 4 Concrete Representation of Datasets ● Points specify where the data is known – specify geometry in ℝN ● Cells allow us to interpolate between points – specify topology of points

Point with known attribute data (i.e. colour ≈ temperature)

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

Taku Komura Data Representation 16 Visualisation : Lecture 4 Cells

● Fundamental building blocks of the shapes

● Various Cell Types – defined by topological dimension – specified as an ordered point list – primary or composite cells

— composite : consists of one or more primary cells

Taku Komura Data Representation 17 Visualisation : Lecture 4 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

Taku Komura Data Representation 18 Visualisation : Lecture 4 One-dimensional cell types

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

Taku Komura Data Representation 19 Visualisation : Lecture 4 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

● – Primary 2D cell type – Definition: ordered list of four points lying in a plane

— constraints: convex + edges must not intersect

Taku Komura Data Representation 20 Visualisation : Lecture 4 Two-dimensional cell types - 2

● Pixel – Primary 2D cell, consisting of 4 points — topologically equivalent to a quadrilateral — constraints: perpendicular edges; axis aligned – numbering is in increasing axis coordinates

● Polygon – Primary 2D cell type – Definition: ordered list of 3 or more points

— constraint: may not self-intersect

Taku Komura Data Representation 21 Visualisation : Lecture 4 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”

Taku Komura Data Representation 22 Visualisation : Lecture 4 Example Applications & Cell Types

● Lines – Segmentation of tissues from images ● Pixels – images, regular height map data ● – Finite element analysis ● Voxels : “3D pixels” – volume data : medical scanners

Taku Komura Data Representation 23 Visualisation : Lecture 4 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) – surface normal, colour (computer graphics) ● Usually categorised into specific types: – scalar (1D) – vector (commonly 2D or 3D; ND in ℝN) – tensor (N dimensional array)

Taku Komura Data Representation 24 Visualisation : Lecture 4 Attribute Data : Scalar

● Single valued data at each location

ozone levels

elevation from reference plane – volume density (MRI) – simplest and most common form of visualisation data

Taku Komura Data Representation 25 Visualisation : Lecture 4 Attribute Data : Vector Data

● Magnitude and direction at each location – 3D triplet of values (i, j, k)

Wind Speed Force / Displacement Magnetic Field – Also Normals – vectors of unit length (magnitude = 1)

Taku Komura Data Representation 26 Visualisation : Lecture 4 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 — Rank 0 is a scalar — Rank 1 is a vector — Rank 2 is a matrix — Rank 3 is a regular 3D array

● Tensor visualisation is an active area of research – covered later in lectures

Taku Komura Data Representation 27 Visualisation : Lecture 4 Types of Dataset

● Defined by structure type: regular or irregular

● If there is a mathematical relationship between the point & cell positions it is regular – if the points are regular, the geometry is regular – if the cells are regular, the topology is regular ● Regular data can be implicitly represented

– saves storage and computation (e.g. grid based representation)

Taku Komura Data Representation 28 Visualisation : Lecture 4 Regular : Structured Points

● Points & cells arranged in a regular grid – axis aligned grid – for line elements (1D), pixels (2D) or voxels (3D) – e.g. images (2D), medical scanners (3D) – regular topology and geometry

— uniform, equally spaced, axis aligned cells

Taku Komura Data Representation 29 Visualisation : Lecture 4 Regular : Rectilinear Grid

● Points and cells arranged in a regular grid – topology is regular – geometry is partially regular – points are arranged along the axes but the spacing may vary – e.g. log-scale

Taku Komura Data Representation 30 Visualisation : Lecture 4 Example : Global climate model

● Could use a regular latitude-longitude grid to represent earth?

Taku Komura Data Representation 31 Visualisation : Lecture 4 Example : earth is a sphere

● Problem : as we approach the pole, cell size gets smaller

Taku Komura Data Representation 32 Visualisation : Lecture 4 Example : Rectilinear Grid

Scale latitude line spacing so that cells represent an equal area of the globe.

Taku Komura Data Representation 33 Visualisation : Lecture 4 Semi-Regular : Structured Grid

● Regular topology Tetrahedral Hexahedral ● Irregular geometry (completely)

● surface triangulations are structured grids – common topology = triangles

● Not all polygon surfaces have regular topology

Taku Komura Data Representation 34 Visualisation : Lecture 4 Irregular : Unstructured Points

● Points irregularly located in space ● No topology ● No structured geometry – e.g. sparse measurements of temperature etc. ● Difficult to visualise – e.g. unstructured point clouds – Can be from 3D scanners Need surface reconstruction from unstructured points clouds

— topology recovery

Taku Komura Data Representation 35 Visualisation : Lecture 4 Irregular : Unstructured grid

● Both topology and geometry unstructured – can range from 0D to 3D topologies – general, flexible, inefficient to store – e.g. finite element analysis

— Using higher resolution grids near where precise simulation is needed, rough grids for regions less important

Taku Komura Data Representation 36 Visualisation : Lecture 4 Polygonal Data - rendering

● Consists of points, lines, polygons – graphics primitives – irregular geometry – irregular topology

– Usually irregular (modelled by humans, or sticking regular data together) – used for rendering

Taku Komura Data Representation 37 Visualisation : Lecture 4 Summary

● Need for topology in data ● Datasets : structure + value – structure = topology & geometry – value = attribute data – cell types in visualisation pipeline ● Types of Attribute Data – scaler, vector, tensor ● Types of Dataset – regular or irregular

Taku Komura Data Representation 38