Data Representation in Visualisation

Transcription

1 Data Representation in Visualisation Visualisation Lecture 4 Taku Komura Institute for Perception, Action & Behaviour School of Informatics Taku Komura Data Representation 1

2 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 dimension D (e.g. 2D / 3D) Taku Komura Data Representation 2

3 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

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

5 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

6 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

7 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

8 How to interpolate over a rectangle? Visualisation : Lecture 4 Will the red color dot on the right lower corner affect the color of the point near the left top? Taku Komura Data Representation 8

9 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

10 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

11 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 Data Representation 11 With interpolation

12 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

13 Data Representation Data objects : structure + value referred to as datasets Abstract Structure - Topology - Geometry Data Attributes Consists of Concrete (e.g. VTK) Cells, points (grid) Scalars, vectors Normals, texture coordinates, tensors etc Taku Komura Data Representation 13

14 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

15 Structure of Data Structure has 2 main parts topology : determines interpolation required for visualisation shape of data geometry OR instantiation of the topology specific position of points in geometric space Taku Komura Data Representation 15

16 Concrete Representation of Datasets Points specify where the data is known specify geometry in R N 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

17 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

18 Vertex Zero-dimensional cell types 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

19 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

20 Two-dimensional cell types - 1 Triangle Primary 2D cell type Definition: counter-clockwise ordering of 3 points Triangle strip order of the points specifies the direction of the surface normal Composite 2D cell consisting of a strip of triangles Definition: ordered list of n+2 points Quadrilateral 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

21 Pixel Two-dimensional cell types - 2 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

22 Three-dimensional cell types Tetrahedron Definition: list of 4 non-planar points Six edges, four faces Hexahedron Definition: ordered list of 8 points Voxel six quadrilateral faces, 12 edges, 8 vertices constraint: edges and faces must not intersect Topologically equivalent to Hexahedron constraint: each face is perpendicular to a coordinate axis 3D pixels Taku Komura Data Representation 22

23 Example Applications & Cell Types Lines Segmentation of tissues from images Pixels images, regular height map data Quadrilaterals Finite element analysis Voxels : 3D pixels volume data : medical scanners Taku Komura Data Representation 23

24 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 R N ) tensor (N dimensional array) Taku Komura Data Representation 24

25 Attribute Data : Scalar Single valued data at each location elevation from reference plane ozone levels volume density (MRI) simplest and most common form of visualisation data Taku Komura Data Representation 25

26 Attribute Data : Vector Data Magnitude and direction at each location 3D triplet of values (i, j, k) Force / Displacement Wind Speed Magnetic Field Also Normals vectors of unit length (magnitude = 1) Taku Komura Data Representation 26

27 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

28 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

29 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

30 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

31 Example : Global climate model Could use a regular latitude-longitude grid to represent earth? Taku Komura Data Representation 31

32 Example : earth is a sphere Problem : as we approach the pole, cell size gets smaller Taku Komura Data Representation 32

33 Example : Rectilinear Grid Visualisation : Lecture 4 Scale latitude line spacing so that cells represent an equal area of the globe. Taku Komura Data Representation 33

34 Semi-Regular : Structured Grid Regular topology Irregular geometry (completely) Tetrahedral Hexahedral surface triangulations are structured grids common topology = triangles Not all polygon surfaces have regular topology Taku Komura Data Representation 34

35 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

36 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

37 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

38 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