DATA VISUALIZATION. Lecture 10--Scientific Visualization

Transcription

1 DATA VISUALIZATION Lecture 10--Scientific Visualization Visualization Process Scalar 1D Data Scalar 2D Data Lin Lu

2 Contents The Visualization Process General Classification Scheme Overview of Visualization Techniques Visualization Techniques -One Dimensional Scalar Data Visualization Techniques -Two Dimensional Scalar Data 2D Interpolation - Scattered Data Surface Views

3 A Simple Example This table shows the observed oxygen levels in the flue gas, when coal undergoes combustion in a furnace TIME (mins) OXYGEN (%)

4 Visualizing the Data - but is this what we want to see?

5 Estimating behaviour between the data - but is this believable?

6 Now it looks believable but something is wrong

7 At least this is credible..

8 What Have We Learnt? It is not only the data that we wish to visualize - it is also the bits inbetween! The data are samples from some underlying field which we wish to understand First step is to create from the data a best estimate of the underlying field - we shall call this a MODEL This needs to be done with care and may need guidance from the scientist The process of fitting a continuous curve (or surface, or volume) through given data is known as INTERPOLATION

9 Data Enrichment This process is sometimes called data enrichment or enhancement If data is sparse, but accurate, we INTERPOLATE to get sufficient data to create a meaningful representation of our model If sparse, but in error, we may need to APPROXIMATE

10 The Visualization Process Overall the Visualization Process can be divided into four logical operations: DATA SELECTION: choose the portion of data we want to analyse DATA ENRICHMENT: interpolating, or approximating raw data - effectively creating a model MAPPING: conversion of data into a geometric representation RENDERING: assigning visual properties to the geometrical objects (eg colour, texture) and creating an image

11 Back to the Simple Example Data Select Enrich Map Render Extract part of data we are interested in Interpolate to create model Select a line graph as technique and create line segments from enriched data Draw line segments on display in suitable colour, line style and width

12 Classification of mapping techniques The mapping stage is where we decide which visualization technique to apply to our enriched data There are a bewildering range of these techniques - how do we know which to choose? First step is to classify the data into sets and associate different techniques with different sets.

13 Back to the Simple Example The underlying field is a function F(x) F represents the oxygen level and is the DEPENDENT variable x represents the time and is the INDEPENDENT variable It is a one dimensional scalar field because the independent variable x is 1D the dependent variable F is a scalar value

14 General classification scheme The underlying field can be regarded as a function of many variables: say F(x) where F and x are both vectors: F = (F 1, F 2,... F m ) x = (x 1, x 2,... x n ) The dimension is n The dependent variable can be scalar (m=1) or vector (m>1)

15 A Simple Notation This leads to a simple classification of data as: E n S/V So the simple example is of type: E 1 S Flow within a volume can be classed as: E 3 V 3

16 Exercise Can you give suitable techniques for the following classes: E S 1 E S 2 E S 3 E V3 3

17 OVERVIEW OF VISUALIZATION TECHNIQUES Different techniques for different types of data

18 Scalar 1D data: E S 1 Scalar: 1 value 1D: value is measured in terms of 1 other variable The humble graph! Clear overlap here between SciVis and InfoVis A nice example of web-based visualization.

19 Scalar 2D Data: E S 2 Here is yesterday s temperature over USA Can you use a 1D technique for this sort of data? Can you improve this visualization?

20 Scalar 2D Data: E S 2 Here is a surface view of the tsunami For movies, see: html movie ENV

21 Scalar 3D Data: E S 3 As dimension increases, it becomes harder to visualize on a 2D surface Here we see a lobster within resin.. where the resin is represented as semi-transparent Technique known as volume rendering Image from D. Bartz and M. Meissner

22 Vector 2D Data:E V2 2 This is a flow field in two dimensions Simple technique is to use arrows.. What are the strengths and weaknesses of this approach? During the module, we will discover better techniques for this

23 Vector 3D Data:E V3 3 This is flow in a volume Arrows become extremely cluttered Here we are tracing the path of a particle through the volume

24 VISUALIZATION TECHNIQUES SCALAR 1D DATA

25 1D Interpolation -The Problem f x Given (x 1,f 1 ), (x 2,f 2 ), (x 3,f 3 ), (x 4,f 4 ) - estimate the value of f at other values of x - say, x*. Suppose x*=1.75

26 Nearest Neighbour f x Take f-value at x* as f-value of nearest data sample. So if x* = 1.75, then f estimated as 3

27 Linear Interpolation f x Join data points with straight lines- read off f-value corresponding to x*.. in the case that x*=1.75, then f estimated as 2.5

28 Linear Interpolation - Doing the Calculation Suppose x* lies between x 1 and x 2. Then apply the transformation: t = (x*-x 1 )/(x 2 -x 1 ) so that t goes from 0 to 1. f(x*) = (1-t) f 1 + t f 2 t=(1.75-1)/(2-1)=0.75 f(1.75)=0.25*1+0.75*3 =2.5 The functions j(t)=1-t and k(t)=t are basis functions. OR, saving a multiplication: f(x*) = f 1 + t ( f 2 -f 1 ) f(1.75)=1+0.75*(3-1) =2.5

29 Nearest Neighbour and Linear Interpolation Nearest Neighbour Very fast : no arithmetic involved Continuity : discontinuous value Bounds : bounds fixed at data extremes Linear Interpolation Fast : one multiply, one divide Continuity : value only continuous, not slope (C 0 ) Bounds : bounds fixed at data extremes

30 Drawing a Smooth Curve Rather than straight line between points, we create a curve piece f 1 f 2 g 1 g 2 We estimate the slopes g 1 and g 2 at the data points, and construct curve which has these values and these slopes at end-points x 1 x 2

31 Slope Estimation Slopes estimated as some average of the slopes of adjacent chords - eg to estimate slope at x2 f 2 f 1 g 2 g 2 usually arithmetic mean (ie average) of f 3 f 2 -f 1 )/(x 2 -x 1 ) x 1 x 2 x 3

32 Piecewise Cubic Interpolation Once the slopes at x 1 and x 2 are known, this is sufficient to define a unique cubic polynomial in the interval [x 1,x 2 ] f 1 g 1 f(x) = c 1 (x) * f 1 + c 2 (x) * f 2 + h*(d 1 (x) * g 1 -d 2 (x) * g 2 ) f 2 g 2 x 1 x 2 c i (x), d i (x) are cubic Hermite basis functions, h = x 2 x 1.

33 Cubic Hermite Basis Functions Here they are: Again set t = (x - x 1 )/(x 2 x 1 ) c 1 (t) = 3(1-t) 2-2(1-t) 3 c 2 (t) = 3t 2-2t 3 d 1 (t) = (1-t) 2 -(1-t) 3 d 2 (t) = t 2 -t 3 Check the values at x = x 1, x 2 (ie t=0,1)

34 Coal data - cubic interpolation

35 Piecewise Cubic Interpolation More computation needed than with nearest neighbour or linear interpolation. Continuity: slope continuity (C 1 ) by construction - and cubic splines will give second derivative continuity (C 2 ) Bounds: bounds not controlled generally - eg if arithmetic mean used in slope estimation...

36 Shape Control However special choices for slope estimation do give control over shape If the harmonic mean is used 1/g 2 = 0.5 ( 1/ 1 + 1/ 2 ) then we find that f(x) lies within the bounds of the data

37 Coal data keeping within the bounds of the data

38 Rendering Line Graphs The final rendering step is straightforward We can assume that the underlying graphics system will be able to draw straight line segments Thus the linear interpolation case is trivial For curves, we do an approximation as sequence of small line segments

39 VISUALIZATION TECHNIQUES SCALAR 2D DATA

40 等值线定义 二维标量场可以看成定义于某一面上的一维标量函数, F=F(x,y) 值为 F t 的等值线定义为所有满足 F(x i,y i )=F t 的点 (x i,y i ) 组成 标量数据定义的网格不同, 也导致了不同的等值线抽取算法的不同 : 正规化网格,Regular grid, 正交网格 非正规化网格,Irregular grid, 三角形网格 按网格处理次序分, 可分为网格序列法 网格无关法

41 2D Interpolation - Rectangular Grid Suppose we are given data on rectangular grid: f given at each grid point; data enrichment fills out the empty spaces by interpolating values within each cell

42 Nearest Neighbour Interpolation Straightforward extension from 1D: take f-value from nearest data sample No continuity Bounds fixed at data extremes

43 Bilinear Interpolation Consider one grid rectangle: suppose corners are at (0,0), (1,0), (1,1), (0,1)... ie a unit square values at corners are f 00, f 10, f 11, f 01 How do we estimate value at a point (x,y) inside the f 00 f 10 square?

44 Bilinear Interpolation f 01 f 11 (x,y) f 00 f 10 We carry out three 1D interpolations: (i) interpolate in x-direction between f 00,f 10 ; and f 01,f 11 (ii) interpolate in y-direction Exercise: Show this is equivalent to calculating - f(x,y) = (1-x)(1-y)f 00 +x(1-y)f 10 +(1-x)yf 01 + xyf 11

45 Piecewise Bilinear Interpolation Apply within each grid rectangle Fast Continuity of value, not slope (C 0 ) Bounds fixed at data extremes

46 Contour Drawing Contouring is very common technique for 2D scalar data Isolines join points of equal value sometimes with shading added How can we quickly and accurately draw these isolines?

47 An Example As an example, consider this data: Where does the zero level contour go?

48 Intersections with sides The bilinear interpolant is linear along any edge - thus we can predict where the contour will cut the edges (just by simple proportions)

49 Simple Approach A simple approach to get the contour inside the grid rectangle is just to join up the intersection points 10-5 Question: Does this always work? 1-2 Try an example where one pair of opposite corners are positive, other pair negative

50 Return to Example Consider this data: Where does the zero level contour go? Can we draw a straight line which will approximate the contour line?

51 Joining Intersections Directly It is tempting to approximate by joining intersections with straight lines:

52 Ambiguity But this does not always work - look at this data: Try it - it is ambiguous!

53 What is the Problem? We need to worry about the behaviour of the interpolant inside the grid square The contour of the bilinear interpolant is NOT a straight line it is a curve This is curve of: f(x,y) = (1-x)(1-y)f 00 +x(1-y)f 10 +(1-x)yf 01 + xyf 11 = 0 BUT how can we draw it? -2 Joining intersections with straight lines was only an approximation drawing the contour of bilinear interpolant will resolve ambiguity

54 Tracking Contours We can track the contour in small steps through the grid rectangle starting from intersection with the edges Take a step, probe at equal distance to either side, then predict next point Current point on contour Next point on contour 0.9 Probes BUT THIS IS SLOW!! 0.9 Can we find an alternative which uses straight lines, but resolves the ambiguous case?

55 Implementing Rectangle-based Contouring For a rectangle, there will be 2 4 = 16 cases There are 4 configurations All same sign (no contour) 3 same sign (one contour piece) 2 adjacent with same sign (one contour piece) 2 opposite with same sign (two pieces, but ambiguous function has a saddle) Note: the ambiguity can be resolved by looking at value at saddle point: Saddle value = (f 00 f 11 -f 01 f 10 )/ (f 00 +f 11 -f 01 +f 10 )

56 Solution by Decomposing Cell Another possibility is to split cell into four triangles Within a triangle, we can fit a linear model F(x,y) = a + bx +cy How do we split? How do we calculate a,b,c? What is the gain? f 1 f 2 f 3

57 Cell Decomposition Problem of drawing the curved lines has been circumvented by decomposing cell into four pieces within which the contours are well defined straight lines 10-5 How might we estimate value at centre?

58 2D Interpolation - Scattered Data Often the data will be given, not on a regular grid, but at scattered locations: f given at each marked point Approach: (i) triangulate (ii) build interpolant in each triangle (iii) draw contours

59 Triangulation Triangulation is the process of forming a grid of triangles from the data points How can we construct the triangulation?

60 Tessellation We solve the DUAL problem: Suppose a wolf is stationed at each data point. Each wolf is equally powerful and dominates the territory closest to its own base What are the territories dominated by each wolf?

61 Tessellation - Two or Three Wolves

62 Tessellation - Two or Three Wolves

63 Dirichlet Tessellation The resulting tessellation is known as the Dirichlet or Voronoi tessellation Given the Dirichlet tessellation for N points P 1, P 2,... P N there is an algorithm for constructing the tessellation when an extra point is added

64 Dirichlet Tessellation P 2 Tessellation for P 1, P 2, P 3 P 3 P 1 Q Point Q added

65 Dirichlet Tessellation P 2 P 3 P 1 Q

66 Dirichlet Tessellation P 2 P 3 P 1 Q

67 Dirichlet Tessellation Determine polygon containing Q - here D3, surrounding P3 Construct perpendicular bisector of P3Q and find intersection with D3 - this becomes point of modified tessellation Determine adjacent polygon - here D2 Repeat the above two steps until D3 is reached again, or there is no intersection Remove all vertices and edges interior to the new polygon

68 Delaunay Triangulation P 2 P 3 P 1 Q

69 Delaunay Triangulation Triangulation formed by joining points whose territories share a common boundary in the tessellation This has the nice property that it avoids long skinny triangles See the nice applets at: Delaunay.html Note the empty circle property of the Delaunay triangulation

70 Contouring from Triangulated Data The final step is to contour from the triangulated data Easy because contours of linear interpolant are straight lines see earlier

71 Implementing Triangle-based Contouring Each vertex can be positive or negative (ignore zero for now) This gives 2 3 = 8 possible cases but there are only 2 distinct configurations No contour (all same sign) Contour (2 of one sign, 1 of the other) Implementation: Determine which of 8 cases Select code for the appropriate configuration Vertex can be zero,how? f 1 All same sign f 2 f 3 f 1 Two same sign f 2 f 3

72 Surface Views A different mapping technique for 2D scalar data is the surface view. Here a surface is created in 3D space, the height representing the scalar value Construction is quite easy - suppose we have a rectangular grid

73 Constructing a Surface View - 1

74 Constructing a Surface View - 2 Surface created as pair of triangles per grid rectangle. Rendering step is then display of triangles.

75 Examples - with added contours

76 Image Plots A further mapping technique for 2D data is the image plot There are three variants: dot array : draw a dot at each data point, coloured according to the value (very fast, but low quality)

77 Image Plots Grid lines:

78 Image Plots Areas:

79 Cross Sections Another option is to look at a cross-section through the data For example, if x and y are the independent variables, we could fix y and look at f in terms of just x then repeat for different y this reduces the E S 2 problem to a sequence of E S 1 problems