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10.1 Vis_04 Data Visualization Lecture 10 Flow Visualization – Part 2 - Image-based Methods - Critical Point Methods

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10.2 Vis_04 Flow Visualization - Texture Effects n A new class of image-based methods attempts to visualize flow as a texturing effect n Most successful for 2D flow - and also for flow over surfaces in 3D n Methods include: – spot noise – line integral convolution - lic

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10.3 Vis_04 Spot Noise for Flow Visualization n Spots of random size and intensity drawn in a plane give a texture effect Texture defined as an intensity function f: f( x ) = a i h( x - x i ) where x i is random position, a i is random scale (zero mean), and h is the spot function - zero everywhere except for small area (here circular) one spotmany spotsspot texture

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10.4 Vis_04 Spot Noise for Flow Visualization n Different textures result from different spot shapes n Aligning the shape of the spot with the direction of flow gives a good visualization effect n In direction of flow, scale proportional to ( 1 + |v | ), |v| = velocity magnitude n At 90 degrees to flow, scale proportional to 1 / ( 1 + | v | )

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10.5 Vis_04 Spot Noise Example

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10.6 Vis_04 Flow Over a Surface Wall friction displayed using oil and paint - wind evaporates oil and paint leaves white traces Numerical simulation of flow, visualized using spot noise

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10.7 Vis_04 Spot Noise Example

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10.8 Vis_04 Spot Noise Movie

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10.9 Vis_04 Learning More about Spot Noise n Spot noise has been developed by researchers in the Netherlands – van Wijk and de Leeuw – see – Thanks to Wim de Leeuw for the images used in these slides – Thanks to Jack van Wijk for the movie –

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10.10 Vis_04 Line Integral Convolution (LIC) n Essence of method is: – consider a white noise texture, T(x,y) – for each pixel, set its intensity as a function (eg average) of values of T along a short streamline segment through the pixel – this has effect of correlating the resulting pixel values along streamlines, so a sense of the flow direction is obtained white noise flow lines LIC

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10.11 Vis_04 LIC Example Flow over surface of car - from CIRA, Italy Italian Aerospace Research Centre

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10.12 Vis_04 LIC Example Flow underneath car - from CIRA, Italy

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10.13 Vis_04 LIC Movie

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10.14 Vis_04 LIC Developments - Oriented LIC n Original LIC shows direction of flow but not orientation (ie -> or <- ) n Oriented LIC uses a sparse texture and a weighting of samples along streamline to give orientation effect

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10.15 Vis_04 Image-based Methods over Surfaces

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10.16 Vis_04 Learning More about LIC and Image-based Flow Vis n Original LIC – B Cabral and C Leedom, Imaging Vector Fields Using Line Integral Convolution, SIGGRAPH93, ACM Computer Graphics, pp , 1993 n Oriented LIC – R Wegenkittl and E Groller – n Image-based flow visualization generally – Jack van Wijk – thanks to Jack for the surface based movies

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10.17 Vis_04 Vector Field Topology n This approach aims to visualize only the significant features of a flow field n It identifies critical points – points where velocity magnitude is zero – point of repulsion, attraction or a saddle point – streamlines from critical points divide space into regions of similar behaviour

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10.18 Vis_04 Characterising a Critical Point n Let u = velocity in x; v = velocity in y n Look at Jacobian matrix: du / dx du / dy dv / dx dv / dy The critical points are characterised by the eigenvalues of this matrix: a 1 + i b 1 a 2 - i b 2 partial derivatives

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10.19 Vis_04 Characterising a Critical Point n Sign of real part indicates: – repulsiona 1, a 2 positive – attractiona 1, a 2 negative – saddlea 1, a 2 opposite signs – centrea 1, a 2 zero n Imaginary part indicates rotation of flow about critical point: – no rotationb 1, b 2 zero(node) – rotationb 1,b 2 non-zero (focus)

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10.20 Vis_04 Attachment and Detachment Points n There are also critical points along surfaces, where streamlines start ( detachment points) or end ( attachment points) n The flow field topology is produced by: – identifying critical points – drawing streamlines from detachment or attachment points and saddles (4 from saddles)... to repulsors and attractors – drawing streamlines to/from critical points that exit boundary

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10.21 Vis_04 Vector Field Topology n In 3D, similar analysis can be carried out - we get stream surfaces separating flow field into uniform regions n Reading: – J Helman and L Hesselink, Representation and Display of Vector Field Topology in Fluid Flow Data Sets, in Visualization in Scientific Computing, IEEE Press 1990 – other/topology

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10.22 Vis_04 Flow Topology on Surface

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10.23 Vis_04

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