Advection methods comparison Stream-ribbon � We really would like to see vorticities, I.e. places were the flow twists. � A point primitive or an icon can hardly convey this � idea: trace neighboring particles and connect them with polygons � shade those polygons appropriately and one will detect twists 4/30/2003 R. Crawfis, Ohio State Univ. 4/30/2003 R. Crawfis, Ohio State Univ. 52 53 Stream-ribbon Stream-tube � Problem - when flow diverges � Generate a stream-line and connect � Solution: Just trace one streamline and a circular crossflow sections along the constant size vector with it: stream-line 4/30/2003 R. Crawfis, Ohio State Univ. 4/30/2003 R. Crawfis, Ohio State Univ. 54 55 Stream Ribbons Streamribbons � First of two other line icons ... � Vorticity � Narrow surfaces between two adjacent streamlines � Formally ⌧ Vorticity ω - measure of rotation of vector field � Streamwise vorticity v . ω Ω = |v| . | ω| � where v - instantaneous velocity � Built from a front with only two particles � Flow divergence � Reflect flow divergence through changing width, and � Measure of the spread of the flow � Vortices are shown in the degree of twist in the ribbon 4/30/2003 R. Crawfis, Ohio State Univ. 56 4/30/2003 R. Crawfis, Ohio State Univ. 57 1
Streamtube example Streamtube example • 6 streamlines • 6 streamlines • Streamline added for comparison • Alternative generation • Stream polygon •Rotation of edges � vorticity •Area of polygon � divergence 4/30/2003 R. Crawfis, Ohio State Univ. 4/30/2003 R. Crawfis, Ohio State Univ. 58 59 Streamballs Stream-balls � Another way to get around diverging stream- � Stream balls: lines � Display stream lines as chains of balls and � simply put implicit surface primitives at use their particle traces - at places where they are radius and close they’ll merge elegantly ... � Color to encode scalar values 4/30/2003 R. Crawfis, Ohio State Univ. 4/30/2003 R. Crawfis, Ohio State Univ. 60 61 Global Texture Techniques 2D vector field visualization � Two approaches: � We want to visualize a function � Advection-based Ω � 2 F : I R ⌧ Streamline placement � with F given only at certain vertices � Icon placement or smearing ⌧ Spot Noise x F ↔ = ij F F ⌧ Vector Kernel ij y F ij ⌧ Line Integral Convolution ⌧ Line Bundles 4/30/2003 R. Crawfis, Ohio State Univ. 62 4/30/2003 R. Crawfis, Ohio State Univ. 63 2
Mappings - Hedgehogs, Glyphs 2D vector field visualization � analogous to tufts or vanes from � First idea experimental flow visualization � Visualize the two scalar fields Fx and Fy � clutter the image real quick � As the two components are normally not independent, this usually provides no insight. � maybe ok for 2D � Second idea � not very informative � Data is supposed to represent the direction of moving particles � Visualize the “flow“ � Do it by means of glyphs or particle tracing 4/30/2003 R. Crawfis, Ohio State Univ. 4/30/2003 R. Crawfis, Ohio State Univ. 64 65 2D vector field visualization 2D vector field visualization � Pros and Cons of glyphs � Glyphs � Simple � We display the vectors using arrows as � 3D effects geometric primitives � Heavy load in the graphics subsystem � Draw an arrow at every vertex of the grid � Inherent occlusion effects ⌧ Length corresponds to magnitude of velocity � Poor results if magnitude of velocity changes ⌧ Direction corresponds to flow direction fast � Use arrows of constant length and color code magnitude 4/30/2003 R. Crawfis, Ohio State Univ. 4/30/2003 R. Crawfis, Ohio State Univ. 66 67 Mappings - Hedgehogs, Glyphs Global techniques � Put “icons” at certain glyphs places in the flow - Display the entire flow field in a single picture - Minimum user intervention � e.g. arrows - represent - Example: Hedgehogs (global arrow plots) direction & magnitude � other primitives are possible oriented lines vortex 4/30/2003 R. Crawfis, Ohio State Univ. 68 4/30/2003 R. Crawfis, Ohio State Univ. 69 3
Advection-Based Textures 2D vector field visualization � Streamline placement � Example Banks and Turk 4/30/2003 R. Crawfis, Ohio State Univ. 4/30/2003 R. Crawfis, Ohio State Univ. 70 71 Streamline Placement Icon Placement � Hedgehogs or arrow plots: uniform grid Movie 4/30/2003 R. Crawfis, Ohio State Univ. 4/30/2003 R. Crawfis, Ohio State Univ. 72 73 Icon Placement More artistic � Image-filling and randomized David Laidlaw Banks and Turk 4/30/2003 R. Crawfis, Ohio State Univ. 74 4/30/2003 R. Crawfis, Ohio State Univ. 75 4
Arrows Arrows Arrows Arrows � But in 3D it suffers from perception � Arrows can be used successfully in 3D as problems: follows: � by slicing the volume, and attaching arrows (with shadow effects) to the slice plane - this Is it this? gives a hedgehog effect � by giving more spatial cues - drawing arrows as true 3D objects but clutter again a or this? problem! {BTW - Eulerian or Lagrangian?} Of course the picture quickly gets cluttered too 4/30/2003 R. Crawfis, Ohio State Univ. 4/30/2003 R. Crawfis, Ohio State Univ. 76 77 Example of a complex Glyph Vector Kernel � An image kernel moves in a discrete and jittered path left to right and top to bottom across the image screen. � The kernel samples the scalar and vector fields. � The vector field uses a weighted probability of drawing an anti-aliased line across the kernel. � The line is oriented in the vector field direction. � The line may also be semi-transparent. � The probability is proportional to the vector magnitude. 4/30/2003 R. Crawfis, Ohio State Univ. 4/30/2003 R. Crawfis, Ohio State Univ. 78 79 Spot Noise for Flow Visualization Vector Kernel Spot Noise for Flow Visualization � Spots of random size and intensity drawn in a plane give a texture effect one spot many spots spot texture Texture defined as an intensity function f: f( x ) = Σ 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) 4/30/2003 R. Crawfis, Ohio State Univ. 80 4/30/2003 R. Crawfis, Ohio State Univ. 81 5
Spot Noise for Flow Visualization Spot Noise Example � Different textures result from different spot shapes � Aligning the shape of the spot with the direction of flow gives a good visualization effect � In direction of flow, scale proportional to ( 1 + |v | ) , |v| = velocity magnitude � At 90 degrees to flow, scale proportional to 1 / ( 1 + | v | ) 4/30/2003 R. Crawfis, Ohio State Univ. 4/30/2003 R. Crawfis, Ohio State Univ. 82 83 Spot Noise Spot Noise � examples of white noise: � Uses small motion blurred particles to � set of random values on a grid visualize flows on stream surfaces � Poisson point process - a set of randomly scaled � Particles represented as ellipses with their delta functions randomly placed (dart throwing) long axes oriented along the direction of the � variation of the data visualization can be flow realized via variation of the spot: � I.e. we multiply our kernel h with an ( ) ( ( ( ) ) ) ∑ = − f x a h m d x , x x k k k amplitude and add a phase shift! k d - data value � Hence - we convolve a spot kernel in spatial m - parameter mapping domain with a random sequence (white noise) 4/30/2003 R. Crawfis, Ohio State Univ. 4/30/2003 R. Crawfis, Ohio State Univ. 84 85 Rendering - Spot Noise Rendering - Spot Noise � bla Different size Different profiles 4/30/2003 R. Crawfis, Ohio State Univ. 86 4/30/2003 R. Crawfis, Ohio State Univ. 87 6
Rendering - Spot Noise Spot Noise for Flow Visualization Spot Noise for Flow Visualization scalar gradients � If velocity direction varies rapidly, � Scalar - use +-shape result is not very successful for positive values, x- shape for negative values � change the size of the spot according to the norm of the gradient � vector data - use an Improvement achieved by ‘bending’ spot along the streamline ellipse shaped spot in flow Velocity potential the direction of the flow ... 4/30/2003 R. Crawfis, Ohio State Univ. 4/30/2003 R. Crawfis, Ohio State Univ. 88 89 Flow Over a Surface Spot Noise Example Numerical simulation of flow, visualized using spot noise Wall friction displayed using oil and paint - wind evaporates oil and paint leaves white traces 4/30/2003 R. Crawfis, Ohio State Univ. 4/30/2003 R. Crawfis, Ohio State Univ. 90 91 Spot Noise Movie Learning More about Spot Noise � Spot noise has been developed by researchers in the Netherlands � van Wijk and de Leeuw � see http://www.cwi.nl/~wimc/spotnoise.html � Thanks to Wim de Leeuw for the images used in these slides 4/30/2003 R. Crawfis, Ohio State Univ. 92 4/30/2003 R. Crawfis, Ohio State Univ. 93 7
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