Scientific Visualization Advanced Algorithms Anders Hast tisdag 24 september 13 Informationsteknologi Outline Dividing Cubes Carpet Plots Clipping (Marching Triangles) Vector Algorithms Tensors Data Extraction Triangle Strips Shading Edge Decimation Delaunay Multidimensional Visualizaion (Parallel Axis) Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Dividing Cubes Generates points (not triangles or lines) A large number of points looks like a surface i.e. density of points must be euqal to or greater than screen resolution Motivation Rendering Points is much faster than rendering Polygons Specialized Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 hardware exists... Informationsteknologi Result Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 http://www.csee.umbc.edu/~ebert/693/THu/dividingcubes.html Informationsteknologi Algorithm (procedural) Normals are computed from gradients No topology No Scan Conversion Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Normals The Gradient is simply the difference Fu=(P1 - P0) A normal can always be found using the cross product of two vectors! N = Fu X Fv = (-Fu, -Fv,1) u N N Fu Informationsteknologi Fv P1 P0 Fu Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 v Informationsteknologi The Cross Product The cross product computes a vector orthogonal to both the vectors involved N A B Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Calculating Normal Vectors for Triangles Use the Cross product P0 P1 P2 Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Vertex Normals Gouraud But introduced one simple method there are many possible approaches Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Recursive Algorithm Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Subdivide until the size of the subvoxel is less than or euqal to the screen resolution Can generate far more points than the procedural method Informationsteknologi More Examples (Computer Graphics) Marc Alexa et al. Laser scanners produce point clouds.... Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Splatting Old technique... Let each point correspond to a footprint, which is projected onto the screen through Ray casting Accumulate footprints in screen buffer Phong Splatting Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Carpet Plots 2D Data sets with associated scalar data Warp in the direction of the surface normal depending on the scalar Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Clipping (Marching Triangles) Clipping i used in Computer Graphics to clip parts of triangles that lie outside the view Frustum New Triangles are created in the process Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Eight cases Create new triangles along the cut Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Vector Algorithms Streamlines can be used to depict particle paths in a vector field Others are Stream Ribbons Stream Tubes These can depict other carachteristics of the data due to their more complex shapes e.g. Stream Vorticity Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Stream Vorticity Vorticity is the measure of rotation of the vector field Can be depicted by a ribbon The Vorticity is visible in the places where the ribbon tvist Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Stream Tubes Colour can depict other variables in Streamlines i.e. Not only direction, but one more variable like speed, pressure or heat, etc... When using Streamtubes also thickness/radius can depict yet another variable Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Tensors Nothing to be afraid of... It is simply a multidimensional array! order = dimensions 0 Scalar 1 Vector 2 Matrix 3 Cube etc Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Data Extraction Necessary when we want to visualize portions of the data Or when it is necessary to reduce the size of the data Huge datasets take time to visualize Therefore it can be necessary to sample the data set Compare to an image which is downscaled in size The smaller image still contains the same ”image” Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Methods Subsampling Extract every n’th point of data vtkExtractGrid vtkExtractVOI Thresholding Extract points based on data attributes < == <= >= < > Probing Resampling The result is a new data set with the topology an geometric structure of the probe Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Triangle Strips Makes Rendering Faster Methods like Marching Cubes creates MANY polygons However, todays GPU’s are fast... really fast! Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Shading of Polygons The flat polygons can appear smooth using some shading algorithm The polygon/vertex Normal is used Phong Gouraud Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Diffuse Light/ Lambert’s Law of Cosines N L Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi The problem with Normals Original faceted Model Will be smoother when rendered... Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Naive Normal Computation Normals are computed from the mesh Normal Polygon A Vertex Normal Normal Polygon B Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Split-Edge Normals The edge is plit into two vertices sharing the same position Polygon A Vertex Normal for B Vertex Normal for A Polygon B The model must contain information about these edges Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Edge Decimation Some Edges can be removed in order to obtain a simpler mesh But sharp edges you want to preserve must not be altered Delete points in flat neighbourhoods Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Delaunay Triangulation Can be used to triangulate point clouds Find empty Circum Circles Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Voronoi Diagram Connects centers of circumcircles Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Voronoi Diagram Any point in a cell is closer to the triangle vertex inside the cell than to any other vertex outside the cell Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Multidimensional Visualization Sometimes data points contain MANY data Example: How to choose a car? New? Strong? Fast? Large? Beautiful? How to Visualize many parameters? Use Parallel Axis Coordinates Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Car Example Each car have 5 parameters (in this case) Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 http://eagereyes.org/techniques/parallel-coordinates Informationsteknologi Brushing Keep 80-82 only Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Change Keep 70-72 Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Correlation How to interpret the lines? There can be important Correlations! http://tex.stackexchange.com/questions/73800/tikz-drawing-the-same-data-with-scatter-plots-and-parallel-coordinates Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13 Informationsteknologi Conclusion There are many Powerful techniques for Visualizing data Choose the one that fits the purpose of your visualiation Do not show too many data in every point Use Parallel Coordinate axis visualization when you have many variables in each data point Institutionen för informationsteknologi | www.it.uu.se tisdag 24 september 13
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