Lecture overview. Visualisatie BMT. Fundamental algorithms. Visualization pipeline. Structural classification - 1. Structural classification - 2

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1 Visualisatie BMT Fundamental algorithms Arjan Kok Lecture overview Classification of algorithms Scalar algorithms Vector algorithms Tensor algorithms Modeling algorithms 1 2 Visualization pipeline Raw Data Data Enrichment/Enhancement Fundamental algorithms Algorithm Method that transforms data from one representation to another representation Visualization algorithms Derived Data Visualization Mapping Abstract Visualization Object Rendering Displayable Image 3 Algorithm classification Structure Which effect(s) has the transformation on the dataset topology Type What is the type of the input dataset 4 Structural classification Geometric transformations Change geometry Do not change topology Examples: rotate, translate, scale points of dataset 2. Topological transformations Change topology Do not change geometry Example: change from polygonal data to unstructured grid Structural classification Attribute transformations Change data attributes Add data attributes Do not change structure (geometry, topology) Example: derive vector magnitude 4. Combined transformations Change both structure (geometry and/or topology) and attributes Example: create contour lines 5 6 1

2 Data type classification 1. Scalar algorithms 2. Vector algorithms 3. Tensor algorithms 4. Modeling algorithms Generate topology, geometry, and/or attributes Algorithms on combined data Scalar algorithms Scalars are single data values associated with each point and/or cell of a dataset temperature = 10.0 pressure = 1200 temperature = 20.3 temperature = Scalar algorithms Color mapping Contouring Scalar generation Color mapping Color mapping Maps scalar data to color Use color lookup table (LUT) Scalar converted to index in LUT 9 10 Color mapping Color mapping Scalar S S < min, index = 0 S > max, index = n - 1 index = n (S min)/(max min) LUT RGB 0 RGB 1 RGB 2 RGB 3 RGB 4 Color (RGB) More general solutions: Transfer function Expression that maps scalar to color specification Continuous version of LUT RGB n-1 Use transparency (alpha) value

3 Color mapping Requirements to color table (transfer function) Should enhance important features Should avoid enhancing unimportant features Should be appealing to observer Should use representation that is known to observer Color mapping A Gray scale table B Blue to red table C Red to blue D Special table that accentuates transitions in data Contouring Contouring Contouring Display regions in dataset with requested scalar (contour) value Generates isolines (2D) or isosurfaces (3D) Basic contouring Basic contouring Contouring algorithms Select a scalar (contour) value Use interpolation to get the attribute values between the cell nodes (on edges) Detect an edge intersection and tracks this contour as it moves across cell boundaries The contour itself is tracked until it closes back on itself or exits a dataset boundary If only one contour exists, then stop If more contours exist, check every edge in the dataset

4 Marching squares Marching squares Computes contour lines in 2D data set Treats each cell independently Assumption: Contour can pass a cell in a finite number of ways, depending on inside/outside relationship with scalar value inside: vertex value larger than scalar outside: vertex value smaller than scalar Table (case table) contains all possible topological states Marching squares Marching squares For every cell 1. Calculate inside/outside state of each vertex of cell 2. Create an index by storing binary state of each vertex in a separate bit 3. Use index to look up the topological state of the cell in a case table 4. Calculate the contour location (via linear interpolation) for each edge in the case table P = (1 u) P(i) + u P(i+1) u = (S D(i)) / (D(i + 1) D(i)) 5. Connect contour locations by line P(i) P P(i+1) 21 Problems Contour ambiguity (cases 5 and 10) 22 3D version of Marching Squares Computes contour surfaces in 3D data set Generates triangles instead of lines Extra step: generate surface normals Classification inside/outside Creation index: 2 8 = 256 cases By using symmetry reduced to 15 topologically different cases

5 Example: case 4 Index = Triangle 1 = e1, e9, e4 Triangle 2 = e6, e7, e Example: case 9 Index = Triangle 1 = e4, e7, e11 Triangle 2 = e1, e7, e4 Triangle 3 = e1, e6, e7 Triangle 4 = e1, e10, e6 For each edge in case table, find vertex intersection position by linear interpolation P = (1 u) P(i)+ u P(i+1) u = (S D(i)) / (D(i + 1) D(i)) P(i) P P(i+1) Normals For each vertex, find the vertex normals from the gradient of the data values D(i + 1, j,k) D(i 1, j,k) x D(i, j+ 1,k) D(i, j 1,k) N(i, j,k) = y D(i, j,k + 1) D(i, j,k 1) z Interpolate normals for edge positions N = (1 u) N(i) + u N(i+1) Marching Cubes Problems Ambiguous cases not easily solved Danger of generating holes

6 Marching Cubes Examples Contouring Marching squares Scalar generation Scalar visualization methods simple and effective Sometimes data is not in a form convenient for color mapping and/or contouring E.g. not scalar (e.g vector), mathematical or relation Convert dataset into form we can use Scalar generation General form Map elevation to range [p l, p h ] Elevation scalar s i at point (x i, y i, z i ) computed by normalized dot product: Use scalars with color mapping Example: terrain data 2D dataset At every (x,y) point a z-value gives the elevation above sea level Use z-value as attribute Vector algorithms Vector 2 or 3 dimensional representation of direction and magnitude (e.g. speed, force) force = (1,1) speed = (3,2) force = (2,1) Vector algorithms Hedgehogs Warping Oriented glyphs Displacement plots Time animation Stream lines force = (0, -0.5)

7 Hedgehogs Hedgehogs Draw an oriented (, scaled) line for each vector Line begins at cell node and is oriented along the vector components (v x, v y, v z ) Lines may be colored according to magnitude or other scalar value(s) Arrows may be added to indicate direction of vector Oriented glyphs Hedgehogs and oriented glyphs Instead of line, use a glyph Glyph is 2D or 3D geometric representation E.g. triangle, cone, 3D arrow Hedgehogs and oriented glyphs Problems Projection on 2D Clutter Scaling problems Warping Vector data often associated with motion in form velocity or displacement Warp (deform) geometry according to vector field Each point in space is displaced according to force Geometry modification

8 Displacement plots Shows motion of an object in direction perpendicular to surface normal The displacement (scalar) is computed by the dot product between surface normal and vector Scalar generation Displacement plot Image shows warping and displacement plot at the same time Application: Study of vibration Show regions of motion