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Data Representation Scientific Visualization Rm Rn Data representation Structures and Classification domain X independent variables scientific data computer graphics & visualization data values dependent variables ⊆ Rn+m Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Data Representation - Discrete representations - The objects we want to visualize are often ‘continuous’ continuous’ - But in most cases, the visualization data is given only at - discrete locations in space and/or time Discrete structures consist of samples, from which grids/meshes consisting of cells are generated computer graphics & visualization Data Representation - Classification of visualization techniques according to - Dimension of the domain of the problem (independent parameters) - Type and dimension of the data to be visualized (dependent parameters) Primitives in multi dimensions dimension 0D 1D 2D 3D cell mesh points lines (edges) triangles, quadrilaterals (rectangles) tetrahedra, prisms, hexahedra polyline(– polyline(–gon) 2D mesh 3D mesh or grid Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Course WS 05/06 – Scientific Visualization computer graphics & visualization Data Representation G 3D F determined by the positions of sample points - Domain is characterized by 1D C D 0D A B 1D 2D - Dimension - Influence - Structure – these are of special interest in this course E 2D Examples: H 3D computer graphics & visualization Domain - The (geometric) shape of the domain is dimension of data type mD Prof. Dr. R. Westermann – Computer Graphics and Visualization Group nD dimension of domain A: gas station along a road B: map of cholera in London C: temperature along a rod D: height field of a continent E: 2D air flow F: 3D air flow in the atmosphere G: stress tensor in a mechanical part H: ozone concentration in the atmosphere Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Course WS 05/06 – Scientific Visualization computer graphics & visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization 1 Domain - Influence of data points - Values at sample points influence the data distribution in a certain region around these samples To reconstruct the data at arbitrary points within the domain, the influence of all samples has to be calculated - Point influence - Local influence - Global influence - Construct a region around each sample point that covers - - Only influence on point itself - VoronoiVoronoi-diagram CellCell-wise interpolation (see later in course) - Each sample might influence any other point within the domain Material properties for whole object Scattered data interpolation Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Course WS 05/06 – Scientific Visualization computer graphics & visualization Domain - Scattered data interpolation Prof. Dr. R. Westermann – Computer Graphics and Visualization Group the domain is computed Weighting functions determine the support of each sample point - The interpolation problem - What is an RBF - Radial basis functions simulate decreasing influence with increasing distance from samples terms of numerical operations A weighted sum of translations of a radially symmetric basis function i =1 Course WS 05/06 – Scientific Visualization computer graphics & visualization Radial Basis Functions - The basis function φ is a real function of positive real r , where r is the radius from the origin Various choices for the basic function - Linear φ (r ) = r Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Given interpolation values f ( f1 , f 2 ,... f N ), we seek the weights λi such that the RBF satisfies s ( xi ) = f i - i = 1,..., N Interpolation condition yields - 2 A ⋅λ = f where ( ) Ai , j = φ xi − x j , r 2 + c 2 (fitting to topographical data) - Course WS 05/06 – Scientific Visualization computer graphics & visualization Radial Basis Functions - - Gaussian φ (r ) = exp(− cr ) (mainly used in neural networks) Prof. Dr. R. Westermann – Computer Graphics and Visualization Group - ( ) Approximate a realreal-valued function f ( x ) by s x given the set of values f ( f1 , f 2 ,... f N ) at the points X = {x1 , x2 ,...x N } ⊂ ℜ d - where λi is a realreal-valued weight, x − x i denote the Euclidean norm between point x and point xi and φ is a basis function Course WS 05/06 – Scientific Visualization - Multiquadric φ (r ) = - - s(x ) is the Radial Basis Function of the form N s (x ) = ∑ λiφ ( x − xi ) x ∈ ℜd - Schemes might be nonnon-interpolating and expensive in Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization Radial Basis Functions - At each point the weighted average of all sample points in - all points that are closer to that sample than to every other sample Each point within a certain region gets assigned the value of the sample point - Only within a certain region - - Domain - VoronoiVoronoi-diagram i , j = 1,..., N Solving the linear system determines the value of c and λ , thus obtaining the RBF Course WS 05/06 – Scientific Visualization computer graphics & visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization 2 Domain - Example Data Structures - Requirements: - Convenience of access - Space efficiency - Lossless vs. lossy - Portability - Radial basis functions with increasing support - binary – less portable, more space/time efficient - text – human readable, portable, less space/time efficient - Definition - If points are arbitrarily distributed and no connectivity exists between them, the data is called scattered - Otherwise, the data is composed of cells bounded by grid lines - Topology specifies the structure (connectivity) of the data Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Course WS 05/06 – Scientific Visualization computer graphics & visualization Data Structures - Some definitions concerning topology and geometry - In topology qualitative questions about geometrical Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization Data Structures - Topology - Properties of geometric shapes that remain unchanged even when under distortion structures are the main concern. - Does it have any holes in it - Is it all connected together - Can it be separated into parts - Underground map shows how lines a connected, but not how far one station is from another Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Same geometry (vertex positions) Same topology of surface Different topology of triangles Course WS 05/06 – Scientific Visualization computer graphics & visualization Data Structures - Topologically equivalent - Things that can be transformed into each other by stretching and squeezing, without tearing or sticking together bits which were previously separated Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization Data Structures - Grid types - Grids differ substantially in the simplicial elements (or cells) they are constructed from and in the way the inherent topological information is given topologically equivalent Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Course WS 05/06 – Scientific Visualization computer graphics & visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization 3 Data Structures - Structured and unstructured grids can be - distinguished by the way the elements or cells meet Structured grids - Have a regular topology and regular / irregular geometry - Unstructured grids - Have irregular topology and geometry Data Structures - Cartesian or equidistant grids - Structured grid - Cells and points are numbered sequentially with respect to increasing X, then Y, then Z, or vice versa Number of points = Nx• Nx•Ny• Ny•Nz Number of cells = (Nx(Nx-1)• 1)•(Ny(Ny-1)• 1)•(Nz(Nz-1) Y 3 P[i,j,(k)] 2 1 dy 0 0 Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group 1 2 3 dx = dy = dz X Course WS 05/06 – Scientific Visualization computer graphics & visualization Data Structures - Cartesian grids - Vertex positions are given implicitly from [i,j,k]: - P[i,j,k].x = origin + i • dx - P[i,j,k].y = origin + j • dy - P[i,j,k].z = origin + k • dz Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization Data Structures - Uniform grids - Similar to Cartesian grids - Consist of equal cells but with different resolution in at least one dimension ( dx ≠ dy (≠ (≠ dz)) - Global vertex index I[i,j,k] = k• k•Ny• Ny•Nx + j• j•Nx + i - dy k = l / (Ny• (Ny•Nx) j = (l % (Ny• (Ny•Nx)) / Nx i = (l % (Ny• (Ny•Nx) % Nx) - Spacing between grid points is constant in each - Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group dimension -> same indexing scheme as for Cartesian grids Most likely to occur in applications where the data is generated by a 3D imaging device providing different sampling rates in each dimension Course WS 05/06 – Scientific Visualization computer graphics & visualization Data Structures - Rectilinear grids - Topology is still regular but irregular spacing between grid points - NonNon-linear scaling of positions along either axis - Spacing, x_coord[L], y_coord[M], z_coord[N], must be stored explicitly - Topology is still implicit Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization Data Structures - Curvilinear grids - Topology is still regular but irregular spacing between grid points - Positions are nonnon-linearly transformed x_coord[L,M,N] y_coord[L,M,N] z_coord[L,M,N] - Topology is still implicit - Geometric structure M might result in concave grids j i N Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Course WS 05/06 – Scientific Visualization computer graphics & visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization 4 Data Structures - Multigrids Data Structures - Characteristics of structured grids - Focus in arbitrary areas to avoid wasted detail - “blow up” up” regions of interest, i.e. finer grid - Easier to compute with - Can be stored in a 2D / 3D array - Arbitrary samples can be directly accessed by indexing - Often composed of sets of connected parallelograms (hexahedra) - Cells might be distorted with respect to transformations - May require more elements or badly shaped elements in order to precisely cover the underlying domain - Topology is represented implicitly by n-vector of dimensions - Geometry is represented explicitly by an array of points - Every interior point has the same number of neighbors Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Course WS 05/06 – Scientific Visualization computer graphics & visualization Data Structures - Characteristics of structured grids Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization Data Structures - If no implicit topological (connectivity) information - Topological information is implicitly coded is given the grids are termed unstructured grids - Direct access to adjacent elements at random - Cartesian, uniform, and rectilinear grids are necessarily convex - Visibility ordering of elements is given implicitly - Curvilinear grids reveal a much more flexible alternative to model arbitrarily shaped objects - But makes the sorting of elements a complex procedure - Unstructured grids are often computed using quadtrees - (recursive domain partitioning for data clustering), or by triangulation of points sets The task is often to create a grid from scattered points - Characteristics of unstructured grids - Grid points and connectivity must be stored - Dedicated data structures needed to allow for efficient traversal and thus data retrieval - Often composed of triangles or tetrahedra - Less elements are needed to cover the domain Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Course WS 05/06 – Scientific Visualization computer graphics & visualization Data Structures - Typical implementation of structured grids Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization Data Structures - Unstructured grids - Composed of arbitrarily positioned and connected elements DataType *data = new DataType[Nx• DataType[Nx•Ny• Ny•Nz]; val = data[i• data[i•(Ny• (Ny•Nz) + j• j•Nz + k]; - Can be composed of one unique element type or they can … code for geometry … - Triangle meshes in 2D and tetrahedral grids in 3D are be hybrid most common Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Course WS 05/06 – Scientific Visualization computer graphics & visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization 5 Data Structures - Typical implementations of unstructured grids Data Structures - - Direct form Coords for vertex 1 x1,y1,(z1) x2,y2,(z2) x3,y3,(z3) x2,y2,(z2) x3,y3,(z3) x4,y4,(z4) face 1 face 2 struct face float verts[3][2] DataType val; 2D struct face float verts[3][3] DataType val; 3D vertex list face list x1,y1,(z1) x2,y2,(z2) x3,y3,(z3) x4,y4,(z4) 1,2,3 1,2,4 3,2,4 ... - Course WS 05/06 – Scientific Visualization Indexed face set More efficient than direct approach in terms of memory requirements But still have to do global search to find local information (i.e. what faces share an edge) Course WS 05/06 – Scientific Visualization computer graphics & visualization Data Structures - Typical implementations of unstructured grids - WingedWinged-edge data structure [Baumgart 1975] Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization Data Structures - WingedWinged-edge data structure - EdgeEdge-based data structure, allows to answer queries - Faces sharing an edge - Faces sharing a vertex - Walk around edges of face previous right edge vertex end Indirect form ... - - Additionally store the data values - Problems: storage space, redundancy next left edge - Coords for vertex 1 ... Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Typical implementations of unstructured grids - Stores for every vertex a pointer to an edge arbitrary edge that is incident to it face - Stores for every face a pointer to an partner edge on its boundary vertex start previous left edge counterclockwise orientation - Implicit assumption: - Every edge has at most two faces which meet at edge Ù twotwo-manifold topology next right edge Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Course WS 05/06 – Scientific Visualization computer graphics & visualization Data Structures - Hybrid grids Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization Data Structures - Combination of different grid types Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Course WS 05/06 – Scientific Visualization computer graphics & visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization 6 Data Structures Data Structures Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Course WS 05/06 – Scientific Visualization computer graphics & visualization Data Structures - Example Course WS 05/06 – Scientific Visualization computer graphics & visualization Data Structures - Example Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization Data Structures - Example Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization Data Structures - Example Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Course WS 05/06 – Scientific Visualization computer graphics & visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization 7 Data Structures - Example Data Structures - Example Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Course WS 05/06 – Scientific Visualization computer graphics & visualization Data Values - Characteristics of data values - Qualitative - NonNon-metric - Ordinal (order along a scale) - Nominal (no order) discretization) Dimension (number of components) Error (variance) Structure of the data - Quantitative - Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization Data Values - Range of values - Range of values - Data type (scalar, vector, tensor data; kind of - Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Metric scale Discrete Continuous Course WS 05/06 – Scientific Visualization computer graphics & visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization Data Values Data Classification - Classification of Bergeron & Grienstein 1989: Data types - - Scalar data is given by a function f(x1,...,xn):R ):Rn→R with n independent variables xi Vector data represent direction and magnitude and is given by a n-tupel (f1,...,fn) with fk=fk(x1,...,xn), with 1≤ 1≤ k ≤ n Tensor data for a tensor of level k is given by ti1,i2,… (x , … ,x ) 1 n i1,i2,…,ik Structure of the data - Sequential (in the form of a list) Relational (as table) Hierarchical (tree structure) Network structure - On arbitrary positions (L0m) - On a line (L1m) - On a surface (L2m) - On a (uniform) 3D grid (L3m) - On a (uniform) n-dimensional grid (Lnm) - Important aspects of data and grid types are missing Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group - Lnm m-dimensional data on an n-dimensional grid - Examples for m-dimensional data Course WS 05/06 – Scientific Visualization computer graphics & visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization 8 Data Classification Data Classification Classification according to Brodlie 1992: Classification via fiber bundles of Butler 1989: - Underlying Field: domain of the data - Visualizing entity (E) - E is a function defined by domain and range of - Fiber bundle: data - Independent variables: dimension and influence - Dependent variables: dimension and data type - base space: independent variables - fiber space: dependent variables - Definition of sections in fiber space - Connection to differential geometry Dependent variables - Examples E5Sn or EV3[3] fiber space Dimension of independent variables Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group base space fiber bundle computer graphics & visualization Data Classification Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Specification according to Wong 1997: Data Classification - Example: Dimension of the data values: dependent variables v Dimension of domain: independent variables d - Bergeron & - Data with n independent variables and m - Brodlie ES{0} - Butler base = set, fiber = float:[-∞, ∞] - Wong 0d1v ndmv Course WS 05/06 – Scientific Visualization Grienstein L01 Course WS 05/06 – Scientific Visualization computer graphics & visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Data Classification - Example: Data Classification - Example: - Bergeron & - Bergeron & Ordered set of points with scalar values Grienstein computer graphics & visualization Unordered set of points with scalar values dependent variables: Prof. Dr. R. Westermann – Computer Graphics and Visualization Group section Course WS 05/06 – Scientific Visualization computer graphics & visualization Scalar volume data set on a uniform grid Grienstein L01 L31 - Brodlie ES[0] - Brodlie ES3 - Butler base = ordered set, fiber = float:[-∞, ∞] - Butler base = 3D-reg-grid, fiber = char:[0,255] - Wong 0d1v - Wong 3d1v Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group Course WS 05/06 – Scientific Visualization computer graphics & visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization 9 Data Classification - Example: Data Classification - Example: - Bergeron & - Bergeron & Flow data on a curvilinear grid Grienstein 3D volume with 3 scalar and 2 vector data values Grienstein L39 L33 - Brodlie EV33 - Brodlie - Butler base = 3D-curvilin-grid, fiber = float3:[-∞, ∞]3 - Butler - Wong 3d3v - Wong Course WS 05/06 – Scientific Visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group E3S2V33 base = 3D-reg-grid, fiber = float x float x float x float3 x float3 3d9v Course WS 05/06 – Scientific Visualization computer graphics & visualization Prof. Dr. R. Westermann – Computer Graphics and Visualization Group computer graphics & visualization 10