Introduction to Visualization and Computer Graphics

Transcription

1 Introduction to Visualization and Comuter Grahics DH2320, Fall 2015 Prof. Dr. Tino Weinkauf Introduction to Visualization and Comuter Grahics Grids and Interolation

2 Next Tuesday No lecture next Tuesday!

3 Introduction to Visualization and Comuter Grahics DH2320, Fall 2015 Prof. Dr. Tino Weinkauf Grids and Interolation Structured Grids Unstructured Grids

4 Digital Data

5 Scientific Data

6 Data Reresentation R n R m satial domain X data values indeendent variables deendent variables

7 Data Reresentation In most cases, the visualization data reresent a continuous real object, e.g., an oscillating membrane, a velocity field around a body, an organ, human tissue, etc. This object lives in an n-dimensional sace - the domain Usually, the data is only given at a finite set of locations, or samles, in sace and/or time Remember imaging rocesses like numerical simulation and CTscanning, note similarity to ixel images We call this a discrete structure, or a discrete reresentation of a continuous object

8 Discrete reresentations Data Reresentation We usually deal with the reconstruction of a continuous real object from a given discrete reresentation Discrete structures consist of oint samles Often, we build grids/meshes that connect neighboring samles

9 Discrete reresentations Data Reresentation We usually deal with the reconstruction of a continuous real object from a given discrete reresentation? Discrete structures consist of oint samles Often, we build grids/meshes that connect neighboring samles

10 Data Reresentation Grid terminology 0D: grid vertex (grid oint) 1D: grid line 2D: grid face 3D: grid voxel grid vertices grid lines grid faces grid cell: largest-dimensional element in a grid 2D: grid face 3D: grid voxel

11 Grids and Interolation Data Connectivity There are different tyes of grids: Structured grids connectivity is imlicitly given. Block-structured grids combination of several structured grids Unstructured grids connectivity is exlicitly given. Hybrid grids combination of different grid tyes

12 Grids Structured grids Structured refers to the imlicitly given connectivity between the grid vertices We distinguish different tyes of structured grids regarding the imlicitly or exlicitly given coordinate ositions of the grid vertices uniform grid imlicitly given coordinates rectilinear grid semi-imlicitly given coordinates curvilinear grid exlicitly given coordinates

13 Grids and Interolation Structured grids Number of grid vertices: D x, D y, D z We can address every grid vertex with an index tule (i, j, k) 0 i < D x 0 j < D y 0 k < D z i, j = 2,3 i, j = 2,3 i, j = 2,3 D y D y D y D x uniform grid imlicitly given coordinates D x rectilinear grid semi-imlicitly given coordinates D x curvilinear grid exlicitly given coordinates

14 Grids and Interolation Structured grids Number of grid vertices: D x, D y, D z We can address every grid cell with an index tule (i, j, k) 0 i < D x 1 0 j < D y 1 0 k < D z 1 Number of cells: D x 1 D y 1 (D z 1) D y D y D y D x uniform grid imlicitly given coordinates D x rectilinear grid semi-imlicitly given coordinates D x curvilinear grid exlicitly given coordinates

15 Regular or uniform grids Grids and Interolation very imortant Cells are rectangles or rectangular cuboids of the same size All grid lines are arallel to the axes To define a uniform grid, we need the following: Bounding box: (x min, y min, z min ) - (x max, y max, z max ) Number of grid vertices in each dimension: D x, D y, D z Cell size: d x, d y, d z

16 Grids and Interolation Regular or uniform grids Well suited for image data (medical alications) Coordinate cell is very simle and chea Global search is good enough; local search not required Coordinate of a grid vertex:

17 Grids and Interolation Cartesian grid Secial case of a uniform grid: d x = d y = d z Consists of squares (2D), cubes (3D)

18 Grids and Interolation Rectilinear grids Cells are rectangles of different sizes All grid lines are arallel to the axes Vertex locations are inferred from ositions of grid lines for each dimension: XLoc = {0.0, 1.5, 2.0, 5.0, } YLoc = {-1.0, 0.3, 1.0, 2.0, } ZLoc = {3.0, 3.5, 3.6, 4.1, } Coordinate cell still quite simle

19 Grids and Interolation Curvilinear grids Vertex locations are exlicitly given XYZLoc = {(0.0, -1.0, 3.0), (1.5, 0.3, 3.5), (2.0, 1.0, 3.6), } Cells are quadrilaterals or cuboids Grid lines are not (necessarily) arallel to the axes 2D curvilinear grid 3D curvilinear grids

20 Grids and Interolation Curvilinear grids Coordinate cell: Local search within last cell or its immediate neighbors Global search via quadtree/octree 2D curvilinear grid 3D curvilinear grids

21 Grids and Interolation Block-structured grids combination of several structured grids DFG-funded SFB 557 Erik Wassen, TU Berlin, Germany 2008

22 Visualization Scenarios Demands on data storage, an examle: Ahmed body Block-structured grid with 52 blocks Each block is a curvilinear grid 17 million grid cells in total Temoral resolution: a article needs time stes from front to back of the Ahmed body DFG-funded SFB 557 Erik Wassen, TU Berlin, Germany 2008

23 Visualization Scenarios Demands on data storage, an examle: 17 million grid cells x time stes x 7 variables X 8 bytes er double = 8.66 terra bytes (60.62 TB for total time stes) DFG-funded SFB 557 Erik Wassen, TU Berlin, Germany 2008 Do not save every time ste, not every variable, and not every block.

24 Grids and Interolation Unstructured grids 2D unstructured grid consisting of triangles 3D unstructured grid consisting of tetrahedra (from TetGen user manual)

25 Grids and Interolation Unstructured grids Vertex locations and connectivity exlicitly given Linear interolation within a triangle/tetrahedron using barycentric coordinates Coordinate triangle/tetra: Local search within last triangle/tetra or its immediate neighbors Global search via quadtree/octree 3D unstructured grid consisting of tetrahedra (from TetGen user manual)

26 Grids and Interolation How to store unstructured grids? Different requirements: Efficient storage bytes er face / bytes er vertex Efficient access of face / vertex roerties (e.g., osition) Efficient traversal e.g., neighboring face, 1-ring of a vertex,... Requirements are cometing

27 Grids and Interolation Face set Store faces 3 ositions no connectivity ( match ositions ) Examle: STL very simle structure (too simle, unractical!) easily ortable

28 Grids and Interolation Shared vertex vertex table stores ositions triangle table stores indices into vertices No exlicit connectivity Examles: OFF, OBJ, PLY Quite simle and efficient Enables efficient oerations on static meshes

29 Grids and Interolation Face-based connectivity vertices store osition face reference faces store 3 vertex references references to 3 neighboring faces

30 Grids and Interolation Edge-based connectivity vertex stores osition reference to 1 edge edge stores references to 2 vertices 2 faces 4 edges face stores reference to 1 edge

31 Halfedge based connectivity Half-edge based connectivity vertex stores osition reference to 1 half-edge half-edge stored references to 1 vertex 1 face 1, 2, or 3 half-edges face stores reference to 1 half-edge

32 Grids and Interolation Half-edge based connectivity

33 Grids and Interolation Half-edge based connectivity: Traversal Building blocks Vertex to (outgoing) halfedge half-edge to next (revious) halfedge half-edge to neighboring half-edge half-edge to face half-edge to start (end) vertex Examle: Traverse around vertex (1-ring) enumerate vertices/faces/half-edges

34 1-ring traversal Start at vertex

35 1-ring traversal Start at vertex Outgoing halfedge

36 1-ring traversal Start at vertex Outgoing halfedge Oosite halfedge

37 1-ring traversal Start at vertex Outgoing halfedge Oosite halfedge Next half-egde

38 1-ring traversal Start at vertex Outgoing halfedge Oosite halfedge Next half-egde Oosite...

39 1-ring traversal Start at vertex Outgoing halfedge Oosite halfedge Next half-egde Oosite... Next......

40 Half-edge based libraries CGAL Comutational geometry Free for non-commercial use Oen Mesh Mesh rocessing Free, LGPL license gmu (gmu-lite) rorietary, directed edges

41 Grids and Interolation Unstructured grids 2D unstructured grid consisting of quads Source: htts://

42 Grids and Interolation Hybrid grids combination of different grid tyes 2D hybrid grid

43 Introduction to Visualization and Comuter Grahics DH2320, Fall 2015 Prof. Dr. Tino Weinkauf Grids and Interolation Linear, Bilinear, Trilinear Interolation in Structured Grids Gradients Linear Interolation in Unstructured Grids

44 Interolation A grid consists of a finite number of samles The continuous signal is known only at a few oints (data oints) In general, data is needed in between these oints By interolation we obtain a reresentation that matches the function at the data oints Reconstruction at any other oint ossible?

45 Interolation Simlest aroach: Nearest-Neighbor Interolation Assign the value of the nearest grid oint to the samle.

46 Interolation Linear Interolation (in 1D domain) Domain oints, scalar function General: Secial Case: Basis Coefficients

47 Interolation Linear Interolation (in 1D domain) Samle values Continuity (discontinuous first derivative) Use higher order interolation for smoother transition, e.g., cubic interolation

48 Interolation Interolation in 2D, 3D, 4D, Bi-Linear Bi-Cubic Tensor Product Interolation Perform linear / cubic interolation in each x,y,z direction searately

49 Interolation Bilinear Interolation very imortant 2D, bi-linear interolate twice in x direction and then once in y direction

50 Grids and Interolation Examle: Bi-linear interolation in a 2D cell Reeated linear interolation

51 Interolation Trilinear Interolation 3D, tri-linear interolate four times in x direction, twice in y direction, and once in z direction

52 Function Derivatives Function Derivative Estimation Called Gradients for multidimensional functions Have a lot of imortant alications (e.g., normal for volume rendering, critical oint classification for vector field toology ) vector of artial derivatives Describes direction of steeest ascend

53 Function Derivatives Two ways to estimate gradients: Direct derivation of interolation formula Finite differences schemes

54 Field Function Derivatives Field Function Derivatives, Bi-Linear derive this interolation formula constant in x direction constant in y direction

55 Field Function Derivatives Problem of exact linear function differentiation: discontinuous gradients (Piecewise) linear function Gradient Solution: Use higher order interolation scheme (cubic) Use finite difference estimation

56 Finite Differences Finite Differences Schemes Aly Taylor series exansion around samles Continuous function Linear function Taylor exansion Forward difference Backward difference

57 Finite Differences Finite Differences Schemes Difference Central difference Central differences have higher aroximation order than forward / backward differences

58 Finite Differences Schemes, Higher order derivatives Finite Differences 1D Examle, linear interolation (Piecewise) linear function Central differences

59 Interolation in Triangles Piecewise Linear Interolation in Triangle Meshes

60 Interolation in Triangles Linear Interolation in a Triangle There is exactly one linear function that satisfies the interolation constraint A linear function can be written as Polynomial can be obtained by solving the linear system Linear in and Interolated values along any ray in the lane sanned by the triangle are linear along that ray

61 Barycentric Coordinates Barycentric Coordinates: Planar case: Barycentric combinations of 3 oints Area formulation: 1 : 3, with )),, ( ( )),, ( (, )),, ( ( )),, ( (, )),, ( ( )),, ( ( area area area area area area 1 2 3

62 Barycentric Coordinates Barycentric Coordinates: Linear formulation: ) (

63 Barycentric Coordinates 3, with :

64 Barycentric Interolation Barycentric Interolation in a Triangle very imortant The linear function of a triangle can be comuted at any oint as with (Barycentric Coordinates) This also holds for the coordinate of the triangle: Can be used to solve for unknown coefficients :

65 Barycentric Interolation Barycentric Interolation in a Triangle Solution of (e.g. Cramer's rule): ith Inside triangle criteria

66 Barycentric Interolation Barycentric Interolation in a Tetrahedron Analogous to the triangle case

67 Barycentric Interolation Gradient of a linearly interolated function in a triangle/tetrahedron Constant!