Fundamental Computer Graphics or the discretization of lines and polygons. Torsten Möller Simon Fraser University

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1 Fundamental Computer Graphics or the discretization of lines and polygons Torsten Möller Simon Fraser University

2 Overview 1D lines in 2D space Cartesian lattices: Bresenham General N-D lattice: Ibáñez Topological issues: separability, minimality 2D planes in 3D space Cartesian lattices BCC lattices 2

3 Discretization Rasterization / Voxelization Fundamental operation in graphics: Discrete representation of continuous world Continuous world modeled with Points Lines Planes (triangles) Curves and surfaces Raster-displays are ubiquitous Texture hardware (3D raster) ubiquitous Acquisition of real models (medical, scientific data) typically on raster 3

4 Discretization Continuous Discrete 2D 3D 4

5 1D lines - Bresenham Original line rasterization based on Cartesian lattices Bresenham quasi-standard (other efficient algorithms proposed) Based on DDA - digital differential analyzers Figured out the dominant direction Use an incremental algorithm in this direction to determine valid pixels NE Q M M NE ME P(xp,yp) E 5

6 Ibáñez algorithm Extension to general N-dimensional lattices by Ibáñez et al. (2001) Select connectivity (neighborhood) Select optimal Vector basis / dominant direction Project onto an orthogonal subspace 6

7 Overview 1D lines in 2D space Cartesian lattices: Bresenham General N-D lattice: Ibáñez Topological issues: separability, minimality 2D planes in 3D space Cartesian lattices BCC lattices 7

8 Topological Issues Notion of connectedness is not straight forward in discrete domain Depending on neighborhood Cartesian lattice 4 neighbourhood or 1-neighborhood 8 neighbourhood or 0-neighborhood 8

9 Topological Issues Notion of connectedness is not straight forward in discrete domain Depending on neighborhood Hexagonal lattice 6 neighbourhood or 0/1-neighborhood 9

10 Definitions Lattice point P is typically a 0-dimensional entity Pixel / Voxel V: Voronoi cell of this lattice point In some context identical to the lattice point k-neighborhood of V: set of voxels sharing a k-dimensional (or higher) face with V 10

11 Definitions k-path: list of voxels, that s made up of only k-neighbours Example - 0-path, but no 1-path 11

12 Definitions k-path: list of voxels, that s made up of only k-neighbours Example: 1-path 12

13 Minimality and Separability Separability: A line/plane or surface L is k-separable if there is no k-path crossing it Example: curve is 1-separable, but not 0-separable 13

14 Minimality and Separability Separability: A line/plane or surface L is k-separable if there is no k-path crossing it Example: curve is 0-separable 14

15 Minimality and Separability Separability: A line/plane or surface L is k-separable if there is no k-path crossing it Minimality: A line/plane or surface L is k-minimal if the removal of any of its voxels will produce a k-path crossing it (also called k-tunnel) Example: curve is 0/1-minimal 15

16 Minimality and Separability Separability: A line/plane or surface L is k-separable if there is no k-path crossing it Minimality: A line/plane or surface L is k-minimal if the removal of any of its voxels will produce a k-path crossing it (also called k-tunnel) Example: curve is neither 0- nor 1-minimal 16

17 Minimality and Separability Separability: A line/plane or surface L is k-separable if there is no k-path crossing it Minimality: A line/plane or surface L is k-minimal if the removal of any of its voxels will produce a k-path crossing it (also called k-tunnel) Example: curve is 0-minimal, but not 1-minimal 17

18 Minimality and Separability Separability: A line/plane or surface L is k-separable if there is no k-path crossing it Minimality: A line/plane or surface L is k-minimal if the removal of any of its voxels will produce a k-path crossing it (also called k-tunnel) Example: curve is 0/1-minimal 18

19 1D lines (in 2D space) revisited Assuming a normalized plane equation L=Ax+By+D Relate discretization to thickness, I.e. thicken the surface in the direction of the line normal. 1-separable line 0-separable line 19

20 1D lines (in 2D space) revisited Assuming a normalized plane equation L=Ax+By+D Relate discretization to thickness, I.e. thicken the surface in the direction of the line normal. -t < Ax+By+D < t convolution with a box filter of appropriate size With appropriate t this can be proven to create minimal and separable discretizations of lines How thick? Depends on lattice structure and neighborhood structure 20

21 1-separable lines in 2D Cartesian Allow 0,1-neighbors N According to Huang et al: t = max( d i " N) = d 1 max i=1,3 Guarantees 1-separable, 1-minimal lines i=1,3 cos# i ( ) α 3 d 3 α 1 d 1 21

22 0-separable lines in 2D Cartesian Allow only 1-neighbors - needs a thicker line N According to Huang et al: t = max i= 2,4 d i " N ( ) = d 2 max( ) Guarantees 0-separable, 0-minimal lines i= 2,4 cos# i d d 3 4 α 4 α 1 This can also be expressed as (Widjaya et al): α 3 d 1 α 2 d 2 t = max i=1,2,3,4 d i " N ( ) = max i=1,2,3,4 ( ) d i cos# i 22

23 Separable lines in other 2D lattice structures 0-neighborhood and 1-neighborhood are identical According to Widjaya et al: t = max i=1,2,3 d i " N ( ) = max i=1,2,3 ( ) d i cos# i α 2 N d 3 α 3 d 2 α 1 d 1 23

24 Overview 1D lines in 2D space Cartesian lattices: Bresenham General N-D lattice: Ibáñez Topological issues: separability, minimality 2D planes in 3D space Cartesian lattices BCC lattices 24

25 2D planes (in 3D space) Assuming a normalized plane equation L=Ax+By+Cz+D Relate discretization to thickness, i.e. thicken the surface in the direction of the plane normal. -t < Ax+By+Cz+D < t Also considered as a convolution with a box filter of appropriate size With appropriate t this can be proven to create minimal and separable discretizations of planes How thick? Depends on lattice structure and neighborhood structure 25

26 2D planes (in 3D space) How thick? Depends on lattice structure and neighborhood structure 3 types of neighbors Face (2-neighbor; also 6-neighbor) Edge + face (1-neighbor; also 18-neighbor) Vertex+edge+face (0-neighbor; also 26-neighbor) 26

27 2-separable planes in 3D Cartesian allows 0,1,2-neighborhood According to Huang et al: t = max i=1,2,3 d i " N ( ) = d 1 max( ) Guarantees 2-separable, 2-minimal planes i=1,2,3 cos# i d 3 d 2 d 1 27

28 1-separable planes in 3D Cartesian Need thicker line d 7 According to Widjaya et al: t = max( d i " N) = max i=1...9 i=1...9 Guarantees 1-separable, 1-minimal planes ( ) d i cos# i d 8 d 9 d 3 d 2 d 1 d 6 d 5 d 4 28

29 0-separable planes in 3D Cartesian Only 2-neighbors allowed - need even thicker line According to Huang et al: d 11 d 7 d 10 d 8 d d 9 d d 6 t = max i= ( d i " N) = d 10 max Guarantees 0-separable, 0-minimal planes i= cos# i ( ) This can also be expressed as (Widjaya et al): d 3 d 2 d 1 d 5 d 4 t = max i= d i " N ( ) = max i= ( ) d i cos# i 29

30 Separable planes in 3D BCC Voronoi cell = truncated octahedron 0-, 1-, and 2-neighborhoods are identical 30

31 Separable planes in 3D BCC Voronoi cell = truncated octahedron 0-, 1-, and 2-neighborhoods are identical 2 types of faces shared - hexagons (8) and squares (6) 31

32 Separable planes in 3D BCC According to Widjaya et al: t = max( d i " N) = max i=1...7 i=1...7 ( ) Guarantees separability and minimality d i cos# i 32

33 Principle Direction There is always one lattice direction, that is determining the rasterization - call it principle direction This guides an efficient implementation: Project plane onto a 2D lattice that spans lattice space together with principle direction Do a normal rasterization of a 2D plane on this 2D space Move up each voxel into their proper position 33

34 Algorithm Step 1 Line A is projected on to the base plane direction along the principal direction Principal Direction Base Plane Direction 34

35 Algorithm Step 2 The projected line is then scan converted along the base plane direction Principal Direction Base Plane Direction 35

36 Algorithm Step 3 The voxels chosen is then projected back to the original line along the principal direction Principal Direction Base Plane Direction 36

37 References A. Kaufman. Efficient Algorithms for 3D Scan-conversion of Parametric Curves, Surfaces, and Volumes. Computer Graphics, 21(4): , D. Cohen-Or, A. Kaufman. Fundamentals of surface Voxelization. Graphical models and Image Processing: GMIP, 57(6): , D. Cohen-Or, A. Kaufman. 3D Line Voxelization and Connectivity Control. IEEE Computer Graphics and Applications, 17(6):80-87, J. Huang, R. Yagel, V. Filippov, Yair Kurzion. An Accurate Method for Voxelizing Polygon Meshes. IEEE Symposium on Volume Visualization, pages , S. Fang, H. Chen. Hardware Accelerated Voxelization. Computers and Graphics, 24(3): , L. Ibáñez, Ch. Hamitouche, Ch. Roux. A Vectorial Algorithm for Tracing Discrete Straight Lines in N-dimensional Generalized Grids. In IEEE Transactions on Visualization and Computer Graphics, vol. 7(2):97-108, H. Widjaya, T. Möller, A. Entezari. Voxelization in Common Sampling Lattices. 11th Pacific Graphics Conference on Computer Graphics and Applications, pages ,

38 Thanks NSERC, BC-ASI Haris Widjaya, Reza Entezari 38

Discrete. Continuous. Fundamental Computer Graphics or the discretization of lines and polygons. Overview. Torsten Möller Simon Fraser University

Discrete. Continuous. Fundamental Computer Graphics or the discretization of lines and polygons. Overview. Torsten Möller Simon Fraser University Fundamental Computer Graphics or the discretization of lines and polygons Torsten Möller Simon Fraser University Overview 1D lines in 2D space Cartesian lattices: Bresenham General N-D lattice: Ibáñez