Discrete. Continuous. Fundamental Computer Graphics or the discretization of lines and polygons. Overview. 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 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 Discretization 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 2D 3D Continuous Discrete 3 4
2 1D lines - Bresenham Ibáñez algorithm 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 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 NE Q M M NE ME P(xp,yp) E 5 6 Overview Topological Issues 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 Notion of connectedness is not straight forward in discrete domain Depending on neighborhood Cartesian lattice 7 4 neighbourhood or 1-neighborhood 8 neighbourhood or 0-neighborhood 8
3 Topological Issues Definitions Notion of connectedness is not straight forward in discrete domain Depending on neighborhood Hexagonal lattice 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 6 neighbourhood or 0/1-neighborhood 9 10 Definitions Definitions k-path: list of voxels, that s made up of only k-neighbours Example - 0-path, but no 1-path k-path: list of voxels, that s made up of only k-neighbours Example: 1-path 11 12
4 Minimality and Separability Minimality and Separability Separability: A line/plane or surface L is k-separable if Example: curve is 1-separable, but not 0-separable Separability: A line/plane or surface L is k-separable if Example: curve is 0-separable Minimality and Separability Minimality and Separability Separability: A line/plane or surface L is k-separable if 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 Separability: A line/plane or surface L is k-separable if 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 15 16
5 Minimality and Separability Minimality and Separability Separability: A line/plane or surface L is k-separable if 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 Separability: A line/plane or surface L is k-separable if 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 D lines (in 2D space) revisited 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 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 19 20
6 1-separable lines in 2D Cartesian 0-separable lines in 2D Cartesian Allow 0,1-neighbors N Allow only 1-neighbors - needs a thicker line N According to Huang et al: t = max( d i " N) = d 1 max cos# i i=1,3 Guarantees 1-separable, 1-minimal lines i=1,3 ( ) 3 d 3 1 d 1 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 4 4 This can also be expressed as (Widjaya et al): t = max ( d " N i ) = max i=1,2,3,4 i=1,2,3,4 ( d i cos# i ) 3 d 3 1 d 1 2 d 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 ( d i cos# i ) i=1,2,3 2 N d 3 3 d 2 1 d 1 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 23 24
7 2D planes (in 3D space) 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 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) separable planes in 3D Cartesian 1-separable planes in 3D Cartesian allows 0,1,2-neighborhood Need thicker line According to Huang et al: According to Widjaya et al: d 8 d 9 d 7 d 6 t = max( d i " N) = d 1 max cos# i i=1,2,3 Guarantees 2-separable, 2-minimal planes i=1,2,3 ( ) d 3 d 2 d 1 t = max( d i " N) = max i=1...9 Guarantees 1-separable, 1-minimal planes ( d i cos# i ) i=1...9 d 3 d 2 d 1 d 5 d
8 0-separable planes in 3D Cartesian Only 2-neighbors allowed - need even thicker d11 lined 7 d8 According to Huang et al: d9 d13 d12 d2 t = max ( di " N ) = d10 max (cos# i ) i= Separable planes in 3D BCC i= d3 Guarantees 0-separable, 0-minimal planes This can also be expressed as (Widjaya et al): d d5 1 d10 Voronoi cell = truncated octahedron 0-, 1-, and 2-neighborhoods are identical d6 d4 t = max ( di " N ) = max ( di cos# i ) i= i= Separable planes in 3D BCC 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) According to Widjaya et al: t = max( di " N ) = max( di cos# i ) i=1...7 i=1...7 Guarantees separability and minimality 31 32
9 Principle Direction Algorithm Step 1 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 Line A is projected on to the base plane direction along the principal direction Principal Direction Base Plane Direction Algorithm Step 2 Algorithm Step 3 The projected line is then scan converted along the base plane direction The voxels chosen is then projected back to the original line along the principal direction Principal Direction Base Plane Direction Principal Direction Base Plane Direction 35 36
10 References Thanks 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 , NSERC, BC-ASI Haris Widjaya, Reza Entezari 37 38
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