3D Visualization Techniques: Meshes, Surfaces, and Volumes

1. Lecture 2 : Visualization Basics Bong-Soo Sohn School of Computer Science and Engineering Chung-Ang University

2. Surface Graphics • Objects are explicitely defined by a surface or boundary representation (explicit inside vs outside) • This boundary representation can be given by: • a mesh of polygons : • a mesh of spline patches

3. Surface Graphics : Pros and Cons • Pros : • fast rendering algorithms are available • acceleration in special hardware is relatively easy and cheap (many $200 game boards) • use OpenGL to specify rendering parameters • surface realism can be added via texture mapping • Cons : • discards the interior of the object and just maintains the object’s shell • does not facilitate real-world operations such cutting, slicing, disection • does not enable artificial viewing modes such as semi-transparencies, X-ray • surface-less phenomena such as clouds, fog, gas are hard to model and represent

4. Volume Graphics • Maintains a 3D image representation that is close to the underlying fully-3D object (but discrete)

5. Volume Graphics : Pros and Cons • Pros : • can achieve a level of realism (and ‘hyper-realism’) that is unmatched by surface graphics • allows easy and natural exploration of volumetric datasets • Cons : • has high rendering complexity Rendering of the inside of a human colon volume rendered surface rendered

6. Volumetric Image (3D image, volume) • it is a 3D array of point samples, called voxels (volume elements) • the point samples are located at the grid points • the process of generating a 2D image from the 3D volume is called volume rendering

7. Basics on Differentiation (of Scalar and Vector Function) • Refer to Prof. Han-Wei Shen’s Notes. • Useful for understanding images and gradients

8. Data Acquisition • Scanned/Sampled Data • CT/MRI/Ultrasound • Electron Microscopy • Computed/Simulated Data • Modeled/Synthetic Data

9. Time-Varying Data • Time-Varying Data from Scanning

10. Evolutionary Morphing

11. Imaging Scanners • Scanners can yield both domains and functions on domains • Scanners yielding domains • Point Cloud Scanners: 300μ-800μ • CT, MRI: 10μ-200μ • Light microscopy: 5μ-10μ • Electron microscopy: < 1μ • Ultra microscopy like Cyro EM 50Å-100Å

12. Imaging Techniques • Computed Tomography (CT) • Measures spacially varying X-ray attenuation coefficient • Each slice 1-10mm thick • High resolution , low noise • Good for high density solids • Magnetic Resonance Imaging (MRI) • Measures distribution of mobile hydrogen nuclei by quantifying relaxation times • Moderate noise • Works well with soft tissue • Ultrasound • Handheld probe • Inexpensive, fast, and real-time • High noise with moderate resolution

13. Static Scalar Meshed Dense Time varying data Vector , Tensor Meshless Sparse Various Data Characteristics

14. Data Format • Mesh (Grid) Type • Regular • Rectilinear • Unstructured • Meshless • Mesh type conversion • Meshless to meshed

15. Mesh Types • Mesh taxonomy • Regular static meshes: • There is an indexing scheme, say i,j,k, with the actual positions being determined as i*dx, j*dy, k*dz. • If dx=dy=dz, then, • In 2-D, we get a pixel, and in 3-D, a voxel. dx A 2-D regular rectilinear grid dy

16. Mesh Types • Irregular static meshes: • Rectilinear: • Individual cells are not identical but are rectangular, and connectivity is related to a rectangular grid dx, dy are not constant in grid, but connectivity is similar in topology to regular grids. A 2-D regular rectilinear grid

17. Mesh types (contd) • Curvilinear: • Sometimes called structured grids as the cells are irregular cubes – a regular grid subjected to a non-linear transformation so as to fill a volume or surround an object. A 2-D curvilinear grid

18. Mesh Types (contd) • Unstructured: • Cells are of any shape (tetrahedral), hexahedra, etc with no implicit connectivity • Hybrid: • Combination of curvilinear and unstructured grids. • Dynamic (Time-varying) meshes

19. Triangulations (Delaunay) & Dual Diagrams (Voronoi) Meshless Data  Meshed Data • Union of balls • Triangulation & Dual • Nerve sub-complex

20. Particle Data to Meshes A Weighted point P = ( p, wp ) where p x Power distance from with is the Euclidean distance

21. Power Diagram ( PD ) of a weighted point set Tiling of space into convex regions where ith region ( tile ) are the set of points in nearest to pi in the power distance metric. p1 p2 l1 l2 l Bisector Plane which matches power distance. Regular Triangulation ( RT ) Dual of Power Diagram ( PD ) with an edge of RT for each Bisector Plane of PD

22. Particle Data to Meshes Atomic Centers CPK CPK Alpha-Shape Solvent Accessible Surface (SAS) Power Diagram of SASSolvent Excluded Surface (SES)

23. Molecular Surfaces(Solvent Excluded Surface) toroidal patches + concave patches spherical patches + SES =

24. Field Data • Scalar temperature, pressure, density, energy, change, resistance, capacitance, refractive index, wavelength, frequency & fluid content. • Vector velocity, acceleration, angular velocity, force, momentum, magnetic field, electric field, gravitational field, current, surface normal • Tensor stress, strain, conductivity, moment of inertia and electromagnetic field • Multivariate Time Series

25. Interpolation • Interpolation/Approximation are often used to approximate the data on the domain • In other words, it constructs a continuous function on the domain

26. Linear Interpolation on a line segment p0pp1 The Barycentric coordinates α = (α0 α1) for any point p on line segment <p0 p1>, are given by fp f1 f f0 which yields p = α0p0 + α1 p1 and fp = α0f0 + α1 f1

27. Linear interpolation over a triangle p0 p1 p p2 For a triangle p0,p1,p2, the Barycentric coordinates α = (α0 α1 α2) for point p,

28. Linear interpolant over a tetrahedron Linear Interpolation within a • Tetrahedron (p0,p1,p2,p3) α = αi are the barycentric coordinates of p p3 p p0 p2 p1 fp3 fp fp2 fp0 fp1

29. Other 3D Interpolation • Unit Prism (p1,p2,p3,p4,p5,p6) p1 p2 p3 p p4 p5 p6 Note: nonlinear

30. Other 3D Interpolation • Unit Pyramid (p0,p1,p2,p3,p4) p0 p1 p2 p p3 p4 Note: nonlinear

31. Trilinear Interpolation • Unit Cube (p1,p2,p3,p4,p5,p6,p7,p8) • Tensor in all 3 dimensions p1 p2 p3 p4 p p5 p6 p7 p8 Trilinearinterpolant

32. Comparison • Bicubic vs Bilinear vs nearest point

33. Resampling • Used in image resize or data type conversion • Rectilinear to rectilinear • Unstructured to rectilinear

34. Rendering • Isocontouring (Surface Rendering) • Builds a display list of isovalued lines/surfaces • Volume Rendering • 3D volume primitives are transformed into 2D discrete pixel space

35. Volume Rover Demo

36. Isosurface Visualization • Isosurface (i.e. Level Set ) : • C(w) = { x | F(x) - w = 0 } ( w : isovalue , F(x) : real-valued function ) isosurfacing <medical> < ocean temperature function > < two isosurfaces (blue,yellow) > <bio-molecular> • Surface Topology : • Property that is invariant to continuous deformation (without cutting or gluing), e.g. donut & cup

37. Isocontouring • Popular Visualization Techniques for Scalar Fields 2. Isocontouring [Lorensen and Cline87,…] • Definition of isosurface C(w) of a scalar field F(x) • C(w)={x|F(x)-w=0} , ( w is isovalue and x is domain R3 ) 1.0 1.0 1.0 0.8 0.4 0.3 0.8 0.4 0.3 0.8 0.4 0.3 0.7 0.6 0.75 0.4 0.7 0.6 0.75 0.4 0.7 0.6 0.75 0.4 0.4 0.4 0.4 0.6 0.4 0.8 0.6 0.4 0.8 0.6 0.4 0.8 0.4 0.4 0.4 0.3 0.25 0.3 0.25 0.3 0.25 0.35 0.35 0.35 ( Isocontour in 2D function: isovalue=0.5 ) • Marching Cubes for Isosurface Extraction • Dividing the volume into a set of cubes • For each cubes, triangulate it based on the 2^8(reduced to 15) cases

38. Cube Polygonization Template

39. Volume Rendering • Popular Visualization Techniques for Scalar Fields 1. Volume Rendering [Drebin88,…] C : color C: opacity I I’ C , C Light traversal from back to front I’= C C + (1- C)I <emission> <incoming light> <produced by CCV vistool> • Hardware Acceleration ( 3D Texturing ) [Westermann98] • Slicing along the viewing direction • Put 3D textures on the slice • Interactive color table manipulation

40. Transfer Function • Mapping from density to (color, opacity)

41. Medical applications