Scalar Data. Alark Joshi

Transcription

1 Scalar Data Alark Joshi

2 Announcements Pick two papers to present me your top 3/4 choices. FIFO allotment Contact your clients Blog summaries:

3 Data Grids

4 Topology If points are arbitrarily distributed and there is no connectivity between them, the data is called scattered Otherwise, data is composed of cells bounded by grid lines Topology specifies the connectivity of data Geometry specifies the position of the data

5 Topology Properties of geometric shapes that remain unchanged even when under distortion

6 Topologically equivalent Things that can be transformed into each other by stretching and squeezing, without tearing or sticking together bits which were previously separated

7 Grid Types Grids differ substantially in the cells (building blocks) they are constructed from and in the way the topological information is specified

8 Structured and Unstructured Grids Structured grids have a regular topology and regular/irregular geometry Unstructured grids have irregular topology and geometry

9 Characteristics of Structured Grids Easier to compute with May require more elements or unevenly shaped elements to precisely cover the underlying domain Topology is represented implicitly by an n- vector of dimensions Geometry is represented explicitly by an array of points Every interior point has the same number of neighbors

10 Characteristics of Unstructured Grids If no implicit topological information is given, the grids are called unstructured grids Grid point geometry and connectivity must be stored Dedicated data structures needed to allow for efficient traversal and data retrieval Often composed of triangles or tetrahedra Typically, fewer elements are needed to cover the domain Image credits:

11 Unstructured Grids Can be adapted to capture local features

12 Types of grids Cartesian or equidistant grids Structured grid Number of points = Nx * Ny * Nz

13 Types of grids Uniform grids are similar to Cartesian grids Consist of equal cells but with different resolution in at least one dimension (dx dy dz) Typical example is medical imaging data that consists of slices Slice images with square pixels (dx = dy) Larger slice distance (dz > dx = dy)

14 Types of grids Rectilinear grids Topology is still regular but irregular spacing between grid points Topology is still implicit

15 Types of grids Curvilinear grids Topology is still regular but irregular spacing between grid points Topology is implicit, but vertex positions are explicitly stored

16 Types of grids Multigrids Focus in specific area to avoid unnecessary detail in other areas Finer grid for regions of interest Difficulties at the boundaries between low and high res grids for operations such as interpolation

17 Scalar Data Visualization

18 Basic Strategies Mapping to geometry Function plots Height fields Isolines and isosurfaces Color coding Techniques for 3D scalar data Volume visualization Slicing Visualization method depends heavily on dimensionality of domain

19 Function Plots Function plot for a 1D scalar field Points 1D manifold: line Errors bars possible

20 Gnuplot examples Function Plots

21 Function plots for 2D scalar field Points 2D manifold: surfaces Surface representations Wireframe Hidden lines Shaded surface

22 Function plots for 2D scalar field Shaded surface

23 Isolines Visualization of 2D scalar fields Given a scalar function f: Ω R and a scalar value c R Isoline consists of points x, y f x, y = c} If f() is differentiable and grad(f) 0 then isolines are curves Contour lines Image credits:

24 Isolines

25 Isolines Pixel by pixel contouring Straightforward approach: scan all pixels for equivalence with isovalue Input: f: 1, xmax 1,, ymax R Isovalues I 1, I n and isocolors c 1,, c n Algorithm:

26 Color coding Easy to apply colors to 1D and 2D scalar fields Map color each pixel on a 1D input signal or 2D image

27 Color coding Example: Separate color table to visualize the brain Separate color table to visualize the tissue

28 Volume Visualization Scalar volumetric data Ω ε R 3 R Medical application: CT, MRI, ultrasound, confocal microscopy etc.

29 Volume Visualization Seismic applications: Oil exploration, earthquake prediction, etc.

30 Volume Visualization Create a representation of a 3D scalar dataset Ω ε R 3 R Voxel (volume element) similar to pixel (picture element) Values between grid points are resampled by interpolation

31 Volume visualization Challenges Essential information in the interior Occlusion? Often datasets cannot be described by geometric representation (medical, phenomena such as fire, clouds, etc.) Image credits: Peter Kutz

32 Slicing 2D approach Interactively resample the data on slices perpendicular to the x-, y-, z-axis Use visualization techniques for 2D scalar fields Color coding, isolines, height fields

33 Oblique slicing Also known as Multiplanar Reformatting (MPR) Resample the data on arbitrarily oriented slices Resampling can be done on the CPU or graphics hardware (GPU) using built-in trilinear interpolation

34 Volume visualization Slicing: Display the data mapped to colors, on a slice plane Isosurfacing: Generate opaque/semi opaque surfaces Transparency effects slice Semi-transparent material Iso-surface

35 Direct Volume Rendering Render volume without extracting any surfaces (DVR) Map scalar values to optical properties (color, opacity) Need optical model Solve volume rendering integral for viewing rays into the volume

36 Direct Volume Rendering Volume Rendering Integral where x(λ) = viewing ray parameterized by the distance λ to the viewpoint. c(s(x)) = color assigned in the classification step. Here color and extinction coefficient assigned to each material is taken into consideration. D = maximum distance that the ray traverses before it goes outside the volume τ(s(x)) = extinction coefficient assigned in the classification step

37 Volume Rendering Integral Ci ; Ai ci ; ai Ci+1 ; Ai+1 Ci+1 = Ai*Ci + (1 - Ai)*ai*ci Ai+1 = Ai + (1 - Ai)*ai

38 Ray Casting Goal: numerical approximation of volume rendering integral Resample volume at equi-spaced locations along the ray Reconstruct at continuous location via tri-linear interpolation Approximate integral

39 Interpolation Linear interpolation x = x 1 x x 0 x 1 x 0 + x 0 x 1 x x 1 x 0 Bilinear interpolation P x, y = P 1,1 1 d 1 d + P 1,2 d 1 d + P 2,1 d 1 d + P 2,2 d d

40 Trilinear Interpolation [Levoy]

41 Levoy Interpolation

42 Interpolation Nearest Neighbor Binary Trilinear Interpolation Smooth/Weighted

43 Direct Volume Rendering Marc Levoy. Display of Surfaces from Volume Data. IEEE Comput. Graph. Appl. 8(3), 29-37, One of the first papers that said it is not necessary to fit geometric primitives to sampled data Create images directly from volumetric data Fast and simple technique Identified a volume rendering pipeline Discussed the gradient operator for shading

44 Volume Rendering Pipeline Acquired values Data preparation Prepared values shading Voxel colors Ray-tracing / resampling classification Voxel opacities Ray-tracing / resampling Sample colors compositing Sample opacities Image Pixels

45 Shading In Computer Graphics, the normal of a triangle/quad serves as an indication of the orientation of that surface All the lighting computations require a surface normal I total k a I ambient # lights i 1 I i k d Nˆ Lˆ k V ˆ Rˆ s n shiny

46 Shading No normals in volumetric data Use gradients as normals

47 Classification Extract features in a dataset Histogram can be a helpful aid Non trivial due to irregularities in intensity of structure and acquisition devices Often requires tedious manipulation of transfer functions

48 Transfer Functions Color table for volume visualization Map raw voxel intensity values into color, opacity etc.

49 Classification Examples of four different transfer functions

50 Novel contributions Translation translates rays vs. data set Classification novel, uses gradient Shading interpolates colors/opacities vs. normals (gouraud) Interpolation tri-linear, central difference Compositing image order (ray-casting) back-to-front

51 Blog comments Reed find it interesting that main reason why this form of visualization became necessary was for MRI and CT scanning technology Tim he could have performed more rendering experiments to provide more data on completion time. Archana creating a mechanism to display weak and fuzzy surfaces Peter impressed with the author s scientific approach

52 Volume Rendering Robert A. Drebin, Loren Carpenter, and Pat Hanrahan. Volume rendering. SIGGRAPH Comput. Graph. 22, 4 (June 1988), Classify volume into material percentages several materials per voxel are possible Calculate colors and opacities for each voxel Calculate a density for each voxel Determine the gradient from the density volume

53 Drebin, et. al. Determine a surface strength for each voxel from the density volume Using the color and opacity volume, the gradient volume and the strength volume, compute a shaded volume. Transform and view this volume

54 Drebin, et. al. Original CT Data Fat Tissue Bone Density Color and Opacity Gradient NX NY NZ Strength Shaded Transformed Final Image

55 Drebin, et. al. Material Classification Use a probability, rather than a threshold. Bayesian estimate Zone centered We know the x-ray absorptions of the materials (bone,...) Air Tissue Fat Bone

56 Drebin, et. al. Work backwards, given x-ray absorption, I, what is the probability of it being air? Of bone?, P(I) = i P i (I) I => percent of material (our goal) P i (I) => probability distribution that material i has an intensity I. The x-ray absorption of a homogenous material. Known a priori

57 Drebin, et. al. Bayesian Estimate i (I) = P i (I) / P j (I) for all n materials. I (I) = 1

58 Drebin, et. al. Surface Extraction Each material has a density, both a real and an artificial one for visualization. D(x,y,z) = P i (I)(x,y,z) I N = D, and normalize For noisy data, blur D.

59 Drebin, et. al. Lighting absorption emitting surface scattering and absorption Compositing Front over Surface over Back I N L S I

60 Blog comments Reed grasp on the way surfaces are estimated, and the way colors and densities are managed. Peter rather than simply showing images, the paper presents a convincing comparison of images created using different processes. Josh The usage of matte volumes is also interesting in that even at this early stage in volume rendering it providing a means of extracting particular regions of a volume or even merging two regions