Scalar Data

Alark Joshi

Announcements

• Pick two papers to present

– Email me your top 3/4 choices.

– FIFO allotment

• Contact your clients

• Blog summaries: http://cs.boisestate.edu/~alark/cs564/participants.html

Data Grids

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

Topology

• Properties of geometric shapes that remain unchanged even when under distortion

Topologically equivalent

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

Grid Types

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

Structured and Unstructured Grids

• Structured grids have a regular topology and regular/irregular geometry

• Unstructured grids have irregular topology and geometry

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

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: http://geolab.larc.nasa.gov/GridTool/Training/VGRID/

Unstructured Grids

• Can be adapted to capture local features

Types of grids

• Cartesian or equidistant grids

– Structured grid

– Number of points = Nx * Ny * Nz

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)

Types of grids

• Rectilinear grids

• Topology is still regular but irregular spacing between grid points

• Topology is still implicit

Types of grids

• Curvilinear grids

– Topology is still regular but irregular spacing between grid points

– Topology is implicit, but vertex positions are explicitly stored

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

Scalar Data Visualization

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

Function Plots

• Function plot for a 1D scalar field

– Points

– 1D manifold: line

– Errors bars possible

Function Plots

• Gnuplot examples

Function plots for 2D scalar field

• Points

• 2D manifold: surfaces

• Surface representations

– Wireframe

– Hidden lines

– Shaded surface

Function plots for 2D scalar field

• Shaded surface

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: http://giscommons.org/?page_id=18

Isolines

Isolines

• Pixel by pixel contouring

• Straightforward approach: scan all pixels for equivalence with isovalue

• Input: 𝑓: 1, … 𝑥𝑚𝑎𝑥 ∗ 1, … , 𝑦𝑚𝑎𝑥 → 𝑅

• Isovalues I1,…In and isocolors c1,…, cn

• Algorithm:

Color coding

• Easy to apply colors to 1D and 2D scalar fields

– Map color each pixel on a 1D input signal or 2D image

Color coding

• Example:

– Separate color table to visualize the brain

– Separate color table to visualize the tissue

Volume Visualization

• Scalar volumetric data Ω 𝜖 𝑅3 → 𝑅

• Medical application: CT, MRI, ultrasound, confocal microscopy etc.

Volume Visualization

• Seismic applications: Oil exploration, earthquake prediction, etc.

Volume Visualization

• Create a representation of a 3D scalar dataset Ω 𝜖 𝑅3 → 𝑅

• Voxel (volume element) similar to pixel (picture element)

• Values between grid points are resampled by interpolation

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

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

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

Volume visualization

• Slicing: Display the data mapped to colors, on a slice plane

• Isosurfacing: Generate opaque/semi opaque surfaces

• Transparency effects

Semi-transparent material

Iso-surface

slice

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

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

Volume Rendering Integral

Ci+1 = Ai*Ci + (1 - Ai)*ai*ci

Ai+1 = Ai + (1 - Ai)*ai

Ci ; Ai

ci ; ai

Ci+1 ; Ai+1

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

Interpolation

• Linear interpolation 𝑥 = 𝑥1 ∗𝑥−𝑥0

𝑥1−𝑥0

+ 𝑥0 ∗𝑥1−𝑥

𝑥1−𝑥0

• 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′

Trilinear Interpolation [Levoy]

Levoy Interpolation

Interpolation

Nearest Neighbor Binary

Trilinear Interpolation Smooth/Weighted

Direct Volume Rendering

• Marc Levoy. Display of Surfaces from Volume Data. IEEE Comput. Graph. Appl. 8(3), 29-37, 1988.

• 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

Acquired values

Data preparation

Prepared values

classification

Voxel colors

Ray-tracing / resampling Ray-tracing / resampling

Sample colors

compositing

Voxel opacities

Sample opacities

Image Pixels

Volume Rendering Pipeline

shading

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

lights

i

n

sdiambientatotal

shiny

RVkLNkIIkI#

1

ˆˆˆˆ

Shading

• No normals in volumetric data

• Use gradients as normals

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

Transfer Functions

• Color table for volume visualization

– Map raw voxel intensity values into color, opacity etc.

Classification

• Examples of four different transfer functions

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

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”

Volume Rendering

• Robert A. Drebin, Loren Carpenter, and Pat Hanrahan. Volume rendering. SIGGRAPH Comput. Graph. 22, 4 (June 1988), 65-74

• 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

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

Drebin, et. al.

Original CT Data

Fat Tissue Bone

Density Color and Opacity

Strength

Gradient NX NY NZ

Shaded

Transformed

Final Image

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 Fat

Tissue Bone

Drebin, et. al.

• Work backwards, given x-ray absorption, I, what is the probability of it being air? Of bone?, …

– P(I) = i Pi(I)

• I => percent of material (our goal)

• Pi(I) => probability distribution that material i has an intensity I. – The x-ray absorption of a homogenous material.

– Known a priori

Drebin, et. al.

• Bayesian Estimate

i(I) = Pi(I) / Pj(I) for all n materials. I (I) = 1

Drebin, et. al.

• Surface Extraction

• Each material has a density, both a real and an artificial one for visualization.

• D(x,y,z) = Pi(I)(x,y,z) I

• N = D, and normalize

• For noisy data, blur D.

Drebin, et. al.

• Lighting

– absorption

– emitting

– surface scattering and absorption

• Compositing

– Front over Surface over Back

L

N

I I

S

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”