CS 655 Advanced Computer Graphics

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CS 655 Advanced Computer Graphics

• Volume Modeling and Rendering

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Goal

• To gain meaningful insight into multi-dimesional
  data
• Sometimes termed Scientific Visualization or
  Visualization of Scientific Data (ViSC)
• Employs methods of
• Computer graphics
• Image processing
• Computer vision
• others

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Data Acquisition

• Data can be gathered from
• Observations
• Simulations
• Data format
• 2D
• 3D
• Scalar
• Tensor
• Multi-valued
• etc.

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Visualizing Data

• Several techniques for visualizing the data
• Graphs
• Charts
• Contour plots
• Surface renderings
• Volume representations
• Problem
• Too much data!
• Data acquisition
• Supercomputers
• Satellite/Spacecraft scanners
• Radio astronomy telescopes
• MRI scanners
• etc.

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Applications

• Scientific Visualization used in
• Computational Fluid Dynamics
• Finite element analysis
• Astrophysics
• Space exploration
• Molecular Modeling
• Mathematics
• Medical imaging
• Weather forecasting
• etc.

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Data

• We typically have data values for some function
• over a 3D grid

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Volume Data Geometry

• Data can be in any type of 3D geometry
• Cartesian
• Regular
• Rectilinear
• Curvilinear
• Block Structured
• Hybrid
• Unstructured

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Volume Data Geometry

• Cartesian
• (i, j, k)
• Data elements are cubic and axis aligned

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Volume Data Geometry

• Regular
• (idx, jdy, kdz)
• Data elements are rectangular and axis aligned,
  but not necessarily cubic

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Volume Data Geometry

• Rectilinear
• (xi, yj, zk)
• Data elements are rectangular and axis aligned,
  but may vary in size

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Volume Data Geometry

• Curvilinear
• (xi,j,k, yi,j,k, zi,j,k)
• Non rectangular cells
• Usually hexahedral or rectangular
• Cells fit a volume or wrap around an object
  (often used in Computational Fluid Dynamics)

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Volume Data Geometry

• Block Structured
• The combination of several curvilinear systems in
  the same data volume
• Used to overcome topology restrictions of a
  single curvilinear system

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Volume Data Geometry

• Unstructured
• (xi, yj, zk)
• No geometric constraints are imposed
• Cells may be any shape

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Volume Data Geometry

• Hybrid
• A combination of structured and unstructured

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Data

• Often we have data with more than 3 dimensions
• How do we visualize that data?
• Aliasing can be a problem
• If the data comes from a simulation, we may be
  able to oversample
• If the data is collected, we just have to use
  what we have

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Displaying Information

• Given the 3D grid of data, how do we display the
  data so that it gives information to the viewer?
• Several approaches have been used
• Contours over 2D slices
• Isosurface rendering
• Volume rendering

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Connect the Contours Method

• Take slices of the 3D volume and locate contours
  for
• for each zk slice

slice zk2
slice zk1
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Connect the Contours Method

• Join the contours between slices with triangular
  polygon patches

slice zk2
slice zk1
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Problem with Connect the Contours

• Ambiguities

slice zk
Should this be interpreted as
or as
We dont know we need user input
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Isosurface Rendering

• Extension of contouring through 2D cells

to 3D voxels
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Isosurface Rendering

• Now we generate surface sections
• Can model surfaces with
• Polygon Patches
• Curved surface sections such as bicubic patches
• Polygon patches are typically used
• Fast
• Can be shaded to produce smooth surfaces

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Marching Cubes Algorithm

• An isosurface rendering technique presented by
  Lorensen and Cline in 1987
• Idea
• Given a threshhold value, obtain polygon
  isosurface patches over a 3D grid of cells by
  processing the cells one by one across the grid
• Tests for isosurface intersections with cells are
  reduced by
• Tabulating possible types of slices an isosurface
  can make across a cell
• Marching through cells, determining the
  isosurface topology across each cell by matching
  the cell pattern to one in the table

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Marching Cubes

• Idea
• Take a threshold value fs
• The values at each node fijk must be either
• For all adjacent vertices with opposite
  relationship to fs, the surface defined by fs
  must pass through the edge connecting those
  vertices.

fijk
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Marching Cubes

• Possible configurations
• All 8 fijk fs
• All 8 fijk lt fs
• One fijk fs, all others lt fs
• Two adjacent fijk fs, all others lt fs
• Two diagonal, in the same XY plane
• Two diagonal, in the same XZ plane
• etc.

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Marching Cubes

• Since each cell has 8 vertices, there are 28
  256 possible ways the vertex function can be
  satisfied across the cube
• The configurations have many symmetries
• Interchange of vertex state
• e.g., we get the same results if one fijk fs
  or if one fijklt fs
• Rotational Symmetry
• This reduces the number of possibilities to 15

is equivalent to
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Marching Cubes Cell Patterns
Pattern 0 all fijk lt fs or all fijk fs 0
polygons
Pattern 1 one fijk lt fs or one fijk fs 1
polygon
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Marching Cubes Cell Patterns
Pattern 3 two diagonal fijk lt fs or two
diagonal fijk fs in the same plane 2 polygons
Pattern 2 two adjacent fijk lt fs or two
adjacent fijk fs 2 polygons
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Marching Cubes Cell Patterns
Pattern 5 three vertices fijk lt fs or three
vertices fijk fs in the same plane 3 polygons
Pattern 4 two diagonal fijk lt fs or two
diagonal fijk fs not in the same plane 2
polygons
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Marching Cubes Cell Patterns
Pattern 7 three fijk lt fs or three fijk fs
not in the same plane 3 polygons
Pattern 6 three fijk lt fs or three fijk
fs two in the same plane 3 polygons
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Marching Cubes Cell Patterns
Pattern 9 four fijk lt fs any three in the same
plane 4 polygons
Pattern 8 four fijk lt fs in the same plane 2
polygons
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Marching Cubes Cell Patterns
Pattern 11 four fijk lt fs as shown 4 polygons
Pattern 10 four fijk lt fs two pairs on
opposing edges 4 polygons
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Marching Cubes Cell Patterns
Pattern 13 four fijk lt fs on opposite
corners 4 polygons
Pattern 12 four fijk lt fs as shown 4 polygons
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Marching Cubes Cell Patterns
Pattern 14 four fijk lt fs as shown 4 polygons
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Marching Cubes Vertex summary

• Vertex Summary
• 1 pattern with 0 or 8 vertices fs
• 1 pattern with 1 or 7 vertices fs
• 3 patterns with 2 or 6 vertices fs
• 3 pattern with 3 or 5 vertices fs
• 7 pattern with 4 vertices fs

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Marching Cubes Triangle summary

• Each isosurface intersection through a cell is
  divided into 0 4 triangles
• 0 or 8 vertices fs 0 triangles
• 1 or 7 vertices fs 1 triangle
• 2 or 6 vertices fs 2 triangles
• 3 or 5 vertices fs 3 triangles
• 4 vertices fs 2 or 4 triangles

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Marching Cubes

• We are also making the assumption that the
  isosurface does not intersect any cell edge more
  than once, i.e., we cant have

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Edge Table for Marching Cubes

• Label all of the edges and vertices of each cube
  as follows

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Edge Table for Marching Cubes

• Maintain an edge table that specifies which edges
  are intersected for a given index
• For each cube, set up an eight bit code that
  serves as the index into the table
• Indexing into the table gives all edge
  intersections for that cube
• index
• Each bit of the index specifies whether
• the vertex is fs or lt fs

V8
V7
V6
V5
V4
V3
V2
V1
V7
E7
V8
E12
E11
V3
E3
V4
E6
E8
E2
E5
E4
V5
V6
E10
E9
E1
V1
V2
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Marching Cubes Algorithm

• Process cells across successive slices. For each
  cell do
• Determine the binary cell code
• Index into the edge intersection table to
  determine which edges are intersected
• Calculate actual intersection positions along the
  edges using linear interpolation
• Calculate normals at cell vertices and
  interpolate to get triangle vertex normals
• Render

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Calculating Normals

• Since the surface to be drawn is constant, i.e.,
• then the gradient component along the surface
  tangential direction is zero.
• Thus, the direction of the gradient is normal to
  the surface
• The gradient is the derivative of the function f
  ,
• with

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Computing the Normal

• We estimate the gradient using central finite
  differences
• First, estimate gradient at the cube vertices
• Next, linearly interpolate to get gradient at
  intersection
• Central finite differences use neighbor values to
  approximate gradients

We then divide the gradient by its length to get
a unit normal vector
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Computing the Normal
(i, j, k)
(i-1, j, k)
(i1, j, k)
If fi-1, j, k 36 fi, j, k 63
fi1, j, k 82 then
Then the x component of the normal at (i, j, k)
is given by
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Computing the normal
1. Compute gradients at cube vertices
2. Linearly interpolate gradients to get
normals at the intersection points
3. Render the triangle(s)
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Sample Marching Cubes Images

• Isosurfaces extracted from (a) MRI Head dataset
  at a75 and (b) MRI Brain dataset at a30 (from
  Zhang and Newman)

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• From Lorensen, Marching Through the Visible
  Man, Proceedings of IEEE Visualization 1995

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• From Pekar et. al, Fast Detection of Meaningful
  Isosurface for Volume Visualization, in
  Proceedings of IEEE Visualization 2001

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Problem with Marching Cubes

• Ambiguity
• Cases 3, 6, 7, 10, 12, and 13 are ambiguous.
• For example, case 3

or
Could be
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Problem with Marching Cubes

• The ambiguities can cause holes in the resultant
  surface
• To remove ambiguities
• User input
• Difficult, tedious
• Use surrounding cell data to pick which option to
  use

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Volume Rendering

• Display interior information of a 3D data set
  instead of simply showing isosurfaces
• Can extract particular values or properties, or
  can accumulate values
• SIGGRAPH 88 articles by
• Sabella
• Upson and Keeler
• Drebin and Carpenter

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Volume Rendering

• Basic Approach
• Cast rays from viewing position through pixel
  into the data volume

Eye
Image plane
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Volume Rendering

• Follow the ray through the volume of data, return
  back
• The first data hit
• A particular data value
• An accumulation of data as the ray passes through
  the volume
• For example
• Seismic Data
• Along each ray, determine position and magnitude
  of maximum value
• Encode the info using grey scale or color range
  and assign the corresponding intensity to the
  pixel
• Transparent Surface Layers
• e.g., tissue around bone
• Store multivalued data (density, opacity,
  texture, etc.)
• Accumulate intensities along path until either
• opacity 1 or the ray exits the data

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Ray Processing

• Intersection calculations with faces of data
  cells (voxels)
• need an efficient voxel traversal scheme

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Ray Processing

• Data Calculations
• Interpolate to intersection positions or
• Average over voxel or
• Search for particular values
• Initial ray entry position is on one of the
  bounding faces of the data grid
• easy to find first voxel

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Ray-Exit Intersection Calculations

• Given a ray with voxel entry position po and unit
  direction vector traveling through a voxel
  with outward normals Nk
• Potential exit faces are those for which

N2
N3
N1
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Ray Intersection Calculations

• If the data grid is aligned with the screen
  coordinate axes,
• Then all we have to do is check the sign of each
  component of to determine the three candidate
  exit planes

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Ray Intersection Calculations

• Exit position on each candidate plane can be
  expressed as
• We can determine Sk from the plane equation
• Thus,
• And the smallest Sk gt 0 identifies the exit face

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Ray Intersection Calculations

• If the Nk are aligned with the coordinate axes,
  this can be simplified
• e.g., if one candidate plane is (1, 0, 0), then
  for that plane
• Where Po (xo, yo, zo), u (ux, uy, uz) and xk
  is the value of the right boundary face for the
  voxel
• Other voxel traversal schemes are possible
• e.g. Can take trial exit plane as the one
  perpendicular to the direction of largest
  component of u

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Example
N2
P0 (0, 1, 0)

• u (1, -1, 0)
• Cube sides (-1, 1)

N3
N1
Nk are axis aligned, so
So N1 is the exit face
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Examples
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Volume Rendering

• Examples

Severe Storm