Reconstruction from Voxels (GATE-540)

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Reconstruction from Voxels(GATE-540)
Dr.Çagatay ÜNDEGER Instructor Middle East
Technical University, GameTechnologies General
Manager SimBT Inc. e-mail cagatay_at_undeger.com
Reference William E. Lorensen and Harvey E. Cline
Game Technologies Program Middle East Technical
University Spring 2010
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Outline

• Reconstruction

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Goals

• Develop 3D Analysis Algorithms
• Reconstruction
• Segmentation
• Feature Detection
• Labeling
• Matching
• Classification
• Retrielval
• Recognition
• Clustering

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Voxel

• A voxel (volumetric pixel)
• A volume element, representing a value on a
  regular grid in three dimensional space.
• Analogous to a pixel, which represents 2D image
  data in a bitmap

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Voxel Data

• Do not typically have their position (their
  coordinates) explicitly encoded along with their
  values.
• Stores
• Binary data
• empty / full
• Float data
• density / color / distance to surface

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Voxel vs Polygon

• Polygons
• Often explicitly represented by the coordinates
  of their vertices.
• Able to efficiently represent simple 3D
  structures with lots of empty or
  homogeneously-filled space.
• Voxels
• Good at representing regularly-sampled spaces
  that are non-homogeneously filled.
• Have a limited resolution, as precise data is
  only available at the center of each cell.

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Usage of Voxel

• Frequently used in the visualization and analysis
  of medical and scientific data.
• Some volumetric displays use voxels to visualize
  models and describe their resolution in 3D
  dimension (512512256 voxels).

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Usage of Voxel

• In games and simulations,
• Used for representation of terrain containing
  overhangs and caves.
• Concave features cannot be represented by
  heightmaps.

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Visualizing Voxels

• Visualization
• Direct volume rendering
• Extraction of polygon iso-surfaces which follow
  the contours of given threshold values.
• The marching cubes algorithm is often used for
  iso-surface extraction.

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Iso-Surface

• A three-dimensional analog of an iso-contour.
• A surface that represents points of a constant
  value (e.g. pressure, temperature, velocity,
  density) within a volume of space.

Iso-surface
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Marching Cubes

• Marching Cubes is an algorithm which creates
  triangle models of constant density surfaces
  (iso-surfaces) from 3D medical data.

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Medical Data Acquisition

• Computed Tomography (CT)
• Magnetic Resonance (MR) Imagining (MRI)
• Single-Photon Emission Computed Tomography
  (SPECT)
• Each scanning process results in two dimensional
  slices of data.

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Data Slices
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Marching Cubes Extraction

• Extracts surfaces from adjacent pairs of data
  slices using cubes.
• Cubes march through the pair of slices until
  the entire surface of both slices has been
  examined.

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Marching Cubes Overview

• Load slices.
• Create a cube from pixels on adjacent slices.
• Find vertices on the surfaces.
• Determine the intersection edges.
• Interpolate the edge intersections.
• Calculate vertex normals.
• Output triangles and normals.

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How Are Cubes Constructed

• Uses identical squares of four pixels connected
  between adjacent slices.
• Each cube vertex is examined to see if it lies
  on or off the surface.

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How Are The Cubes Used

• Pixels on the slice surfaces determine 3D
  surfaces.
• 256 surface permutations, but only 15 unique
  patterns.
• A normal is calculated for each triangle vertex
  for rendering.

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15 Unique Patterns
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Triangle Creation

• Determine triangles contained by a cube.
• Determine which cube edges are intersected.
• Interpolate intersection point using pixel
  density.
• Calculate unit normals for each triangle vertex
  using the gradient vector.

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Determining Triangles

• An index to a pre-calculated array of 256
  possible polygon configurations (28 256) within
  the cube
• Treat each of the 8 scalar values (cube corners)
  as a bit in an 8-bit integer.
• 8 scalars (8 bits) determine the actual index to
  the polygon configuration array.

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Determining Inside/Outside

• Select a iso-value that the surface will pass
  through.
• If the scalar's value is higher than the
  iso-value then
• The appropriate bit is set to one (inside)
• If it is lower then
• The appropriate bit is set to zero (outside)

iso-value 0.3
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Determining Intersections

• Determine intersection points to iso-suface by
  interpolation.

iso-value 0.3
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Determining Intersections

• Gradient of the scalar field at each grid point
  is also the normal vector of a hypothetical
  iso-surface passing from that point.

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Determining Intersections

• Interpolate these normals along the edges of each
  cube to find the normals of the generated
  vertices.

N
N
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Grid Resolution

• Variations can increase/decrease surface density.

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Examples
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Marching Squares

• 2D version of Marching Cubes
• Aims at drawing lines between interpolated values
  along the edges of a square considering given
  weights of the corners and a reference value.

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Conclusion

• Marching Squires / Marching Cubes provides a
  simple algorithm to translate a series of 2D
  medical scans into 2D / 3D objects.