Volume Visualization

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Volume Visualization The Marching Cubes
Algorithm

• Jesse Stewart

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Overview

• Volume visualization techniques used to
  visualize three-dimensional data sets. In
  biomedicine, this usually involves a model of a
  human patient, which can aide physicians in
  recognizing and studying disease and physiology
  in general.
• Many rendering techniques have been developed
  over the past 20 years. Faster graphics
  hardware and massive storage media have afforded
  highly interactive and detailed visualization
  systems.

Visible Human Project
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Biomedical Data Sources

• CT Scan (computer aided tomography) an x-ray
  imaging technique. May involve a liquid-born
  contrast material which helps highlight important
  structures.
• PET Scan (positron emission tomography) -
  involves the use of a radioactive isotope
  creates image layers a few mm thick. Useful for
  studying metabolic processes as well as anatomy.
• MRI (magnetic resonance imaging) uses a
  magnetic field combined with pulses of RF energy
  to excite hydrogen nuclei in the specimen.
  Provides superior contrast resolution compared to
  other methods.
• Others Ultrasound, SPECT (similar to CT)
• Simulation data also often needs to be visualized
  in 3D.
• Data from multiple sources may be combined.

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Assembling Data

• Many cross-sectional MRI or CT scans are usually
  done at once, producing a stack of 2D images.
  Together they form the 3D data set to be
  visualized. Density is often what is measured,
  but the same techniques can be used regardless of
  the actual measurement. Another example is
  temperature.
• Some number of preprocessing steps may be
  performed
• Filtering out noise
• Normalization/Thresholding
• Down-sampling
• This effort hopefully makes the visualization
  process more accurate.

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Example Cross-Sectional Images of the Human Body
Cross section of the head (CT) NLM
Cross-section of the head (MRI) NLM
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Data Representation

• In 3 dimensions, each location is often called a
  voxel (or volumetric pixel). Each voxel contains
  a single sample value (which may represent
  density or some other object property). Other
  properties may also be attached to each voxel
  such as an opacity value.

A 3D grid of voxels
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Two Distinct Methods

• Direct Volume Rendering (DVR) a technique that
  can produce a highly accurate picture of the
  object using a ray casting (called back
  projection) procedure.
• Surface-fitting (SF) by identifying one or more
  iso-surfaces within the dataset, a polygonal mesh
  can be constructed which approximates it.
  Traditional geometric rendering techniques can
  then be used to display these primitives, using
  fast graphics hardware. With the addition of
  shading, a more realistic view of the object can
  be produced.

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DVR Images
VolVis
Standford VolPak

• DV-Rendered images can show many semi-transparent
  parts of the image at once. However, if the
  viewing angle/distance is altered, the ray
  casting process may need to be repeated in its
  entirety, resulting in a slower and less
  interactive experience for the researcher/user.

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Isosurface based images
Volvis
mendel

• Images constructed via surface-fitting can be
  rendered quickly once the mesh is constructed.
  Lighting and shading can be adjusted to provide
  the best view of the object. However, data inside
  the surface(s) is often discarded.

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Isosurfaces

• An isosurface is an arbitrary (not necessarily
  analytical) surface in 3d space that intersects
  voxels having a constant value.

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Finding Contours

• Producing a representation of such a surface is
  similar to locating contours on a topographic
  map.
• While an approximation of a 2D contour might be
  composed of line segments, an approximation to an
  isosurface can be composed of polygons.

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Marching Cubes

• A frequently used method to create a polygonal
  mesh is known as the Marching Cubes algorithm,
  due to Lorenson.
• First introduced at SIGGRAPH in 87, Lorensons
  paper describing this algorithm is considered one
  of the most important in the field of
  visualization.
• Since then, many other researchers have
  discovered improvements and variations on the
  basic algorithm. However, it remains an integral
  part of most visualization software today.
• A description of the algorithm follows.

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Algorithm Outline

• Marching Cubes works in a divide-and-conquer
  fashion by considering groups of 8 adjacent
  voxels at a time. Together they form the corners
  of a cube. Each corner of the cube is tested to
  see whether it lies inside or outside the surface
  to be displayed. This is done by a simple
  comparison against a threshold value determined
  beforehand. For example, bone is usually found to
  be gt500 Hounsfield units in a CT scan, so any
  voxel with a value of at least 500 is inside
  the surface, while those with a lesser value are
  outside.

A cube with voxels at each corner
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Two-dimensional Version

• In a plane, a regions edge can be approximated
  in a similar fashion.
• The corners of each grid square can be tested and
  marked inside or outside. One or more line
  segments can then be added wherever the edge
  appears to be.
• There are 24 16 distinct possibilities in the
  2D case, 2 of which are ambiguous (it is not
  clear which edges to choose).

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Ambiguity

• As mentioned, cases 5 and 10 suggest two possible
  edge choices. This ambiguity can be resolved by
  increasing the resolution of the grid.
• At least in the 2D case, the edges remain closed,
  so the approximation is still composed of
  polygons.

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Three Dimensions

• In the 3D case, there are 28 256 possible
  corner configurations. Each case suggests that
  the surface intersects the cubes edges in a
  particular way.
• When all 8 corners are either inside or outside,
  nothing interesting happens.
• When exactly one corner is inside, a triangle
  whose vertices lie on the three edges adjacent to
  that corner is added to the mesh (a list of
  triangles).
• When more than one corner is inside, up to 4
  triangles may be added.

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Encoding each case

• Instead of laboriously computing which triangles
  should be added every time, a lookup table can be
  used based on the corner information. It is also
  convenient that a byte is just enough to store
  the information.
• Example 00001101 13 means only corners v1, v3,
  v4 are inside.

11111111
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Cube Families

• Accounting for rotational symmetry and inverse
  configurations (where c1c2 256), there are
  only 15 families.

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Choosing Triangle Vertices

• A simple way to define each triangle is to place
  its vertices on the midpoints of cube edges. For
  a more accurate mesh, linear interpolation can be
  used.
• If t is threshold, and A and B are values at
  adjacent corners, the vertex should be placed D
  units along the edge.
• D 1 / ( 1 (a-t)/(t-b) )
• Yet another choice is binary search.

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Ambiguity Again

• Cases 3,6,7,10,12, and 13 can cause problems due
  to ambiguous triangle choices. To resolve this,
  various methods have been proposed, some more
  complex (heuristic) than others. The most
  straightforward method is to compare adjacent
  cubes to ensure neighboring edges meld together
  appropriately.

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Example of Mismatch
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Surface Normals

• Surface normals are computed along with each
  triangle, for use with shading later
• A gradient is computed at each cube corner,
  estimated by central differences along the three
  axes.
• Then, linear interpolation is used once again to
  compute a gradient for the triangle vertices.

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Surface normals
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Variations

• Marching Tetrahedrons fewer cases, eliminates
  ambiguity
• Adaptive

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Other Uses

• Rendering Metaballs

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Interesting Links

• Visible Human Project
• http//www.nlm.nih.gov/research/visible/
• SIGGRAPH Volume Visualization and Rendering
  materials
• http//www.siggraph.org/education/materials/Hype
  rVis/vistech/volume/volume.htm
• VolVix FAQ
• http//www.sjoki.uta.fi/shmhav/med-volviz-faq.t
  xt
• Standfords VolPak software
• http//www-graphics.stanford.edu/software/volpac
  k/
• VolumeJ Java Volume Rendering
• http//bij.isi.uu.nl/vr.htm
• Visible Human Server
• http//visiblehuman.epfl.ch/
• VolViz group homepage
• http//www.volvis.org/
• IEEE Visualization Conference
• http//vis.computer.org
• SIGGRAPH homepage
• http//www.siggraph.org/