Surface Reconstruction from 3D Volume Data

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Surface Reconstructionfrom3D Volume Data
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Problem Definition

• Construct polyhedral surfaces from
  regularly-sampled 3D digital volumes.

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Medical data sources

• CT (Computed Tomography)
• MRI (Magnetic Resonance Imaging)
• SPECT (Single Photon Emission Computed
  Tomography)
• Parallel slices at intervals of 1 to 5mm
• Gray-level images with 8 or 12 bits
• Usually 128x128 to 512 x 512 pixel images

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Assumptions

• Structured data
• Slices are parallel and along Z axis
• The XY plane is sampled uniformly
• The sampling is dense and uniform
• Spatial coherence
• Curvature

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Boundary segmentation

• Gray levels correspond to different bone and
  tissue densities (Hounsfield units)
• Segmentation of different types of organs based
  on predetermined gray values
• Extraction of iso-surfaces surfaces whose pixels
  are within predetermined range

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Example of iso-surface extraction
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Desired properties of the reconstruction

• Agree with the data
• should not create new features
• should contain all existing features
• Correct topology
• Position and rotation invariance
• Surface continuity

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Reconstruction methods

• 2D methods (contour matching)
• extract contours in each image separately and
  then try to connect them to form the surface
• 3D methods (Marching cubes)
• work on consequent images, constructing pieces
  of the surface connecting them

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2D-Based Reconstruction

• 1. 2D contours extraction.
• boundary following (left hand on the wall).
• sequential scanning.
• 2. Surface tiling
• establish correspondences between points

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2D Reconstruction problems

• How to match the points?
• individual 2D contour sampling does not
  necessarily make points correspond
• closest point matching manual corrections
• smoothing contours to reduce number of points
• Topological problems
• self intersection
• inconsistent topology

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2D-based reconstruction problems (2)

• Inherent asymmetry
• the XY plane is treated differently than the
  other planes.
• This cause the algorithm to be dependent of the
  objects orientation.

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3D Based Reconstruction

• Define 3D elements as VOXELS and the connectivity
  between them. Two representations

Boundary element as a vertex
Boundary element as a face
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Block and beveled form duality
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Marching cubes algorithm

• Look at eight pixels in two consecutive slices
• Determine if a surface goes through according to
  the values at its edges
• If a surface goes through, construct the surface
  elements and march on to adjacent voxels

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The Marching Cube
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Cube representation
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Surface topology within a cube
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Surface topologies cases

• A cube with 8 vertices, each is either 1 (inside
  object) or 0 (outside)
• Total of 28 256 cases to examine(one bit to
  each vertex).
• The 8 bits form an 8 bit index to a 256-entries
  edge table.
• Reduce number of entries by symmetry

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Instances of the same case
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14 cases of surface topologies
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Positioning the Surface

• Find the exact location of the surfaces vertices
  on the voxels edges.
• The original vertices gray levels are linearly
  interpolated.
• Higher order interpolation can be used but
  experiments show little improvement.

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The Range 310 lt I lt 200
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Surface normals (1)

• Calculate unit normal for each triangle vertex
  (for shading calculations).
• The gradient Vector g, is the derivative of the
  density function

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Surface normals (2)

• Estimate the gradient vector at the cube
  vertices.
• For each vertex D(i,j,k) - density at
  pixel(i,j) in slice k, length of
  the cube edges. slices distance

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Surface normals (3)

• Linearly interpolate the gradient at the point of
  intersection.
• Dividing the gradient by its length produce the
  unit normal at the vertex.
• On a surface the direction of the gradient vector
  is the normal to this surface.
• Keep only 4 slices in memory in order to
  calculate at all vertices of the cube.

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Marching Cubes Algorithm(Lorenson , Cline 1987)
Calculateindex
Read 4 slicesinto memory
look at table
Form a cube from 2 slices
Interpolate normals
use densitiesfind intersection
Output trianglevertices normals
Calculate a unitnormals
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Efficiency Enhancement

• Take advances of pixels, lines and slice
  coherence.
• 3 new edges need to be interpolated for each
  cube (for interior cubes) instead of 12.
• Decrease the number of triangles by reducing the
  slice resolution - averaging 4 pixels into one.

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Coherence
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Desired properties of the reconstruction

• Agree with the data
• should not create new features
• should contain all existing features
• Correct topology
• Position and rotation invariance
• Surface continuity

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Surface quality

• Non uniform sampling in the X-Y and Zplanes
  causes steps results.

Z 5mm
constructed model
X 0.5mm
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Sampling at a higher rate
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Face ambiguities
a
c
d
b
e
f
g
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Holes
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Dealing with ambiguity (1)

• Supersampling - interpolate to obtain new values
  which are used to disambiguate.
• 1. calculate each face separately
• 2. disambiguate by calculating the value of the
  center of face.

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Dealing with ambiguity (2)

• Voxel Topology - evaluate vertex adjacencies and
  check consistency in order to disambiguate.The
  surface should separate not only on-off
  voxels but also on voxels that are not locally
  adjacent.

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• Create objects that are connected according to
  pre-defined adjacency
• Resolve face ambiguity and objects can be
  unambiguously recover from its surface.
• Advantage No need to hold actual data values
  (gray levels), but binary data.No need to
  interpolate Faster

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Dealing with ambiguity (3)

• Use tetrahedra and not cubes to decide where the
  surface intersects.
• Decompose each cube to 5 tetrahedra with 4
  voxels.

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• No ambiguous cases.
• 14 ways that surface can pass through tetrahedron
  reduced by rotation and complementary symmetry to
  2 cases
• disadvantage higher number ofpolyhedral faces
• recommended preprocessing merging procedure

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Tetrahedra - problems

• Non uniform topology on the voxels. 2 voxels
  are adjacent iff they lay in the same tetrahedron
  T-adjacency.
• Result Not translation invariant - the surface
  construction depends also on the position of the
  objects
• ambiguous surface construction.

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Example 3D - checker board
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Conclusions

• Wildly used in todays applications.
• Problems ambiguity, self intersection
• construct high number of triangles.
• Recommended - surface simplification as
  post-processing step