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Seminar 1
Surface Rendering, Decimation
Presented By Sonali Barua
Date:10/31/2005
Marching Cubes
Algorithm to create Triangle
Models of 3D Medical Data
Information Flow for most 3D
Medical Data
Former Algorithms and Drawbacks
• Surface Contours, Cuberilles (voxels),
Octree, ray casting, shadow graph of
density volume
• Didn’t work as they threw away useful
information in the original Data such as
inter slice connectivity and some
algorithms relied on Motion to give a 3D
effect
Marching Cubes
• Retains the Original 3D Data
• Can be Displayed using standard
rendering Algorithms.
• Uses Divide and Conquer Techniques to
calculate the surface in a logical cube
created from 8 pixels, four each from
adjacent slices.
Approach
• Assign 1 to Vertex if data
value>=value of the
surface being constructed
hence inside the cube.
• Assign 0 to Vertex <
value of surface hence
outside the surface.
• 8 vertices =28 =256 ways
a surface intersects cube.
• Creating table of 256
cases tedious and error
prone
Alternate way
• Using two symmetrical
Properties : Reversing or
Complimentary cases and
Rotational symmetry.
• 14 unique patterns.
• Permutations of these unique
patterns using complementary
and rotational symmetry
produces the 256 cases.
Indexes
• To recreate the 256 cases an 8
bit index is used, each bit
representing a vertex.
• The index serves as a pointer
into an edge table that gives all
edge intersections for a given
cube configuration
• With the help of the knowledge
of the edges which the surface
intersects, the surface is
interpolated along the edge.
• Usually linear interpolation is
enough as it forms sufficient
number of triangles per cube.
Steps in Marching Cubes
• Four Slices of Data are read into Memory
• Scan two slices Create a cube using four neighbors from
each of the slices.
• An index of the Cube is Calculated by Comparing
density values with user specified surface constants.
• Using the Index, list of edges is looked up from the table.
• Linear interpolation is used to calculate the surface edge
intersection using the densities of the vertices.
• Unit Normal at each vertex of the cube is calculated. The
Normal is interpolated to each triangle vertex.
• The triangle vertices and the vertex Normal are the
outputs which is consequently used for Gourand
shading.
Calculating the Gradient Normal of
each triangle vertex
• Estimate the Gradient
Vectors at the cube
vertices using central
differences and linearly
interpolate the gradient at
the point of intersection.
• Gradient/length=unit
Normal
• Unit Normal is
interpolated linearly with
the point of intersection.
• Reason why 4 slices are
kept in Memory.
Enhancements
• Efficiency.
• By getting 9 edges from previous slices
and calculating only 3 edges.
• Save only previous line and pixel
intersections.
• Reduction of Slice resolution using
averaging , though it causes loss of data.
Functional Enhancements
• Use of Boolean Operations for
cutting and Capping of Solid
Models using concepts of
inside, outside and on the
surface using the index values
of the cube.
• Truth Table is drawn.
• When both surfaces are there
then the clipping algorithm by
Sutherland- Hodgman is used.
• Truth table allows for multiple
surface extraction.
Implementation
• C language
• Sun workstations using UNIX, VAX under
VMS and IBM 3081 under IX/370
Results CT
MRI
SPECT
Questions
?????
Decimation of Triangle
Meshes
Algorithm to reduce the number of
triangles rendered.
Reason for the Algorithm
• Rendering speeds and memory
requirements are proportional to the
number of polygons.
• Surface construction Algorithms such as
Marching cubes creates large number of
triangles to render.
Decimation
• Goal: Reduce the total number of Triangles while still
retaining most of the important features
• Applies to Discrete Modeling.
• Approaches can be either Adaptive or Filter based for
synthesizing objects
• Adaptive: Produces more primitives in selected Areas.
• Filter based: Starts with large number of samples and
replaces samples to reduce model size.
Reduced Mesh Requirements
• Must preserve original topology of mesh
including non-manifold forms.
• Good Geometric Approximation to the
original Mesh.
• Hence, New vertices should not be
created.
The Simple Algorithm
• For each pass and each vertex
• Characterize the local vertex geometry and
topology
• Evaluate the decimation criteria
• Triangulate the resulting hole
• Multiple passes are made over all vertices
with adjustments in the decimation criteria
until a termination criteria is met.
Characterizing Local Geometry/topology
• Each vertex is
classified into one of
the 5 types
• If vertex is Complex, it
is not a candidate for
deletion. Rest of the
vertex type are.
• If an angle is greater
than the specified
feature angle than a
feature edge exists.
Decimation Criteria
•
Two Decimation Criteria used:
• Vertex distance to plane d=| n . (v-x)|
• Vertex distance to edge
•
•
•
Simple vertices used the first Criteria.
Boundary and interior edge use the Second Criteria.
To get rid of undesirable feature edges or small triangles caused due to
“noise” in the original mesh, corner and interior edge vertices may use the
first criteria to determine the edges.
Triangulation
• Deletion of vertex causes one or two loops.
• In each loop a triangulation must be created
such that they are non-intersecting and nondegenerate.
• Two Important Characteristics are used:
• If loop cannot be triangulated, vertex is not deleted.
• Due to the star shaped nature of each loop, recursive
splitting will be effective.
• Upon completion of triangulation, original vertex
and it’s triangles are deleted.
Implementation
• It is used as a filter in LYMB/VISAGE
visualization environment.
• Algorithm repeated till a specified
decimation threshold is met.
• Decimation is controlled by slowly
adjusting the distance and feature angle
criterion
• Sometimes Number of iteration is limited
• Triangulation aspect ratio is also modified.
Data Structures
• Uses a vertex –triangle ring hierarchy
structure
• Contains hierarchical pointers from vertex
up to triangles using the vertex and
pointers from triangles down to vertices.
• three lists:
• List of Vertex Coordinates.
• List of Triangle definition.
• List of lists using the each vertex.
Triangulation
• Recursive loop splitting Algorithm
• Each loop divided into 2 halves along a Split
line.
• Split line = a line defined by 2 non-neighboring
vertices in loop
• Each new loop is divided till there is only 3
vertices remaining
• A loop of 3 vertices forms a triangle that is added
to the Mesh.
Continued…..
• Since loops are non-planar and star shaped a split plane is used
• Split Plane= a Plane orthogonal to the average plane that contains
the split line.
• Split plane helps determine the non-overlapping of loops.
• If split plane isn’t able to make two non-overlapping loops then the
vertex and it’s associated triangles are kept in the mesh.
Aspect Ratio
• Each loop can be split more than one way, so
best splitting plane is used.
• To get this a criteria of aspect ratio is used
• Aspect ratio= (min distance of the loop vertices
to the split plane)/length of the split line.
• Best Split plane=max (aspect ratio). Can be
made to be greater than some specified aspect
ratio.
Special Cases
• Closed tetrahedron, Tunnels.
• Remove a vertex to resolve the issue.
Volume Modeling
Terrain Modeling
Questions???
Thank you