CELLULAR AUTOMATA FOR 3D MORPHING

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CELLULAR AUTOMATA FOR 3D MORPHING

• Sudhanshu K Semwal
• Kaushal S Chandrashekar
• Department of Computer Science
• University of Colorado at Colorado Springs

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Outline

• Introduction to Morphing
• Background
• Design of volume morphing algorithms
• Implementation
• Results
• Future work
• Conclusions

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Introduction to Morphing

• Morphing
• Given a source model S and a target model T,
  morphing constructs a series of transformations
  Wt t ? 0,1 such that W0 S and W1 T

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Introduction to Morphing

• Applications of morphing
• Movies
• 3D games
• Study of evolution
• Understanding fetal and plant growth

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Introduction to Morphing

• Types of morphing
• 2 dimensional
• 3 dimensional
• Benefits of 3 dimensional morphing
• Free from lighting and environmental
  considerations
• Real-time camera view changes

6
Background

• Types of 3D morphing
• Polygonal mesh
• Volumetric
• Polygonal mesh morphing
• Uses polygonal or polyhedral meshes as input
• Mesh
• Linear surface composed of polygons described as
  Vertex/Face/Edge/Graph or vertices
• Describes topology of surface

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Background contd

• Benefits
• Very popular format, supported by many modeling
  packages
• Most approaches are of this form
• Drawbacks
• Datasets from medical and geological fields do
  not support polygonal data
• Complex topologies difficult to handle, esp. with
  user control
• Complex implementations reqd for processing and
  rendering

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Background contd.

• Two parts of the morphing problem
• Correspondence problem
• Interpolation problem

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Background contd

• Volumetric morphing
• Uses 3 dimensional volumes as source and target
• Volumes composed of voxels
• Voxels atomic unit of volume, short for volume
  pixel

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Background contd

• Volume defined as a collection of scattered
  voxels, with each voxel being associated with a
  set of values of size S, i.e, the volume V is
  given by
• V (xi, vi ) xi ? R3 , vi ? RS, i 1 .. n
  Chen1995
• Common representation is a 3 dimensional grid
• Each (i,j,k) either empty or containing one value

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Background contd

• Benefits
• Free of restrictions of topology and geometries
• Can be easily be applied to meshes by converting
  them to volumetric data, while the reverse can
  result in topologies difficult to morph.
• Data in the medical, geological and energy fields
  is generated in the volumetric format and needs
  to be morphed directly.
• Do not need a bijective mapping between vertices
  of the source and target formats.
• The simple format allows implementation ease.

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Background contd.

• Drawbacks
• Not as popular as polygonal formats in the
  entertainment industry, fewer test models
  available.
• Computationally intensive to process and render.

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Background contd.

• Cohen-Or1998 Uses distance field transformation
  method
• Uses user-defined control points
• Morphing done using warping and cross-dissolving
• Disadvantage quality of morph dependent on
  control points

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Background contd.

• Problems with current 3D morphing techniques
• Correspondence
• Require user-defined control points
• Exhibit

TO
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Background contd.

• Our goal
• Less restrictive approach
• Minimal or no user control
• Natural looking process

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Background contd

• Cellular Automata
• Dynamic system composed of discrete cells, work
  in discrete steps of time and are characterized
  by local interactions
• Some of the characteristics of CA are
  Wolfram1984
• They utilize a discrete lattice of sites
• Discrete time steps drive the simulation.
• Each site can only take a finite set of values
• Each site evolves according to same deterministic
  rules
• The evolution of a site only depends on the
  neighborhood.

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Design contd

• Characteristics of CA used
• 3 dimensional lattice.
• The dimension of the lattice is that of the
  volume.
• Each cell can either be empty or contain one
  value.
• All cells are equal
• The cellular automaton is non-circular.

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Design contd

• Core increment algorithm
• Source and target volumes overlaid within same CA
• Uses intersection of 2 volumes as base
• Intersection is called the core
• Core incremented step-wise to fill both source
  and target volumes
• At each iteration, each voxel in the core checks
  its surroundings, if source or target voxel
  found, adds that voxel to the core.

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Design contd

• Voxels required to start from core and fill both
  volumes recorded in add-array(target volume) and
  del-array (source volume) as separate sets.
• Process continues until core cannot increment any
  more
• Source model loaded into new volume
• Add and del arrays scanned, former forward and
  latter in reverse
• At each iteration, voxels for that iteration in
  the del-array are removed from the source volume
  and voxels for that iteration in the add-array
  are added
• Source voxels not intersecting with target are
  gradually removed and target voxels are gradually
  added
• Generates a smooth organic quality

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Design contd
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Design contd
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Design contd
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Design contd
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Design contd
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Design contd

• Benefits
• Rules defining behavior of cellular automata are
  simple, hence easy to upgrade or replace.
• Uses 3 dimensional arrays as volumes. Popular
  formats can be used directly.
• No correspondence required between source and
  target models.
• The morph can be controlled because of the add
  and delete arrays containing the information of
  each iteration as separate sets.
• Organic quality of morph

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Design contd

• Disadvantages
• Requires intersection of volumes
• 4 volumes are reqd to be created,
• Source, Target, Core, Source for morphing
  iteration
• Volumes containing source and target have to be
  of same size, no restrictions on models
  themselves
• No attempt to preserve mass
• Color handling not solved, no solution elsewhere

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Design contd

• Bacterial growth model
• Based on papers using CA for bacterial simulation
• Bacteria modeled as individual agents in lattice
  and rules govern its reaction to environment and
  other bacteria
• Each non-empty voxel in source volume is
  considered as a bacterium

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Design contd

• Rules for bacteria
• Non-motile.
• Homogenous
• Have 2 needs, food and the need to reproduce, the
  latter being dependent on the former.
• Will reproduce if it finds food and has space to
  place its offspring by making a copy of itself.
• Creates only one offspring per iteration.
• Will live and reproduce infinitely given enough
  space.
• Will die if food is not present in its current
  position.
• Cooperate to keep each other alive.

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Design contd

• Bacterial Growth Model
• Each voxel in target considered inexhaustible
  source of food
• Target volume overlaid in 3 dimensions over
  source
• Effectively, food is provided to bacteria in
  source
• For each position (i,j,k) in source, if (i,j,k)
  in target is non-empty, then (i,j,k) in source
  has food
• Each bacterium in source checks to see if food
  present in current position

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Design contd

• If food found, and space available, reproduces
  with certain probability p1
• If food not found in current position, and not
  surrounded, dies with certain probability p2
• Bacteria in features that do not intersect with
  target starve and die, thus removing those
  features
• Bacteria are encouraged to grow into features of
  the target volume, thus morphing into that shape

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Design contd
Probability of bacteria dying 0.5 Probability
of bacteria reproducing 1.0
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Design contd
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Design contd
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Design contd

• Benefits
• Extremely simple algorithm, easy to implement.
• The source and target volumes need to be loaded
  only once and morphing can proceed.
• No costly pre-processing steps required.
• Ideal as a quick and dirty method of morphing,
  works well with models of similar sizes

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Design contd

• Disadvantages
• Requires that input volumes be of same
  dimensions. No restriction on the sizes of the
  models themselves.
• Computationally intensive because each voxel has
  to be considered. Can be alleviated to a certain
  extent by use of iso-surfaces of the source
  volume
• Needs intersection of volumes to work.
• No way to control the morphing process.
• No support for handling color

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Implementation

• C used as implementation language
• Visualization ToolKit used for rendering

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Results

• Test cases
• a) Input.bin to Fuel.bin
• 64 X 64 X 64 volume size
• 17,000 non-empty voxels
• Large extent of intersection
• b) Cube256x256x256.bin to aneurism.bin
• 256 x 256 x 256 sized datasets
• 1.1 million non-empty voxels
• Target dissipated through volume, branch like
• Small amount of overlap

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Results contd

• c) Cube256x256x256.bin to MRI-head.bin
• 256 x 256 x 256 sized datasets.
• 7.1 million non-empty voxels
• Source model overlaps and is smaller than target
• Large number of voxel additions
• d) MRI-head.bin to bonsai.bin
• 256 x 256 x 256 sized datasets.
• 7.3 million non-empty voxels
• Source model overlaps and is larger than target
• Large number of voxel deletions

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Bacterial growth model
Volume Dimensions Source non-empty Voxel Count Target non-empty Voxel Count No of iterations Avg Morphing Time per iteration (secs) Total Time taken for morphing (secs) Avg Rendering Time (secs)
a 64 X 64 X 64 4096 13731 32 0.02 0.64 0.76
b 256X256X256 1000000 168948 387 1.16 448.92 3.16
c 256X256X256 1000000 6198649 132 3.38 446.16 2.14
d 256X256X256 6176412 1298598 195 1.33 259.35 2.71
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Results contd

• Observations
• Complexity is n3 where n is the size of one
  dimension of the source or target volume.
• In normal cases, the amount of time taken for
  each iteration as well as the number of
  iterations depend on the size of the volume and
  the number of non-empty voxels within it.

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Results contd

• In case of test (b), the small amount of overlap
  leads to a large amount of additions and
  deletions.
• The performance of this algorithm is better than
  the core increment method due to iso-surfacing
• This algorithm seems to do well in cases where
  there is a large percentage of overlap between
  volumes
• The quality of the morphs is in general worse
  than the core-increment algorithm

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Core increment model
Volume Dimensions Source non-empty Voxel Count Target non-empty Voxel Count No of iterations Avg. Morphing Time Per Iteration (secs) Total Time Taken for Morphing (secs) Avg. Rendering Time (secs)
a. 64 X 64 X 64 4096 13731 32 .0334 1.07 1.27
b. 256X256X256 1000000 168948 248 2.5027 620.67 4.46
c. 256X256X256 1000000 6198649 120 8.7642 1051.71 3.32
d. 256X256X256 6176412 1298598 175 11.24 1966.24 3.33
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Results contd

• Observations
• The complexity of this algorithm is of the order
  of n3.
• The time taken to morph increases linearly with
  the dimension of the volume.
• There is no performance hit even when the overlap
  in test (b) is significantly lower than the
  others.
• Tests suggest that the core increment algorithm
  is more stable with a varied type of source and
  target models than the bacterial growth models.

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Results contd

• Movies of Results

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Future Work

• Handling non-intersecting volumes
• Multi-value volume support
• Parallelization
• Color handling
• Scripting language support

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Conclusion

• Two algorithms for morphing volumetric data
  developed
• Do not require any correspondence information
• Do not require any user control.
• Disparate models can be morphed
• Applications
• Entertainment
• Medicine
• Evolution, geology, education etc

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• Thank you
• Any Questions?

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Introduction to Morphing

• Desirable features of 3D morphing are
• Intermediate transitions should undergo
  continuous 3D transformations
• Should apply to a variety of shapes and
  topologies
• Should be able to use user input, but work well
  without it also

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Background contd.

• Current research
• Lee1999 simplify source and target meshes to
  form coarse base domains
• Since base domains are simple meshes, easy to
  find correspondence
• Allows user defined features

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Background contd.

• Current Research
• He1994 use Direct interpolation
• For a function s(x,y,z) representing the source
  volume in 3D space and a function t(x,y,z)
  representing the target volume
• Ig(x,y,z) (1 g) s(x,y,z) g t(x,y,z) , 0 lt
  g lt 1
• where Ig is a function that represents the
  intermediate volumes.
• Problems with high frequency components

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Background contd

• Volumetric morphing used
• Reason for using volumetric morphing
• Fertile area of research, allowing more room for
  innovation.
• Simple and unrestricted nature allow
  unconventional approaches to be implemented with
  ease.
• The cellular automata based algorithm fits the
  volumetric approach in comparison to other 3D
  techniques such as polygonal morphing

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Background contd.

• Applications
• VLSI design Sarkar2000
• Fault tolerant computing Sarkar2000
• Ecological simulationsBezzi2000
• Medical simulations Sloot2002
• Crowd behavior and social interaction models
  Hamagami2003
• Physics modeling in games Forsyth2002

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Background contd.

• Current Research
• Droun2003 simulate highly deformable substances
• Concept of virtual clay
• 3D CA used with integer in each cell representing
  amount of clay
• Based on input, amount of clay is redistributed
  as model is manipulated

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Background contd.

• Claudia2001 use CA to simulate landslides
• Use the Cellular Automata Network model
• Each component of system represented by a CA and
  interactions between components form network.
• Dependency graph generated to satisfy
  dependencies of one component on another

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Implementation contd

• Input File Format
• Count of rows unsigned char
• Count of columns unsigned char
• Count of depth unsigned char
• Raw data in horizontal slice fashion, with
  (0,0,0) at the top front part of the volume and
  (row-max, col-max, depth-max) at the bottom back
  part unsigned char
• Dimensions have to be lesser than 1024

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Design

• Basis of design
• Search for simplicity Occam's Razor
• Analyzed problem at most basic level, ignoring
  correspondence and interpolation problems
• Looked at volumetric morphing as a collection of
  voxels comprising the source model trying to
  achieve similarity with the voxels in the target
  model
• Realized global control mechanisms not reqd.
• Cellular automata was perfect answer

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Implementation contd

• Supporting Programs
• raw2nativeformat
• raw2nativeformat rows columns depth inputfilename
  outputfilename
• resizevolume
• resizevolume inputfilename rows colums depth
  outputfilename
• createtestcube
• createtestcube volumerows volumecols volumedepth
  cubeStartx cubeStarty cubeStartz width height
  depth valueToFill outputfilename
• volviewer
• volviewer filename

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Implementation contd

• VTK classes used in thesis
• VtkRenderer
• VtkRenderWindow
• VtkPiecewiseFunction
• VtkVolumeProperty
• VtkVolumeRayCastCompositeFunction
• VtkVolume
• VtkColorTransferFunction
• VtkVolumeRayCastMapper
• VtkStructuredPoints
• VtkRenderWindowInteractor
• vtkUnsignedCharArray

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VTK Pipeline
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Class Structure
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Implementation contd

• Classes implemented
• CAVolume
• CAVData
• CAFileReader
• CADisplayEngine
• CABaseMorph
• CACoreIncrementMorph
• CABacterialGrowthMorph