Geometric Algorithms and Applications

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Geometric Algorithms and Applications
Dr. Marina L. Gavrilova
Assistant Professor Dept of Comp. Science,
University of Calgary, AB, Canada, T2N1N4
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My Affiliations

• Co-Head, Biometric Technologies Laboratory,
  sponsored by CFI Grant, ES 221
• Co-Head, SPARCS Laboratory for Spatial Analysis
  and Computational Science, sponsored by GEOIDE,
  ICT 7th floor
• Chair, ICCSA 2002, ICCSA 2003, ICCSA 2004, ICCSA
  2005, International Workshops on Computational
  Geometry and Applications 2000, 2001, 2002, 2003,
  2004, 2005

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My Research Interests

• Applications of computational geometry to
  modeling and simulation of natural processes
• Topology-based approach (Voronoi diagrams and
  Delaunay triangulations)
• Collision detection optimization
• Nearest-neighbor searches
• Terrain modeling and visualization
• Feature transforms and image processing
• Finding a skeleton (medial axis)
• Exact computation
• Computational methods in spatial analysis

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Areas of applications

• Areas of applications
• Fluid and porous system modeling
• GIS (Geographical Information Systems)
• Biometrics
• Visualization
• Navigation
• Image processing

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Software developed

• Computing properties of particle arrangements,
  such as their volume and topology, in a 2D and 3D
  space
• Testing intersections and collisions between
  particles
• Finding a nearest-neighbor in a multi-particle
  system
• Modeling system with cylindrical boundaries on an
  example of porous materials
• GIS terrain rendering, terrain visualization,
  autocorrelation
• Finding a skeleton in 3D coral growth
• Segment and hyperbolic curves intersection
  computation
• Computing distance transform

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Main research directions

• Geometric algorithms and optimization
• Image processing
• Applications in Biometrics
• Applications in GIS

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Part 1. Proximity and Optimization

• Biological systems (plants, corals)
• Granular-type materials (silo, shaker, billiards)
• Molecular systems (fluids, lipid bilayers,
  protein docking)
• GIS terrain modeling

• Proximity problems
• Polygon intersections
• Voronoi diagrams
• Medial axis problem
• Motion planning

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Pool of Data Structures
Dynamic Delaunay triangulation
Spatial subdivisions
Segment trees
K-d trees Interval trees Combination of data
structures
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Generalized Voronoi diagram

• A generalized Voronoi diagram (GVD) for a set of
  objects in space is
• the set of generalized Voronoi regions
• where d(x,P) is a distance function between a
  point x and a site P in the
• d-dimensional space.

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Generalized Delaunay tessellation

• A generalized Delaunay triangulation (GDT) is
  the dual of the generalized Voronoi diagram
  obtained by joining all pairs of sites whose
  Voronoi regions share a common Voronoi edge.

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General metrics

• Generalized distance functions
• Power
• Additively weighted
• Euclidean
• Manhattan
• supremum

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Example VD and DT in power metric
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Example supremum VD and DT

• The supremum weighted Voronoi diagram (left) and
  the corresponding Delaunay triangulation (right)
  for 1000 randomly distributed sites .

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Collision detection optimization

• Problem A set of n moving particles is given in
  the plane or 3D with equations of their motion.
  It is required to detect and handle collisions
  between objects and/or boundaries. Collisions are
  instantaneous and one-on-one only.
• Approach Use dynamic data structures in the
  context of time-step event oriented simulation
  model.
• Data structures implemented are
• dynamic generalized DT
• regular spatial subdivision
• regular spatial tree
• set of segment tree

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The nearest-neighbor problem

• Task To find the nearest-neighbor in a system of
  circular objects Gavrilova 01
• Approach To use generalized Voronoi diagram in
  Manhattan and power metric and k-d tree as a data
  structure.
• The Initial Distribution Generator (IDG) module
• Used to create various input configurations the
  uniform distribution of sites in a square, the
  uniform distribution of sites in a circle, cross,
  ring, degenerate grid and degenerate circle. The
  parameters for automatic generation are the
  number of sites, the distribution of their radii,
  the size of the area, and the type of the
  distribution.
• The Nearest-Neighbour Monitor (NNM) module
• The program constructs the additively weighted
  supremum VD, the power diagram and the k-d tree
  in supremum metric performs series of
  nearest-neighbour searches and displays
  statistics.
• Tests large data sets (10000 particles), silo
  model

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Site configurations

• Six configurations of sites uniform square,
  uniform circle, cross, degenerate grid, ring and
  degenerate circle.

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Application to Silo model

• Silo model Newton-Euler method, power,
  supremum and k-d methods compared, simple and
  efficient solution to a problem. Analysis of
  pressure on cylinder boundaries is performed.

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Study of porous materials in 3d

• Collaborators N.N. Medvedev, V.A.Luchnikov, V.
  P. Voloshin, Russian Academy of Sciences,
  Novosibirsk Luchnikov 01.
• Task To study the properties of the system of
  polydisperse spheres in 3D, confined inside a
  cylindrical container.
• Approach A boundary of a container is considered
  as one of the elements of the system.
• To compute the Voronoi network for a set of balls
  in a cylinder we use the modification of the
  known 3D incremental construction technique,
  discussed in Gavrilova et. al.
• The center of an empty sphere, which moves inside
  the system so that it touches at least three
  objects at any moment of time, defines an edge of
  the 3D Voronoi network.
• Tests porous materials, molecular structures

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Example 3D Euclidean Voronoi diagram

• 3D Euclidean Voronoi diagram hyperbolic arcs
  identify voids empty spaces around items
  obtained by Monte Carlo method.

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Experiments

• The approach was tested on a system representing
  dense packing of 300 Lennard-Jones atoms. The
  largest channels of the Voronoi network occur
  near to the wall of the cylinder. A fraction of
  large channels along the wall is higher for the
  model with the fixed diameter (right) than for
  the model with relaxed diameter (left).

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Part 2. Image processing and Computer Graphics

• Image reconstruction
• Image compression
• Morphing
• Detail enhancement
• Image comparison
• Pattern recognition

• Space partitioning
• Trees
• Geometric data structures
• Feature transform
• Weighted distances
• Medial axis computation

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Coral modeling

• Collaborators J. Kaandorp, University of
  Amsterdam, the Netherlands
• Problem the concept of interacting virtual
  particles whose dynamics give rise to complex
  behaviour, by using cellular automata for
  modeling and distributed simulation. This
  approach is used to model growing biological
  surfaces (corals, polips and sponges).

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Approach

• Approach Use generalized dynamic Delaunay
  triangulation to represent biological models.
  Computing properties of biological models, such
  as volume, area, rate of the growth, and the
  level of interactions between elements can be
  efficiently solved.
• Specific tasks
• Medial axis computation Model rendering Branch
  intersection computation updating topology

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Medial axis transform

• The medial axis, or skeleton of the set D,
  denoted M(D), is defined as the locus of points
  inside D which lie at the centers of all closed
  discs (or spheres) which are maximal in D,
  together with the limit points of this locus.

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Medial axis transform
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Voronoi diagram in 3D
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Pattern Matching

• Aside from a problem of measuring the distance,
  pattern matching between the template and the
  given image is a very serious problem on its own.

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Template Matching approach to Symbol Recognition
Compare an image with each template and see which
one gives the best match (courtesy of Prof. Jim
Parker, U of C)
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Good Match
Most of the pixels overlap means a good match
(courtesy of Prof. Jim Parker, U of C)
Image
Template
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Distance Transform
Given an n x m binary image I of white and black
pixels, the distance transform of I is a map that
assigns to each pixel the distance to the nearest
black pixel (a feature). It can be used for
efficient template matching.
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Part 3. Terrain modeling and GIS

• Terrain visualization
• Terrain modeling
• Urban planning
• City planning
• GIS systems design
• Navigation and tracking problems
• Statistical analysis

• Space partitioning
• Grids
• Distance metrics
• Geometric data structures
• Pattern recognition
• Autocorrelation analysis

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GIS studies - SPARCS Lab

• Collaborators S. Bertazzon, Dept. of Geography,
  C. Gold, Hong Kong Polytechnic, M. Goodchild,
  Santa Barbara
• Problem study or patterns and correlation among
  attributed geographical entities, including
  health, demographic, education etc. statistics.
• Approach pattern analysis using 3D Voronoi
  diagram, spatial statistics and autocorrelation
  using Lp metrics, visualization

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Focus of Research

• Computational methods in spatial analysis
• Investigation of spatio-temporal phenomena
• Application of the spatial analysis to health and
  social sectors
• GIS real-time terrain modeling and visualization
• Autocorrelation studies using weighted distance
  functions
• Methods for selection of Lp norms for city
  planning
• Dynamic grid-based approach to statistical
  analysis of census and population data
• Pattern matching and point pattern analysis using
  computational geometry methods
• Application of the Voronoi diagram and Delaunay
  triangulation methodology to GIS problems

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Terrain models
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Quantitative Map Analysis
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DEM Digital Elevation Model

• Contains only relative Height
• Regular interval
• Pixel color determine height
• Discrete resolution

X
Kluanne National Park
Y
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Non-Photo-Realistic Real-time 3D Terrain
Rendering

• Uses DEM as input of the application
• Generates frame coherent animated view in
  real-time
• Uses texturing, shades, particles etc. for layer
  visualization

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Part 4. Biometrics

• Fingerprint recognition
• Hand geometry
• Face features modeling and aging
• Multimodal biometrics

• Topological approaches
• Medial axis computation
• Delaunay triangulation
• Feature point extraction
• Matching algorithms
• Image processing

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Background

• Biometrics refers to the automatic identification
  of a person based on his/her physiological or
  behavioral characteristics.

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Thermogram vs. distance transform
Thermogram of an ear (Brent Griffith, Infrared
Thermography Laboratory, Lawrence Berkeley
National Laboratory )
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Use of metrics

• Regularity of metric allows to measure the
  distances from some distinct features of the
  template more precisely, and ignore minor
  discrepancies originated from noise and imprecise
  measurement while obtaining the data.
• We presume that the behavioral identifiers, such
  as typing pattern, voice and handwriting styles
  will be less susceptible to improvement using the
  proposed weighted distance methodology than the
  physiological identifiers.

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Geometric algorithms in biometrics

• The methodology is making its way to the core
  methods of biometrics, such as fingerprint
  identification, iris and retina matching, face
  analysis, ear geometry and others (see recent
  works by Xiao, Zhang, Burge.
• The methods are using Voronoi diagram to
  partition the area of a studies image and compute
  some important features (such as areas of Voronoi
  region, boundary simplification etc.) and compare
  with similarly obtained characteristics of other
  biometric data.

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Nearest Neighbor Approach

• Voronoi diagram

• Directions of feature points

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Delaunay Triangulation of Minutiae Points
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(a) Binary Hand
(b) Hand Contour
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Spatial Interpolation using RBF(Radial Basis
Functions)
Deformation in 2D and 3D
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Topology-based solution to generating biometric
information

• Finally, one of the most challenging areas is a
  recently emerged problem of generating biometric
  information, or so-called inverse problem in
  biometrics.
• In order to verify the validity of algorithms
  being developed, and to ensure that the methods
  work efficiently and with low error rates in
  real-life applications, a number of biometric
  data can be artificially created, resembling
  samples taken from live subjects.
• In order to perform this procedure, a variety of
  methods should be used, but the idea that we
  explore is based on the extraction of important
  topological information from the relatively small
  set of samples (such as boundary, skeleton,
  important features etc), applying variety of
  computational geometry methods, and then using
  these geometric samples to generate the adequate
  set of test data.

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Summary

• Thank you and come and join the Lab!