Algorithms and Modern Computer Science

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Algorithms and Modern Computer Science
Dr. Marina L. Gavrilova
Dept of Comp. Science, University of
Calgary, AB, Canada, T2N1N4
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My Research Interests

• Computer modeling and simulation
• Computational geometry
• Image processing
• Visualization
• Voronoi diagram and Delaunay triangulation
• Biometric technologies
• Collision detection optimization
• Terrain modeling and visualization
• Exact computation
• Computational methods in spatial analysis and GIS

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Affiliations

• Co-Founder, Biometric Technologies Laboratory,
  sponsored by CFI Grant, ES 221
• Co-Founder, SPARCS Laboratory for Spatial
  Analysis and Computational Science, sponsored by
  GEOIDE, ICT 7th floor

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Data Structures to be Studied

• Hashing and hash tables
• Trees
• Spatial subdivisions
• Graphs
• Flow networks
• Geometric data structures

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Algorithms to be studies

• Search heuristics
• Encoding and compression techniques
• Linear programming
• Dynamic programming
• Game design techniques
• Randomized algorithms

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Long-Term Goals of Research in Computer Science

• Provide a solution to a problem
• Decrease possibility of an error
• Improve methodology or invent a novel solution
• Make solution more robust
• Make solution more efficient
• Make solution less memory consuming

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Examples of data structures applications in areas
of computer science

• Typical applications
• Heaps for data ordering and faster access in
  operating systems
• K-d trees for multi-dimensional database searches
• B, B, B trees for file accesses
• Geometric data structures for geographical data
  representation and processing
• Compression algorithms for remote access,
  Internet, network transmission and security
• Search heuristics for game strategy
  implementation

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More Advanced Applications

• Data structures in Optimization and Computer
  Simulation
• Data structures in Image Processing and Computer
  Graphics
• Data structures in GIS (Geographical Information
  Systems) and statistical analysis
• Data structures in biometrics

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Part 1. Optimization and Computer Modeling

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

• Space partitioning
• Trees
• Geometric data structures

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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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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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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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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
• Compression
• Search heuristics

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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 mach (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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Template comparison

• The most common methods are based on bit-map
  comparison techniques, scaling, rotating and
  modifying image to fit the template through the
  use of linear operators, and extracting template
  boundaries or skeleton (also called medial axis)
  for the comparison purposes.
• In addition, template comparison methods also
  differ, being based on either pixel to pixel,
  important features positions, or
  boundary/skeleton comparison.

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Distance transform

• Definition 1. 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).
• The distance transform method introduced in
  Gavrilova and Alsuwayel is based on fast scans
  of image in the top-bottom and left-right
  directions using a fast polygonal chain
  maintenance algorithm.
• After the distance transform is build, it can be
  used to visualize proximity information in a form
  of temperature map.
• As the distance from the black pixels (features)
  increases, the color intensity changes.

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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).
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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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Part 3. Social Sciences 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

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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, pattern matching and
  visualization

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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

• Hashing
• Space partitioning
• Trees
• Geometric data structures
• Searching

• Biometric identification
• Biometric recognition
• Biometric synthesis

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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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Conclusion

• Data structures and algorithms studies in the
  course are powerful tools not only for basic
  operation of computer systems and networks but
  also a vast array of techniques for advancing the
  state of the research in various computer science
  disciplines.