Algorithms and Modern Computer Science

Transcription

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

2 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

3 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 7 th floor

4 Data Structures to be Studied Hashing and hash tables Trees Spatial subdivisions Graphs Flow networks Geometric data structures

5 Algorithms to be studies Search heuristics Encoding and compression techniques Linear programming Dynamic programming Game design techniques Randomized algorithms

6 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

7 Examples of data structures applications in areas of computer Typical applications: science Heaps for data ordering and faster access in operating systems K-d 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

8 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

9 Part 1. Optimization and Computer Modeling Space partitioning Trees Geometric data structures Biological systems (plants, corals) Granular-type materials (silo, shaker, billiards) Molecular systems (fluids, lipid bilayers, protein docking) GIS terrain modeling

10 Pool of Data Structures Dynamic Delaunay triangulation P 1 P 1 P 1 P 2 P 2 P 2 P 4 P 4 P 4 P 3 P 3 P 3 INCIRCLE( P, P, P, P ) > INCIRCLE( P, P, P, P ) = INCIRCLE( P, P, P, P ) < Spatial subdivisions k cells Segment trees K-d trees Interval trees Combination of data structures

11 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 on-one 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

12 The nearest-neighbor neighbor problem Task: To find the nearest-neighbor neighbor in a system of circular objects {Gavrilova 01} Approach: To use generalized Voronoi diagram in Manhattan and power metric and k-d 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 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 neighbour searches and displays statistics. Tests: large data sets (10000 particles), silo model

13 Example: supremum VD and DT The supremum weighted Voronoi diagram (left) and the corresponding Delaunay triangulation (right) for 1000 randomly distributed sites.

14 Application to Silo model Silo model: Newton-Euler method, power, supremum and k-k d methods compared, simple and efficient solution to a problem. Analysis of pressure on cylinder boundaries is performed. Silo: Query time vs. Number of Sites (1000 queries) Time (sec.) Suprem Pow er k-d Number of sites

15 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

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

17 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).

18 Part 2. Image processing and Computer Graphics Space partitioning Trees Geometric data structures Compression Search heuristics Image reconstruction Image compression Morphing Detail enhancement Image comparison Pattern recognition

19 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.

20 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)

21 Good Match Most of the pixels overlap means a good match (courtesy of Prof. Jim Parker, U of C) Image Template

22 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.

23 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.

24 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).

25 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.

26 Medial axis transform

27 Voronoi diagram in 3D

28 Part 3. Social Sciences and GIS Space partitioning Grids Distance metrics Geometric data structures Terrain visualization Terrain modeling Urban planning City planning GIS systems design Navigation and tracking problems Statistical analysis

29 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 L p metrics, pattern matching and visualization

30 Terrain models

31 Quantitative Map Analysis Population, Km.

32 DEM: Digital Elevation Model Contains only relative Height Regular interval Pixel color determine height Discrete resolution

33 Non-Photo 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

34 Part 4. Biometrics Hashing Space partitioning Trees Geometric data structures Searching Biometric identification Biometric recognition Biometric synthesis

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

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

37 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.

38 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.

39 Nearest Neighbor Approach Voronoi diagram Directions of feature points

40 Delaunay Triangulation of Minutiae Points

41 (a) Binary Hand (b) Hand Contour

42 Spatial Interpolation using RBF(Radial Basis Functions) Deformation in 2D and 3D

43 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 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.

44 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.