STIT tessellations a mathematical model. for structures generated by sequential division. Werner Nagel, Jena
|
|
|
- Winfred Freeman
- 5 years ago
- Views:
Transcription
1 STIT tessellations a mathematical model for structures generated by sequential division Werner Nagel, Jena
2 Examples of structures that can be modeled by random tessellations Craquelée on a ceramic surface (Photo: G. Weil)
3 Examples of structures that can be modeled by random tessellations Simulated cracks in soil (H.-J. Vogel)
4 Examples of structures that can be modeled by random tessellations Connective tissue in a rat muscle (I. Erzen)
5 Examples of structures that can be modeled by random tessellations Gris Perla.jpg Fissures in granite (D. Nikolayev, S. Siegesmund, S. Mosch, A. Hoffmann)
6 Mathematical models for tessellations Poisson-Voronoi Poisson line
7 Mathematical models for tessellations Poisson-Voronoi? Poisson line Gris Perla.jpg
8 Mathematical models for tessellations Poisson-Voronoi STIT Poisson line Gris Perla.jpg
9 Simulations of STIT tessellations three directions
10 Simulations of STIT tessellations four directions
11 Simulations of STIT tessellations eight directions
12 Simulations of STIT tessellations isotropic model
13 Simulations of STIT tessellations 3d isotropic STIT model (Ohser/Lautensack/Sych)
14 Construction of STIT tessellations: A process of cell divisions/cracks
15 Construction of STIT tessellations: A process of cell divisions/cracks
16 Construction of STIT tessellations: A process of cell divisions/cracks
17 Construction of STIT tessellations: A process of cell divisions/cracks
18 Construction of STIT tessellations: A process of cell divisions/cracks
19 Construction of STIT tessellations: A process of cell divisions/cracks
20 Construction of STIT tessellations: A process of cell divisions/cracks
21 Construction of STIT tessellations (Y t, t > 0)... STIT process in R d determined by Λ... translation invariant measure on (H, H) on the space of hyperplanes in R d Λ =image [ γ l θ ] γ > 0... intensity (i.e. scaling parameter), l... Lebesgue measure, distance from the origin, θ... directional distribution
22 Construction of STIT tessellations (Y t, t > 0)... STIT process in R d determined by Λ... translation invariant measure on (H, H) on the space of hyperplanes in R d Λ =image [ γ l θ ] γ > 0... intensity (i.e. scaling parameter), l... Lebesgue measure, distance from the origin, θ... directional distribution Λ([C])... parameter of the exponential life-time distr. of an individual cell C, [C]... set of hyperplanes that intersect C
23 Construction of STIT tessellations (Y t, t > 0)... STIT process in R d determined by Λ... translation invariant measure on (H, H) on the space of hyperplanes in R d Λ =image [ γ l θ ] γ > 0... intensity (i.e. scaling parameter), l... Lebesgue measure, distance from the origin, θ... directional distribution Λ([C])... parameter of the exponential life-time distr. of an individual cell C, [C]... set of hyperplanes that intersect C Λ [C] = 1 Λ([C]) Λ( [C])... division rule
24 Features of STIT tessellations Among a variety of cell division models, (only) STIT is mathematically tractable. STIT in-between Poisson-Voronoi and Poisson line/plane tessellations in many aspects STIT is a reference model for several crack/fissure or cell division structures, i.e. it can be a good starting point for model fitting The seminal paper: Nagel/Weiss (2005). Crack STIT tessellations characterization of the stationary random tessellations which are stable with respect to iteration. Adv. Appl. Prob. 37.
25 A key issue: Consistency in space For any fixed t > 0 and W W : [Y t constructed in W ] W D = [Yt constructed in W ]
26 A key issue: Consistency in space For any fixed t > 0 and W W : [Y t constructed in W ] W D = [Yt constructed in W ]
27 Consistency in space Intuitively: If a construction is consistent in space, then we do not have edge effects in bounded windows.
28 Consistency in space Intuitively: If a construction is consistent in space, then we do not have edge effects in bounded windows. Moreover: If the construction is consistent in space, then we have the existence of a random tessellation in the whole space R d such that its distribution coincides in any bounded window with the construction (Theorem by Schneider and Weil).
29 Consistency in space Theorem (N. and Weiss 2005) The STIT tessellation process is consistent in space.
30 Consistency in space Theorem (N. and Weiss 2005) The STIT tessellation process is consistent in space. Theorem (N. and Biehler 2015) The STIT tessellation processes are the ONLY cell division processes which are consistent in space. Cell Division process: Given any cell, its life time and division rule is (conditionally) independent from all the other cells.
31 Features of STIT tessellations STIT in-between Poisson-Voronoi and Poisson line/plane tessellations in many aspects Results by N./Weiß, and Redenbach/Thäle
32 Mean values of essential parameters Assumptions: R 3, i.e. 3d case, homogeneous (i.e. spatially stationary) and isotropic (i.e. distribution invariant under rotations) tessellations. Upper index: P... Poisson plane tessellation V... Poisson Voronoi tessellation none... STIT tessellation Rescaled, such that Mean number of cells per unit volume N P 3 = N 3 = N V 3
33 Mean values of essential parameters Then: Per unit volume: Mean 6 N P 0 = N N V 0 number of vertexes (nodes) 4 N P 1 = N N V 1 number of edges 7 3 N P 2 = N N V 2 number of faces S P V = S V S V V total area of faces 2 L P V = L V L V V total length of edges
34 Mean values of essential parameters Typical cell: Mean V P 3 = V 3 = V V 3 volume A P 3 = A A V 3 surface B P 3 = B B V 3 mean width 3 2 U P 3 = U U V 3 total edge length
35 Mean values of essential parameters For the typical cell: Mean N P 30 = 8; N 30 = 24; N V number of nodes N P 31 = 12; N 31 = 36; N V number of edges N P 32 = 6; N 32 = 14; N V number of faces
36 Second order quantities variance edge length 0e+00 2e+07 4e+07 6e+07 Voronoi Plane STIT R R 3 ; variance of total edge length in cube with side-length R
37 Second order quantities g(r) Voronoi Plane STIT r R 3 ; pair correlation function of the PP of nodes (R./Th.)
38 Summary For homogeneous (spatially stationary) tessellations: Poisson-Voronoi tess.: always isotropic Poisson plane tess. and STIT: arbitrary directional distr. Poisson plane tess. and STIT: typical cells identically distr. i.e. the same shape and size distribution of the cells. Poisson plane tess. and STIT: different arrangement of cells Isotropic STIT and Poisson-Voronoi : Isotropic Poisson plane tess.: topol. param. similar significant difference
39 Summary STIT fills a gap between Poisson Voronoi and Poisson plane tessellations. Therefore, STIT should be taken into account, when a tessellation model is adapted to a real structure.
MOMENTS OF THE LENGTH OF LINE SEGMENTS IN HOMOGENEOUS PLANAR STIT TESSELLATIONS
Image Anal Stereol 9;8:69-76 Original Research Paper MOMENTS OF THE LENGTH OF LINE SEGMENTS IN HOMOGENEOUS PLANAR STIT TESSELLATIONS CHRISTOPH THÄLE Department of Mathematics University of Fribourg CH-7
A global construction of homogeneous planar STIT tessellations
A global construction of homogeneous planar STIT tessellations Joseph Mecke, Friedrich-Schiller-Universität Jena, Werner Nagel, Friedrich-Schiller-Universität Jena, Viola Weiß, Fachhochschule Jena Abstract
Length distributions of edges in planar stationary and isotropic STIT tessellations
Length distributions of edges in planar stationary and isotropic STIT tessellations Joseph Mecke, Friedrich-Schiller-Universität Jena, Werner Nagel, Friedrich-Schiller-Universität Jena, Viola Weiß, Fachhochschule
Laguerre Tessellations
Voronoi Tessellation Applications of Voronoi tessellations Laguerre tessellation Voronoi cells vs. Laguerre cells Normal tessellations are LT Claudia Lautensack 1 2 1 University of Applied Sciences, Darmstadt,
2. Draw a non-isosceles triangle. Now make a template of this triangle out of cardstock or cardboard.
Tessellations The figure at the left shows a tiled floor. Because the floor is entirely covered by the tiles we call this arrangement a tessellation of the plane. A regular tessellation occurs when: The
9.913 Pattern Recognition for Vision. Class 8-2 An Application of Clustering. Bernd Heisele
9.913 Class 8-2 An Application of Clustering Bernd Heisele Fall 2003 Overview Problem Background Clustering for Tracking Examples Literature Homework Problem Detect objects on the road: Cars, trucks, motorbikes,
The Capacity of Wireless Networks
The Capacity of Wireless Networks Piyush Gupta & P.R. Kumar Rahul Tandra --- EE228 Presentation Introduction We consider wireless networks without any centralized control. Try to analyze the capacity of
H.Geometry Chapter 7 Definition Sheet
Section 7.1 (Part 1) Definition of: - A mapping of points in a figure to points in a resulting figure - Manipulating an original figure to get a new figure - The original figure - The resulting figure
Lifetime Analysis of Random Event-Driven Clustered Wireless Sensor Networks. Presented by Yao Zheng
Lifetime Analysis of Random Event-Driven Clustered Wireless Sensor Networks Presented by Yao Zheng Contributions Analyzing the lifetime of WSN without knowing the lifetime of sensors Find a accurate approximation
ACT Math and Science - Problem Drill 11: Plane Geometry
ACT Math and Science - Problem Drill 11: Plane Geometry No. 1 of 10 1. Which geometric object has no dimensions, no length, width or thickness? (A) Angle (B) Line (C) Plane (D) Point (E) Polygon An angle
Lecture notes on the simplex method September We will present an algorithm to solve linear programs of the form. maximize.
Cornell University, Fall 2017 CS 6820: Algorithms Lecture notes on the simplex method September 2017 1 The Simplex Method We will present an algorithm to solve linear programs of the form maximize subject
LAUTENSACK ET AL: 3d image analysis of foams using random tessellations
LAUTENSACK ET AL: 3d image analysis of foams using random tessellations Claudia Lautensack Fraunhofer Institut Techno- und Wirtschaftsmathematik (ITWM), Gottlieb-Daimler-Str. 49, D-67663 Kaiserslautern
Fitting and Simulation of Models for Telecommunication Access Networks
Fitting and Simulation of Models for Telecommunication Access Networks Frank Fleischer Joint work with C. Gloaguen, H. Schmidt, V. Schmidt University of Ulm Department of Applied Information Processing
Dalimir Orfanus (IFI UiO + ABB CRC), , Cyber Physical Systems Clustering in Wireless Sensor Networks 2 nd part : Examples
Dalimir Orfanus (IFI UiO + ABB CRC), 27.10.2011, Cyber Physical Systems Clustering in Wireless Sensor Networks 2 nd part : Examples Clustering in Wireless Sensor Networks Agenda LEACH Energy efficient
Advanced Operations Research Techniques IE316. Quiz 1 Review. Dr. Ted Ralphs
Advanced Operations Research Techniques IE316 Quiz 1 Review Dr. Ted Ralphs IE316 Quiz 1 Review 1 Reading for The Quiz Material covered in detail in lecture. 1.1, 1.4, 2.1-2.6, 3.1-3.3, 3.5 Background material
CSG obj. oper3. obj1 obj2 obj3. obj5. obj4
Solid Modeling Solid: Boundary + Interior Volume occupied by geometry Solid representation schemes Constructive Solid Geometry (CSG) Boundary representations (B-reps) Space-partition representations Operations
Voronoi Diagram. Xiao-Ming Fu
Voronoi Diagram Xiao-Ming Fu Outlines Introduction Post Office Problem Voronoi Diagram Duality: Delaunay triangulation Centroidal Voronoi tessellations (CVT) Definition Applications Algorithms Outlines
Digital Image Processing
Digital Image Processing Part 9: Representation and Description AASS Learning Systems Lab, Dep. Teknik Room T1209 (Fr, 11-12 o'clock) achim.lilienthal@oru.se Course Book Chapter 11 2011-05-17 Contents
Surfaces: notes on Geometry & Topology
Surfaces: notes on Geometry & Topology 1 Surfaces A 2-dimensional region of 3D space A portion of space having length and breadth but no thickness 2 Defining Surfaces Analytically... Parametric surfaces
CS 223B Computer Vision Problem Set 3
CS 223B Computer Vision Problem Set 3 Due: Feb. 22 nd, 2011 1 Probabilistic Recursion for Tracking In this problem you will derive a method for tracking a point of interest through a sequence of images.
Non-Parametric Modeling
Non-Parametric Modeling CE-725: Statistical Pattern Recognition Sharif University of Technology Spring 2013 Soleymani Outline Introduction Non-Parametric Density Estimation Parzen Windows Kn-Nearest Neighbor
1 Projective Geometry
CIS8, Machine Perception Review Problem - SPRING 26 Instructions. All coordinate systems are right handed. Projective Geometry Figure : Facade rectification. I took an image of a rectangular object, and
SUPPLEMENTARY FILE S1: 3D AIRWAY TUBE RECONSTRUCTION AND CELL-BASED MECHANICAL MODEL. RELATED TO FIGURE 1, FIGURE 7, AND STAR METHODS.
SUPPLEMENTARY FILE S1: 3D AIRWAY TUBE RECONSTRUCTION AND CELL-BASED MECHANICAL MODEL. RELATED TO FIGURE 1, FIGURE 7, AND STAR METHODS. 1. 3D AIRWAY TUBE RECONSTRUCTION. RELATED TO FIGURE 1 AND STAR METHODS
An Introduction to Geometrical Probability
An Introduction to Geometrical Probability Distributional Aspects with Applications A. M. Mathai McGill University Montreal, Canada Gordon and Breach Science Publishers Australia Canada China Prance Germany
SSLV318 Validation of the tridimentionnelles crack X-FEM catalog
Titre : SSLV318 - Fissure X-FEM rectangulaire dans un cube Date : 14/12/2011 Page : 1/6 SSLV318 Validation of the tridimentionnelles crack X-FEM catalog Abstract: The purpose of this document is validating
Vocabulary. Term Page Definition Clarifying Example. center of dilation. composition of transformations. enlargement. glide reflection.
CHAPTER 12 Vocabulary The table contains important vocabulary terms from Chapter 12. As you work through the chapter, fill in the page number, definition, and a clarifying example. center of dilation Term
Surface Reconstruction with MLS
Surface Reconstruction with MLS Tobias Martin CS7960, Spring 2006, Feb 23 Literature An Adaptive MLS Surface for Reconstruction with Guarantees, T. K. Dey and J. Sun A Sampling Theorem for MLS Surfaces,
Less naive type theory
Institute of Informatics Warsaw University 26 May 2007 Plan 1 Syntax of lambda calculus Why typed lambda calculi? 2 3 Syntax of lambda calculus Why typed lambda calculi? origins in 1930s (Church, Curry)
Simulation of Typical Modulated Poisson-Voronoi Cells with Applications to Telecommunication Network Modelling
Simulation of Typical Modulated Poisson-Voronoi Cells with Applications to Telecommunication Network Modelling Frank Fleischer Joint work with C. Gloaguen, H. Schmidt, V. Schmidt and F. Schweiggert University
Linear Programming and its Applications
Linear Programming and its Applications Outline for Today What is linear programming (LP)? Examples Formal definition Geometric intuition Why is LP useful? A first look at LP algorithms Duality Linear
CS 231A Computer Vision (Fall 2012) Problem Set 3
CS 231A Computer Vision (Fall 2012) Problem Set 3 Due: Nov. 13 th, 2012 (2:15pm) 1 Probabilistic Recursion for Tracking (20 points) In this problem you will derive a method for tracking a point of interest
COMP331/557. Chapter 2: The Geometry of Linear Programming. (Bertsimas & Tsitsiklis, Chapter 2)
COMP331/557 Chapter 2: The Geometry of Linear Programming (Bertsimas & Tsitsiklis, Chapter 2) 49 Polyhedra and Polytopes Definition 2.1. Let A 2 R m n and b 2 R m. a set {x 2 R n A x b} is called polyhedron
Definitions. Topology/Geometry of Geodesics. Joseph D. Clinton. SNEC June Magnus J. Wenninger
Topology/Geometry of Geodesics Joseph D. Clinton SNEC-04 28-29 June 2003 Magnus J. Wenninger Introduction Definitions Topology Goldberg s polyhedra Classes of Geodesic polyhedra Triangular tessellations
Texture Analysis. Selim Aksoy Department of Computer Engineering Bilkent University
Texture Analysis Selim Aksoy Department of Computer Engineering Bilkent University saksoy@cs.bilkent.edu.tr Texture An important approach to image description is to quantify its texture content. Texture
5LSH0 Advanced Topics Video & Analysis
1 Multiview 3D video / Outline 2 Advanced Topics Multimedia Video (5LSH0), Module 02 3D Geometry, 3D Multiview Video Coding & Rendering Peter H.N. de With, Sveta Zinger & Y. Morvan ( p.h.n.de.with@tue.nl
Solid Modelling. Graphics Systems / Computer Graphics and Interfaces COLLEGE OF ENGINEERING UNIVERSITY OF PORTO
Solid Modelling Graphics Systems / Computer Graphics and Interfaces 1 Solid Modelling In 2D, one set 2D line segments or curves does not necessarily form a closed area. In 3D, a collection of surfaces
Example for calculation of clustering coefficient Node N 1 has 8 neighbors (red arrows) There are 12 connectivities among neighbors (blue arrows)
Example for calculation of clustering coefficient Node N 1 has 8 neighbors (red arrows) There are 12 connectivities among neighbors (blue arrows) Average clustering coefficient of a graph Overall measure
The 2-core of a Non-homogeneous Hypergraph
July 16, 2012 k-cores A Hypergraph G on vertex set V is a collection E of subsets of V. E is the set of hyperedges. For ordinary graphs, e = 2 for all e E. The k-core of a (hyper)graph is the maximal subgraph
Computer Graphics Hands-on
Computer Graphics Hands-on Two-Dimensional Transformations Objectives Visualize the fundamental 2D geometric operations translation, rotation about the origin, and scale about the origin Learn how to compose
Depth. Common Classification Tasks. Example: AlexNet. Another Example: Inception. Another Example: Inception. Depth
Common Classification Tasks Recognition of individual objects/faces Analyze object-specific features (e.g., key points) Train with images from different viewing angles Recognition of object classes Analyze
Obtaining the H and T Honeycomb from a Cross-Section of the 16-cell Honeycomb
Bridges 2017 Conference Proceedings Obtaining the H and T Honeycomb from a Cross-Section of the 16-cell Honeycomb Hideki Tsuiki Graduate School of Human and Environmental Studies, Kyoto University Yoshida-Nihonmatsu,
Tessellations. Irena Swanson Reed College, Portland, Oregon. MathPath, Lewis & Clark College, Portland, Oregon, 24 July 2018
Tessellations Irena Swanson Reed College, Portland, Oregon MathPath, Lewis & Clark College, Portland, Oregon, 24 July 2018 What is a tessellation? A tiling or a tessellation of the plane is a covering
CS 231A Computer Vision (Winter 2014) Problem Set 3
CS 231A Computer Vision (Winter 2014) Problem Set 3 Due: Feb. 18 th, 2015 (11:59pm) 1 Single Object Recognition Via SIFT (45 points) In his 2004 SIFT paper, David Lowe demonstrates impressive object recognition
Conjectures concerning the geometry of 2-point Centroidal Voronoi Tessellations
Conjectures concerning the geometry of 2-point Centroidal Voronoi Tessellations Emma Twersky May 2017 Abstract This paper is an exploration into centroidal Voronoi tessellations, or CVTs. A centroidal
A MEASURE OF THE AREA OF A HOMOGENEOUS RANDOM SURFACE IN SPACE*
404 NOTES [Vol. XII, No. 4 where fi is a constant, where a(t), b(t), c{t) are arbitrary functions of t, and where Y = (y b) cos Qt + (z c) sin tit, Z = (y b) sin Qi + (z c) cos 01. Comparison of (3) with
Mathematical Programming and Research Methods (Part II)
Mathematical Programming and Research Methods (Part II) 4. Convexity and Optimization Massimiliano Pontil (based on previous lecture by Andreas Argyriou) 1 Today s Plan Convex sets and functions Types
A Versatile Dependent Model for Heterogeneous Cellular Networks
1 A Versatile Dependent Model for Heterogeneous Cellular Networks Martin Haenggi University of Notre Dame July 7, 1 Abstract arxiv:135.97v [cs.ni] 7 May 13 We propose a new model for heterogeneous cellular
Lesson 2. Investigation. Name: a. Identify the shape of the sign and describe the symmetries
Check Your Understanding Being able to recognize traffic signs by their shape and color is important when Unit driving 6and is often tested on exams for a driver s license. Examine the school crossing
1 The range query problem
CS268: Geometric Algorithms Handout #12 Design and Analysis Original Handout #12 Stanford University Thursday, 19 May 1994 Original Lecture #12: Thursday, May 19, 1994 Topics: Range Searching with Partition
WIRELESS sensor networks (WSNs) have attracted a. Homology-based Distributed Coverage Hole Detection in Wireless Sensor Networks
IEEE/ACM TRANSACTIONS ON NETWORKING 1 Homology-based Distributed Coverage Hole Detection in Wireless Sensor Networks Feng Yan, Member, IEEE, Anaïs Vergne, Philippe Martins, Senior Member, IEEE, Laurent
The Simplex Algorithm
The Simplex Algorithm Uri Feige November 2011 1 The simplex algorithm The simplex algorithm was designed by Danzig in 1947. This write-up presents the main ideas involved. It is a slight update (mostly
AQA GCSE Maths - Higher Self-Assessment Checklist
AQA GCSE Maths - Higher Self-Assessment Checklist Number 1 Use place value when calculating with decimals. 1 Order positive and negative integers and decimals using the symbols =,, , and. 1 Round to
"Unpacking the Standards" 4th Grade Student Friendly "I Can" Statements I Can Statements I can explain why, when and how I got my answer.
0406.1.1 4th Grade I can explain why, when and how I got my answer. 0406.1.2 I can identify the range of an appropriate estimate. I can identify the range of over-estimates. I can identify the range of
Experiment 8 Wave Optics
Physics 263 Experiment 8 Wave Optics In this laboratory, we will perform two experiments on wave optics. 1 Double Slit Interference In two-slit interference, light falls on an opaque screen with two closely
J Linear Programming Algorithms
Simplicibus itaque verbis gaudet Mathematica Veritas, cum etiam per se simplex sit Veritatis oratio. [And thus Mathematical Truth prefers simple words, because the language of Truth is itself simple.]
05 - Surfaces. Acknowledgements: Olga Sorkine-Hornung. CSCI-GA Geometric Modeling - Daniele Panozzo
05 - Surfaces Acknowledgements: Olga Sorkine-Hornung Reminder Curves Turning Number Theorem Continuous world Discrete world k: Curvature is scale dependent is scale-independent Discrete Curvature Integrated
Open problems in convex geometry
Open problems in convex geometry 10 March 2017, Monash University Seminar talk Vera Roshchina, RMIT University Based on joint work with Tian Sang (RMIT University), Levent Tunçel (University of Waterloo)
CPSC 695. Methods for interpolation and analysis of continuing surfaces in GIS Dr. M. Gavrilova
CPSC 695 Methods for interpolation and analysis of continuing surfaces in GIS Dr. M. Gavrilova Overview Data sampling for continuous surfaces Interpolation methods Global interpolation Local interpolation
We can use square dot paper to draw each view (top, front, and sides) of the three dimensional objects:
Unit Eight Geometry Name: 8.1 Sketching Views of Objects When a photo of an object is not available, the object may be drawn on triangular dot paper. This is called isometric paper. Isometric means equal
3D Computer Vision II. Reminder Projective Geometry, Transformations. Nassir Navab. October 27, 2009
3D Computer Vision II Reminder Projective Geometr, Transformations Nassir Navab based on a course given at UNC b Marc Pollefes & the book Multiple View Geometr b Hartle & Zisserman October 27, 29 2D Transformations
Anatomical Descriptions That Compute Functional Attributes
Anatomical Descriptions That Compute Functional Attributes Goal: To write a description of an anatomical structure that leads directly to the calculation of its functional attributes. For instance, an
Using Markov decision processes to optimise a non-linear functional of the final distribution, with manufacturing applications.
Using Markov decision processes to optimise a non-linear functional of the final distribution, with manufacturing applications. E.J. Collins 1 1 Department of Mathematics, University of Bristol, University
Reversible Hyperbolic Triangulations
Reversible Hyperbolic Triangulations Circle Packing and Random Walk Tom Hutchcroft 1 Joint work with O. Angel 1, A. Nachmias 1 and G. Ray 2 1 University of British Columbia 2 University of Cambridge Probability
In what follows, we will focus on Voronoi diagrams in Euclidean space. Later, we will generalize to other distance spaces.
Voronoi Diagrams 4 A city builds a set of post offices, and now needs to determine which houses will be served by which office. It would be wasteful for a postman to go out of their way to make a delivery
Imperfections and Errors (Chapter 6)
Imperfections and Errors (Chapter 6) EC4630 Radar and Laser Cross Section Fall 011 Prof. D. Jenn jenn@nps.navy.mil www.nps.navy.mil/jenn AY011 1 Imperfections and Errors Imperfections and errors are deviations
Common Core Vocabulary and Representations
Vocabulary Description Representation 2-Column Table A two-column table shows the relationship between two values. 5 Group Columns 5 group columns represent 5 more or 5 less. a ten represented as a 5-group
Stereo Vision. MAN-522 Computer Vision
Stereo Vision MAN-522 Computer Vision What is the goal of stereo vision? The recovery of the 3D structure of a scene using two or more images of the 3D scene, each acquired from a different viewpoint in
Parametric Distributions of Connection Lengths for the Efficient Analysis of Fixed Access Network
Ann. Telecommunn. manuscript No. (will be inserted by the editor) Parametric Distributions of Connection Lengths for the Efficient Analysis of Fixed Access Network Catherine Gloaguen Florian Voss Volker
Lecture 4: Geometric Algorithms (Convex Hull, Voronoi diagrams)
Advanced Algorithms Fall 2015 Lecture 4: Geometric Algorithms (Convex Hull, Voronoi diagrams) Faculty: K.R. Chowdhary : Professor of CS Disclaimer: These notes have not been subjected to the usual scrutiny
Scope and Sequence for the New Jersey Core Curriculum Content Standards
Scope and Sequence for the New Jersey Core Curriculum Content Standards The following chart provides an overview of where within Prentice Hall Course 3 Mathematics each of the Cumulative Progress Indicators
THIELE CENTRE for applied mathematics in natural science
THIELE CENTRE for applied mathematics in natural science Directional Analysis of Digitized D Images by Configuration Counts P Gutkowski, E B V Jensen and M Kiderlen Research Report No 01 January 004 Directional
CS 523: Computer Graphics, Spring Shape Modeling. Differential Geometry of Surfaces
CS 523: Computer Graphics, Spring 2011 Shape Modeling Differential Geometry of Surfaces Andrew Nealen, Rutgers, 2011 2/22/2011 Differential Geometry of Surfaces Continuous and Discrete Motivation Smoothness
NO more pertinent introduction to this paper could be
# 15691831 1 Parametric Distance Distributions for Fixed Access Network Analysis and Planning Catherine Gloaguen, Member, IEEE, Florian Voss, Volker Schmidt Abstract The access network displays the important
7. The Gauss-Bonnet theorem
7. The Gauss-Bonnet theorem 7.1 Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed
Course Number: Course Title: Geometry
Course Number: 1206310 Course Title: Geometry RELATED GLOSSARY TERM DEFINITIONS (89) Altitude The perpendicular distance from the top of a geometric figure to its opposite side. Angle Two rays or two line
Teaching diary. Francis Bonahon University of Southern California
Teaching diary In the Fall 2010, I used the book Low-dimensional geometry: from euclidean surfaces to hyperbolic knots as the textbook in the class Math 434, Geometry and Transformations, at USC. Most
Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem
Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem Jan Lellmann, Frank Lenzen, Christoph Schnörr Image and Pattern Analysis Group Universität Heidelberg EMMCVPR 2011 St. Petersburg,
Prague 8, Czech Republic
acta physica slovaca vol. 56 No. 4, 439 444 August 2006 MORPHOLOGY OF POLYMER NANOCOMPOSITES S. Novák 1,a, R. Hrach 2,a,b a J. E. Purkyně University, Faculty of Science, České mládeže 8, 400 96 Ústí nad
To be a grade 1 I need to
To be a grade 1 I need to Order positive and negative integers Understand addition and subtraction of whole numbers and decimals Apply the four operations in correct order to integers and proper fractions
Click the mouse button or press the Space Bar to display the answers.
Click the mouse button or press the Space Bar to display the answers. 9-4 Objectives You will learn to: Identify regular tessellations. Vocabulary Tessellation Regular Tessellation Uniform Semi-Regular
Contents. Preface... VII. Part I Classical Topics Revisited
Contents Preface........................................................ VII Part I Classical Topics Revisited 1 Sphere Packings........................................... 3 1.1 Kissing Numbers of Spheres..............................
Glossary of dictionary terms in the AP geometry units
Glossary of dictionary terms in the AP geometry units affine linear equation: an equation in which both sides are sums of terms that are either a number times y or a number times x or just a number [SlL2-D5]
COMS 4771 Support Vector Machines. Nakul Verma
COMS 4771 Support Vector Machines Nakul Verma Last time Decision boundaries for classification Linear decision boundary (linear classification) The Perceptron algorithm Mistake bound for the perceptron
Computing the Betti Numbers of Arrangements. Saugata Basu School of Mathematics & College of Computing Georgia Institute of Technology.
1 Computing the Betti Numbers of Arrangements Saugata Basu School of Mathematics & College of Computing Georgia Institute of Technology. 2 Arrangements in Computational Geometry An arrangement in R k is
Level lines based disocclusion
Level lines based disocclusion Simon Masnou Jean-Michel Morel CEREMADE CMLA Université Paris-IX Dauphine Ecole Normale Supérieure de Cachan 75775 Paris Cedex 16, France 94235 Cachan Cedex, France Abstract
2D rendering takes a photo of the 2D scene with a virtual camera that selects an axis aligned rectangle from the scene. The photograph is placed into
2D rendering takes a photo of the 2D scene with a virtual camera that selects an axis aligned rectangle from the scene. The photograph is placed into the viewport of the current application window. A pixel
First-passage percolation in random triangulations
First-passage percolation in random triangulations Jean-François Le Gall (joint with Nicolas Curien) Université Paris-Sud Orsay and Institut universitaire de France Random Geometry and Physics, Paris 2016
Statistical modelling of the patterns of planar sections of prostatic capillaries on the basis of Strauss hard-core processes
TITLE Statistical modelling of the patterns of planar sections of prostatic capillaries on the basis of Strauss hard-core processes AUTHORS Torsten Mattfeldt 1 Stefanie Eckel 2 Frank Fleischer 3 Volker
Computer Vision I - Appearance-based Matching and Projective Geometry
Computer Vision I - Appearance-based Matching and Projective Geometry Carsten Rother 01/11/2016 Computer Vision I: Image Formation Process Roadmap for next four lectures Computer Vision I: Image Formation
A triangle that has three acute angles Example:
1. acute angle : An angle that measures less than a right angle (90 ). 2. acute triangle : A triangle that has three acute angles 3. angle : A figure formed by two rays that meet at a common endpoint 4.
SAMPLING AND THE MOMENT TECHNIQUE. By Sveta Oksen
SAMPLING AND THE MOMENT TECHNIQUE By Sveta Oksen Overview - Vertical decomposition - Construction - Running time analysis - The bounded moments theorem - General settings - The sampling model - The exponential
Octonion multiplication and Heawood s map
Octonion multiplication and Heawood s map Bruno Sévennec arxiv:0.0v [math.ra] 29 Jun 20 June 30, 20 Almost any article or book dealing with Cayley-Graves algebra O of octonions (to be recalled shortly)
Algebraic Geometry of Segmentation and Tracking
Ma191b Winter 2017 Geometry of Neuroscience Geometry of lines in 3-space and Segmentation and Tracking This lecture is based on the papers: Reference: Marco Pellegrini, Ray shooting and lines in space.
LESSON PLAN
LESSON PLAN WWW.MINDWARE.COM In the Ocean Classroom Activities Four suggested activities are outlined here that make use of the puzzle pieces along with the enclosed worksheets. When discussing symmetry
Helpful Hint When you are given a frieze pattern, you may assume that the pattern continues forever in both directions Notes: Tessellations
A pattern has translation symmetry if it can be translated along a vector so that the image coincides with the preimage. A frieze pattern is a pattern that has translation symmetry along a line. Both of
Collision Detection. Jane Li Assistant Professor Mechanical Engineering & Robotics Engineering
RBE 550 MOTION PLANNING BASED ON DR. DMITRY BERENSON S RBE 550 Collision Detection Jane Li Assistant Professor Mechanical Engineering & Robotics Engineering http://users.wpi.edu/~zli11 Euler Angle RBE
Digital Image Processing. Image Enhancement (Point Processing)
Digital Image Processing Image Enhancement (Point Processing) 2 Contents In this lecture we will look at image enhancement point processing techniques: What is point processing? Negative images Thresholding
On-Line Computer Graphics Notes CLIPPING
On-Line Computer Graphics Notes CLIPPING Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science University of California, Davis 1 Overview The primary use of clipping in
Shape Modeling and Geometry Processing
252-0538-00L, Spring 2018 Shape Modeling and Geometry Processing Discrete Differential Geometry Differential Geometry Motivation Formalize geometric properties of shapes Roi Poranne # 2 Differential Geometry
The Ultimate Maths Vocabulary List
The Ultimate Maths Vocabulary List The 96 Words Every Pupil Needs to Know by the End of Year 6 KS1 & KS2 How to Use This Resource An essential building block in pupil s understanding of maths is their