DOCSLIB.ORG

Basic Geometry and Topology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Basic Geometry and Topology Stephan Stolz September 7, 2015 Contents 1 Pointset Topology 1 1.1 Metric spaces . .1 1.2 Topological spaces . .5 1.3 Constructions with topological spaces . .6 1.3.1 Subspace topology . .6 1.3.2 Product topology . .7 1.3.3 Quotient topology. .8 1.4 Properties of topological spaces . 12 1.4.1 Hausdorff spaces . 13 1.4.2 Compact spaces . 13 1.4.3 Connected spaces . 15 1 Pointset Topology 1.1 Metric spaces We recall that a map f : Rm ! Rn between Euclidean spaces is continuous if and only if 8x 2 X 8 > 0 9δ > 0 8y 2 X d(x; y) < δ ) d(f(x); f(y)) < , (1.1) where p 2 2 d(x; y) = jjx − yjj = (x1 − y1) + ··· + (xn − yn) 2 R≥0 is the Euclidean distance between two points x; y in Rn. Example 1.2. (Examples of continuous maps.) 2 1. The addition map a: R ! R, x = (x1; x2) 7! x1 + x2; 2 2. The multiplication map m: R ! R, x = (x1; x2) 7! x1x2; The proofs that these maps are continuous are simple estimates that you probably remember from calculus. Since the continuity of all the maps we'll look at in these notes is proved by expressing them in terms of the maps a and m, we include the proofs of continuity of a and m for completeness. 1 1 POINTSET TOPOLOGY 2 2 Proof. To prove that the addition map a is continuous, suppose x = (x1; x2) 2 R and > 0 2 are given. We claim that for δ := /2 and y = (y1; y2) 2 R with d(x; y) < δ we have d(a(x); a(y)) < and hence a is a continuous function. To prove the claim, we note that p 2 2 d(x; y) = jx1 − y1j + jx2 − y2j and hence jx1 − y1j ≤ d(x; y), jx1 − y1j ≤ d(x; y). It follows that d(a(x); a(y)) = ja(x) − a(y)j = jx1 + x2 − y1 − y2j ≤ jx1 − y1j + jx2 − y2j ≤ 2d(x; y) < 2δ = . To prove that the multiplication map m is continuous, we claim that for δ := minf1; /(jx1j + jx2j + 1)g 2 and y = (y1; y2) 2 R with d(x; y) < δ we have d(m(x); m(y)) < and hence m is a continuous function. The claim follows from the following estimates: d(m(y); m(x)) = jy1y2 − x1x2j = jy1y2 − x1y2 + x1y2 − x1x2j ≤ jy1y2 − x1y2j + jx1y2 − x1x2j = jy1 − x1jjy2j + jx1jjy2 − x2j ≤ d(x; y)(jy2j + jx1j) ≤ d(x; y)(jx2j + jy2 − x2j + jx1j) ≤ d(x; y)(jx1j + jx2j + 1) < δ(jx1j + jx2j + 1) ≤ n n Lemma 1.3. The function d: R × R ! R≥0 has the following properties: 1. d(x; y) = 0 if and only if x = y; 2. d(x; y) = d(y; x) (symmetry); 3. d(x; y) ≤ d(x; z) + d(z; y) (triangle inequality) Definition 1.4. A metric space is a set X equipped with a map d: X × X ! R≥0 with properties (1)-(3) above. A map f : X ! Y between metric spaces X, Y is continuous if condition (1.1) is satisfied. an isometry if d(f(x); f(y)) = d(x; y) for all x; y 2 X; Two metric spaces X, Y are homeomorphic (resp. isometric) if there are continuous maps (resp. isometries) f : X ! Y and g : Y ! X which are inverses of each other. Example 1.5. An important class of examples of metric spaces are subsets of Rn. Here are particular examples we will be talking about during the semester: 1 POINTSET TOPOLOGY 3 n n n n n 1. The n-disk D := fx 2 R j jxj ≤ 1g ⊂ R , and Dr := fx 2 R j jxj ≤ rg, the n-disk of radius r > 0. The dilation map n n D −! Dr x 7! rx n n is a homeomorphism between D and Dr with inverse given by multiplication by 1=r. However, these two metric spaces are not isometric for r 6= 1. To see this, define the diameter diam(X) of a metric space X by diam(X) := supfd(x; y) j x; y 2 Xg 2 R≥0 [ f1g: n For example, diam(Dr ) = 2r. It is easy to see that if two metric spaces X, Y are n n isometric, then their diameters agree. In particular, the disks Dr and Dr0 are not isometric unless r = r0. 2. The n-sphere Sn := fx 2 Rn+1 j jxj = 1g ⊂ Rn+1. 3. The torus T = fv 2 R3 j d(v; C) = rg for 0 < r < 1. Here 2 2 3 C = f(x; y; 0) j x + y = 1g ⊂ R is the unit circle in the xy-plane, and d(v; C) = infw2C d(v; w) is the distance between v and C. 4. The general linear group n n GLn(R) = fvector space isomorphisms f : R ! R g n ! f(v1; : : : ; vn) j vi 2 R ; det(v1; : : : ; vn) 6= 0g n n n2 = finvertible n × n-matricesg ⊂ R × · · · × R = R | {z } n Here we think of (v1; : : : ; vn) as an n × n-matrix with column vectors vi, and the bijection is the usual one in linear algebra that sends a linear map f : Rn ! Rn to the matrix (f(e1); : : : ; f(en)) whose column vectors are the images of the standard basis n elements ei 2 R . 5. The special linear group n n2 SLn(R) = f(v1; : : : ; vn) j vi 2 R ; det(v1; : : : ; vn) = 1g ⊂ R 6. The orthogonal group n n O(n) = flinear isometries f : R ! R g n n2 = f(v1; : : : ; vn) j vi 2 R ; vi's are orthonormalg ⊂ R n We recall that a collection of vectors vi 2 R is orthonormal if jvij = 1 for all i, and vi is perpendicular to vj for i 6= j. 1 POINTSET TOPOLOGY 4 7. The special orthogonal group n2 SO(n) = f(v1; : : : ; vn) 2 O(n) j det(v1; : : : ; vn) = 1g ⊂ R 8. The Stiefel manifold n k n Vk(R ) = flinear isometries f : R ! R g n kn = f(v1; : : : ; vk) j vi 2 R ; vi's are orthonormalg ⊂ R Example 1.6. The following maps between metric spaces are continuous. While it is pos- sible to prove their continuity using the definition of continuity, it will be much simpler to prove their continuity by `building' these maps using compositions and products from the continuous maps a and m of Example 1.2. We will do this below in Lemma 1.22. 1. Every polynomial function f : Rn ! R is continuous. We recall that a polynomial function is of the form f(x ; : : : ; x ) = P a xi1 ····· xin for a 2 . 1 n i1;:::;in i1;:::;in 1 n i1;:::;in R n2 2. Let Mn×n(R) = R be the set of n × n matrices. Then the map Mn×n(R) × Mn×n(R) −! Mn×n(R)(A; B) 7! AB given by matrix multiplication is continuous. Here we use the fact that a map to the n2 product Mn×n(R) = R = R × · · · × R is continuous if and only if each component map is continuous (see Lemma 1.21), and each matrix entry of AB is a polynomial and hence a continuous function of the matrix entries of A and B. Restricting to the invertible matrices GLn(R) ⊂ Mn×n(R), we see that the multiplication map GLn(R) × GLn(R) −! GLn(R) is continuous. The same holds for the subgroups SO(n) ⊂ O(n) ⊂ GLn(R). −1 3. The map GLn(R) ! GLn(R), A 7! A is continuous (this is a homework problem). The same statement follows for the subgroups of GLn(R). n p 2 2 The Euclidean metric on R given by d(x; y) = (x1 − y1) + ··· + (xn − yn) for x; y 2 Rn is not the only reasonable metric on Rn. Another metric on Rn is given by n X d1(x; y) = jxi − yij: (1.7) i=1 The question arises whether it can happen that a map f : Rn ! Rn is continuous with respect to one of these metrics, but not with respect to the other. To see that this doen't happen, it is useful to characterize continuity of a map f : X ! Y between metric spaces X, Y in a way that involves the metrics on X and Y less directly than Definition 1.4 does. This alternative characterization will be based on the following notion of \open subsets" of a metric space. 1 POINTSET TOPOLOGY 5 Definition 1.8. Let X be a metric space. A subset U ⊂ X is open if for every point x 2 U there is some > 0 such that B(x) ⊂ U. Here B(x) = fy 2 X j d(y; x) < g is the ball of radius around x. To illustrate this, lets look at examples of subsets of Rn equipped with the Euclidean n n n n metric. The subset Dr = fv 2 R j jjvjj ≤ rg ⊂ R is not open, since for for a point v 2 Dr n with jjvjj = r any open ball B(v) with center v will contain points not in Dr . By contrast, n the subset Br(0) ⊂ R is open, since for any x 2 Br(0) the ball Bδ(x) of radius δ = r − jjxjj is contained in Br(0), since for y 2 Bδ(x) by the triangle inequality we have d(y; 0) ≤ d(y; x) + d(x; 0) < δ + jjxjj = (r − jjxjj) + jjxjj = r: Lemma 1.9.
Recommended publications
  • Projective Geometry: a Short Introduction

    Projective Geometry: a Short Introduction

    Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g.
  • I-Sequential Topological Spaces∗

    I-Sequential Topological Spaces∗

    Applied Mathematics E-Notes, 14(2014), 236-241 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ -Sequential Topological Spaces I Sudip Kumar Pal y Received 10 June 2014 Abstract In this paper a new notion of topological spaces namely, I-sequential topo- logical spaces is introduced and investigated. This new space is a strictly weaker notion than the first countable space. Also I-sequential topological space is a quotient of a metric space. 1 Introduction The idea of convergence of real sequence have been extended to statistical convergence by [2, 14, 15] as follows: If N denotes the set of natural numbers and K N then Kn denotes the set k K : k n and Kn stands for the cardinality of the set Kn. The natural densityf of the2 subset≤ Kg is definedj j by Kn d(K) = lim j j, n n !1 provided the limit exists. A sequence xn n N of points in a metric space (X, ) is said to be statistically f g 2 convergent to l if for arbitrary " > 0, the set K(") = k N : (xk, l) " has natural density zero. A lot of investigation has been donef on2 this convergence≥ andg its topological consequences after initial works by [5, 13]. It is easy to check that the family Id = A N : d(A) = 0 forms a non-trivial f g admissible ideal of N (recall that I 2N is called an ideal if (i) A, B I implies A B I and (ii) A I,B A implies B I.
  • MTH 304: General Topology Semester 2, 2017-2018

    MTH 304: General Topology Semester 2, 2017-2018

    MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds..........................
  • General Topology

    General Topology

    General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry).
  • DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic

    DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic

    DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V .
  • • Rotations • Camera Calibration • Homography • Ransac

    • Rotations • Camera Calibration • Homography • Ransac

    Agenda • Rotations • Camera calibration • Homography • Ransac Geometric Transformations y 164 Computer Vision: Algorithms andx Applications (September 3, 2010 draft) Transformation Matrix # DoF Preserves Icon translation I t 2 orientation 2 3 h i ⇥ ⇢⇢SS rigid (Euclidean) R t 3 lengths S ⇢ 2 3 S⇢ ⇥ h i ⇢ similarity sR t 4 angles S 2 3 S⇢ h i ⇥ ⇥ ⇥ affine A 6 parallelism ⇥ ⇥ 2 3 h i ⇥ projective H˜ 8 straight lines ` 3 3 ` h i ⇥ Table 3.5 Hierarchy of 2D coordinate transformations. Each transformation also preserves Let’s definethe properties families listed of in thetransformations rows below it, i.e., similarity by the preserves properties not only anglesthat butthey also preserve parallelism and straight lines. The 2 3 matrices are extended with a third [0T 1] row to form ⇥ a full 3 3 matrix for homogeneous coordinate transformations. ⇥ amples of such transformations, which are based on the 2D geometric transformations shown in Figure 2.4. The formulas for these transformations were originally given in Table 2.1 and are reproduced here in Table 3.5 for ease of reference. In general, given a transformation specified by a formula x0 = h(x) and a source image f(x), how do we compute the values of the pixels in the new image g(x), as given in (3.88)? Think about this for a minute before proceeding and see if you can figure it out. If you are like most people, you will come up with an algorithm that looks something like Algorithm 3.1. This process is called forward warping or forward mapping and is shown in Figure 3.46a.
  • Math 535 - General Topology Additional Notes

    Math 535 - General Topology Additional Notes

    Math 535 - General Topology Additional notes Martin Frankland September 5, 2012 1 Subspaces Definition 1.1. Let X be a topological space and A ⊆ X any subset. The subspace topology sub on A is the smallest topology TA making the inclusion map i: A,! X continuous. sub In other words, TA is generated by subsets V ⊆ A of the form V = i−1(U) = U \ A for any open U ⊆ X. Proposition 1.2. The subspace topology on A is sub TA = fV ⊆ A j V = U \ A for some open U ⊆ Xg: In other words, the collection of subsets of the form U \ A already forms a topology on A. 2 Products Before discussing the product of spaces, let us review the notion of product of sets. 2.1 Product of sets Let X and Y be sets. The Cartesian product of X and Y is the set of pairs X × Y = f(x; y) j x 2 X; y 2 Y g: It comes equipped with the two projection maps pX : X × Y ! X and pY : X × Y ! Y onto each factor, defined by pX (x; y) = x pY (x; y) = y: This explicit description of X × Y is made more meaningful by the following proposition. 1 Proposition 2.1. The Cartesian product of sets satisfies the following universal property. For any set Z along with maps fX : Z ! X and fY : Z ! Y , there is a unique map f : Z ! X × Y satisfying pX ◦ f = fX and pY ◦ f = fY , in other words making the diagram Z fX 9!f fY X × Y pX pY { " XY commute.
  • Math 395: Category Theory Northwestern University, Lecture Notes

    Math 395: Category Theory Northwestern University, Lecture Notes

    Math 395: Category Theory Northwestern University, Lecture Notes Written by Santiago Can˜ez These are lecture notes for an undergraduate seminar covering Category Theory, taught by the author at Northwestern University. The book we roughly follow is “Category Theory in Context” by Emily Riehl. These notes outline the specific approach we’re taking in terms the order in which topics are presented and what from the book we actually emphasize. We also include things we look at in class which aren’t in the book, but otherwise various standard definitions and examples are left to the book. Watch out for typos! Comments and suggestions are welcome. Contents Introduction to Categories 1 Special Morphisms, Products 3 Coproducts, Opposite Categories 7 Functors, Fullness and Faithfulness 9 Coproduct Examples, Concreteness 12 Natural Isomorphisms, Representability 14 More Representable Examples 17 Equivalences between Categories 19 Yoneda Lemma, Functors as Objects 21 Equalizers and Coequalizers 25 Some Functor Properties, An Equivalence Example 28 Segal’s Category, Coequalizer Examples 29 Limits and Colimits 29 More on Limits/Colimits 29 More Limit/Colimit Examples 30 Continuous Functors, Adjoints 30 Limits as Equalizers, Sheaves 30 Fun with Squares, Pullback Examples 30 More Adjoint Examples 30 Stone-Cech 30 Group and Monoid Objects 30 Monads 30 Algebras 30 Ultrafilters 30 Introduction to Categories Category theory provides a framework through which we can relate a construction/fact in one area of mathematics to a construction/fact in another. The goal is an ultimate form of abstraction, where we can truly single out what about a given problem is specific to that problem, and what is a reflection of a more general phenomenom which appears elsewhere.
  • 9.2.2 Projection Formula Definition 2.8

    9.2.2 Projection Formula Definition 2.8

    9.2. Intersection and morphisms 397 Proof Let Γ ⊆ X×S Z be the graph of ϕ. This is an S-scheme and the projection p :Γ → X is projective birational. Let us show that Γ admits a desingularization. This is Theorem 8.3.44 if dim S = 0. Let us therefore suppose that dim S = 1. Let K = K(S). Then pK :ΓK → XK is proper birational with XK normal, and hence pK is an isomorphism. We can therefore apply Theorem 8.3.50. Let g : Xe → Γ be a desingularization morphism, and f : Xe → X the composition p ◦ g. It now suffices to apply Theorem 2.2 to the morphism f. 9.2.2 Projection formula Definition 2.8. Let X, Y be Noetherian schemes, and let f : X → Y be a proper morphism. For any prime cycle Z on X (Section 7.2), we set W = f(Z) and ½ [K(Z): K(W )]W if K(Z) is finite over K(W ) f Z = ∗ 0 otherwise. By linearity, we define a homomorphism f∗ from the group of cycles on X to the group of cycles on Y . This generalizes Definition 7.2.17. It is clear that the construction of f∗ is compatible with the composition of morphisms. Remark 2.9. We can interpret the intersection of two divisors in terms of direct images of cycles. Let C, D be two Cartier divisors on a regular fibered surface X → S, of which at least one is vertical. Let us suppose that C is effective and has no common component with Supp D.
  • Random Projections for Manifold Learning

    Random Projections for Manifold Learning

    Random Projections for Manifold Learning Chinmay Hegde Michael B. Wakin ECE Department EECS Department Rice University University of Michigan [email protected] [email protected] Richard G. Baraniuk ECE Department Rice University [email protected] Abstract We propose a novel method for linear dimensionality reduction of manifold mod- eled data. First, we show that with a small number M of random projections of sample points in RN belonging to an unknown K-dimensional Euclidean mani- fold, the intrinsic dimension (ID) of the sample set can be estimated to high accu- racy. Second, we rigorously prove that using only this set of random projections, we can estimate the structure of the underlying manifold. In both cases, the num- ber of random projections required is linear in K and logarithmic in N, meaning that K < M ≪ N. To handle practical situations, we develop a greedy algorithm to estimate the smallest size of the projection space required to perform manifold learning. Our method is particularly relevant in distributed sensing systems and leads to significant potential savings in data acquisition, storage and transmission costs. 1 Introduction Recently, we have witnessed a tremendous increase in the sizes of data sets generated and processed by acquisition and computing systems. As the volume of the data increases, memory and processing requirements need to correspondingly increase at the same rapid pace, and this is often prohibitively expensive. Consequently, there has been considerable interest in the task of effective modeling of high-dimensional observed data and information; such models must capture the structure of the information content in a concise manner.
  • Lecture 16: Planar Homographies Robert Collins CSE486, Penn State Motivation: Points on Planar Surface

    Lecture 16: Planar Homographies Robert Collins CSE486, Penn State Motivation: Points on Planar Surface

    Robert Collins CSE486, Penn State Lecture 16: Planar Homographies Robert Collins CSE486, Penn State Motivation: Points on Planar Surface y x Robert Collins CSE486, Penn State Review : Forward Projection World Camera Film Pixel Coords Coords Coords Coords U X x u M M V ext Y proj y Maff v W Z U X Mint u V Y v W Z U M u V m11 m12 m13 m14 v W m21 m22 m23 m24 m31 m31 m33 m34 Robert Collins CSE486, PennWorld State to Camera Transformation PC PW W Y X U R Z C V Rotate to Translate by - C align axes (align origins) PC = R ( PW - C ) = R PW + T Robert Collins CSE486, Penn State Perspective Matrix Equation X (Camera Coordinates) x = f Z Y X y = f x' f 0 0 0 Z Y y' = 0 f 0 0 Z z' 0 0 1 0 1 p = M int ⋅ PC Robert Collins CSE486, Penn State Film to Pixel Coords 2D affine transformation from film coords (x,y) to pixel coordinates (u,v): X u’ a11 a12xa'13 f 0 0 0 Y v’ a21 a22 ya'23 = 0 f 0 0 w’ Z 0 0z1' 0 0 1 0 1 Maff Mproj u = Mint PC = Maff Mproj PC Robert Collins CSE486, Penn StateProjection of Points on Planar Surface Perspective projection y Film coordinates x Point on plane Rotation + Translation Robert Collins CSE486, Penn State Projection of Planar Points Robert Collins CSE486, Penn StateProjection of Planar Points (cont) Homography H (planar projective transformation) Robert Collins CSE486, Penn StateProjection of Planar Points (cont) Homography H (planar projective transformation) Punchline: For planar surfaces, 3D to 2D perspective projection reduces to a 2D to 2D transformation.
  • 3D to 2D Bijection for Spherical Objects Under Equidistant Fisheye Projection

    3D to 2D Bijection for Spherical Objects Under Equidistant Fisheye Projection

    Computer Vision and Image Understanding 125 (2014) 172–183 Contents lists available at ScienceDirect Computer Vision and Image Understanding journal homepage: www.elsevier.com/locate/cviu 3D to 2D bijection for spherical objects under equidistant fisheye projection q ⇑ Aamir Ahmad , João Xavier, José Santos-Victor, Pedro Lima Institute for Systems and Robotics, Instituto Superior Técnico, Lisboa, Portugal article info abstract Article history: The core problem addressed in this article is the 3D position detection of a spherical object of known- Received 6 November 2013 radius in a single image frame, obtained by a dioptric vision system consisting of only one fisheye lens Accepted 5 April 2014 camera that follows equidistant projection model. The central contribution is a bijection principle Available online 16 April 2014 between a known-radius spherical object’s 3D world position and its 2D projected image curve, that we prove, thus establishing that for every possible 3D world position of the spherical object, there exists Keywords: a unique curve on the image plane if the object is projected through a fisheye lens that follows equidis- Equidistant projection tant projection model. Additionally, we present a setup for the experimental verification of the principle’s Fisheye lens correctness. In previously published works we have applied this principle to detect and subsequently Spherical object detection Omnidirectional vision track a known-radius spherical object. 3D detection Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction 1.2. Related work 1.1. Motivation Spherical object position detection is a functionality required by innumerable applications in the areas ranging from mobile robot- In recent years, omni-directional vision involving cata-dioptric ics [1], biomedical imaging [3,4], machine vision [5] to space- [1] and dioptric [2] vision systems has become one of the most related robotic applications [6].