DOCSLIB.ORG

• Rotations • Camera Calibration • Homography • Ransac

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
• 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. Can you think of any problems with this approach? procedure forwardWarp(f,h, out g): For every pixel x in f(x) 1. Compute the destination location x0 = h(x). 2. Copy the pixel f(x) to g(x0). Algorithm 3.1 Forward warping algorithm for transforming an image f(x) into an image g(x0) through the parametric transform x0 = h(x). Rotations Linear transformations that preserve distances and angles Definition: an orthogonal transformation perserves dot products T T n n n a Tb = F (a) F (b)whereF (a)=Aa, a Rn ,A Rn n⇥ a b = T (a)T (b)whereT (a)=Aa, a 2R ,A 2R ⇥ 2 2 T T T T a b = a A Ab A A = I [can conclude by setting a,b = coordinate vectors] () Defn: A is a rotation matrix if ATA = I, det(A) = 1 Defn: A is a reflection matrix if ATA = I, det(A) = -1 2D Rotations cos ✓ sin ✓ R = sin ✓ −cos ✓ 1 DOF 3D Rotations X r11 r12 r13 X R Y = r r r Y 2 3 2 21 22 233 2 3 Z r31 r32 r33 Z 4 5 4 5 4 5 Think of as change of basis where ri = r(i,:) are orthonormal basis vectors r2 r1 rotated coordinate frame r3 How many DOFs? 3 = (2 to point r1 + 1 to rotate along r1) 7 Shears 1 hxy hxz 0 h 1 h 0 A˜ = 2 yx yz 3 hzx hzy 10 6 00017 6 7 4 Shears y into5 x 7 3D Rotations 8 LotsRotations of parameterizations that try to capture 3 DOFs Helpful• 3D Rotations one for fundamentally vision: axis-angle more complex representation than in 2D! • 2D: amount of rotation! Represent a 3D rotation with a unit vector that represents the axis of • 3D: amount and axis of rotation rotation, and an angle of rotation about that vector -vs- 2D 3D 8 05-3DTransformations.key - February 9, 2015 Recall: cross-product Dot product: a b = a b cos✓ · || || || || Cross product: ijk a2 a3 a1 a3 a1 a2 a1 a2 a3 = i j + k b2 b3 − b1 b3 b1 b2 b b b 1 2 3 Cross product 0 a a b − 3 2 1 matrix: a b = ˆab = a3 0 a1 b2 ⇥ 2 a a −0 3 2b 3 − 2 1 3 4 5 4 5 Approach ! R3, ! =1 2 || || ✓ x Approach ! R3, ! =1 2 || || x ? ✓ x k x 1. Write as x as sum of parallel and perpindicular component to omega 2. Rotate perpindicular component by 2D rotation of theta in plane orthogonal to omega R = I +ˆw sin ✓ +ˆwwˆ(1 cos ✓) − [Rx can simplify to cross and dot product computations] Exponential map ! R3, ! =1 2 || || x ? ✓ x k x R =exp(ˆv), where v = !✓ 1 = I +ˆv + vˆ2 + ... 2! [standard Taylor series expansion of exp(x) @ x=0 as 1 + x + (1/2!)x2 +…] Implication: we can approximate change in position due to a small rotation as v x, where v = !✓ ⇥ Agenda • Rotations • Camera calibration • Homography • Ransac Perspective projection y x (X,Y,Z) (x,y,1) z COP [right-handed coordinate system] f x = X Z f y = Y Z Perspective projection revisited x f 00 X λ y = 0 f 0 Y 213 20013 2Z 3 4 5 4 5 4 5 Given (X,Y,Z) and f, compute (x,y) and lambda: λx = fX λ = Z λx fX x = = λ Z Special case: f = 1 Natural geometric intuition: • 3D point is obtained by scaling ray pointed at image coordinate • Scale factor = true depth of point (X,Y,Z) (x,y,1) COP x X Z y = Y 213 2Z 3 4 5 4 5 [Aside: given an image with a focal length ‘f’, resize by ‘1/f’ to obtain unit-focal-length image] Homogenous notation x X y Y 2z3 ⇠ 2Z 3 4x5 4X5 y Y 2z3 ⌘ 2Z 3 4 5 4 5 For now, think of above as shorthand notation for x X λ s.t. λ y = Y 9 2z3 2Z 3 4 5 4 5 Camera projection X x f 00 r r r t 11 12 13 x Y λ y = 0 f 0 r r r t 21 22 23 y 2Z 3 213 20013 2r r r t 3 31 32 33 z 6 1 7 6 7 4 5 Camera4 instrinsic matrix5 K 4 Camera extrinsics 53D point in (can include skew & non-square pixel size) (rotation and translation) world4 coordinates5 r2 r1 camera r3 T world coordinate frame λx Aside: homogenous notation is shorthand for x = λ Fancier intrinsics xs = sxx } non-square pixels ys = syy x0 = xs + ox } shifted origin y0 = ys + oy y skewed image axes x”=x0 + s✓y0 ✓ x sx s✓ ox f 00 fsx fs✓ ox K = 0 sy oy 0 f 0 = 0 fsy oy 2 0013 20013 2 0013 4 5 4 5 4 5 Notation [Using Matlab’s rows x columns] X x fs fs o r r r t x ✓ x 11 12 13 x Y λ y = 0 fs o r r r t y y 21 22 23 y 2Z 3 213 2 0013 2r r r t 3 31 32 33 z 6 1 7 4 5 4 5 4 5 6 7 X 4 5 Y = K3 3 R3 3 T3 1 2 3 ⇥ ⇥ ⇥ Z ⇥ ⇤ 6 1 7 6 7 X 4 5 Y = M3 4 2 3 ⇥ Z 6 1 7 6 7 4 5 Claims (without proof): 1. A 3x4 matrix ‘M’ can be a camera matrix iff det(M) is not zero 2. M is determined only up to a scale factor Notation (more) X X Y Y M3 4 2 3 = A3 3 b3 1 2 3 ⇥ Z ⇥ ⇥ Z 6 1 7 ⇥ ⇤ 6 1 7 6 7 6 7 4 5 X 4 5 = A3 3 Y + b3 1 ⇥ 2Z 3 ⇥ 4 5 T T m1 a1 b1 T T M = m2 ,A= a2 ,b= b2 2 T 3 2 T 3 2 3 m3 a3 b3 4 5 4 5 4 5 Applying the projection matrix 1 x = ( XY Za + b ) λ 1 1 ⇥ ⇤ 1 y = ( XY Za + b ) λ 2 2 ⇥ ⇤ λ = XY Za3 + b3 ⇥ ⇤ Set of 3D points that project to x = 0: XY Za1 + b1 =0 Set of 3D points that project to y = 0: ⇥XY Z⇤ a2 + b2 =0 ⇥ ⇤ Set of 3D points that project to x = inf or y = inf: XY Za3 + b3 =0 ⇥ ⇤ Rows of the projection matrix describe the 3 planes defined by the image coordinate system a3 a1 y COP a2 x image plane Other geometric properties (x,y) COP (X,Y,Z) What’s set of (X,Y,Z) points that project to same (x,y)? X x 1 1 Y = λw + b where w = A− y ,b= A− b 2Z 3 213 − 4 5 4 5 What’s the position of COP / pinhole? X X 1 A Y + b =0 Y = A− b 2Z 3 ) 2Z 3 − 4 5 4 5 Affine Cameras x = XY Za + b T 1 1 m3 = 0001 y = XY Za + b ⇥ ⇤ ⇥ ⇤ 2 1 ⇥ ⇤ Image coordinates (x,y) are an affine function of world coordinates (X,Y,Z) Affine transformations = linear transformations plus an offset • Example: Weak-perspective projection model • Projection defined by 8 parameters • Parallel lines are projected to parallel lines • The transformation can be written as a direct linear transformation Geometric Transformations Euclidean (trans + rot) Affine: preserves Projective: preserves lengths + angles parallel lines preserves lines Projective Affine Euclidean Agenda • Rotations • Camera calibration • Homography • Ransac Calibration: Recover M from scene points P1,..,PN and the corresponding projections in the image plane p1,..,pN Find M that minimizes the distance between the actual points in the image, pi, and their predicted projections MPi Problems: • The projection is (in general) non-linear • M is defined up to an arbitrary scale factor PnP = Perspective n-Point The math for the calibration procedure follows a recipe that is used in many (most?) problems involving camera geometry, so it’s worth remembering: Write relation between image point, projection matrix, and point in space: pi ≡ MPi m T P m T P Write non-linear relations 1 i 2 i ui = vi = between coordinates: T T m3 Pi m3 Pi T T m1 Pi − (m3 Pi ) ui = 0 Make them linear: T T m2 Pi − (m3 Pi ) vi = 0 ⎡m1 ⎤ ⎡PT 0 − u PT ⎤ Write them in i i i m = 0 m = ⎢m ⎥ matrix form: ⎢ PT T ⎥ ⎢ 2 ⎥ ⎣ 0 i − vi Pi ⎦ ⎣⎢m3 ⎦⎥ T T ⎡ P1 0 − u1P1 ⎤ ⎢ T T ⎥ Put all the relations for all the ⎢ 0 P1 − v1P1 ⎥ points into a single matrix: ⎢ ! ! ! ⎥m = 0 ⎢ T T ⎥ ⎢PN 0 − uN PN ⎥ ⎢ T T ⎥ (vector of 0’s) ⎣ 0 PN − vN PN ⎦ In noise-free case: Lm = 0 What about noisy case? min Lm 2 m 2=1 || || || || Min right singular vector of L (or eigenvector of LTL) Is this the right error to minimize? If not, what is? Ideal error Pi (u1,v1) P1 y MPi (ui,vi) z 2 2 ⎛ m ⋅ P ⎞ ⎛ m ⋅ P ⎞ ⎜u 1 i ⎟ ⎜v 2 i ⎟ x Error(M) = ⎜ i − ⎟ + ⎜ i − ⎟ ⎝ m3 ⋅ Pi ⎠ ⎝ m3 ⋅ Pi ⎠ Initialize nonlinear optimization with “algebraic” solution Radial Lens Distortions Radial Lens Distortions No Distortion Barrel Distortion Pincushion Distortion Correcting Radial Lens Distortions Before After http://www.grasshopperonline.com/barrel_distortion_correction_software.html Overall approach Minimize reprojection error: Error(M,k’s) Initialize with algebraic solution (approaches in literature based on various assumptions) Revisiting homographies Place world coordinate frame on object plane X x f 00 r r r t 11 12 13 x Y λ y = 0 f 0 r r r t 21 22 23 y 2 0 3 213 20013 2r r r t 3 31 32 33 z 6 1 7 4 5 4 5 4 5 6 7 4 5 Projection of planar points X x f 00 r r r t 11 12 13 x Y λ y = 0 f 0 r r r t 21 22 23 y 2 0 3 213 20013 2r r r t 3 31 32 33 z 6 1 7 4 5 4 5 4 5 6 7 4 5 f 00 r11 r12 tx X = 0 f 0 r r t Y 2 3 2 21 22 y3 2 3 001 r31 r32 tz 1 4 5 4 5 4 5 fr11 fr12 ftx X = fr fr ft Y 2 21 22 y3 2 3 r31 r32 tz 1 4 5 4 5 Convert between 2D location
Load more

Recommended publications
  • Globally Optimal Affine and Metric Upgrades in Stratified Autocalibration
    Globally Optimal Affine and Metric Upgrades in Stratified Autocalibration Manmohan Chandrakery Sameer Agarwalz David Kriegmany Serge Belongiey [email protected] [email protected] [email protected] [email protected] y University of California, San Diego z University of Washington, Seattle Abstract parameters of the cameras, which is commonly approached by estimating the dual image of the absolute conic (DIAC). We present a practical, stratified autocalibration algo- A variety of linear methods exist towards this end, how- rithm with theoretical guarantees of global optimality. Given ever, they are known to perform poorly in the presence of a projective reconstruction, the first stage of the algorithm noise [10]. Perhaps more significantly, most methods a pos- upgrades it to affine by estimating the position of the plane teriori impose the positive semidefiniteness of the DIAC, at infinity. The plane at infinity is computed by globally which might lead to a spurious calibration. Thus, it is im- minimizing a least squares formulation of the modulus con- portant to impose the positive semidefiniteness of the DIAC straints. In the second stage, the algorithm upgrades this within the optimization, not as a post-processing step. affine reconstruction to a metric one by globally minimizing This paper proposes global minimization algorithms for the infinite homography relation to compute the dual image both stages of stratified autocalibration that furnish theoreti- of the absolute conic (DIAC). The positive semidefiniteness cal certificates of optimality. That is, they return a solution at of the DIAC is explicitly enforced as part of the optimization most away from the global minimum, for arbitrarily small .
    [Show full text]
  • 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.
    [Show full text]
  • Robot Vision: Projective Geometry
    Robot Vision: Projective Geometry Ass.Prof. Friedrich Fraundorfer SS 2018 1 Learning goals . Understand homogeneous coordinates . Understand points, line, plane parameters and interpret them geometrically . Understand point, line, plane interactions geometrically . Analytical calculations with lines, points and planes . Understand the difference between Euclidean and projective space . Understand the properties of parallel lines and planes in projective space . Understand the concept of the line and plane at infinity 2 Outline . 1D projective geometry . 2D projective geometry ▫ Homogeneous coordinates ▫ Points, Lines ▫ Duality . 3D projective geometry ▫ Points, Lines, Planes ▫ Duality ▫ Plane at infinity 3 Literature . Multiple View Geometry in Computer Vision. Richard Hartley and Andrew Zisserman. Cambridge University Press, March 2004. Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992 . Available online: www.cs.cmu.edu/~ph/869/papers/zisser-mundy.pdf 4 Motivation – Image formation [Source: Charles Gunn] 5 Motivation – Parallel lines [Source: Flickr] 6 Motivation – Epipolar constraint X world point epipolar plane x x’ x‘TEx=0 C T C’ R 7 Euclidean geometry vs. projective geometry Definitions: . Geometry is the teaching of points, lines, planes and their relationships and properties (angles) . Geometries are defined based on invariances (what is changing if you transform a configuration of points, lines etc.) . Geometric transformations
    [Show full text]
  • Feature Matching and Heat Flow in Centro-Affine Geometry
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 093, 22 pages Feature Matching and Heat Flow in Centro-Affine Geometry Peter J. OLVER y, Changzheng QU z and Yun YANG x y School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA E-mail: [email protected] URL: http://www.math.umn.edu/~olver/ z School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P.R. China E-mail: [email protected] x Department of Mathematics, Northeastern University, Shenyang, 110819, P.R. China E-mail: [email protected] Received April 02, 2020, in final form September 14, 2020; Published online September 29, 2020 https://doi.org/10.3842/SIGMA.2020.093 Abstract. In this paper, we study the differential invariants and the invariant heat flow in centro-affine geometry, proving that the latter is equivalent to the inviscid Burgers' equa- tion. Furthermore, we apply the centro-affine invariants to develop an invariant algorithm to match features of objects appearing in images. We show that the resulting algorithm com- pares favorably with the widely applied scale-invariant feature transform (SIFT), speeded up robust features (SURF), and affine-SIFT (ASIFT) methods. Key words: centro-affine geometry; equivariant moving frames; heat flow; inviscid Burgers' equation; differential invariant; edge matching 2020 Mathematics Subject Classification: 53A15; 53A55 1 Introduction The main objective in this paper is to study differential invariants and invariant curve flows { in particular the heat flow { in centro-affine geometry. In addition, we will present some basic applications to feature matching in camera images of three-dimensional objects, comparing our method with other popular algorithms.
    [Show full text]
  • Estimating Projective Transformation Matrix (Collineation, Homography)
    Estimating Projective Transformation Matrix (Collineation, Homography) Zhengyou Zhang Microsoft Research One Microsoft Way, Redmond, WA 98052, USA E-mail: [email protected] November 1993; Updated May 29, 2010 Microsoft Research Techical Report MSR-TR-2010-63 Note: The original version of this report was written in November 1993 while I was at INRIA. It was circulated among very few people, and never published. I am now publishing it as a tech report, adding Section 7, with the hope that it could be useful to more people. Abstract In many applications, one is required to estimate the projective transformation between two sets of points, which is also known as collineation or homography. This report presents a number of techniques for this purpose. Contents 1 Introduction 2 2 Method 1: Five-correspondences case 3 3 Method 2: Compute P together with the scalar factors 3 4 Method 3: Compute P only (a batch approach) 5 5 Method 4: Compute P only (an iterative approach) 6 6 Method 5: Compute P through normalization 7 7 Method 6: Maximum Likelihood Estimation 8 1 1 Introduction Projective Transformation is a concept used in projective geometry to describe how a set of geometric objects maps to another set of geometric objects in projective space. The basic intuition behind projective space is to add extra points (points at infinity) to Euclidean space, and the geometric transformation allows to move those extra points to traditional points, and vice versa. Homogeneous coordinates are used in projective space much as Cartesian coordinates are used in Euclidean space. A point in two dimensions is described by a 3D vector.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Automorphisms of Classical Geometries in the Sense of Klein
    Automorphisms of classical geometries in the sense of Klein Navarro, A.∗ Navarro, J.† November 11, 2018 Abstract In this note, we compute the group of automorphisms of Projective, Affine and Euclidean Geometries in the sense of Klein. As an application, we give a simple construction of the outer automorphism of S6. 1 Introduction Let Pn be the set of 1-dimensional subspaces of a n + 1-dimensional vector space E over a (commutative) field k. This standard definition does not capture the ”struc- ture” of the projective space although it does point out its automorphisms: they are projectivizations of semilinear automorphisms of E, also known as Staudt projectivi- ties. A different approach (v. gr. [1]), defines the projective space as a lattice (the lattice of all linear subvarieties) satisfying certain axioms. Then, the so named Fun- damental Theorem of Projective Geometry ([1], Thm 2.26) states that, when n> 1, collineations of Pn (i.e., bijections preserving alignment, which are the automorphisms of this lattice structure) are precisely Staudt projectivities. In this note we are concerned with geometries in the sense of Klein: a geometry is a pair (X, G) where X is a set and G is a subgroup of the group Biy(X) of all bijections of X. In Klein’s view, Projective Geometry is the pair (Pn, PGln), where PGln is the group of projectivizations of k-linear automorphisms of the vector space E (see Example 2.2). The main result of this note is a computation, analogous to the arXiv:0901.3928v2 [math.GR] 4 Jul 2013 aforementioned theorem for collineations, but in the realm of Klein geometries: Theorem 3.4 The group of automorphisms of the Projective Geometry (Pn, PGln) is the group of Staudt projectivities, for any n ≥ 1.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Download Download
    Acta Math. Univ. Comenianae 911 Vol. LXXXVIII, 3 (2019), pp. 911{916 COSET GEOMETRIES WITH TRIALITIES AND THEIR REDUCED INCIDENCE GRAPHS D. LEEMANS and K. STOKES Abstract. In this article we explore combinatorial trialities of incidence geome- tries. We give a construction that uses coset geometries to construct examples of incidence geometries with trialities and prescribed automorphism group. We define the reduced incidence graph of the geometry to be the oriented graph obtained as the quotient of the geometry under the triality. Our chosen examples exhibit interesting features relating the automorphism group of the geometry and the automorphism group of the reduced incidence graphs. 1. Introduction The projective space of dimension n over a field F is an incidence geometry PG(n; F ). It has n types of elements; the projective subspaces: the points, the lines, the planes, and so on. The elements are related by incidence, defined by inclusion. A collinearity of PG(n; F ) is an automorphism preserving incidence and type. According to the Fundamental theorem of projective geometry, every collinearity is composed by a homography and a field automorphism. A duality of PG(n; F ) is an automorphism preserving incidence that maps elements of type k to elements of type n − k − 1. Dualities are also called correlations or reciprocities. Geometric dualities, that is, dualities in projective spaces, correspond to sesquilinear forms. Therefore the classification of the sesquilinear forms also give a classification of the geometric dualities. A polarity is a duality δ that is an involution, that is, δ2 = Id. A duality can always be expressed as a composition of a polarity and a collinearity.
    [Show full text]
  • 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.
    [Show full text]