DOCSLIB.ORG

CHOW's LEMMA 1. Projective Morphisms in Order to State The

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

CHOW'S LEMMA HANSHENG DIAO 1. Projective Morphisms In order to state the Chow's Lemma, we need the notion of pro- jective morphisms of schemes. Here we adopt the definition used in Hartshorne's Algebraic Geometry. n Notation 1. • For any ring A, let PA denote the projective n- space over A; i.e., n PA := Proj A[x0; x1; :::; xn]: n • For any scheme X and any positive integer n, define PX to be the projective n-space over X; i.e., n := n × X: PX PZ Spec Z Definition 2. • A morphism f : X ! Y of schemes is said to n be projective if it factors into a closed embedding X ! PY for n some n, followed by the projection PY ! Y . • A morphism f : X ! Y of schemes is said to be quasi-projective if it factors into an open embedding X ! X0 followed by a projective morphism X0 ! Y . Proposition 3. (1) A closed embedding is projective. (2) A composition of projective morphisms is projective. (3) Projectiveness is stable under base change. (4) A product of projective morphisms are projective. Proposition 4. (1) Open and closed embeddings are quasi-projective. (2) Every quasi-affine morphism of finite type is quasi-projective. (3) A composition of quasi-projective morphisms is quasi-projective. (4) Quasi-projectiveness is stable under base change. (5) A product of quasi-projective morphisms are quasi-projective. Proposition 5. (1) Every projective morphism is quasi-projective and proper. (2) If Y is a quasi-compact scheme, then every proper quasi-projective morphism f : X ! Y is projective. 1 2 HANSHENG DIAO For the proof of Proposition 3, 4, and 5, please refer to Hartshorne II.4 and EGA II.5.4. In the proof of Chow's Lemma, we need the following notion of scheme-theoretic image of a morphism. Proposition 6. (1) Let f : X ! Y be a morphism of schemes such that f∗(OX ) is a quasi-coherent OY -module. Then there exists a minimal closed subscheme Y 0 of Y such that f factors through the injection j : Y 0 ! Y . Moreover, Y 0 is homeomorphic to f(X) as topological subspaces in Y . This Y 0 is called the scheme-theoretic image of f : X ! Y in Y . (2) Suppose X is a subscheme of Y and the embedding i : X,! Y is quasi-compact. Then the scheme-theoretic image of i is just the closure X of X in Y , and the embedding i factors into an open embedding X,! X followed by a closed embedding X ! Y . For the proof of Proposition 6, please refer to EGA I.9.5. 2. Chow's Lemma By Proposition 5, every projective morphism is proper. On the other hand, Chow's lemma tells us that a proper morphism is not too far from a projective one. Theorem 7 (Chow's Lemma). Let f : X ! S be a proper morphism of schemes and S is noetherian. Then there exists a scheme X0 and a morphism g : X0 ! X over S such that X0 is projective over S, and there is an open dense subset U ⊆ X such that g induces an isomorphism g−1(U) !∼ U. Proof. Step 1: We can reduce to the case X irreducible. Since S is noetherian and f : X ! S is proper, X is also noe- therian (because X is locally noetherian and quasi-compact). So we can write X as union of finitely many irreducible components; say Sn X = i=1 Xi. Since Xi are closed in X, the maps fi : Xi ! S re- 0 main proper. Suppose we can find Xi and Ui for each map Xi ! S 0 Fn 0 satisfying the conditions in Chow's Lemma. We take X = i=1 Xi Fn S 0 and take U = i=1(Uin j6=i Xj). Then X ! S is also projective and −1 Fn −1 ∼ Fn g (U) = i=1 gi (Ui) ! i=1 Ui = U. CHOW'S LEMMA 3 Step 2: Now assume X is irreducible. Since S is noetherian, we can find a finite affine open covering d d [ [ S = Si = Spec Ai: i=1 i=1 Since f is of finite type, we can find, for each i, a finite affine open covering mi mi −1 [ [ f (Si) = Ui;j = Spec Bi;j j=1 i=1 such that Bi;j is a finitely generated Ai-algebra. Note that Ui;j ! Si are affine and of finite type, thus quasi-projective. We also know that the open embeddings Si ,! S are quasi-projective. ~ Hence fi;j = fjUi;j : Ui;j ! S is also quasi-projective. Step 3: Construction of X0 Relabel fUi;jg by fV1; :::; Vng. For each k 2 f1; 2; :::; ng, we have quasi- projective morphism fk = fjVk : Vk ! S. In particular, fk factors into an open embedding φk : Vk ! Pk followed by a projective morphism Pk ! S. Tn Put U = k=1 Vk. Since X is irreducible, U must be dense open in X. Let φ = (φ1; :::; φn)S : U ! P = P1 × P2 × · · · × Pn: S S S Let qk : P ! Pk be the natural projection onto the k-th factor. We have a natural map P ! S, which is projective because products of projective morphisms is projective. Now define Ψ = (j; φ): U ! X ×P S where j : U,! X is the natural embedding. Note that Ψ : U ! X × P factors into the diagonal map 4 : U ! S U × U × · · · × U followed by an open embedding U × U × · · · × U,! S S S S S S X × P1 × · · · × Pn. Clearly, 4 is a closed embedding. So Ψ is an em- S S S bedding. Let X0 = Ψ(U) be the scheme-theoretic image os Ψ in X ×P . S In particular, we have an open embedding 0 : U,! X0 and a closed embedding τ : X0 ! X × P . S Step 4: Construction of g : X0 ! X. Let p1 : X × P ! X and p2 : X × P ! P be the natural projections. S S 0 Let g = p1 ◦ τ : X ! X. Let p : X × P ! S be the composition of p1 S 4 HANSHENG DIAO and f : X ! S. Let ρ = f ◦ g : X0 ! S. Then we have the following commutative diagram: 0 τ p U > X0 > X × P 1 > X S ρ f p > _ S < Step 5: We show that g−1(U) !∼ U. Clearly, p−1(U) = U × P . Let U 0 = g−1(U) = τ −1(U × P ). Note that S S 0 0 g ◦ = p1 ◦ (τ ◦ ) = p1 ◦ Ψ = j : U,! X: So U = 0(U) ⊆ g−1(U) = U 0. We have a natural open embedding 0 0 0 i : U,! U . The composition of i : U,! U and τjU 0 : U ! U × P is S simply the graph morphism Γφ : U ! U × P , which is a closed embed- S ding because P is separated over S. So the scheme-theoretic image of Γφ is just U. On the other hand, the scheme-theoretic image of Γφ (as a topological space) is the closure of U in U × P , which is just U 0. So S U ' U 0 = g−1(U), as desired. Step 6: Finally, we show that ρ : X0 ! S is projective. p Note that ρ : X0 ! S is the composition of η : X0 !τ X × P !2 P and S a projective morphism P ! S. So we only need to show that η is a closed embedding. We also notice that p2 : X × P ! P is a closed map because X ! S S is proper. So η = p2 ◦ τ is closed. It remains to show that η is an embedding. We have the following diagram: 0 0 τ p1 U > X > X × P > X < Vk S p2 η φk > _ qk < P > Pk CHOW'S LEMMA 5 0 −1 0 0 −1 00 −1 Let Vk = φk(Vk), Wk = qk (Vk), Uk = g (Vk), and Uk = η (Wk). 0 0 Then fUkg is an open covering of X . 0 00 Claim: Uk ⊆ Uk . Proof of the claim: It suffices to show that the following diagram is commutative: ηjU0 0 k Uk > P gjU0 qk k _ _ φk Vk > Pk 0 0 0 0 Note that Uk is an open subscheme of X containing U . Since U is 0 0 0 dense in X , U is also dense in Uk. If we compose the above diagram 0 0 with the embedding U = U ! Uk, we obtain the following diagram: φ U > P jk qk _ _ φk Vk > Pk where jk : U,! Vk is the natural embedding. This diagram is obviously commutative. So the original diagram is also commutative 0 0 because U is dense in Uk. 00 0 By the claim, fUk g is also an open covering of X . Note that fWkg 0 forms an open covering of η(X ). So we only need to show that ηj 00 : Uk 00 Uk ! Wk is an embedding. Consider the composition φ−1 qk 0 k uk : Wk ! Vk ! Vk ,! X: Since X is separated over S, the graph morphism Γuk : Wk ! X × Wk S is a closed embedding. Namely, Tk = Γuk (Wk) is a closed subscheme of X × Wk. S 6 HANSHENG DIAO 0 Claim: U ⊆ Tk. 0 Proof of the claim: Let vk : U ,! X × Wk be the natural em- S 0 bedding. We need to find a morphism wk : U ! Wk such that vk = Γuk ◦ wk. By the definition of fibre product, this is equivalent 0 0 to show p1 ◦ vk = uk ◦ p2 ◦ vk. Since : U ! U is an isomorphism, it suffices to check that p1 ◦ = uk ◦ p2 ◦ .
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.
  • Fundamental Groups of Schemes

    Fundamental Groups of Schemes

    Fundamental Groups of Schemes Master thesis under the supervision of Jilong Tong Lei Yang Universite Bordeaux 1 E-mail address: [email protected] Chapter 1. Introduction 3 Chapter 2. Galois categories 5 1. Galois categories 5 §1. Definition and elementary properties. 5 §2. Examples and the main theorem 7 §2.1. The topological covers 7 §2.2. The category C(Π) and the main theorem 7 2. Galois objects. 8 3. Proof of the main theorem 12 4. Functoriality of Galois categories 15 Chapter 3. Etale covers 19 1. Some results in scheme theory. 19 2. The category of étale covers of a connected scheme 20 3. Reformulation of functoriality 22 Chapter 4. Properties and examples of the étale fundamental group 25 1. Spectrum of a field 25 2. The first homotopy sequence. 25 3. More examples 30 §1. Normal base scheme 30 §2. Abelian varieties 33 §2.1. Group schemes 33 §2.2. Abelian Varieties 35 §3. Geometrically connected schemes of finite type 39 4. G.A.G.A. theorems 39 Chapter 5. Structure of geometric fundamental groups of smooth curves 41 1. Introduction 41 2. Case of characteristic zero 42 §1. The case k = C 43 §2. General case 43 3. Case of positive characteristic 44 (p0) §1. π1(X) 44 §1.1. Lifting of curves to characteristic 0 44 §1.2. the specialization theory of Grothendieck 45 §1.3. Conclusion 45 ab §2. π1 46 §3. Some words about open curves. 47 Bibliography 49 Contents CHAPTER 1 Introduction The topological fundamental group can be studied using the theory of covering spaces, since a fundamental group coincides with the group of deck transformations of the asso- ciated universal covering space.
  • 4. Coherent Sheaves Definition 4.1. If (X,O X) Is a Locally Ringed Space

    4. Coherent Sheaves Definition 4.1. If (X,O X) Is a Locally Ringed Space

    4. Coherent Sheaves Definition 4.1. If (X; OX ) is a locally ringed space, then we say that an OX -module F is locally free if there is an open affine cover fUig of X such that FjUi is isomorphic to a direct sum of copies of OUi . If the number of copies r is finite and constant, then F is called locally free of rank r (aka a vector bundle). If F is locally free of rank one then we way say that F is invertible (aka a line bundle). The group of all invertible sheaves under tensor product, denoted Pic(X), is called the Picard group of X. A sheaf of ideals I is any OX -submodule of OX . Definition 4.2. Let X = Spec A be an affine scheme and let M be an A-module. M~ is the sheaf which assigns to every open subset U ⊂ X, the set of functions a s: U −! Mp; p2U which can be locally represented at p as a=g, a 2 M, g 2 R, p 2= Ug ⊂ U. Lemma 4.3. Let A be a ring and let M be an A-module. Let X = Spec A. ~ (1) M is a OX -module. ~ (2) If p 2 X then Mp is isomorphic to Mp. ~ (3) If f 2 A then M(Uf ) is isomorphic to Mf . Proof. (1) is clear and the rest is proved mutatis mutandis as for the structure sheaf. Definition 4.4. An OX -module F on a scheme X is called quasi- coherent if there is an open cover fUi = Spec Aig by affines and ~ isomorphisms FjUi ' Mi, where Mi is an Ai-module.
  • Nakai–Moishezon Ampleness Criterion for Real Line Bundles

    Nakai–Moishezon Ampleness Criterion for Real Line Bundles

    NAKAI{MOISHEZON AMPLENESS CRITERION FOR REAL LINE BUNDLES OSAMU FUJINO AND KEISUKE MIYAMOTO Abstract. We show that the Nakai{Moishezon ampleness criterion holds for real line bundles on complete schemes. As applications, we treat the relative Nakai{Moishezon ampleness criterion for real line bundles and the Nakai{Moishezon ampleness criterion for real line bundles on complete algebraic spaces. The main ingredient of this paper is Birkar's characterization of augmented base loci of real divisors on projective schemes. Contents 1. Introduction 1 2. Preliminaries 2 3. Augmented base loci of R-divisors 3 4. Proof of Theorem 1.4 4 5. Proof of Theorem 1.3 5 6. Proof of Theorem 1.5 7 7. Proof of Theorem 1.6 8 References 9 1. Introduction Throughout this paper, a scheme means a separated scheme of finite type over an alge- braically closed field k of any characteristic. We call such a scheme a variety if it is reduced and irreducible. Let us start with the definition of R-line bundles. Definition 1.1 (R-line bundles). Let X be a scheme (or an algebraic space). An R-line bundle (resp. a Q-line bundle) is an element of Pic(X) ⊗Z R (resp. Pic(X) ⊗Z Q) where Pic(X) is the Picard group of X. Similarly, we can define R-Cartier divisors. Definition 1.2 (R-Cartier divisors). Let X be a scheme. An R-Cartier divisor (resp. a Q-Cartier divisor) is an element of Div(X)⊗Z R (resp. Div(X)⊗Z Q) where Div(X) denotes the group of Cartier divisors on X.
  • • 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].
  • Chapter V. Fano Varieties

    Chapter V. Fano Varieties

    Chapter V. Fano Varieties A variety X is called Fano if the anticanonical bundle of X is ample. Thus Fano surfaces are the same as Del pezzo surfaces. The importance of Fano varieties in the theory of higher dimensional varieties is similar to the sig­ nificance of Del Pezzo surfaces in the two dimensional theory. The interest in Fano varieties increased recently since Mori's program predicts that every uniruled variety is birational to a fiberspace whose general fiber is a Fano variety (with terminal singularities). From this point of view it is more important to study the general prop­ erties of Fano varieties with terminal singularities than to understand the properties of smooth Fano varieties. At the moment, however, we know much more about smooth Fano varieties, and their theory should serve as a guide to the more subtle questions of singular Fano varieties. Fano varieties also appear naturally as important examples of varieties. In characteristic zero every projective variety which is homogeneous under a linear algebraic group is Fano (1.4), and their study is indispensable for the theory of algebraic groups. Also, Fano varieties have a very rich internal geometry, which makes their study very rewarding. This is one of the reasons for the success of the theory of Fano threefolds. This is a beautiful subject, about which I say essentially nothing. Section 1 is devoted to presenting the basic examples of Fano varieties and to the study of low degree rational curves on them. The largest class of examples are weighted complete intersections (1.2-3); these are probably the most accessible by elementary methods.