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1 A camera is modeled as a map from a space pt (X,Y,Z) to a pixel (u,v) by ‘homogeneous coordinates’ have been used to ‘treat’ translations ‘multiplicatively’ in matrices (translation was ‘additive’ in vectors), but this is not the full story … Motivation This will be re-used in computer graphics, computer vision and robotics!

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2 Basic geometric concepts to understand Affine, Euclidean geometries (inhomogeneous coordinates) projective geometry (homogeneous coordinates) plane at infinity: affine geometry (absolute conic: Euclidean geometry) NB: some of the content in this part is only optional!!!, don’t worry

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3 Introduction to projective geometry Intuitive ideas from projective geometry (Formal definition of projective spaces)

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4 Naturally everything starts from the known vector space add two vectors multiply any vector by any scalar zero vector – origin finite basis Intuitive introduction

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5 Vector space to affine: isomorph, one-to-one vector to Euclidean as an enrichment: scalar prod. affine to projective as an extension: add ideal elements Pts, lines, parallelism Angle, distances, circles Pts at infinity

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6 Algebraic extension to pts at infinity: introduction of homogeneous coordiantes Points at infinity: Rq: the homogeneous coordinates are not unique, up to a scale.

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7 The direction d is a pt at infinity: On a plane, Can we see the pts at infinity?

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8 a projective space is an affine space + some pts at infinity a projective space is a space of ‘homogeneous coordinates’ or Provisional summary

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9 ((Formal) definition of projective geometry) Given K=R or C, can be defined as the nonzero equivalent classes determined by the relation ~ on If there is non-zero real number such that Any element of the equivalent class will be called the homogeneous coordinates of the point.

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10 Definition: a pt x is said to be linearly dependent on a set of pts if A projective space is nothing but a quotient space (space of equivalent classes): A space of homogeneous coordinates Basic structure: linear dependence of points

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11 P2 and R2 Relation between Pn (homo) and Rn (in-homo): Rn --> Pn, extension, embedded in Pn --> Rn, restriction,

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12 One example of construction of projective line by quotient space

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13 Examples of projective spaces Projective plane P2 Projective line P1 Projective space P3

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14 Pts are elements of P2 Projective plane 4 pts determine a projective basis 3 ref. Pts + 1 unit pt to fix the scales for ref. pts Relation with R2, (x,y,0), line at inf., (0,0,0) is not a pt Pts at infinity: (x,y,0), the line at infinity Space of homogeneous coordinates (x,y,t) Pts are elements of P2

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15 Line equation: Lines: Linear combination of two algebraically independent pts Operator + is ‘span’ or ‘join’

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16 Point/line duality: Point coordinate, column vector A line is a set of linearly dependent points Two points define a line Line coordinate, row vector A point is a set of linearly dependent lines Two lines define a point What is the line equation of two given points? ‘line’ (a,b,c) has been always ‘homogeneous’ since high school!

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17 Given 2 points x1 and x2 (in homogeneous coordinates), the line connecting x1 and x2 is given by Given 2 lines l1 and l2, the intersection point x is given by NB: ‘cross-product’ is purely a notational device here.

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18 Compute the intersection point of two lines, each defined by two points

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19 Conics: a curve described by a second-degree equation 3*3 symmetric matrix 5 d.o.f 5 pts determine a conic affine classification with pts at inf the line tangent to a conic at a pt dual conic pole and polar one numerical example Conics

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20 Tangent to a conic at a pt x on C is given by l=Cx Dual conic (in line coordinates) is given by l^T C^{-1} l = 0 Polar of a pt x is l = C x and is also a tangent on C from x

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21 Line at infinity

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22 Projective line Finite pts: Infinite pts: how many? Topology? A basis by 3 pts Fundamental inv: cross-ratio Homogeneous pair (x1,x2)

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23 Euclidean coordinate: the distance Affine coordinate: the ratio of the distances (x-a/a-o) Projective coordinate: the ratio of the ratio of the distances (cross-ratio, double ratio) ((x-a)/(a-o)) / ((x-b)(b-o))

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24 Pts, elements of P3 Relation with R3, plane at inf. lines: linear comb of 2 pts, but 3*4 matrix, complicated …back later planes: linear comb of 3 pts Basis by 4 (ref pts) +1 pts (unit) quadrics: two classes---ruled and unruled (topology of P3) Plane equation:... Line equation? Projective space P3

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25 planes In practice, take SVD Homework: compute plane normal vector?

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26 How many d.o.f??? 6 2*2 minors, Two lines intersect in space iff (Plucker coordinates of lines in P3)

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27 (Quadric surfaces) Ruled: hyperboloid of one sheet, 1,1,-1,-1---topo torus Unruled: sphere, ellipsoid, hyperboloid and paraboloid: 1,1,1,-1 ---- topo sphere

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28 Key points Homo. Coordinates are not unique 0 represents no projective pt finite points embedded in proj. Space (relation between R and P) pts at inf. (x,0) missing pts, directions hyper-plane (co-dim 1): dualily between u and x,

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29 2D general Euclidean transformation: 2D general affine transformation: 2D general projective transformation: Introduction to transformation

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30 Projective transformation = collineation = homography Consider all functions All linear transformations are represented by matrices A Note: linear but in homogeneous coordinates!

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31 (n+1)*(n+1) -1 d.o.f. all projective properties are left invariant by A all transformations form a group GL(n,R) Check the most important one: linear dependency, i.e. lines into lines as line is just a span Starting pt for new investigation: Klein’s Erlangen program Inversely, we may also prove that any 1-1 transf. Preserving lines is a linear trans in homogeneous coord. Properties N+2 pts to determine a trans. = a proj. basis

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32 on pts, lines and conics: Transforms contravariantly Co-variantly to preserve incidence Co-variantly NB: co-,contra-variance is w.r.t. the basis trans. Transpose is of no importance, il accommodates row/column vectors Some numerical examples of transformation on P2 (Some examples of transformations)

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33 How to compute (canonical or standard) coordinates?--- affine case by definition, vector(x4-x1) = a vector(x2-x1) + b vector(x3-x1) by canonical transformation, x1->(0,0), x2->(1,0), x3->(0,1), get transfromation A, then Ax4 Given 4 pts, x1, x2, x3, x4, find the affine coord of x4 w.r.t. x1, x2 and x3: How to solve Ax=b?

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34 Canonical projective coordinates? Given 5 pts, x1, x2, x3, x4, x5find the affine coord of x5 w.r.t. x1, x2, x3, x4: By canonical transformation: How to solve Ax=0?

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35 A transformation between 2 spaces?

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36 Exercise Compute the transformation from (0,0,1), (1,0,1), (0,1,1) and (1,1,1) into (0,0,1), (1,1/4,1),(0,1,1) and (1,3/4,1)

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37 (Geometry as an invariant theory of transformation groups) projective geom. GL(n,R) cross-ratio affine geom. Subgroup A(n,R) ratio Euclidean geom. Subgroup E(n,R) distance Hierarchy of geometry: All proj. Transformations nicely form a group! Each geometry is associated with a (sub)group!

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38 Affine transformation is a projective one which leaves the line at inf. invariant: x3=x3’=0 Example of dim 2 From projective to affine:

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39 Similarity transformation is an affine one which leaves the circular pts I and J invariant What are the circular points? From affine to euclidean

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40 Intuitive introduction of circular pts The pair of circular points The line at infinity of a usual plane Circular points

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41 Affine transformation leaves the plane at inf. invariant Similarity (euclidean) leaves the absolute conic (globally, not point-wise) invariant What is the absolute conic? Example of transformation in P3

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42 (Absolute conic) A space conic on the plane at infinity: In point coordinates: In plane coordinates: –rank 3 space quadric=absolute quadric Euclidean structure in projective space by the absolute conic

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43 The plane at infinity A usual plane in 3D The absolute conic The pair of circular points The line at infinity of a usual plane

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44 Key message from projective geometry for vision ‘abstract camera’ is a projective transformation from P3 to P2, so 3*4 matrix the intrinsic parameters of the camera are the image of the absolute conic!

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45 Summary transformation and geometry group of transformation affine group: hyper-plane at inf. euclidean group: absolute pts

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