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Lecture 2.2 a Brief Introduction to Projective Geometry

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Lecture 2.2 a Brief Introduction to Projective Geometry Lecture 1.3 Basic projective geometry Thomas Opsahl Motivation • For the pinhole camera, the correspondence between observed 3D points in the world and 2D points in the captured image is given by straight lines through a common point (pinhole) • This correspondence can be described by a mathematical model known as “the perspective camera model” or “the pinhole camera model” • This model can be used to describe the imaging geometry of many modern cameras, hence it plays a central part in computer vision 2 Motivation • Before we can study the perspective camera model in detail, we need to expand our mathematical toolbox • We need to be able to mathematically describe the position and orientation of the camera relative to the world coordinate frame • Also we need to get familiar with some basic elements of projective geometry, since this will make it MUCH easier to describe and work with the perspective camera model 3 Introduction • We have seen that the pose of a coordinate frame relative to a coordinate frame , denoted , can be represented as a homogeneous transformation in 2D rrA t AAR t 11 12 Bx AAξ = BB= A∈ B TB r21 r 22 tBx SE (2) 0 1 00 1 Introduction • We have seen that the pose of a coordinate frame relative to a coordinate frame , denoted , can be represented as a homogeneous transformation in 2D and 3D rrrA t 11 12 13 Bx AARt rrrA t AAξ T = BB = 21 22 23 By ∈ SE (3) BBA 0 1 rrr31 32 33 tBz 000 1 Introduction • And we have seen how they can transform points from one reference frame to another if we represent points in homogeneous coordinates x x x x y = = pp= = y ppy y z 1 z 1 • The main reason for representing pose as homogeneous transformations, was the nice algebraic properties that came with the representation Introduction AAB AABA • Euclidean geometry p=ξB⋅= p pRBB pt+ , AAB A A AB AB A – ξξξCBC=⊕=+ ( RC,,t C) ( RR B C R BCtt B) – Complicated algebra A AT ATA ⟼ ξB ( RRC,− CCt ) • Projective geometry AAB AAB p=⋅=ξBB p ppT – = AABξξξ=⊕=AT AB TT 1 CBC C BC AAξ −1 – Simple ⟼ algebra BB T • In the following we will take a closer look at some basic elements of projective geometry that we will encounter when we study the geometrical aspects of imaging – Homogeneous coordinates, homogeneous transformations The projective plane Points How to describe points in the plane? The projective plane Points How to describe points in the plane? Euclidean plane • Choose a 2D coordinate2 frame • Each point correspondsℝ to a unique pair of Cartesian coordinates y x = , = x 2 ∈ ℝ ⟼ 2 The projective plane Points How to describe points in the plane? Euclidean plane x 2 • Choose a 2D coordinate2 frame w • Each point correspondsℝ to a unique pair of Cartesian coordinates y x = , = 2 x Projective plane ∈ℝ ⟼ • Expand coordinate2 frame to 3D 2 y • Each point corresponds to a triple of homogeneous coordinates 1 = , , = x 2 � s.t.� � � � ∈ ℝ ⟼ � � , , = , , \� 0 � � � � � � ∀ ∈ ℝ The projective plane Points Observations 1. Any point = , in the Euclidean plane has a corresponding x = , , 1 2 homogeneous point in the w projective plane � 2. Homogeneous points of the form , , 0 y x does not have counterparts in the x Euclidean plane � � 2 y They correspond to points at infinity and are called ideal points 1 x The projective plane Points Observations 3. When we work with geometrical problems in the plane, we can swap x 2 between the Euclidean representation w and the projective representation x y x 22x ∋=xx =y ∈ x y 1 2 y x x 22 w ∋=xx y = y 1 w w x Example 1. These homogeneous vectors are different numerical representations of the same point in the plane 3 6 − 30 = = =−∈ 2 x 2 4 20 1 2 − 10 2. The homogeneous point 1,2,3 represents the same point as , 2 1 2 2 ∈ 3 3 ∈ ℝ The projective plane Lines How to describe lines in the plane? The projective plane Lines How to describe lines in the plane? Euclidean plane • 3 parameters ,2 , = , ℝ | + + = 0 ∈ ℝ y l x 2 The projective plane Lines How to describe lines in the plane? Euclidean plane • Triple , , 2 \ 2 ℝ w = , | 3 + + = 0 ∈ ℝ y Projective plane l • Homogeneous vector2 = , , x = | = 0 ̃ 2 2 ̃ y � ∈ � 1 l x The projective plane Lines Observations 1. Points and lines in the projective plane have the same representation, we say 2 that points and lines are dual objects in w 2 2. All lines in the Euclidean plane have a y l corresponding line in the projective x plane 2 y 3. The line = 0,0,1 in the projective plane does not have an Euclidean counterpart̃ 1 l x This line consists entirely of ideal points, and is know as the line at infinity The projective plane Lines Properties of lines in the projective plane 1. In the projective plane, all lines intersect, parallel lines intersect at 2 infinity w Two lines and intersect in the point = × y ̃1 ̃2 l 1 2 x 2. The line passing� ̃ through̃ points and 2 is given by 1 y = × � �2 ̃ �1 �2 1 l x Example Determine the line passing through the two points , and , Homogeneous representation of points 25 22 Matrix representation xx=∈=∈4 13 12 of the cross product 11 × × Homogeneous representation of line where ⟼ 0− 14 5 −− 9 3 0 −uu32 def lxx =×= xx =1 0 − 2 13 = 3 = 1 12[ 12]× [u] = uu0 − × 31 −42 0 1 6 2 −uu 0 21 Equation of the line −3xy ++= 20 ⇔ y = 3 x − 2 Example A point at infinity https://en.wikipedia.org/wiki/Projective_plane 21 The projective plane Transformations • Some important transformations – like the action of a pose on points in the plane – happen to be linear in the projective plane and non-linear in the Euclidean plane • The most general invertible transformations of the projective plane are known as homographies – or projective transformations / linear projective transformations / projectivities / collineations Definition A homography of is a linear transformation on homogeneous 3-vectors represented by a homogeneous, non2-singular 3 × 3 matrix = �′ ℎ11 ℎ12 ℎ13 � 21 22 23 �′ ℎ ℎ ℎ � ℎ31 ℎ32 ℎ33 So is unique up to scale, i.e. �=′ \ 0 � ∀ ∈ ℝ The projective plane Transformations • One characteristic of homographies is that they preserve lines, in fact any invertible transformation of that preserves lines is a homography 2 • Examples – Central projection from one plane to another is a homography Hence if we take an image with a perspective camera of a flat surface from an angle, we can remove the perspective distortion with a homography Perspective distortion Without distortion ′ Images from http://www.robots.ox.ac.uk/~vgg/hzbook.html The projective plane Transformations • One characteristic of homographies is that they preserve lines, in fact any invertible transformation of that preserves lines is a homography 2 • Examples – Central projection from one plane to another is a homography – Two images, captured by perspective cameras, of the same planar scene is related by a homography Image 1 Image 2 The projective plane Transformations • One characteristic of homographies is that they preserve lines, in fact any invertible transformation of that preserves lines is a homography 2 • Examples – Central projection from one plane to another is a homography – Two images, captured by perspective cameras, of the same planar scene is related by a homography • One can show that the product of two homographies also must be a homography We say that the homographies constitute a group – the projective linear group 3 • Within this group there are several more specialized subgroups Transformations of the projective plane Transformation of Matrix #DoF Preserves Visualization 2 Translation 2 Orientation 1 + all below Euclidean 3 Lengths 1 + all below Similarity 4 Angles 1 + all below Affine 11 12 13 6 Parallelism, 21 22 23 line at infinity 0 0 1 + all below Homography 11 12 13 8 Straight lines /projective 21 22 23 ℎ31 ℎ32 ℎ33 ℎ ℎ ℎ ℎ ℎ ℎ 26 The projective space • The relationship between the Euclidean space and the projective space is much like the relationship between and 3 3 2 2 ℝ ℝ x • In the projective space x – We represent points in homogeneous coordinates 33 y ∋=xxy = ∈ z = = \ 0 z � � 1 � � � ∀ ∈ ℝ x ̃ ̃ x w � – Points at infinity have last homogeneous coordinate� equal to zero 33y y ∋=xx = ∈ z w – Planes and points are dual objects z w w = | = 0 3 � � � � – The plane at infinity are made upΠ of all points∈ at infinity 27 Transformations of the projective space Transformation of Matrix #DoF Preserves Translation 3 Orientation 1 + all below Euclidean 6 Volumes, volume ratios, lengths 1 + all below Similarity 7 Angles 1 + all below Affine 12 Parallelism of planes, The plane at infinity 11 12 13 14 + all below 021 022 023 124 31 32 33 34 Homography 15 Intersection and tangency of /projective surfaces in contact, straight lines ℎ11 ℎ12 ℎ13 ℎ14 ℎ21 ℎ22 ℎ23 ℎ24 ℎ31 ℎ32 ℎ33 ℎ34 ℎ41 ℎ42 ℎ43 ℎ44 Summary • The projective plane • Additional reading – Homogeneous coordinates2 – Szeliski: 2.1.2, 2.1.3, – Line at infinity – Points & lines are dual • The projective space – Homogeneous coordinates3 – Plane at infinity – Points & planes are dual • Linear transformations of and – Represented by homogeneous2 matrices3 – Homographies Affine Similarities Euclidean Translations ⊃ ⊃ ⊃ ⊃ 29 Summary • The projective plane • Additional reading – Homogeneous coordinates2 – Szeliski: 2.1.2, 2.1.3, – Line at infinity – Points & lines are dual MATLAB WARNING When we work with linear transformations, we • The projective space represent them as matrices that act on points by – Homogeneous coordinates3 right multiplication nn – Plane at infinity T : → – Points & planes are dual x yx= M R Matlab seem to prefer left multiplication instead • Linear transformations of and nn T : → – Represented by homogeneous2 matrices3 T TT x yx= M L – Homographies Affine Similarities Euclidean Translations So if you use built in matlab functions when you ⊃ ⊃ ⊃ work with transformations, be careful!!! ⊃ 30 .
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