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Basic geometric concepts to understand

Basic geometric concepts to understand. Affine, Euclidean geometries ( in homogeneous coordinates) projective geometry (homogeneous coordinates) plane at infinity: affine geometry. Intuitive introduction. Naturally everything starts from the known vector space.

benedict-tyler
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Basic geometric concepts to understand

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  1. Basic geometric concepts to understand • Affine, Euclidean geometries (inhomogeneous coordinates) • projective geometry (homogeneous coordinates) • plane at infinity: affine geometry

  2. Intuitive introduction Naturally everything starts from the known vector space

  3. 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

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

  5. Examples of projective spaces • Projective plane P2 • Projective line P1 • Projective space P3

  6. Projective plane Space of homogeneous coordinates (x,y,t) Pts are elements of P2 Pts are elements of P2 Pts at infinity: (x,y,0), the line at infinity 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

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

  8. 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 Point/line duality: • What is the line equation of two given points? • ‘line’ (a,b,c) has been always ‘homogeneous’ since high school!

  9. 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.

  10. Conics Conics: a curve described by a second-degree equation • 3*3 symmetric matrix • 5 d.o.f • 5 pts determine a conic

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

  12. Projective space P3 • Pts, elements of P3 • Relation with R3, plane at inf. • planes: linear comb of 3 pts • Basis by 4 (ref pts) +1 pts (unit)

  13. planes In practice, take SVD

  14. 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,

  15. Introduction to transformation 2D general Euclidean transformation: 2D general affine transformation: 2D general projective transformation: Colinearity Cross-ratio

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

  17. How to compute transformatins and canonical projective coordinates?

  18. X’ P3 u X u’ u v O P2 Geometric modeling of a camera How to relate a 3D point X (in oxyz) to a 2D point in pixels (u,v)?

  19. Z u Y y X x X f x v O Camera coordinate frame

  20. Z u u y Y y X x o x X f x v v O Image coordinate frame

  21. 5 intrinsic parameters • Focal length in horizontal/vertical pixels (2) • (or focal length in pixels + aspect ratio) • the principal point (2) • the skew (1) one rough example: 135 film In practice, for most of CCD cameras: • alpha u = alpha v i.e. aspect ratio=1 • alpha = 90 i.e. skew s=0 • (u0,v0) the middle of the image • only focal length in pixels?

  22. Z Zw u Y Xw Yw y X x X f x v O World (object) coordinate frame Xw

  23. 6 extrinsic parameters World coordinate frame: extrinsic parameters Relation between the abstract algebraic and geometric models is in the intrinsic/extrinsic parameters!

  24. Finally, we have a map from a space pt (X,Y,Z) to a pixel (u,v) by

  25. What does the calibration give us? It turns the camera into an angular/direction sensor! Normalised coordinates: Direction vector:

  26. Camera calibration Given from image processing or by hand  • Estimate C • decompose C into intrinsic/extrinsic

  27. Decomposition • analytical by equating K(R,t)=P

  28. Pose estimation = calibration of only extrinsic parameters • Given • Estimate R and t

  29. 3-point algebraic method 3 reference points == 3 beacons • First convert pixels u into normalized points x by knowing the intrinsic parameters • Write down the fundamental equation: • Solve this algebraic system to get the point distances first • Compute a 3D transformation

  30. 3D transformation estimation given 3 corresponding 3D points: • Compute the centroids as the origin • Compute the scale • (compute the rotation by quaternion) • Compute the rotation axis • Compute the rotation angle

  31. C D B x_d A x_a O Linear pose estimation from 4 coplanar points Vector based (or affine geometry) method

  32. Midterm statistics 0~59 7 60~69 12 70~79 17 80~89 8 90~99 5 100 2 Total 71.80392157 16.30953047 Q1: 14.98039216 5.82920302 Q2: 12.03921569 6.141533308 Q3: 14.56862745 4.817696138 Q4: 12.35294118 7.638909685 Q5: 14.90196078 7.105645367 Q6: 7.254901961 4.511510334

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