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2-D Geometry

2-D Geometry. The Image Formation Pipeline. Outline. Vector, matrix basics 2-D point transformations Translation, scaling, rotation, shear Homogeneous coordinates and transformations Homography, affine transformation. Notes on Notation (a little different from book).

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2-D Geometry

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  1. 2-D Geometry

  2. The Image Formation Pipeline

  3. Outline • Vector, matrix basics • 2-D point transformations • Translation, scaling, rotation, shear • Homogeneous coordinates and transformations • Homography, affine transformation

  4. Notes on Notation (a little different from book) • Vectors, points: x, v (assume column vectors) • Matrices: R, T • Scalars: x, a • Axes, objects: X, Y, O • Coordinate systems: W, C • Number systems: R, Z • Specials • Transpose operator: xT (as opposed to x0) • Identity matrix: Id • Matrices/vectors of zeroes, ones: 0, 1

  5. Block Notation for Matrices • Often convenient to write matrices in terms of parts • Smaller matrices for blocks • Row, column vectors for ranges of entries on rows, columns, respectively • E.g.: If A is 3 x 3 and :

  6. 2-D Transformations • Types • Scaling • Rotation • Shear • Translation • Mathematical representation

  7. 2-D Scaling

  8. 2-D Scaling

  9. 2-D Scaling sx 1 Horizontal shift proportional to horizontal position

  10. 2-D Scaling sy 1 Vertical shift proportional to vertical position

  11. 2-D Scaling

  12. Matrix form of 2-D Scaling

  13. 2-D Scaling

  14. 2-D Rotation

  15. µ 2-D Rotation

  16. µ 2-D Rotation

  17. µ Matrix form of 2-D Rotation (this is a counterclockwise rotation; reverse signs of sines to get a clockwise one)

  18. µ Matrix form of 2-D Rotation

  19. 2-D Shear (Horizontal)

  20. 2-D Shear (Horizontal) Horizontal displacement proportional to vertical position

  21. 2-D Shear (Horizontal) (Shear factorh is positive for the figure above)

  22. 2-D Shear (Horizontal)

  23. 2-D Shear (Vertical)

  24. 2-D Translation

  25. 2-D Translation

  26. 2-D Translation

  27. 2-D Translation

  28. 2-D Translation

  29. 2-D Translation

  30. Representing Transformations • Note that we’ve defined translation as a vector addition but rotation, scaling, etc. as matrix multiplications • It’s inconvenient to have two different operations (addition and multiplication) for different forms of transformation • It would be desirable for all transformations to be expressed in a common, linear form (since matrix multiplications are linear transformations)

  31. Example: “Trick” of additional coordinate makes this possible • Old way: • New way:

  32. Translation Matrix • We can write the formula using this “expanded” transformation matrix as: • From now on, assume points are in this “expanded” form unless otherwise noted:

  33. Homogeneous Coordinates • “Expanded” form is called homogeneous coordinates or projective space • Technically, 2-D projective space P2 is defined as Euclidean space R3¡ (0, 0, 0)T • Change to projective space by adding a scale factor (usually but not always 1):

  34. Homogeneous Coordinates: Projective Space • Equivalence is defined up to scale, (non-zero for finite points) • Think of projective points in P2 as rays in R3, where z coordinate is scale factor • All Euclidean points along ray are “same” in this sense

  35. Leaving Projective Space • Can go back to non-homogeneous representation by dividing by scale factor and dropping extra coordinate: • This is the same as saying “Where does the ray intersect the plane defined by z = 1”?

  36. Homogeneous Coordinates: Rotations, etc. • A 2-D rotation, scaling, shear or other transformation normally expressed by a 2 x 2 matrix R is written in homogeneous coordinates with the following 3 x 3 matrix: • The non-commutativity of matrix multiplication explains why different transformation orders give different results—i.e., RT  TR In homogeneous form In homogeneous form

  37. Example: Transformations Don’t Commute

  38. Example: Transformations Don’t Commute

  39. Example: Transformations Don’t Commute

  40. Example: Transformations Don’t Commute

  41. 2-D Transformations • Full-generality 3 x 3 homogeneous transformation is called a homography

  42. 2-D Transformations • Full-generality 3 x 3 homogeneous transformation is called a homography Translation components

  43. 2-D Transformations • Full-generality 3 x 3 homogeneous transformation is called a homography Scale/rotation components

  44. 2-D Transformations • Full-generality 3 x 3 homogeneous transformation is called a homography Shear/rotation components

  45. 2-D Transformations • Full-generality 3 x 3 homogeneous transformation is called a homography Homogeneous scaling factor

  46. 2-D Transformations • Full-generality 3 x 3 homogeneous transformation is called a homography When these are zero (as they have been so far), H is an affine transformation

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