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Robert CollinsCSE486, Penn State                        Lecture 13:                     Camera Projection II              ...
Robert CollinsCSE486, Penn State                     Recall: Imaging Geometry                                          W  ...
Robert CollinsCSE486, Penn State                         Imaging Geometry                               Camera Coordinate ...
Robert CollinsCSE486, Penn State                               Imaging Geometry                                           ...
Robert CollinsCSE486, Penn State                             Imaging Geometry                                             ...
Robert CollinsCSE486, Penn State                             Imaging Geometry   Camera      Image (film)                  ...
Robert CollinsCSE486, Penn State                       Forward Projection    World             Camera         Film        ...
Robert CollinsCSE486, Penn State                     Intrinsic Camera Parameters    World               Camera     Film   ...
Robert CollinsCSE486, Penn State                     Intrinsic parameters         • Describes coordinate transformation   ...
Robert CollinsCSE486, Penn State                     Intrinsic parameters (offsets)              film plane               ...
Robert CollinsCSE486, Penn State                               Intrinsic parameters    sometimes one or more coordinate ax...
Robert CollinsCSE486, Penn State                     Intrinsic parameters (scales)  sampling determines how many rows/cols...
Robert CollinsCSE486, Penn State                        Effective Scales: sx and sy                         1  X          ...
Robert CollinsCSE486, Penn State                     Perspective projection matrix   Adding the intrinsic parameters into ...
Robert CollinsCSE486, Penn State                               Note:    Sometimes, the image and the camera coordinate sys...
Robert CollinsCSE486, Penn State                                    Note 2            In general, I like to think of the c...
Robert CollinsCSE486, Penn State                                Huh?           Did he just say it was “a fine” transformat...
Robert CollinsCSE486, Penn State                     Summary : Forward Projection       World                Camera       ...
Robert CollinsCSE486, Penn State                     Summary: Projection Equation                       Film plane   Persp...
Robert CollinsCSE486, Penn State                          Lecture 13/14:                     Intro to Image Mappings
Robert CollinsCSE486, Penn State                     Image Mappings                       Overviewfrom R.Szeliski
Robert CollinsCSE486, Penn State                     Geometric Image Mappings                                 Geometric   ...
Robert CollinsCSE486, Penn State                         Linear Transformations                          (Can be written a...
Robert CollinsCSE486, Penn State                                Translation             y                           y’    ...
Robert CollinsCSE486, Penn State                                    Scale             y                           y’      ...
Robert CollinsCSE486, Penn State                                    Rotation             y                           y’   ...
Robert CollinsCSE486, Penn State                             Euclidean (Rigid)             y                           y’ ...
Robert CollinsCSE486, Penn State                            Partitioned Matriceshttp://planetmath.org/encyclopedia/Partiti...
Robert CollinsCSE486, Penn State                     Partitioned Matrices               2x1     2x2   2x1   2x1           ...
Robert Collins            Another Example (from last time)CSE486, Penn State                     X          r11 r12 r13 tx...
Robert CollinsCSE486, Penn State                     Similarity (scaled Euclidean)             y                          ...
Robert CollinsCSE486, Penn State                                    Affine             y                           y’     ...
Robert CollinsCSE486, Penn State                                    Projective             y                           y’ ...
Robert Collins                Summary of 2D TransformationsCSE486, Penn State
Robert Collins                Summary of 2D TransformationsCSE486, Penn State Euclidean
Robert Collins                Summary of 2D TransformationsCSE486, Penn State Similarity
Robert Collins                Summary of 2D TransformationsCSE486, Penn State     Affine
Robert Collins                Summary of 2D TransformationsCSE486, Penn State Projective
Robert CollinsCSE486, Penn State                  Summary of 2D Transformationsfrom R.Szeliski
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Lecture13

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Lecture13

  1. 1. Robert CollinsCSE486, Penn State Lecture 13: Camera Projection II Reading: T&V Section 2.4
  2. 2. Robert CollinsCSE486, Penn State Recall: Imaging Geometry W Object of Interest in World Coordinate System (U,V,W) V U
  3. 3. Robert CollinsCSE486, Penn State Imaging Geometry Camera Coordinate Y System (X,Y,Z). X Z • Z is optic axis f • Image plane located f units out along optic axis • f is called focal length
  4. 4. Robert CollinsCSE486, Penn State Imaging Geometry W Y y X Z x V U Forward Projection onto image plane. 3D (X,Y,Z) projected to 2D (x,y)
  5. 5. Robert CollinsCSE486, Penn State Imaging Geometry W Y y X Z x V u U v Our image gets digitized into pixel coordinates (u,v)
  6. 6. Robert CollinsCSE486, Penn State Imaging Geometry Camera Image (film) World Coordinates Coordinates W Coordinates Y y X Z x V u U v Pixel Coordinates
  7. 7. Robert CollinsCSE486, Penn State Forward Projection World Camera Film Pixel Coords Coords Coords Coords U X x u V Y y v W Z We want a mathematical model to describe how 3D World points get projected into 2D Pixel coordinates. Our goal: describe this sequence of transformations by a big matrix equation!
  8. 8. Robert CollinsCSE486, Penn State Intrinsic Camera Parameters World Camera Film Pixel Coords Coords Coords Coords U X x u V Y y v W Z Affine Transformation
  9. 9. Robert CollinsCSE486, Penn State Intrinsic parameters • Describes coordinate transformation between film coordinates (projected image) and pixel array • Film cameras: scanning/digitization • CCD cameras: grid of photosensors still in T&V section 2.4
  10. 10. Robert CollinsCSE486, Penn State Intrinsic parameters (offsets) film plane pixel array (projected image) ox (0,0) u (col) oy x v (row) (0,0) y X Y u  f  ox v  f  oy Z Z ox and oy called image center or principle point
  11. 11. Robert CollinsCSE486, Penn State Intrinsic parameters sometimes one or more coordinate axes are flipped (e.g. T&V section 2.4) film plane pixel array ox (0,0) u (col) oy y v (row) x (0,0) X Y u  f  ox v  f  oy Z Z
  12. 12. Robert CollinsCSE486, Penn State Intrinsic parameters (scales) sampling determines how many rows/cols in the image film scanning resolution pixel array C cols x R rows CCD analog resample
  13. 13. Robert CollinsCSE486, Penn State Effective Scales: sx and sy 1 X 1 Y u s f  ox v  f  oy x Z sy Z Note, since we have different scale factors in x and y, we don’t necessarily have square pixels! Aspect ratio is sy / sxO.Camps, PSU
  14. 14. Robert CollinsCSE486, Penn State Perspective projection matrix Adding the intrinsic parameters into the perspective projection matrix: X   x  f / s x 0 ox 0    y    0 f / sy oy Y  0    Z   z   0    0 1 0    1   To verify: x’ u 1 X 1Y z’ u s f  ox v  f  oy y’ x Z sy Z v z’O.Camps, PSU
  15. 15. Robert CollinsCSE486, Penn State Note: Sometimes, the image and the camera coordinate systems have opposite orientations: [the book does it this way] X X  f  ( u  o x ) s x  x  f / s x 0  ox 0   Z  y    0  f / sy  oy Y  0 Y    Z  f  ( v  o y )s y  z  0    0 1 0    1 Z  
  16. 16. Robert CollinsCSE486, Penn State Note 2 In general, I like to think of the conversion as a separate 2D affine transformation from film coords (x,y) to pixel coordinates (u,v): X  u’  13 a11 a12 xa f 0 0 0   v’  ya   0 a21 a22 23 f Y  0 0 w’    Z  0 0 z1    0   0 1 0    1   Maff Mproj u = Mint PC = Maff Mproj PC
  17. 17. Robert CollinsCSE486, Penn State Huh? Did he just say it was “a fine” transformation? No, it was “affine” transformation, a type of 2D to 2D mapping defined by 6 parameters. More on this in a moment...
  18. 18. Robert CollinsCSE486, Penn State Summary : Forward Projection World Camera Film Pixel Coords Coords Coords Coords U X x u V Mext Y Mproj Maff y v W Z U Mext X Mint u V Y v W Z U u M V m11 m12 m13 m14 v W m21 m22 m23 m24 m31 m31 m33 m34
  19. 19. Robert CollinsCSE486, Penn State Summary: Projection Equation Film plane Perspective World to camera to pixels projection Maff Mproj Mext Mint M
  20. 20. Robert CollinsCSE486, Penn State Lecture 13/14: Intro to Image Mappings
  21. 21. Robert CollinsCSE486, Penn State Image Mappings Overviewfrom R.Szeliski
  22. 22. Robert CollinsCSE486, Penn State Geometric Image Mappings Geometric image transformation transformed image (x,y) (x’,y’) x’ = f(x, y, {parameters}) y’ = g(x, y, {parameters})
  23. 23. Robert CollinsCSE486, Penn State Linear Transformations (Can be written as matrices) Geometric image transformation transformed image (x,y) (x’,y’) x’ x y’ = M(params) y 1 1
  24. 24. Robert CollinsCSE486, Penn State Translation y y’ transform 1 ty 0 0 1 x tx x’ equations matrix form
  25. 25. Robert CollinsCSE486, Penn State Scale y y’ transform S 1 0 0 0 1 x 0 S x’ equations matrix form
  26. 26. Robert CollinsCSE486, Penn State Rotation y y’ transform 1  ) 0 0 1 x x’ equations matrix form
  27. 27. Robert CollinsCSE486, Penn State Euclidean (Rigid) y y’ transform  ) 1 ty 0 0 1 x tx x’ equations matrix form
  28. 28. Robert CollinsCSE486, Penn State Partitioned Matriceshttp://planetmath.org/encyclopedia/PartitionedMatrix.html
  29. 29. Robert CollinsCSE486, Penn State Partitioned Matrices 2x1 2x2 2x1 2x1 1x1 1x2 1x1 1x1 matrix form equation form
  30. 30. Robert Collins Another Example (from last time)CSE486, Penn State X r11 r12 r13 tx U Y r21 r22 r23 ty V Z r31 r32 r33 tz W 1 0 0 0 1 1 3x1 3x3 3x1 3x1 PC R T PW 1x1 = 1x3 1x1 1x1 1 0 1 1 PC = R P W + T
  31. 31. Robert CollinsCSE486, Penn State Similarity (scaled Euclidean) y y’ S transform  ) 1 ty 0 0 1 x tx x’ equations matrix form
  32. 32. Robert CollinsCSE486, Penn State Affine y y’ transform 1 0 0 1 x x’ equations matrix form
  33. 33. Robert CollinsCSE486, Penn State Projective y y’ transform 1 0 0 1 x x’ Note! equations matrix form
  34. 34. Robert Collins Summary of 2D TransformationsCSE486, Penn State
  35. 35. Robert Collins Summary of 2D TransformationsCSE486, Penn State Euclidean
  36. 36. Robert Collins Summary of 2D TransformationsCSE486, Penn State Similarity
  37. 37. Robert Collins Summary of 2D TransformationsCSE486, Penn State Affine
  38. 38. Robert Collins Summary of 2D TransformationsCSE486, Penn State Projective
  39. 39. Robert CollinsCSE486, Penn State Summary of 2D Transformationsfrom R.Szeliski

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