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Robert Collins
CSE486, Penn State




                         Lecture 16:
                     Planar Homographies
Robert Collins
CSE486, Penn State         Motivation:
                     Points on Planar Surface

                        y

                            x
Robert Collins
CSE486, Penn State
                     Review : Forward Projection
       World                Camera             Film            Pixel
       Coords               Coords             Coords          Coords
         U                    X                  x                u
         V           Mext     Y      Mproj              Maff
                                                 y                v
         W                    Z

          U                   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
Robert Collins

            World to Camera Transformation
CSE486, Penn State



                                   PC PW
                                              W

                     Y

                 X
                                      U
         R                Z                         V
                     C
     Rotate to                   Translate by - C
     align axes                  (align origins)

                         P C = R ( PW - C )
                             = R PW + T
Robert Collins
CSE486, Penn State

                     Perspective Matrix Equation
       X                       (Camera Coordinates)
  x= f
       Z
  y= f
       Y                                           X 
       Z               x'  f        0      0 0  
                       y ' =  0     f         Y 
                                              0 0
                                                Z 
                       z'  0
                                    0      1 0  
                                                  1
                                                    
                               p = M int ⋅ PC
Robert Collins
CSE486, Penn State
                         Film to Pixel Coords
            2D affine transformation from film
            coords (x,y) to pixel coordinates (u,v):
                                                   X 
        u’           a11 a12 xa
                            ' 13  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
Robert Collins

               Projection of Points on Planar Surface
CSE486, Penn State



   Perspective
   projection
                             y
                                 Film coordinates
                                  x




                                                    Point
                                                    on plane
                     Rotation +
                     Translation
Robert Collins
CSE486, Penn State
                     Projection of Planar Points
Robert Collins

                Projection of Planar Points (cont)
CSE486, Penn State




                                          Homography H
                                          (planar projective
                                          transformation)
Robert Collins

                Projection of Planar Points (cont)
CSE486, Penn State




                                        Homography H
                                        (planar projective
                                        transformation)




       Punchline: For planar surfaces, 3D to 2D perspective
       projection reduces to a 2D to 2D transformation.

       Punchline2: This transformation is INVERTIBLE!
Robert Collins
CSE486, Penn State
                     Special Case : Frontal Plane
        What if the planar surface is perpendicular to
        the optic axis (Z axis of camera coord system)?

         Then world rotation matrix simplies:
Robert Collins
CSE486, Penn State
                          Frontal Plane

       So the homography for a frontal plane simplies:




                     Similarity Transformation!
Robert Collins
CSE486, Penn State
                     Convert to Pixel Coords
              Internal camera   a11 a12 a13   v   pixels
              parameters        a21 a22 a23
                                0 0 1
                                                       u
Robert Collins

                     Planar Projection Diagram
CSE486, Penn State




  Here’s where
  transformation
  groups get useful!
Robert Collins

                     General Planar Projection
CSE486, Penn State




                                       H1


                                H
                                       A1
Robert Collins

                     General Planar Projection
CSE486, Penn State




                                       H1


                                H-1
                                       A1
Robert Collins

                     Frontal Plane Projection
CSE486, Penn State




                                       S1


                               A
                                       A1
Robert Collins

                     Frontal Plane Projection
CSE486, Penn State




                                       S1


                               A-1
                                       A1
Robert Collins

                     General Planar Projection
CSE486, Penn State




                           H1       H2




                                H
Robert Collins
CSE486, Penn State
                     Summary: Planar Projection
                                                      Pixel coords
                     Internal                     u
                     params

 Perspective                               v
 projection
                            y
                                          Homography
                                x




                                                      Point on plane
                         Rotation + Translation
Robert Collins
              Applying Homographies to Remove
CSE486, Penn State

                    Perspective Distortion




                                                           from Hartley & Zisserman


                     4 point correspondences suffice for
                         the planar building facade
Robert Collins
CSE486, Penn State
                     Homographies for
                     Bird’s-eye Views




                                        from Hartley & Zisserman
Robert Collins
CSE486, Penn State
                     Homographies for Mosaicing




                                                  from Hartley & Zisserman
Robert Collins
CSE486, Penn State
                       Two Practical Issues

              How to estimate the homography given
                four or more point correspondences
                (will derive L.S. solution now)


              How to (un)warp image pixel values to
                produce a new picture (last class)
Robert Collins
CSE486, Penn State
                     Estimating a Homography
          Matrix Form:




          Equations:
Robert Collins
CSE486, Penn State
                     Degrees of Freedom?



      There are 9 numbers h11,…,h33 , so are there 9 DOF?
      No. Note that we can multiply all hij by nonzero k
      without changing the equations:
Robert Collins
CSE486, Penn State
                      Enforcing 8 DOF
   One approach: Set h33 = 1.




   Second approach: Impose unit vector constraint




     Subject to the
     constraint:
Robert Collins
CSE486, Penn State
                     L.S. using Algebraic Distance
         Setting h33 = 1



         Multiplying through by denominator



         Rearrange
Robert Collins
CSE486, Penn State
                     Algebraic Distance, h33=1 (cont)
                             2N x 8        8x1   2N x 1
     Point 1

     Point 2

     Point 3

     Point 4

 additional
 points
Robert Collins
CSE486, Penn State
                     Algebraic Distance, h33=1 (cont)
         Linear                  2Nx8     8x1          2Nx1
         equations               A h              =     b

                          8x2N     2Nx8    8x1         8x2N         2Nx1
        Solve:
                          AT A h                  = AT               b
                             8x8          8x1                 8x1

                         (AT A) h                 = (AT b)
                                                  -1
                         h       =   (AT         A) (AT b)

         Matlab:          h = Ab
Robert Collins
CSE486, Penn State
                     Caution: Numeric Conditioning

   R.Hartley: “In Defense of the Eight Point Algorithm”

   Observation: Linear estimation of projective transformation
      parameters from point correspondences often suffer from poor
      “conditioning” of the matrices involves. This means the solution
      is sensitive to noise in the points (even if there are no outliers).

   To get better answers, precondition the matrices by performing
   a normalization of each point set by:
   • translating center of mass to the origin
   • scaling so that average distance of points from origin is sqrt(2).
   • do this normalization to each point set independently
Robert Collins
CSE486, Penn State
                     Hartley’s PreConditioning
  PointSet1          H = T2-1 S2H?Hnorm S1 T1
                                -1
                                                    PointSet2


                              Translate
            T1                center of mass        T2
                              to origin




                            Scale so average
            S1              point dist is sqrt(2)   S2


                          Estimate Hnorm
Robert Collins
CSE486, Penn State
                     A More General Approach

         What might be wrong with setting h33 = 1?


         If h33 actually = 0, we can’t get the right answer.
Robert Collins
CSE486, Penn State
                     Algebraic Distance, ||h||=1
         ||h|| = 1



         Multiplying through by denominator



         Rearrange
                                                   =0
                                                   =0
Robert Collins

                 Algebraic Distance, ||h||=1 (cont)
CSE486, Penn State




                          2N x 9         9x1   2N x 1


    4
     P
     O
     I
     N
     T
     S




 additional
 points
Robert Collins

                 Algebraic Distance, ||h||=1 (cont)
CSE486, Penn State




         Homogeneous                2Nx9   9x1          2Nx1
         equations                  A h            =     0
                     9x2N    2Nx9   9x1          9x2N   2Nx1
  Solve:
                     AT A h                = AT          0
                            9x9     9x1          9x1

                     (AT A) h              = 0
                     SVD of ATA = U               D     UT

                     Let h be the column of U (unit eigenvector)
                     associated with the smallest eigenvalue in D.
                     (if only 4 points, that eigenvalue will be 0)

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Lecture16

  • 1. Robert Collins CSE486, Penn State Lecture 16: Planar Homographies
  • 2. Robert Collins CSE486, Penn State Motivation: Points on Planar Surface y x
  • 3. Robert Collins CSE486, Penn State Review : Forward Projection World Camera Film Pixel Coords Coords Coords Coords U X x u V Mext Y Mproj Maff y v W Z U 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
  • 4. Robert Collins World to Camera Transformation CSE486, Penn State PC PW W Y X U R Z V C Rotate to Translate by - C align axes (align origins) P C = R ( PW - C ) = R PW + T
  • 5. Robert Collins CSE486, Penn State Perspective Matrix Equation X (Camera Coordinates) x= f Z y= f Y X  Z  x'  f 0 0 0    y ' =  0 f Y  0 0    Z   z'  0    0 1 0    1   p = M int ⋅ PC
  • 6. Robert Collins CSE486, Penn State Film to Pixel Coords 2D affine transformation from film coords (x,y) to pixel coordinates (u,v): X  u’ a11 a12 xa  ' 13 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
  • 7. Robert Collins Projection of Points on Planar Surface CSE486, Penn State Perspective projection y Film coordinates x Point on plane Rotation + Translation
  • 8. Robert Collins CSE486, Penn State Projection of Planar Points
  • 9. Robert Collins Projection of Planar Points (cont) CSE486, Penn State Homography H (planar projective transformation)
  • 10. Robert Collins Projection of Planar Points (cont) CSE486, Penn State Homography H (planar projective transformation) Punchline: For planar surfaces, 3D to 2D perspective projection reduces to a 2D to 2D transformation. Punchline2: This transformation is INVERTIBLE!
  • 11. Robert Collins CSE486, Penn State Special Case : Frontal Plane What if the planar surface is perpendicular to the optic axis (Z axis of camera coord system)? Then world rotation matrix simplies:
  • 12. Robert Collins CSE486, Penn State Frontal Plane So the homography for a frontal plane simplies: Similarity Transformation!
  • 13. Robert Collins CSE486, Penn State Convert to Pixel Coords Internal camera a11 a12 a13 v pixels parameters a21 a22 a23 0 0 1 u
  • 14. Robert Collins Planar Projection Diagram CSE486, Penn State Here’s where transformation groups get useful!
  • 15. Robert Collins General Planar Projection CSE486, Penn State H1 H A1
  • 16. Robert Collins General Planar Projection CSE486, Penn State H1 H-1 A1
  • 17. Robert Collins Frontal Plane Projection CSE486, Penn State S1 A A1
  • 18. Robert Collins Frontal Plane Projection CSE486, Penn State S1 A-1 A1
  • 19. Robert Collins General Planar Projection CSE486, Penn State H1 H2 H
  • 20. Robert Collins CSE486, Penn State Summary: Planar Projection Pixel coords Internal u params Perspective v projection y Homography x Point on plane Rotation + Translation
  • 21. Robert Collins Applying Homographies to Remove CSE486, Penn State Perspective Distortion from Hartley & Zisserman 4 point correspondences suffice for the planar building facade
  • 22. Robert Collins CSE486, Penn State Homographies for Bird’s-eye Views from Hartley & Zisserman
  • 23. Robert Collins CSE486, Penn State Homographies for Mosaicing from Hartley & Zisserman
  • 24. Robert Collins CSE486, Penn State Two Practical Issues How to estimate the homography given four or more point correspondences (will derive L.S. solution now) How to (un)warp image pixel values to produce a new picture (last class)
  • 25. Robert Collins CSE486, Penn State Estimating a Homography Matrix Form: Equations:
  • 26. Robert Collins CSE486, Penn State Degrees of Freedom? There are 9 numbers h11,…,h33 , so are there 9 DOF? No. Note that we can multiply all hij by nonzero k without changing the equations:
  • 27. Robert Collins CSE486, Penn State Enforcing 8 DOF One approach: Set h33 = 1. Second approach: Impose unit vector constraint Subject to the constraint:
  • 28. Robert Collins CSE486, Penn State L.S. using Algebraic Distance Setting h33 = 1 Multiplying through by denominator Rearrange
  • 29. Robert Collins CSE486, Penn State Algebraic Distance, h33=1 (cont) 2N x 8 8x1 2N x 1 Point 1 Point 2 Point 3 Point 4 additional points
  • 30. Robert Collins CSE486, Penn State Algebraic Distance, h33=1 (cont) Linear 2Nx8 8x1 2Nx1 equations A h = b 8x2N 2Nx8 8x1 8x2N 2Nx1 Solve: AT A h = AT b 8x8 8x1 8x1 (AT A) h = (AT b) -1 h = (AT A) (AT b) Matlab: h = Ab
  • 31. Robert Collins CSE486, Penn State Caution: Numeric Conditioning R.Hartley: “In Defense of the Eight Point Algorithm” Observation: Linear estimation of projective transformation parameters from point correspondences often suffer from poor “conditioning” of the matrices involves. This means the solution is sensitive to noise in the points (even if there are no outliers). To get better answers, precondition the matrices by performing a normalization of each point set by: • translating center of mass to the origin • scaling so that average distance of points from origin is sqrt(2). • do this normalization to each point set independently
  • 32. Robert Collins CSE486, Penn State Hartley’s PreConditioning PointSet1 H = T2-1 S2H?Hnorm S1 T1 -1 PointSet2 Translate T1 center of mass T2 to origin Scale so average S1 point dist is sqrt(2) S2 Estimate Hnorm
  • 33. Robert Collins CSE486, Penn State A More General Approach What might be wrong with setting h33 = 1? If h33 actually = 0, we can’t get the right answer.
  • 34. Robert Collins CSE486, Penn State Algebraic Distance, ||h||=1 ||h|| = 1 Multiplying through by denominator Rearrange =0 =0
  • 35. Robert Collins Algebraic Distance, ||h||=1 (cont) CSE486, Penn State 2N x 9 9x1 2N x 1 4 P O I N T S additional points
  • 36. Robert Collins Algebraic Distance, ||h||=1 (cont) CSE486, Penn State Homogeneous 2Nx9 9x1 2Nx1 equations A h = 0 9x2N 2Nx9 9x1 9x2N 2Nx1 Solve: AT A h = AT 0 9x9 9x1 9x1 (AT A) h = 0 SVD of ATA = U D UT Let h be the column of U (unit eigenvector) associated with the smallest eigenvalue in D. (if only 4 points, that eigenvalue will be 0)