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Pairwise alignment

  • Procrustes Algorithm [Golub & VanLoan]

  • Given two sets of matching points, compute R

pi’ = R pi

with 3D rays

p i = N ( x i , y i , z i ) = N ( u i - u c , v i - v c , f ) A = Σ i p i p i T = Σ i p i p i T R T = U S V T = ( U S U R = V UT T ) R T V T = U T R T

Do we have to project onto a plane?

mosaic PP

Cylindrical projection

Y Z

X

unit cylinder

  • Map 3D point (X,Y,Z) onto cylinder

  • Convert to cylindrical coordinates

  • Convert to cylindrical image

coordinates

unwrapped cylinder

cylindrical image

Rotation about vertical axis

  • What if our camera rotates on a tripod?

  • What’s the structure of H?

Full Panoramas

  • What if you want a 360° field of view?

mosaic Projection Cylinder

Cylindrical Projection

Y

X

8

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