Recall: Imaging Geometry. Lecture 13: Camera Projection II. Imaging Geometry. Imaging Geometry. Imaging Geometry. Imaging Geometry

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1 Recall: Iaging Geoetr Lectre 3: Caera Projection II Reading: T& Section 2.4 Object o Interest in orld Coordinate Sste (,,) Iaging Geoetr Iaging Geoetr Caera Coordinate Sste (,,). is optic ais Iage plane located nits ot along optic ais is called ocal length Forward Projection onto iage plane. 3D (,,) projected to 2D (,) Iaging Geoetr Iaging Geoetr Caera Coordinates Iage (il) Coordinates orld Coordinates Or iage gets digitized into piel coordinates (,) Piel Coordinates

2 Forward Projection Intrinsic Caera Paraeters orld Caera Fil Piel orld Caera Fil Piel e want a atheatical odel to describe how 3D orld points get projected into 2D Piel coordinates. Aine Transoration Or goal: describe this seqence o transorations b a big atri eqation! Intrinsic paraeters Intrinsic paraeters (osets) Describes coordinate transoration between il coordinates (projected iage) and piel arra Fil caeras: scanning/digitization CCD caeras: grid o photosensors il plane (projected iage) o o (,) piel arra (,) (col) (row) still in T& section 2.4 o o o and o called iage center or principle point il plane Intrinsic paraeters soeties one or ore coordinate aes are lipped (e.g. T& section 2.4) o o o (,) (,) (row) piel arra (col) o Intrinsic paraeters (scales) sapling deterines how an rows/cols in the iage il CCD scanning resoltion analog resaple piel arra C cols R rows 2

3 O.Caps, PS Eectie Scales: s and s o s s o Note, since we hae dierent scale actors in and, we don t necessaril hae sqare piels! Aspect ratio is s / s O.Caps, PS Perspectie projection atri Adding the intrinsic paraeters into the perspectie projection atri: To eri: ' / s ' z' z z / s s o o o s o Note: Soeties, the iage and the caera coordinate sstes hae opposite orientations: [the book does it this wa] ( o ) s ( o ) s ' / s ' z' / s o o Note 2 In general, I like to think o the conersion as a separate 2D aine transoration ro il coords (,) to piel coordinates (,): w a' 3 ' z' a a 2 a 2 a 22 a 23 M a M proj = M int P C = M a M proj P C Hh? Sar : Forward Projection Did he jst sa it was a ine transoration? No, it was aine transoration, a tpe o 2D to 2D apping deined b 6 paraeters. orld Caera Fil M et M proj M a Piel More on this in a oent... M et M int M

4 Sar: Projection Eqation Fil plane to piels Perspectie projection orld to caera Lectre 3/4: Intro to Iage Mappings M a M proj M et M int M Iage Mappings Oeriew iage Geoetric Iage Mappings Geoetric transoration transored iage (,) (, ) = (,, {paraeters}) = g(,, {paraeters}) ro R.Szeliski iage Linear Transorations (Can be written as atrices) Geoetric transoration transored iage Translation transor (,) (, ) t t = M(paras) eqations atri or 4

5 Scale Rotation transor S transor ) S eqations atri or eqations atri or Eclidean (Rigid) Partitioned Matrices transor ) t t eqations atri or Partitioned Matrices CSE486, Penn Another State Eaple (ro last tie) r 2 r 3 r 2 r 23 r 22 r r 3 r 32 r 33 t t t z 2 atri or 3 P C P 3 R T = eqation or P C = R P + T 5

6 Siilarit (scaled Eclidean) Aine transor ) S transor t t eqations atri or eqations atri or Projectie Sar o 2D Transorations transor Note! eqations atri or Sar o 2D Transorations Sar o 2D Transorations Eclidean Siilarit 6

7 Sar o 2D Transorations Sar o 2D Transorations Aine Projectie Sar o 2D Transorations ro R.Szeliski 7