Jean Ponce (ponce@di.ens.fr) http://www.di.ens.fr/~ponce - - PowerPoint PPT Presentation

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Jean Ponce (ponce@di.ens.fr) http://www.di.ens.fr/~ponce - - PowerPoint PPT Presentation

Aug 10, 2023 12 likes •424 views

Jean Ponce (ponce@di.ens.fr) http://www.di.ens.fr/~ponce Equipe-projet WILLOW ENS/INRIA/CNRS UMR 8548 Laboratoire dInformatique Ecole Normale Suprieure, Paris Outline Motivation: Making mosaics Perspetive and weak perspective

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Jean Ponce (ponce@di.ens.fr) http://www.di.ens.fr/~ponce Equipe-projet WILLOW ENS/INRIA/CNRS UMR 8548 Laboratoire d’Informatique Ecole Normale Supérieure, Paris

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Outline

  • Motivation: Making mosaics
  • Perspetive and weak perspective
  • Coordinates changes
  • Intrinsic and extrensic parameters
  • Affine registration
  • Projective registration
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Feature-based alignment outline

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Feature-based alignment outline

Extract features

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Feature-based alignment outline

Extract features Compute putative matches

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Feature-based alignment outline

Extract features Compute putative matches Loop:

  • Hypothesize transformation T (small group of putative

matches that are related by T)

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Feature-based alignment outline

Extract features Compute putative matches Loop:

  • Hypothesize transformation T (small group of putative

matches that are related by T)

  • Verify transformation (search for other matches consistent

with T)

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Feature-based alignment outline

Extract features Compute putative matches Loop:

  • Hypothesize transformation T (small group of putative

matches that are related by T)

  • Verify transformation (search for other matches consistent

with T)

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2D transformation models

Similarity (translation, scale, rotation) Affine Projective (homography)

Why these transformations ???

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Pinhole perspective equation

NOTE: z is always negative..

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Affine models: Weak perspective projection

is the magnification.

When the scene relief is small compared its distance from the Camera, m can be taken constant: weak perspective projection.

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Affine models: Orthographic projection When the camera is at a (roughly constant) distance from the scene, take m=1.

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Analytical camera geometry

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Coordinate Changes: Pure Translations

OBP = OBOA + OAP , BP = AP + BOA

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Coordinate Changes: Pure Rotations

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Coordinate Changes: Rotations about the z Axis

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A rotation matrix is characterized by the following properties:

  • Its inverse is equal to its transpose, and
  • its determinant is equal to 1.

Or equivalently:

  • Its rows (or columns) form a right-handed
  • rthonormal coordinate system.
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Coordinate changes: pure rotations

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Coordinate Changes: Rigid Transformations

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Pinhole perspective equation

NOTE: z is always negative..

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The intrinsic parameters of a camera Normalized image coordinates Physical image coordinates Units: k,l : pixel/m f : m α,β : pixel

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The intrinsic parameters of a camera Calibration matrix The perspective projection equation

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The extrinsic parameters of a camera

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Perspective projections induce projective transformations between planes

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Weak-perspective projection Paraperspective projection

Affine cameras

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Orthographic projection Parallel projection

More affine cameras

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Weak-perspective projection model

r

(p and P are in homogeneous coordinates)

p = A P + b

(neither p nor P is in hom. coordinates)

p = M P

(P is in homogeneous coordinates)

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Affine projections induce affine transformations from planes

  • nto their images.
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Affine transformations

An affine transformation maps a parallelogram onto another parallelogram

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Fitting an affine transformation

Assume we know the correspondences, how do we get the transformation?

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Fitting an affine transformation

Linear system with six unknowns Each match gives us two linearly independent equations: need at least three to solve for the transformation parameters

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Beyond affine transformations

What is the transformation between two views of a planar surface? What is the transformation between images from two cameras that share the same center?

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Perspective projections induce projective transformations between planes

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Beyond affine transformations

Homography: plane projective transformation (transformation taking a quad to another arbitrary quad)

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Fitting a homography

Recall: homogenenous coordinates

Converting to homogenenous image coordinates Converting from homogenenous image coordinates

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Fitting a homography

Recall: homogenenous coordinates Equation for homography:

Converting to homogenenous image coordinates Converting from homogenenous image coordinates

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Fitting a homography

Equation for homography:

3 equations, only 2 linearly independent 9 entries, 8 degrees of freedom (scale is arbitrary)

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Direct linear transform

H has 8 degrees of freedom (9 parameters, but scale is arbitrary) One match gives us two linearly independent equations Four matches needed for a minimal solution (null space

  • f 8x9 matrix)

More than four: homogeneous least squares

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Application: Panorama stitching

Images courtesy of A. Zisserman.

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