Projective Transformations for Image Transition Animations

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Projective Transformations for Image Transition
Animations

• Planar surfaces are related
• by projective transformation

TzuYen Wong, Peter Kovesi Amitava Datta School
of Computer Science Software Engineering The
University of Western Australia
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Objective 1

• Inputs 2D images quadrilateral pairs
• Output smooth transition
• Novelty perspective correctness
• Approach decomposing homography
• Issue Normalisation

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Objective 2

• Inputs 2D shapes multiple quadrilateral pairs
• Output smooth transition
• Novelty good shape interpolation with few
  primitives
• Approach force equilibrium scheme for alignment

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Motivation

• Image transition animations usually use affine
  transformations on triangle patches. (6 DOF)
• Eg. image morphing, shape interpolation

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Motivation

• Image transition animations usually use affine
  transformations on triangle patches. (6 DOF)
• Eg. image morphing, shape interpolation
• Perspective effect is not modelled (requires 8
  DOF)
• Approximated results
• large number of triangles

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Motivation

• Image transition animations usually use affine
  transformations on triangle patches. (6 DOF)
• Eg. image morphing, shape interpolation
• Perspective effect is not modelled (requires 8
  DOF)
• Approximated results
• large number of triangles
• Propose projective transformations (8 DOF) on
  quadrilateral image patches as a solution

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Previous work Affine Transformation

• Matrix animation and polar decomposition
  (Shoemake 1992)
• Shortcomings of direct interpolations of vertices
  or matrix elements shrinking, unpredictable
• Polar decomposition
• Affine matrix rotation scaling translation
• Interpolate the angle of rotation produces good
  result

K. Shoemake and T. Duff. Matrix animation and
polar decomposition. In Proceedings of the
conference on Graphics interface 92, pages
258264, San Francisco, CA, USA, 1992. Morgan
Kaufmann Publishers Inc.
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Previous work Affine Transformation

• Matrix animation and polar decomposition
  (Shoemake 1992)
• Interpolation using
• Vertex positions
• Matrix elements
• rotation angle
• scaling translation

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Previous work Affine Transformation

• As-rigid-as-possible shape interpolation
  (Alexa 2000)
• Close form solution for least distortion affine
  interpolation of multiple triangles
• Triangles disconnect from each other in rigid
  interpolation
• Find the vertices of connected triangles which
  have transformation matrices that are closest to
  the transformation matrices of the vertices of
  disconnected rigid interpolation

M. Alexa, D. Cohen-Or, and D. Levin.
As-rigid-as-possible shape interpolation. In K.
Akeley, editor, Siggraph 2000, Computer Graphics
Proceedings, pages 157164. ACM
Press/Addison-Wesley Publishing Co., 2000.
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Previous work Affine Transformation

• As-rigid-as-possible shape interpolation
  (Alexa 2000)
• Close form solution for least distortion affine
  interpolation of multiple triangles
• Quality of the transition strongly depends on
  avoiding having irregular triangles through
  complex isomorphic dissections.

M. Alexa, D. Cohen-Or, and D. Levin.
As-rigid-as-possible shape interpolation. In K.
Akeley, editor, Siggraph 2000, Computer Graphics
Proceedings, pages 157164. ACM
Press/Addison-Wesley Publishing Co., 2000.
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Previous work Projective Transformation

• Other projectively correct interpolation
• 2D View morphing (Seitz 1996, Xiao 2004)

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Previous work Projective Transformation

• Other projectively correct interpolation
• 2D View morphing (Seitz 1996, Xiao 2004)
• Prewarp for parallel view interpolation
• Postwarp is manual or with affine approximation

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Previous work Projective Transformation

• Other projectively correct interpolation
• 2D View morphing (Seitz 1996, Xiao 2004)
• Prewarp for parallel view interpolation
• Postwarp is manual or with affine approximation
• 3D graphics
• Interpolate planar surfaces in 3D
• Required 3D information

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Method overview

• Source quadrilateral Target quadrilateral
• Decomposition of projective matrix, H
  (homography)
• perspective rotation scaling translation
• Last row of homography vanishing line vector

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Projective matrix decomposition

• Normalised Homography decomposed into
• Perspective
• Axis alignment
• Scaling
• Rotation
• Translation

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Projective matrix decomposition

• Normalised Homography decomposed into
• Perspective
• Axis alignment
• Scaling
• Rotation
• Translation

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Projective matrix decomposition

• Normalised Homography decomposed into
• Perspective
• Axis alignment
• Scaling
• Rotation
• Translation
• Singular Value Decomposition (SVD)

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Projective matrix Interpolation

• Interpolation with Identity matrix
• Perspective - Interpolation of vanishing line
  vector
• Find the axis of rotation and interpolate angle

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Projective matrix Interpolation

• Interpolation with Identity matrix
• Perspective - Interpolation of vanishing line
  vector
• Find the axis of rotation and interpolate angle
• Rotation - Interpolation of angle (q)
• Scaling Translation - linear interpolation
• Sa (1-a)I aS
• Axis alignment - no interpolation

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Normalisation issues

• Vanishing line normalisation
• Arbitrary scaled homography hH
• Set , unit magnitude vanishing line
• Avoid having two scaling interpolations as the
  combination is non-linear and creates visual
  artefact

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Normalisation issues

• Vanishing line normalisation
• Arbitrary scaled homography hH
• Set , unit magnitude vanishing line
• Avoid having two scaling interpolations as the
  combination is non-linear and creates visual
  artefact
• Example
• Scaling of P 1.0 ? 2.0
• Scaling of S 1.0 ? 0.5
• Combination 1.0 ? 1.0

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Normalisation issues

• Vanishing line normalisation
• Arbitrary scaled homography hH
• Set , unit magnitude vanishing line
• Avoid having two scaling interpolations as the
  combination is non-linear and creates visual
  artefact
• Example
• Scaling of P 1.0 ? 1.2 ? 1.4 ? 1.6 ?
  1.8 ? 2.0
• Scaling of S 1.0 ? 0.9 ? 0.8 ? 0.7 ?
  0.6 ? 0.5
• Combination 1.0 ? 1.0

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Normalisation issues

• Vanishing line normalisation
• Arbitrary scaled homography hH
• Set , unit magnitude vanishing line
• Avoid having two scaling interpolations as the
  combination is non-linear and creates visual
  artefact
• Example
• Scaling of P 1.0 ? 1.2 ? 1.4 ? 1.6 ?
  1.8 ? 2.0
• Scaling of S 1.0 ? 0.9 ? 0.8 ? 0.7 ?
  0.6 ? 0.5
• Combination 1.0 ? 1.08 ? 1.12 ? 1.12 ? 1.08 ?
  1.0
• 1 gt1 1

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Normalisation issues

• Translation and scale of source quadrilateral
  affect the vanishing line vector
• Translation normalisation of source quadrilateral
  
• The source points are translated to be centered
  at the origin before computing the homography
• Scale normalisation of source quadrilateral
• Only excessive scaling creates undesirable
  results

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Normalisation issues

• Examples
• (a) Perspectively correct transition
• (b) Source translated away from origin (75 of
  object size)
• (c) Source scaled down (coordinate magnitudes
  0.1)
• (d) Vanishing line vector with magnitude 10.

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Shape interpolation with multiple quadrilaterals

• Replace affine interpolation of triangles
• physical correctness of projective transformation
  
• smaller number of primitives
• Disconnected boundaries among vertices of
  different intermediate quadrilaterals
• Force-equilibrium scheme to enforce boundary
  connectivity

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Shape interpolation with multiple quadrilaterals

• Force on each quad

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Shape interpolation with multiple quadrilaterals

• Force on each quad
• At force equilibrium,
• all Fi 0
• With Nq equations,
• C Teq Fp (close form solution)

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Results
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Results

• Non-rigid object shape interpolation
• a) Projective interpolation with quadrilaterals
• b) Alexas as-rigid-as-possible affine
  interpolation with triangles.

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Results

• Non-rigid object shape interpolation
• a) Projective interpolation with quadrilaterals
• b) Alexas as-rigid-as-possible affine
  interpolation with triangles.

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Results

• Rigid object shape interpolation
• a) Projective interpolation with quadrilaterals
• b) Alexas as-rigid-as-possible affine
  interpolation with triangles.

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Results

• Projective interpolation applied on a real image
• 5 pairs of separate homographies for 5 planar
  surfaces

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Results

• Projective interpolation applied on a real image
• 5 pairs of separate homographies for 5 planar
  surfaces

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Results

• Projective shape interpolation
• da Vincis Vitruvian Man to Coyote

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Results

• Projective shape interpolation
• da Vincis Vitruvian Man to Coyote

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Conclusions

• Enable perspectively correct interpolation of
  image patches, without 3D information.
• Improve interpolation results by normalisation
  of homography and source points position
• Enable quadrilaterals to be used as primitives
  for shape interpolation and morphing applications.

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Future Work

• Optimisation of the source position for optimal
  interpolation
• Exploration of applications of this technique
  besides shape interpolation and image morphing.

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Projective Transformations for Image Transition
Animations
Enable perspectively correct interpolation of
image patches, without 3D information Improve
interpolation results by normalisation of
homography and source points position Enable
quadrilaterals to be used as primitives for shape
interpolation and morphing applications
TzuYen Wong, Peter Kovesi Amitava Datta