Geometric Image Transformations and Image Morphing

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Geometric Image Transformations and Image Morphing
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Geometric Transforms Euclidean Transforms
y
y
T
x
x
Translation Rotation
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Mathematical representation of Euclidean Transform
Translation (T)
Rotation (R)
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• Properties of Euclidean Transforms
• Preserve distances
• Preserve angles

Homogeneous form
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Geometric Transforms Affine Transforms
y
y
T
x
x
Euclidean Transform reflection, scaling,
shearing
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Example affine transform
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Mathematical representation
6 parameters
Affine properties Preserve parallel lines
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Geometric Transforms Projective Transforms
Projection the mapping of points from N-D space
to M-D subspace. (MltN)
There are only 8 free-parameters. Why?
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Mathematical representation
Which can be expressed as the following rational
linear equation
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Relation of the geometric transforms
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Application

• Given the input image I and geometric transform
  T, how to generate the output image?
• Given the input image I and geometrically
  transformed image I , how to calculate the
  transform T?
• Given the geometrically transformed image I and
  the transform T, how to restore the original
  image?

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Geometrically transform image ( with
Interpolation techniques)
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How to estimate the geometric transform
parameters?
Known input image, output image
Unknown Geometric transform T
For affine transform, how may pairs of
corresponding pixels do we need to determine the
geometric transform?
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Affine transform has 6 parameters. 3 pair of
points gt 6 equations.
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How about projective transform? 8 parameters
need 4 pairs of corresponding pixels.
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Image Warping
Image warping using control points (tiepoints)
Geometric distorted image using NN interpolation
and restored result.
Geometric distorted image using bilinear
interpolation and restored result.
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