3D Geometry for Computer Graphics

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3D Geometry forComputer Graphics

• Aligning Shapes

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Notations

• Denote our data points by x1, x2, , xn ? Rd

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The origin of the new axes

• The origin is zero-order approximation of our
  data set (a point)
• It will be the center of mass
• It can be shown that
• You should move all the shapes so their center of
  mass (m) will be at the origin

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Scatter matrix

• Denote yi xi m, i 1, 2, , n
• where Y is d?n matrix with yk as columns (k 1,
  2, , n)
• S is the result of multiplying the input points
  coordinates with themselves

Y
YT
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Scatter matrix - eigendecomposition

• S is symmetric
• S has eigendecomposition S V?VT
• this decomposition can be achived by using the
  function SVD on S.

S
v1
?1
?2
v2

v2
v1
vd
?d
vd
The eigenvectors form orthogonal basis
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Principal components

• Eigenvectors that correspond to big eigenvalues
  are the directions in which the data has strong
  components ( large variance).
• If the eigenvalues are more or less the same
  there is no preferable direction.
• Note the eigenvalues are always non-negative.
  Think why

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Principal components

• Theres no preferable direction
• S looks like this
• Any vector is an eigenvector

• There is a clear preferable direction
• S looks like this
• ? is close to zero, much smaller than ?.

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How to use what we got

• For finding the principal axis of a shape, we
  simply compute the axes defined by the
  eigenvectors of S. The origin is at the mean
  point m.
• In order to align all the shape the same way, you
  should rotate the shape such that the bigger axis
  (the one with the bigger eigenvalue) is aligned
  with the X axis and the the other one with the Y
  axis. Keep in mine to align them such that the
  positive value will agree with the positive
  directions of the original axes.

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Scale Normalization

• Scale normalization can be achived by scaling
  each shape so that their biggest eigen value will
  be equal to 1 (i.e. scale by 1/E where is is the
  largest eighevalue).