Contents

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• Lecture 4
• Contents
• Numerical shape descriptors
• Simple scalar descriptors
• Convex hull
• Moment based descriptors

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Eccentricity A/B

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Elongatedness A/B area/(2d)2 where d is the
number of erosions necessary to remove the
structure Rectangularity area/(AB) Triangularit
y could also be defined

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Compactness (region_border_length)2/area 16
for a square 4p for a circle Compactness in
digital images depends on whether the border
length is the outer or inner boundary.

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Convex hull

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Disussion so far Eccentricity, elongatedness,
rectangularity, convex hull all depend on a few
pixels near the decisive border sections. This
means poor noise tolerance. Compactness depend
on the definition of region_border_length.
Compactness is not rotation invariant due to
aliasing effects

• These drawbacks are reduced in case of
• subpixel definition of border points
• use of regional descriptors cartesian moments

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f has 4 solutions between 0 and 2p. Reduction to
two solutions between 0 and 2p can be achieved by
solving the equations cos2f
A( m2,0, - m0,2 )
sin2f A m1,1 An algorithm giving one angle
between -p/2 and p/2 f ½ atan2(
m1,1, m2,0 -m 0,2 )
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• f has 4 solutions between 0 and 2p. Reduction to
  two solutions between 0 and 2p can be achieved by
  solving the equations
• cos2f A( m2,0 - m0,2 )
• sin2f A m1,1
• In code giving an angle between -p/2 and p/2
• Phi 0.5atan2(mStar11,mStar20-mStar02)
• This gives the angle between the x-axis and the
  elongated principal axis. The remaining 180
  degree ambiguity can only be removed using higher
  order rotated moments (defined later), e.g. h1,2
  (f ) lt 0.

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Exercise Prove that h1,1 (f ) 0 if
tan2f m1,1/(m2,0 -m0,2)
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Rotation invariants

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Noise tolerance of rotation invariants
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The weakness of central moments Low sensitivity
near the centre-of-mass. Remedy Weighted moments
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Central second order moments as fast detectors
for co-linearity
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Project related to moments Using graytone
interpolation, one can derive a subpixel-defined
region border. One can assign an arc length Ds to
each border pixel. Compare the two types of
descriptors 1. Central moments over a
region 2. Border moments defined as mbi,j
S (x - xo)i (y - yo) jDs xo S x Ds /
SDs yo S y Ds /SDs Are they equally
suitable for generation rotation
invariant descriptors? Note If subpixel borders
are available, then the border moments are much
faster to calculate than region moments.