Contents

1

• Contents
• Numerical shape descriptors, other than moments
  (SHB 6.3.1)
• Hough transformations (SHB 5.2.6)
• Texture analysis (SHB 14, 14.1-2)

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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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Finding straght lines Hough methods for straight
lines or edges Principle A straight line in real
space corresponds to a point in Hough space
(a-b-space or f-r-space) Features in real space
cast votes in Hough space


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Hough methods 1. Conventional Hough method 2.
Randomised Hough method 3. Gradient Hough
methods ___________________________________ Conve
ntional A. In case of edges detect edges B.
Perform thresholding giving a binary bitmap C.
For each set pixel, increment the Hough space
along relevant line or curve (a straight line
in a-b-space, a sin/cos-curve in f-r-space) D.
Find peaks in Hough space, each peak to be
identified as a straight line or edge in real
space

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Real space Hough space

When defining r and f, it is best to let the
origin be the centre of the image. Signed r , 0
lt f lt p

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Randomised Hough method for edges or lines A.
In case of edges detect edges B. Perform
thresholding giving a binary bitmap C. Repeat
Pick at random two set pixels
Increment in Hough space at the point
corresponding to a line through the two pixels D.
Apply a suitable stop criteria E. Find peaks in
Hough space, each peak to be identified as
a straight line or edge in real space

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Gradient Hough method for straight edges

• Speed optimisation is achieved by
• using a 32 bit integer real space image
  utilising the full dynamic range
• using a 32 bit Hough space
• quantising the gradient magnitude strongly
• argtan calls are cancelled for pixels with zero
  gradient after quantisation

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Finding circles of arbitrary radius The Hough
space is 3-dimensional with coordinates (xc, yc,
r) where (xc, yc) is the circle center and r is
the radius. (x-xc)2 (y-yc)2
r2 is the formula for an arbitrary circle in x-y
space It is a formula for a circular cone in
xc-yc-r-space Finding arbitrary ellipses The
Hough space is 5-dimensional with coordinates
(xc, yc, a, b,q) where (xc, yc) is the ellipse
center, and a,b are length of principal axes,
and q is the rotation angle ((x-xc)cos q
(y-yc)sin q)/a)2 ((y- yc)cos q - (x-xc)sin
q)/b)2 r2 The hypersurface in 5-dimensional
xc yc,a,b,q-space using x and y as parameters has
no special name


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Gradient Hough for finding centres of circular
edges or segments of circular edges wiht
approximate a priori knowledge about radius The
Hough space has the same dimension and units as
the real space A. Copy the image to a float image
(floating point grey tone) B. Blur the image C.
For each pixel Find gradient,
Increment in Hough space along a straight line
through the pixel with a slope defined by
the gradient. Use a priory knowledge on the
range of the circle radius and sign of
gradient D. Find peaks in Hough space to be
identified as centres of circles or circle
segments

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• Ridge detector
• Assume the Taylor expansion of the graytone
• F(x,y) a0 ax (x-x0) ay (y-y0)
• axx (x-x0)2 ayy (y-y0)2 axy (x-x0)(y-y0)
• Estimates of the 6 parameters can be obtained by
  using the greytone at (x0, y0) and 8 surrounding
  pixels.
• Gradient sqrt(ax2 ay2)
• B. Eigen values l1 , l2 with l1 lt l2 of the
  matrix
• axx axy
• axy ayy
• describe the second order behaviour. Downward
• curved greytone landscape has negative l
  values.
• A ridge has low gradient, negative value of l2 ,
  and
• l1ltlt l2

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Texture analysis
Can we classify regions wringles (forhead and
cheeks), hair, dress, uniform background?

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Statistical approach to texture
analysis Co-occurrence matrix Matrix
M0..cmax-10..cmax-1 of floating point
elements Graytones c(x,y) 0 .. cmax-1 Input
is the considered N N subimage and a
displacement vector D connecting a pixel pair
under consideration Pseudo code Set all elements
of M to zero Repeat for all pixel positions P1
P2 P1D Increment matrix element
Mc(P1) c(P2) by (1/N2) Usually cmax is
between 4 and 16, and N 32-64. Each
displacement vector D defines a co-occurrence
matrix. Usually D is (1,1), (1,0) or analogous
vectors


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• Numerical texture descriptors
• M is normalized, that
• Si Sj Mij 1
• 1) Maximum probability max(Mij)
• 2) Element difference moment of order k
• Si Sj (i-j)k Mij
• 3) Inverse element difference moment of order k
• Si Sj Mij/(i-j)k
• 4) Uniformity Si Sj Mij2
• 5) Entropy -Si Sj Mij log(Mij)

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Spectral approach to texture analysis Let C(kx,
ky) be the Fourier transform of c(x,y) using a N
x N subimage (N is a power of 2 so that FFT can
be used) kx and ky is in the range -1/(2N). .
1/(2N) Let A(ik) be average amplitudes of C
along rings of radius k sqrt(kx2ky2) (ik)Dk
and width Dk, where ik is an integer. Radial
moments of order q mq S ik A(ik)ikq are good
descriptors for texture analysis. C(0,0) is the
average graytone, so mq/C(0,0) is invariant to
scaling of the greytone.