Moments - University at Buffalo
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Transcript Moments - University at Buffalo
Moments
describe the image content (or distribution) with
respect to its axes
M p,q i( x, y).x y dxdy
p
q
D
M 0, 0
M
0,1
area of the object
, M1,0
center of mass
Centralized Moments
Moments are not invariant geometric transformations
To achieve invariance under translation
M
c
p ,q
i ( x, y ).(x x) ( y y ) dxdy
p
D
x
M 1,0
M 0, 0
,y
M 0,1
M 0, 0
q
Hu Moments
Hu described a set of 6 moments that are rotation, scaling, translation
invariant
Hu Moments(contd.)
In addition he described a 7th invariant that is
skew invariant
Other invariants are
Legendre Moments
Complex Zernike Moments
Image Reconstruction
Unless we have all Nmax moments, the image cannot be
reconstructed.
The top order moments are good approximations of the images
0-8
2-12
Hough Transform
Procedure to find occurrences of a shape”in
an image
Assumes the “shape” can be described in
some parametric form
Points in image correspond to a family of
parametric solutions
A voting scheme is used to determine the
correct parameters
Accumulator Space
A line in the cartesian space is a point in the
hough space
Create an accumulator whose axis are the
parameters
Set all values to zero
We “discretize” the parameter space
Parameter are quantized to fit into the finite p-space
For each edge point, votes for appropriate
parameters in the accumulator
Increment this value in the accumulator
Line Detection
all possible lines going through P
Parametric form y = mx + c
r x cos y sin
Line Detection (contd.)
Line Detection (example)
Circle Detection
Consider a 2D circle
It can be parameterized as:
r 2 = (x-a) 2 + (y-b)2
Assume an image point was part of a
circle, it could belong to a unique family of
circles with varying parameters:
a, b, r