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