Moments

dxdyyxyxiM q

D

pqp ).,(,

0,0M area of the object

0,11,0 ,MM center of mass

describe the image content (or distribution) with respect to its axes

Centralized Moments

0,0

1,0

0,0

0,1

,

,

)()).(,(

M

My

M

Mx

dxdyyyxxyxiM qp

D

cqp

Moments are not invariant geometric transformations

To achieve invariance under translation

Hu MomentsHu 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 ReconstructionUnless we have all Nmax moments, the image cannot be reconstructed.The top order moments are good approximations of the images

2-12

0-8

Hough TransformProcedure 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 spaceCreate 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

sincos yxr

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