Moments area of the object center of mass  describe the image content (or distribution) with respect to its axes

Centralized Moments  Moments are not invariant geometric transformations  To achieve invariance under translation

Hu Moments Hu described a set of 6 moments that are rotation, scaling, translation invariant

Hu Moments(contd.) In addition he described a 7 th invariant that is skew invariant Other invariants are  Legendre Moments  Complex Zernike Moments

Image Reconstruction Unless we have all N max moments, the image cannot be reconstructed. The top order moments are good approximations of the images

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

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