Mellin polar coordinate moment and its affine invariance

The moment-based method is a fundamental approach to the extraction of affine invariants. However, only integer-order traditional moments can be used to construct affine invariants. No invariants can be constructed by moments with an order lower than 2. Consequently, the obtained invariants are sensitive to noise. In this paper, the moment order is generalized from integer to non-integer. However, the moment order cannot simply be generalized from integer to non-integer to achieve affine invariance. The difficulty of this generalization lies in the fact that the angular factor owing to shearing in the affine transform can hardly be eliminated for non-integer order moments. In order to address this problem, the Mellin polar coordinate moment (MPCM) is proposed, which is directly defined by a repeated integral. The angular factor can easily be eliminated by appropriately selecting a repeated integral. A method is provided for constructing affine invariants by means of MPCMs. The traditional affine moment invariants (AMIs) can be derived in terms of the proposed MPCM. Furthermore, affine invariants constructed with real-order (lower than 2) MPCMs can be derived using the proposed method. These invariants may be more robust to noise than AMIs. Several experiments were conducted to evaluate the proposed method performance.