5. Conclusions
In this paper, the MPCM has been introduced for the construction of affine invariants, which can be viewed as the generalization of the traditional moment. The order of the traditional moment is generalized from integer to non-integer. In order to deal with the angular factor α(θ) owing to shearing in affine transform, a repeated integral is directly employed to define the MPCM. The angular factor can easily be eliminated by using an appropriate repeated integral. As a result, non-integer order moments can be used to construct affine invariants. Based on the MPCM, a method for constructing affine invariants was provided. AMIs derived from traditional moments can be constructed by means of the proposed method with MPCMs. Furthermore, affine invariants can be constructed by any order MPCMs. Consequently, invariants constructed by real-order (less than 2) MPCMs can be derived. The experimental results demonstrated that these invariants are more robust to noise. Although the order of the traditional moment has been generalized from integer to non-integer, the selection of an appropriate order is a problem that should be addressed, and will form our research direction in the future. From Figs. 4(b) and 5, among others, we can observe that the robustness to noise of the constructed invariants first increases and then decreases when s1 or s2 varies from 1 to 4. This may be owing to the normalization by M4(2, 1, 0, 0) in Eq. (32). The further modification of MPCM is an additional research direction.
