abstract

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.

 Introduction

Images of an object captured from different viewpoints are often subject to perspective distortions [1–۴]. If the object is small compared to the camera-to-scene distance, the perspective effect becomes negligible and the affine model provides a reasonable approximation of the projective model [2,5]. Therefore, the extraction of affine invariant features plays an important role in object recognition and registration [6–۱۰]. This method has been used extensively in numerous fields, such as shape recognition and retrieval [11,12], watermarking [13], aircraft identification [14,15], texture classification [16], and image registration [17,18]. The moment-based method is the most commonly used technique for the extraction of affine invariant features. However, only integer-order moments can be used to construct affine invariants [4]. It has been reported that high-order moments are sensitive to noise [19]. Hence, in practice, a moment of the lowest possible order should be used [20]. For similarity transform (including only translation, scaling, and rotation), Fourier Mellin descriptors [21] can be viewed as the invariants constructed by complex number order moments. However, similarity transform is only a spe cial case of affine transform [4]. Thus far, an order of moments for constructing affine invariants can only be an integer. In this paper, we consider generalizing the moment order from integer to noninteger, and construct affine invariants by means of the proposed moment.