Combined Scale and Translation Invariants of Krawtchouk Moments

This paper derives scale, translation invariants as well as combined scale and translation invariants of Krawtchouk moments. Unmodified Krawtchouk moments are used for scale invariants whereas modified Krawtchouk moments are used for obtaining translation invariants because to retain the orthogonality property of discrete weighted Krawtchouk polynomials that is lost due to translation. Scale invariants are obtained from the Krawtchouk moments using the standard procedure of expressing the scaled moments independent of scale factors. In order to obtain the modified moments for translation invariants, the discrete weighted Krawtchouk polynomials are assumed to be periodic with period equal to the given number of data points and these are shifted to the centroid of the image. It is shown that the modified moments are invariant to translation. Further, a mathematical expression is derived for combined scale and translation invariance. In order to test the derived invariants, invariance and face recognition problems are attempted. First problem was tested using two standard images, where as the second problem was tested using ORL database images.