The discrete Gaussian-Hermite moment (DGHM) is a global feature representation method that can be applied to square images. We propose a modified DGHM (MDGHM) method and an MDGHM-based scale-invariant feature transform (MDGHM-SIFT) descriptor. In the MDGHM, we devise a movable mask to represent the local features of a non-square image. The complete set of non-square image features are then represented by the summation of all MDGHMs. We also propose to apply an accumulated MDGHM using multi-order derivatives to obtain distinguishable feature information in the third stage of the SIFT. Finally, we calculate an MDGHM-based magnitude and an MDGHM-based orientation using the accumulated MDGHM. We carry out experiments using the proposed method with six kinds of deformations. The results show that the proposed method can be applied to non-square images without any image truncation and that it significantly outperforms the matching accuracy of other SIFT algorithms.