Local Geometric Deformations in the DHT Domain With Applications

Local rotation, translation, and scaling of the image domain represent a basic toolkit in adaptive image processing, such as image registration, template matching, local invariant feature detection, and super-resolution imaging, among others. In this paper, it is shown how the local rotation, scaling, and translations can be performed in the discrete Hermite transform (DHT) domain. As the DHT satisfies the generalized steerability property, basic geometric operations are expressed as linear mappings in the DHT domain and hence can facilitate the solution of many image processing problems. The local rotation and scaling were previously shown in the continuous domain using the Hermite Transform, the former is used here as a good approximation for discrete images, whereas the latter is extended to a discrete domain. In addition, the local translation operation is fully developed in the discrete domain. The application of these three operations is illustrated with three exemplar applications including: 1) mathematical morphology; 2) template matching; and 3) depth from defocus. The simple yet effective methods presented in the paper indicate that local image decompositions satisfying the steerability property, such as the DHT, are desirable for solving a number of interesting image processing problems.