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, super-resolution imaging, among others. In this article, 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.

The multiscale hermite transform for local orientation analysis.

The efficient representation of local differential structure at various resolutions has been a matter of great interest for adaptive image processing and computer vision tasks. In this paper, we derive a multiscale model to represent natural images based on the scale-space representation: a model that has an inspiration in the human visual system. We first derive the one-dimensional case and then extend the results to two and three dimensions. The operators obtained for analysis and synthesis stages are derivatives of the Gaussian smoothing kernel, so that, for the two-dimensional case, we can represent them either in a rotated coordinate system or in terms of directional derivatives. The method to perform the rotation is efficient because it is implemented by means of the application of the so-called generalized binomial filters. Such a family of discrete sequences fulfills a number of properties that allows estimating the local orientation for several image structures. We also define the discrete counterpart in which the coordinate normalization of the continuous case is approximated as a subsampling of the discrete domain.