Local Feature Detection by Higher Order Riesz Transforms on Images

An important issue in image analysis is the detection of features in images. Problems occur, when the images contain noise or are rotated. An example for a rotation invariant feature is the apex angle. However, apex angle detection is only expedient, on a special class of images, therefore in this thesis only images with up to two overlaid intrinsically one dimensional structures are considered. Images belonging to this class also allow the calculation of the local main orientation. Together, local main orientation and apex angle provide access to the computation of both orientations and vice versa. The estimation of the local main orientation and apex angle are done by applying Riesz transforms up to third order to a local neighborhood of the image. In order to accomplish that, a new interpretation of the Riesz transform is studied, which in turn provides a new interpretation of the monogenic signal and the monogenic curvature tensor. Another direct result of the new interpretation of the Riesz transform is the ability to detect the rotation angle between two otherwise identical images. In order to implement Riesz transforms up to the third order they need to be combined with a suitable bandpass filter. The Laplacian of Poisson allows the development of new filter kernels for the second and third order Riesz transforms. The implementation of this method is compared to an implementation of a well-known method using image gradients and computing the orientations of up to two overlaid structures that has been developed in Lübeck.