We present a novel approach to the detection and characterization of edges, ridges, and blobs in two-dimensional images which exploits the symmetry properties of directionally sensitive analyzing functions in multiscale systems that are constructed in the framework of alpha-molecules. The proposed feature detectors are inspired by the notion of phase congruency, stable in the presence of noise, and by definition invariant to changes in contrast. We also show how the behavior of coefficients corresponding to differently scaled and oriented analyzing functions can be used to obtain a comprehensive characterization of the geometry of features in terms of local tangent directions, widths, and heights. The accuracy and robustness of the proposed measures are validated and compared to various state-of-the-art algorithms in extensive numerical experiments in which we consider sets of clean and distorted synthetic images that are associated with reliable ground truths. To further demonstrate the applicability, we show how the proposed ridge measure can be used to detect and characterize blood vessels in digital retinal images and how the proposed blob measure can be applied to automatically count the number of cell colonies in a Petri dish.
@cite_61: The present thesis deals with amplitude and sign decompositions of images via complex wavelets. We develop the monogenic curvelet transform for the directional decomposition of images into amplitude and sign. We use the amplitude information for the development and the analysis of an algorithm for the separation of edges in images, and the sign information for a new edge detection method. Eventually, we draw a connection between amplitude and sign representations and the perception of optical illusions.
@cite_36: The paper discusses the theory behind the dual-tree transform, shows how complex wavelets with good properties can be designed, and illustrates a range of applications in signal and image processing. The authors use the complex number symbol C in CWT to avoid confusion with the often-used acronym CWT for the (different) continuous wavelet transform. The four fundamentals, intertwined shortcomings of wavelet transform and some solutions are also discussed. Several methods for filter design are described for dual-tree CWT that demonstrates with relatively short filters, an effective invertible approximately analytic wavelet transform can indeed be implemented using the dual-tree approach.
While contrast-invariance is indeed a highly desirable feature in many edge and ridge detection tasks, localizing a property of the frequency components in the time representation of a signal comes at a price. Indeed, it was already observed in @cite_61 and @cite_36 that replacing isotropic wavelets with anisotropically scaled analyzing functions in deteriorates the detection of features in images rather than improving it. This should come as a surprise, as one would assume that anisotropically scaled elements would be better suited for detecting typically anisotropic features such as edges or ridges. Finding a way to bring together the intuitions behind the phase congruency measure @math and modern constructions of anisotropic analyzing functions was in fact one of the main factors motivating the present work.