Nonparametric Techniques for Analyzing Directional Structure in Space-Time Random Fields
Final Report Abstract
The detection and analysis of directional structure in images is crucial to many applications since one-dimensional patterns often correspond to important image features such as object contours or trajectories. Detecting one-dimensional patterns and estimating their orientation is particularly important; for instance, in the detection of ship wakes in Synthetic Aperture Radar (SAR) images, optical flow estimation and its application to myocardial motion estimation, the analysis of texture by estimating multidimensional orientation, the efficient coding of local differential structures in images, image encoding, labeling and reconstruction, or the analysis of superimposed directional patterns, which may occur in X-ray projection imaging. Many of the techniques developed for orientation estimation assume that there is indeed a directional structure to be estimated. In the absence of such a directional structure, they will therefore still produce an estimated direction – which would be meaningless. In order to address this problem, several measures for the degree of directionality have been defined in the literature. However, having a normalized measure for the degree of directionality alone may not help us classify a structure as directional or nondirectional: Is a structure with degree of directionality equal to, say, 0.6 directional or not? A meaningful answer to this question can only be given based on sound statistical arguments, and a specified null model. Moreover, having a threshold based on statistical arguments is important to automatically classify images, which is essential when dealing with large amounts of data, as happens, for instance, in the detection of ship wakes in the ocean. In order to derive a threshold, we model the image as a stationary random field and then phrase our problem as a hypothesis test: “Is there a directional structure or not?” Deciding this question requires not only a measure for directionality but also a threshold, based on the statistics of the image, above which a structure can indeed be regarded as directional. So far, there had been little work on detecting and analyzing directional structures in random fields, as a large fraction of the work in statistics has focussed on isotropic random fields. In this project, we have developed a novel approach based on the random monogenic signal. The monogenic signal enables the unique decomposition of a two-dimensional real image into a local amplitude, a local orientation, and a local phase. While the monogenic signal has received a lot of attention, we were the first to provide a thorough discussion of the second-order statistical properties of the random monogenic signal. Our approach allowed us to define a measure for the degree of unidirectionality and to provide a detailed statistical analysis, which may be used to test an image for the presence of unidirectional components. Such a statistical test enables the automatic processing and classification of large volumes of data, which is of importance, for instance, in the earth sciences.
Publications
- “Detecting directionality in random fields using the monogenic signal,” IEEE Transactions on Information Theory, vol. 60, no. 10, pp. 6491–6510, 2014
S. C. Olhede, D. Ramirez and P. J. Schreier
(See online at https://doi.org/10.1109/TIT.2014.2342734)
