Detection and Localization of Random Signals

Object detection and localization are common tasks in image analysis. Correlation based detection algorithms are known to work well, when dealing with objects with known geometry in Gaussianly distributed additive noise. In the Bayes' view, correlation is linearly related to the logarithm of the probability density, and optimal object detection is obtained by the integral of the exponentiated squared correlation under appropriate normalization. Correlation with a model is linear in the input image, and can be computed effectively for all possible positions of the model using Fourier based linear filtering techniques. It is therefore interesting to extend the application to objects with many but small degrees of freedom in their geometry. These geometric variations deteriorate the linear correlation signal, both regarding its strength and localization with multiple peaks from a single object. Localization is typically preferred over detection, and Bayesian localization may be obtained as local integration of the probability density. In this work, Gaussian kernels of the exponentiated correlation are studied, and the use of Linear Scale-Space allows us to extend the Bayes detection with a well-posed localization, to extend the usage of correlation to a larger class of shapes, and to argue for the use of mathematical morphology with quadratic structuring elements on correlation images.

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