We consider a problem intimately related to the creation of maxima under Gaussian blurring: the number of modes of a Gaussian mixture in D dimensions. To our knowledge, a general answer to this question is not known. We conjecture that if the components of the mixture have the same covariance matrix (or the same covariance matrix up to a scaling factor), then the number of modes cannot exceed the number of components. We demonstrate that the number of modes can exceed the number of components when the components are allowed to have arbitrary and different covariance matrices. We will review related results from scale-space theory, statistics and machine learning, including a proof of the conjecture in 1D. We present a convergent, EM-like algorithm for mode finding and compare results of searching for all modes starting from the centers of the mixture components with a brute-force search. We also discuss applications to data reconstruction and clustering.
|Title of host publication||Scale Space Methods in Computer Vision|
|Subtitle of host publication||4th International Conference, Scale Space 2003 Isle of Skye, UK, June 10–12, 2003 Proceedings|
|Publisher||Springer Berlin Heidelberg|
|Number of pages||16|
|Publication status||Published - 2003|
|Name||Lecture Notes in Computer Science|