Multidimensional Median Filters for Finding Bumps
Jefirey C. Miecznikowski, Kimberly F. Sellers, William F. Eddy · 2009
One of the main topics within the fleld of mathematical morphology is the detection of objects, e.g. spots in an image. In general, spot detection or bump hunting in high-dimensional data is a well studied problem. Here, we propose a novel spot detection method based on the smoothing decomposition, Data = Smooth + Rough (Velleman, 1980). By studying the rough from a cross shaped smoother, we will show how to locate spots or \bumps in multi-dimensional images. The essential feature of our novel method is the shape of the window. Usual practice is to choose a window which is (hyper-) cubical or (hyper-) spherical. We have found that by choosing the window to be (hyper-)crossical (i.e. shaped like a multi-dimensional cross), the resulting \rough is also shaped like a cross centered on the local maxima. We choose to use the median as the summary statistic over the pixels in the smoothing window. Further, we show that the choice of a nonlinear summary statistic, such as the median, is critical in the ability to distinguish the spots in the image. We demonstrate a few properties of this procedure and apply it to a variety of images from various flelds, including genetic and proteomic data analysis. The supplemental materials provided contain the supporting theory used for this manuscript.