Two-parameter Persistence for Images via Distance Transform

The distance transform of a binary image is a classic tool in computer vision and it has been widely used in the field of Topological Data Analysis (TDA) to study porous media. A common practice is to convert grayscale images to binary ones to apply the distance transform. In this work, by considering the threshold decomposition of a grayscale image, we prove that threshold decomposition and distance transform together to formulate a two-parameter filtration. This would offer the TDA community a concrete example to apply multi-parameter persistence on digital image analysis. We demonstrate our method on the firn dataset.