Homological spanning forest framework for 2D image analysis
The fast growing area of computational (algebraic) topology needs algorithms well-suited to concrete situations. In the topological analysis of a digital picture, a problem is recognizing and taking into account the objects depicted in the image. The main contribution of this paper, in my opinion, is the flexibility of the method with respect of objects of interest. A Homological Spanning Forest (HSF) plays in homology more or less the role that a spanning tree plays for computing the fundamental group of a cell complex. HSFs have apparently been conceived for homology computation in the case of a 2D image, but their definition is quite general and they are ready for higher dimensional applications. HSFs are defined as directed subgraphs of the 1-skeleton of a cell complex associated with the pixel structure of a raster image, satisfying a covariance relation between the boundary operator and the discrete vector field determined by the directed edges. The paper is mainly devoted to introducing and studying the construction and reduction of an HSF out of a 2D raster image; the reduction is necessary for simplifying the homology computation by putting generators in evidence (much in the line of thought of \textit{R. González-Diaz} et al. [Discrete Appl. Math. 157, No. 3, 490--499 (2009; Zbl 1168.68045)]). Moreover, it can be performed while isolating particular objects of the image (Regions Of Interest) and fitting to them. The article is self-contained, detailed and well supported by meaningful examples. Unfortunately some typos, some less clear passages and the absence of colour in the printed pictures, sometimes make reading a bit difficult.
scientific article; zbMATH DE number 6119070
scientific article