Tree-Based Morse Regions: A Topological Approach to Local Feature Detection

This paper introduces a topological approach to local invariant feature detection motivated by Morse theory. We use the critical points of the graph of the intensity image, revealing directly the topology information as initial interest points. Critical points are selected from what we call a tree-based shape-space. In particular, they are selected from both the connected components of the upper level sets of the image (the Max-tree) and those of the lower level sets (the Min-tree). They correspond to specific nodes on those two trees: 1) to the leaves (extrema) and 2) to the nodes having bifurcation (saddle points). We then associate to each critical point the largest region that contains it and is topologically equivalent in its tree. We call such largest regions the tree-based Morse regions (TBMRs). The TBMR can be seen as a variant of maximally stable extremal region (MSER), which are contrasted regions. Contrarily to MSER, TBMR relies only on topological information and thus fully inherit the invariance properties of the space of shapes (e.g., invariance to affine contrast changes and covariance to continuous transformations). In particular, TBMR extracts the regions independently of the contrast, which makes it truly contrast invariant. Furthermore, it is quasi-parameter free. TBMR extraction is fast, having the same complexity as MSER. Experimentally, TBMR achieves a repeatability on par with state-of-the-art methods, but obtains a significantly higher number of features. Both the accuracy and robustness of TBMR are demonstrated by applications to image registration and 3D reconstruction.