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

Abstract

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.

A Tutorial on Well-Composedness

“…The link between topological persistence [53] and the tree of shapes [34,63] could also be investigated; see for example the work of Xu et al [191], which detects local invariant features in images thanks to Morse theory and the tree of shapes. On a related theme, many problems in image analysis, digital processing and shape optimization can be expressed as variational problems involving the discretization of some discrete discrete differential operators [46,121,73] such as normal vectors or curvatures [128,129] of digital surfaces which are of prime importance in geometry processing.…”

OpenMVG: Open Multiple View Geometry

“…• Affine invariant regions: Tree-Based Morse Regions (TBMR) [23], Maximally Stable Extremal Regions (MSER) [24].…”

Partition and Inclusion Hierarchies of Images: A Comprehensive Survey

“…A whole variety of connected operators can be realized by replacing the Max-tree with other component trees, and this fundamental extension to the framework of connected operators was studied in [52]. Recently, Xu and co-authors [85,139] have proposed defining the filtering in shape space, an image whose adjacency relation is given by the parental relations in its original Max-tree. A novel way of filtering and node selection is also proposed by Perret and Collet [114], which puts Mathematical Morphology in relation with stochastic models, relying on a Markovian algorithm to classify the nodes of a tree.…”