# 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. Specificallythey 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 (TBMR). TBMR can be seen as a variant of 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 (em 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. ExperimentallyTBMR achieves a repeatability on par with state-of-the-art methods, but obtains a significantly higher number of features. Both the accuracy and the robustness of TBMR are demonstrated by applications to image registration and 3D reconstruction.

## Bibtex (lrde.bib)

@Article{	  xu.14.itip,
author	= {Yongchao Xu and Thierry G\'eraud and Pascal Monasse and
Laurent Najman},
title		= {Tree-Based Morse Regions: A Topological Approach to Local
Feature Detection},
journal	= {IEEE Transactions on Image Processing},
year		= {2014},
volume	= {23},
number	= {12},
pages		= {5612--5625},
month		= dec,
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. Specifically,
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 (TBMR). TBMR can be seen as a
variant of 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 ({\em 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 the robustness of TBMR are
demonstrated by applications to image registration and 3D
reconstruction.}
}