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Computational topology approach for pattern recognition in 2D images
Extended abstract
Nikolay Makarenko, Irina Knyazeva, Fedor Urtiev
Central (Pulkovo) Astronomical Observatory, St-Petersburg, Russia
[email protected]
We applied methods of computational topology for three different tasks connected to the
image processing analysis. The sets of methods that were used included persistent homology and
persistent landscapes estimation by images. We used characteristics extracted from the topology
as an additional feature for image description. Additionally, for geometrical identification
persistent topological features of images we use an approach based on computation of MorseSmale complex. In future we plan to change the state of Gaussian smoothing of noisy data by
persistence based smoothing. We used this approach for Solar data, FMRI brain data and
remote sensing images.
The first application came from Solar physics. The problem was formulated as follows. In
so called Active Regions of the Sun, or regions with very big magnetic field relative to the Sun
in whole, from time to time occurs large amounts of energy releases so called Solar flares. It is
believed that solar flares connected to the changes in magnetic field, but this relation is not
explicit. For analysis time sequence of high resolution 2D magnetograms of active regions is
available. The questing is whether it is possible to extract some patterns which could be
connected with the solar flares? There are plenty of works where different characteristics were
offered, but without any obvious success. At first we start from computing Minkowsky
functional by the level sets of magnetogram and analyzing them in time course. We have found
that this curve demonstrate different complex dynamic, but we couldn't extract any specific
pattern preceding solar flares. After that we start to compute persistent homology and analyze
different characteristics of persistence diagram, such as different moments of persistence,
moments of distribution and others. Finally, we find that the cumulative sum length of barcodes
and mean length of barcodes demonstrates strong grows preceding flares. We connect this with
the physical effect of new magnetic flux which appeared before the flares which caused
increasing in complexity. Also, we tried to extract Morse-Smale cells from each magnetogram
and track their evolution in time, we find that changes very strong independently of the Flares
process. Now we are working under Morse Smale simplification of magnetogram. Another one
direction where we think that topological approach could be useful is Solar data is vector field
analysis. Now there are time series of Solar vector field dealt with good discrete in time and
space. Analysis of these data at this time restricted by the analysis of module values. We would
like to apply topological approach for vector field such as Conley index estimation, but this is
also work in progress.
The second application is the analysis of FMRI brain data. We have sets of images
corresponding to active and resting states of the brain. It is known that active and resting state
differs only in several parts of images and this changes very small, about 1.5%. . The classical
approach based on estimation correlation of real date with the expected response. The main
problem of this approach is that we know about brain response only for simple action, in case of
complex tasks or a mix of many tasks it is hard to build a model. Our task was on simple action
to find activated part of the brain. We computed persistence landscapes by active and resting sets
of images and find that there are difference for Betti1. After that we locate pixels which caused
the difference, specifically where new long barcode arise and find rather good agreement with
the classical approach used in the analysis of FMRI data.
The third application is analysis of remote sensing date. We have the set of different type
of textures. For each type of texture we have a high resolution image. Also, there are remote
sensing images where can be many different textures, our task is segmentation of images of
texture type. We find that persistence diagram and persistence landscapes for different texture
also differs. Also, we try to combine the multifractal approach with topological for segmentation
improvement. The results which we have rather promising, but this work also in progress now.
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