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.