Physics of auroral phenomena : proceedings of the 37th Annual seminar, Apatity, 25 - 28 February, 2014 / [ed. board: A. G. Yahnin, N. V. Semenova]. - Апатиты : Изд-во Кольского научного центра РАН, 2014. - 125 с. : ил., табл.

*Physics o fAuroral Phenomena", Proc. XXXVII A nnual Seminar, Apatity, pp. 42-45, 2014 © Kola Science Centre, Russian Academy of Science, 2014 Polar Geophysical Institute APPLICATION OF MATHEMATICAL MORPHOLOGY TO DATA OF GROUND-BASED AURORAL OBSERVATIONS Boris V. Kozelov (Polar Geophysical Institute, Apatity, Murmansk region, 184209 Russia) Abstract. Methods of mathematical morphology (MM) have been actively developed due to the practical need to solve problems of automatic recognition of objects on TV images, in cartography and computer vision. Ground-based optical auroral observations have a number of features (strong aspect-angle distortion, low contrast, lack of a clear boundary, etc.) that must be considered when spatial and temporal structure of the auroral glow are analyzing by the MM methods. The report discusses the basic MM definitions and application of the MM methods to obtain the spatio- temporal characteristics of pulsating auroral arcs. 1. Introduction Mathematical morphology is a theory which gives useful tools aimed to analysis of plane and spatial shapes. It widely applied to analysis of different data sets [1-3]. Mathematical morphology stands somewhat apart from traditional linear image processing, since the basic operations of morphology are non-linear in nature, and thus make use of a totally different type of algebra than the linear algebra. So, it can be used in complement to more traditional methods. The basic idea of the MM in simplest binary case is to probe an image in each pixel with a simple, pre­ defined shape, drawing conclusions on how this shape fits or misses the local shapes in the image. This simple pre-defmed shape is called the structuring element. Different combinations of the procedure with different structuring elements and procedures from set theory used to define all MM operations. This report presents basic MM definitions, illustrates the main operations by elementary examples and compares with other known numerical methods. Then we show an example of application of the MM methods to obtain characteristics of pulsating auroral arcs. 2. Mathematical background The theoretical foundations of morphological image processing lies in set theory and the mathematical theory of order (mathematics of lattices). In practical image processing, it is sufficient to know that morphology can be applied to a finite set E if i) we can partially order its elements, (where the ordering is denoted by “5”), i.e., for all x, y, z e E: x< x x < y , y < x => x = y x < y , y < z => x < z ii) each non-empty subset of E has a maximum and minimum. For example, any finite set o f real or integer numbers is a suitable set E. The ordering “<” can be defined as in ordinary calculus (2 < 3, 3 S 8, etc.). The maximum and minimum are also defined in the usual sense (e.g., max{7, 1, 3} = 7). This means we can apply morphology not only to binary images, but and to grey­ valued one (or subsets of images), because the collection of grey values can be viewed as a finite set E with ordering, maximum, and minimum well defined. All MM operations are constructed by basics operations named translation, dilation and erosion, see definitions in Fig.l illustrated by simple binary grid examples. Here A is a plane set under study and В is the structuring element. Translation: Лг = { q|ae,4, q = a + z } Dilation: A 0 В = {a + b|aeA , b e 6} = ~ UbSB-^b Erosion: А в В= { г \ В г ^ А } = [ ) 2 е ц А2 KIM A® В IBB в »■ Й AeB ■ 1 ■■ ■ ■■ в ■■■ ■ X ■ Hi] 1 Figure 1. Definition o f basics mathematical morphology operations. Combination o f dilation and erosion give us the opening and closing operations: opening: X o В = (X e В) © В (1) closing: X » B = ( X ® B ) e B (2) Sequential applications opening and closing with the same structuring element can be used to eliminate small-scale noise structures (so-called open-close filter). To separate structures in same range o f scales we can use two different structuring elements, Bn and Ba. In this case we should firstly use an open-close filter with a small-size structuring element, followed by an open-close filter with a structuring element of larger size: F,(X) = (X o Bn) • Bn - (((X о Bn) • Вn) о Вa) • Ba (3) In this way we can implement an alternating sequential filter, which gradually eliminates the structure components from smallest to largest. 42