WP1 Advanced reflectional symmetry
WP2 Rotational, axial, centre symmetry, groups of symmetry
WP3 More general types of symmetry
WP4 Symmetry of continuously described objects
WP5 Approximate symmetry
WP6 Equivalences of finite sets of points
WP7 Exact projective (and other) equivalences between algebraic surfaces in 3-space
WP8 Approximate projective equivalences of special algebraic varieties
WP9 Approximate symmetries of perturbed surfaces
WP10 Initial study of reflectional symmetry detection on EO data
WP11 Symmetry-aware feature extraction in EO data fusion
WP12 Integration of symmetries into semantic segmentation and object recognition in EO data fusion
WP13 Validation of symmetry utilization in EO data applications

Implementation of WP1 – WP5 (O1)

100%. UWB-CSE mostly. New methods for the detection of local reflection [1] and rotational [19] symmetries, also applicable to approximate symmetries and corrupted data, have been developed and published. The latter has also been applied to contour simplification in the generation of maps from 3D point clouds [2]. A metric for the assessment of the object’s symmetry has been developed [20]. Progress has also been made in the detection of reflection symmetry with respect to the non-planar symmetry surface. A method for detecting symmetry polylines in polygons [21] represents an additional generalization. User tests for the assessment of approximate symmetry in EO data have been performed (the paper is in the second round of peer review in the journal). UM FERI has developed its own method for the detection of approximate global reflection symmetries, which is suitable for engineering use due to its speed and flexibility. We have obtained a European patent for it [26].

Implementation of WP6 – WP9 (O2)

100%. UWB-M mostly. We have developed and proved a number of important mathematical concepts and implemented algorithms based on them. Methods for the approximate reconstruction of non-exact curves using perturbed polynomials and matrix representation for symmetry estimation have been developed [3, 4, 22]. An algorithm for the detection of global symmetries of finite point sets has been designed, applicable to discrete curves and point clouds [5, 23]. Coopearation between mathematicians and computer scientists led to the exploitation of symmetries of trigonometric curves via Laplacian smoothing [6, 7]. An algorithm for the recognition of rotational, axial, reflection and central symmetries of implicitly defined algebraic surfaces was developed [8, 9]. The decomposition of the symmetries was generalized to discrete surfaces in 3D space, proposing a unifying pattern for the computation of symmetries [10].

Implementation of WP10 – WP13 (O3)

100%. UM FERI mostly. In WP10, we have developed and implemented a Geographic Information System (GIS) that supports the integration of different symmetry detection algorithms and the treatment of detected symmetries as additional features [28]. It is based on the Gemma Fusion Suite (GFS) SDK. In [11], we proposed several speed-ups and new functionalities of the global reflection symmetry detection method [29] from a granted European patent [26]. We also developed a local reflection symmetry detection method adapted to EO data [12, 24, 30]. From it, we also derived the local rotational symmetry detection (WP2) [31]. In WP11, we extracted target objects from the LIDAR point clouds and then used the global symmetry detection for classification purposes. In this way, the water height of Lake Cerknica was predicted based on Sentinel-2 multispectral imagery [13, 25]. For the determination of the boundary of the intermittent lake, we used our own region fusion algorithm [27]. The use of reflection symmetries [32] in railway line detection has significantly reduced the detection of false positives. As part of the second pilot to develop some preliminary steps towards tree species classification [33], we have developed a rich test database of trees with different predefined canopy shapes and symmetry planes [14]. In [15, 16], we have discussed different criteria for optimizing tree pruning and forest growth prediction, which are also useful for tree species determination. In classification, however, we we need not only the individual stronger symmetries, but also their structural relationships. In [17] we introduced a method for geometric object characterization that uses a multi-sweeping paradigm to represent the object structure consisting of chains of axes of local symmetries, and in [18, 34] we presented a hierarchical data structure for objects of a pre-segmented scene. The results of the project were also used in an application for prostate cancer detection using reflection symmetry [35], which was developed in collaboration with the University Clinical Centre (UKC) Maribor.