On Data Structures from Symmetry Sets of 2D Shapes

This technical report consists of 4 chapters that elaborate on technical report nr. 36, and focus on 2+1D Symmetry Sets: 2D Symmetry Sets in either a one-parameter family of pertubations, or radius space, or a multi-scale context. These chapters are included in article format, since they are all either published, or submitted, or intended to be submitted to conferences. Therefore, some overlap in text and / or figures occurs. They are inspired by the results of the research written down in the deliverable 10. One part of this deliverable was presented at ECCV 2004 (focussing on the data structure induced by the Symmetry Set), while a second part was presented at S+SSPR 2004 (concerning alternative representations of Symmetry Set, e.g.\ the pre-Symmetry Set). Chapter 1 focusses on the changes that can occur in the Pre-Symmetry Set. They follow directly from possible changes of the Symmetry Set. It allows descriptions by means of a dynamic 2+1D Symmetry Set. This chapter has been presented at ICPR 2004 and is the result of close collaboration with Ole Fogh Olsen (ITU) and Peter Giblin (Liverpool). Chapter 2 describes a possible application of the usage of the transitions, viz.\ amending the pre-Symmetry Set. It is a way to remove small details in the (pre-)Symmetry Set. This chapter is the result of collaboration with Ole Fogh Olsen (ITU). Chapter 3 describes a fast and elegant extraction of the Medial Axis once the (pre-)Symmetry Set is known. It uses the radius space as described in deliverable 10. This chapter is the result of collaboration with Ole Fogh Olsen (ITU). Chapter 4 describes a general multi-scale frame work, where the pre-Symmetry Set is embedded in a mean curvature motion structure - the intrinsic heat equation for shapes. It yields a hierarchical structure suitable for describing and comparing shapes. It is the result of collaboration with Ole Fogh Olsen (ITU), Peter Giblin (Liverpool), and Dirk Siersma (Mathematics, Utrecht).