Homotopic Affine Transformations in the 2D Cartesian Grid

Abstract

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Building the Topological Tree of Shapes from the Tree of Shapes

“…This is motivated by the strong links that exist between them. In [19], we introduced the unifying notion of a complete tree of shapes, that provides a continuum between both structures. The complete tree of shapes contains the nodes of the min-and maxtrees, and allows to derive the tree of shapes by a decreasing (reversible) homeomorphism.…”

“…Definition 6 (Topological tree of shapes [19]) The topological tree of shapes of F is the tree T H = (H, ◁ H ). (See Fig.…”

“…This led to a construction scheme of the topological tree of shapes from the tree of shapes, via the complete tree of shapes. A first non-optimal algorithm was proposed in [19].…”

“…In this article, we propose an algorithm for building the topological tree of shapes that improves the one of [19]. It aims to minimize the space cost of intermediate structures by avoiding the storage of redundant information.…”

Construction of Fast and Accurate 2D Bijective Rigid Transformation

“…We can also consider rigid motions from quasi-shear transforms [4,5,6], or reflections [7,8]. For specific applications, we can even look for an approximation of the rotation preserving the homotopy for subsets of Z 2 [9].…”