Nanoscale patchworks

On Patchworks and Hierarchies

Motivated by questions in biological classification, we discuss some elementary combinatorial and computational properties of certain set systems that generalize hierarchies, namely, ‘patchworks’, ‘weak patchworks’, ‘ample patchworks’ and ‘saturated patchworks’ and also outline how these concepts relate to an apparently new ‘duality theory’ for cluster systems that is based on the fundamental c...

Patchworks and metablocks enumeration

Finite Curvature Continuous Polar Patchworks

We present an algorithm for completing a C surface of up to degree bi-6 by capping an n-sided hole with polar layout. The cap consists of n tensor-product patches, each of degree 6 in the periodic and degree 5 in the radial direction. To match the polar layout, one edge of these patches is collapsed. We explore and compare with alternative constructions, based on more pieces or using total-degr...

UNROOTED SUPERTREES Limitations, traps, and phylogenetic patchworks

Whereas biologists might think of rooted trees as the natural, or even the only, way to display phylogenetic relationships, this is not the case for a mathematician, to whom rooted and unrooted trees are graph-theoretical constructions that can be transformed easily into one another. An unrooted tree contains the same information as its rooted counterpart with the single exception that it does ...

PatchWorks: Example-Based Region Tiling for Image Editing

Filling regions of an image in a seamless manner is an important task for editing digital images. We propose to address it by using an example-based image generation technique. Such techniques have recently proved very effective in the context of unconstrained texture synthesis. We show that they can also be surprisingly effective to fill large regions in images, even within complex mixes of te...