Combinatorial Properties and Recognition of Unit Square Visibility Graphs.

Unit square visibility graphs (USV) are described by axis-parallel visibility between unit squares placed in the plane. If the squares are required to be placed on integer grid coordinates, then USV become unit square grid visibility graphs (USGV), an alternative characterisation of the well-known rectilinear graphs. We extend known combinatorial results for USGV and we show that, in the weak case (i.e., visibilities do not necessarily translate into edges of the represented combinatorial graph), the area minimisation variant of their recognition problem is NP-hard. We also provide combinatorial insights with respect to USV, and as our main result, we prove their recognition problem to be NP-hard, which settles an open question.

Depth potential function for folding pattern representation, registration and analysis.

Some surfaces present folding patterns formed by juxtapositions of ridges and valleys as, for example, the cortical surface of the human brain. The fundamental problem with ridges is to find a correspondence among and analyze the variability among them. Many techniques to achieve these goals exist but use scalar functions. Depth maps are used to efficiently project the geometry of folds into a scalar function in the case where a natural projection plane exists. However, in most cases of curved surfaces, there is no natural projection plane to represent folding patterns. This paper studies the problem of shape matching and analysis of folding patterns by extending the notion of depth maps when no natural projection plane exists. The novel depth measure is called a depth potential function. The depth potential function integrates the information known from the curvature of the surface into a point-of-view invariant representation. The main advantage of the depth potential function is that it is computed by solving a time independent Poisson equation. The Poisson equation endows our surface representation with a significant computational advantage that makes it orders of magnitude faster to compute compared with other available surface representations. The method described in this paper was validated using both synthetic surfaces and cortical surfaces of human brain acquired by magnetic resonance imaging. On average, the improvement in shape matching when using the depth potential was of 11%, which is considerable.

Magnetic self-assembly of three-dimensional surfaces from planar sheets.

This report describes the spontaneous folding of flat elastomeric sheets, patterned with magnetic dipoles, into free-standing, 3D objects that are the topological equivalents of spherical shells. The path of the self-assembly is determined by a competition between mechanical and magnetic interactions. The potential of this strategy for the fabrication of 3D electronic devices is demonstrated by generating a simple electrical circuit surrounding a spherical cavity.

Tuning and comparing spatial normalization methods.

Spatial normalization is a key process in cross-sectional studies of brain structure and function using MRI, fMRI, PET and other imaging techniques. A wide range of 2D surface and 3D image deformation algorithms have been developed, all of which involve design choices that are subject to debate. Moreover, most have numerical parameters whose value must be specified by the user. This paper proposes a principled method for evaluating design choices and choosing parameter values. This method can also be used to compare competing spatial normalization algorithms. We demonstrate the method through a performance analysis of a nonaffine registration algorithm for 3D images and a registration algorithm for 2D cortical surfaces.

Dissections: self-assembled aggregates that spontaneously reconfigure their structures when their environment changes.

This Communication describes a new strategy for the design of adaptive structures based on reconfigurable mesoscale self-assembly. Several sets of millimeter-scale objects have been designed that can self-assemble into two different, regular aggregates at the interface between an aqueous solution and perfluorodecalin; the choice between the two aggregates is determined by the density of the aqueous phase.

Using hierarchical self-assembly to form three-dimensional lattices of spheres.

This paper describes an approach to the fabrication of three-dimensional (3-D) structures of millimeter-scale spherical beads having a range of lattices-tetragonal, cubic, and hexagonal-using hierarchical self-assembly. The process has five steps: (i) metal-coated beads are packed in a rod-shaped cavity in an elastomeric polymer (poly(dimethylsiloxane), PDMS); (ii) the beads are embedded in a second polymer (PDMS or polyurethane, PU) using a procedure that leaves the parts of the beads in contact with the PDMS exposed; (iii) the exposed areas of the beads are coated with a solder having a low melting point; (iv) the polymer rods-with embedded beads and exposed solder drops-are suspended in an approximately isodense medium (an aqueous solution of KBr) and allowed to self-assemble by capillary interactions between the drops of molten solder; and (v) the assembly is finished by several procedures, including removing the beads from the polymer matrix by dissolution, filling the voids left with another material, and dissolving the matrix. The confinement of the beads in regular structures in polymer rods makes it possible to generate self-assembled structures with a variety of 3-D lattices; the type of the lattice formed can be controlled by varying the size of the beads, and the size and shape of the cross-section of the rods.

Fabrication of Micro-Chain Mail by Simultaneous, Patterned Electrodeposition on a Plane and Multiple Cylinders.

Modeling with paper clips? No, a general method for the fabrication of complex three-dimensional (3D) microstructures is reported. The 3D structure is decomposed into substructures that can be patterned onto separate substrates and then these substrustures are joined to form the desired microstructure. The picture shows an optical micrograph of a free-linked nickel chain mail.