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The median filter scheme is an elegant, monotone discretization of the level set formulation of motion by mean curvature. It turns out to evolve every level set of the initial condition precisely by another class of methods known as threshold dynamics. Median filters are, in other words, the natural level set versions of threshold dynamics algorithms. Exploiting this connection, we revisit median filters in light of recent progress on the threshold dynamics method. In particular, we give a variational formulation of, and exhibit a Lyapunov function for, median filters, resulting in energy based unconditional stability properties. The connection also yields analogues of median filters in the multiphase setting of mean curvature flow of networks. These new multiphase level set methods do not require frequent redistancing, and can accommodate a wide range of surface tensions. 

Abstract Control over the copy number and nanoscale positioning of quantum dots (QDs) is critical to their application to functional nanomaterials design. However, the multiple non-specific binding sites intrinsic to the surface of QDs have prevented their fabrication into multi-QD assemblies with programmed spatial positions. To overcome this challenge, we developed a general synthetic framework to selectively attach spatially addressable QDs on 3D wireframe DNA origami scaffolds using interfacial control of the QD surface. Using optical spectroscopy and molecular dynamics simulation, we investigated the fabrication of monovalent QDs of different sizes using chimeric single-stranded DNA to control QD surface chemistry. By understanding the relationship between chimeric single-stranded DNA length and QD size, we integrated single QDs into wireframe DNA origami objects and visualized the resulting QD-DNA assemblies using electron microscopy. Using these advances, we demonstrated the ability to program arbitrary 3D spatial relationships between QDs and dyes on DNA origami objects by fabricating energy-transfer circuits and colloidal molecules. Our design and fabrication approach enables the geometric control and spatial addressing of QDs together with the integration of other materials including dyes to fabricate hybrid materials for functional nanoscale photonic devices.