We propose an algebraic framework for generalized graph Laplacians which unifies the study of resistor networks, the critical group, and the eigenvalues of the Laplacian and adjacency matrices. Given a graph with boundary $G$ together with a generalized Laplacian $L$ with entries in a commutative ring $R$, we define a generalized critical group $\Upsilon_R(G,L)$. We relate $\Upsilon_R(G,L)$ to spaces of harmonic functions on the network using the Hom, Tor, and Ext functors of homological algebra. We study how these algebraic objects transform under combinatorial operations on the network $(G,L)$, including harmonic morphisms, layer-stripping, duality, and symmetry. In particular, we use layer-stripping operations from the theory of resistor networks to systematize discrete harmonic continuation. This leads to an algebraic characterization of the graphs with boundary that can be completely layer-stripped, an algorithm for simplifying computation of $\Upsilon_R(G,L)$, and upper bounds for the number of invariant factors in the critical group and the multiplicity of Laplacian eigenvalues in terms of geometric quantities.

Spatial de-interlacing is an essential part of motion adaptive de-interlacing used for reconstructing missing lines in cases of fast motion detection. Common spatial de-interlacing algorithms often produce artifacts in the output image, especially along edges with flat horizontal angles. In this paper we introduce a new method for spatial de-interlacing based on the dynamic time warping (DTW) procedure. The DTW algorithm finds an alignment between two original consecutive lines, and then, the missing line between them is reconstructed based on this alignment. This method preserves the smoothness of the original image edges and produces a high quality progressive image.

We present an authenticate graphical barcode symbology, and describe a system for its reliable encoding and decoding is presented. The graphical barcode is generated by modulating halftone patterns according to the message to be embedded in the barcode during the rendering of a base image. The generated barcode has the appearance of the base image, and when it is printed and scanned back in, the message can be retrieved. A graphical indicium is a graphical barcode embedded with a digitally signed message such that the integrity and authenticity of the message can be verified.