Programmable Self-Assembly: Constructing Global Shape using Biologically-inspire

In this thesis I present a language for instructing a sheet of identically-programmed, flexible, autonomous agents (``cells'') to assemble themselves into a predetermined global shape, using local interactions. The global shape is described as a folding construction on a continuous sheet, using a set of axioms from paper-folding (origami). I provide a means of automatically deriving the cell program, executed by all cells, from the global shape description. With this language, a wide variety of global shapes and patterns can be synthesized, using only local interactions between identically-programmed cells. Examples include flat layered shapes, all plane Euclidean constructions, and a variety of tessellation patterns. In contrast to approaches based on cellular automata or evolution, the cell program is directly derived from the global shape description and is composed from a small number of biologically-inspired primitives: gradients, neighborhood query, polarity inversion, cell-to-cell contact and flexible folding. The cell programs are robust, without relying on regular cell placement, global coordinates, or synchronous operation and can tolerate a small amount of random cell death. I show that an average cell neighborhood of 15 is sufficient to reliably self-assemble complex shapes and geometric patterns on randomly distributed cells. The language provides many insights into the relationship between local and global descriptions of behavior, such as the advantage of constructive languages, mechanisms for achieving global robustness, and mechanisms for achieving scale-independent shapes from a single cell program. The language suggests a mechanism by which many related shapes can be created by the same cell program, in the manner of D'Arcy Thompson's famous coordinate transformations. The thesis illuminates how complex morphology and pattern can emerge from local interactions, and how one can engineer robust self-assembly.

Vector-Based Integration of Local and Long-Range Information in Visual Cortex

Integration of inputs by cortical neurons provides the basis for the complex information processing performed in the cerebral cortex. Here, we propose a new analytic framework for understanding integration within cortical neuronal receptive fields. Based on the synaptic organization of cortex, we argue that neuronal integration is a systems--level process better studied in terms of local cortical circuitry than at the level of single neurons, and we present a method for constructing self-contained modules which capture (nonlinear) local circuit interactions. In this framework, receptive field elements naturally have dual (rather than the traditional unitary influence since they drive both excitatory and inhibitory cortical neurons. This vector-based analysis, in contrast to scalarsapproaches, greatly simplifies integration by permitting linear summation of inputs from both "classical" and "extraclassical" receptive field regions. We illustrate this by explaining two complex visual cortical phenomena, which are incompatible with scalar notions of neuronal integration.

Analysis of Differential and Matching Methods for Optical Flow

Several algorithms for optical flow are studied theoretically and experimentally. Differential and matching methods are examined; these two methods have differing domains of application- differential methods are best when displacements in the image are small (<2 pixels) while matching methods work well for moderate displacements but do not handle sub-pixel motions. Both types of optical flow algorithm can use either local or global constraints, such as spatial smoothness. Local matching and differential techniques and global differential techniques will be examined. Most algorithms for optical flow utilize weak assumptions on the local variation of the flow and on the variation of image brightness. Strengthening these assumptions improves the flow computation. The computational consequence of this is a need for larger spatial and temporal support. Global differential approaches can be extended to local (patchwise) differential methods and local differential methods using higher derivatives. Using larger support is valid when constraint on the local shape of the flow are satisfied. We show that a simple constraint on the local shape of the optical flow, that there is slow spatial variation in the image plane, is often satisfied. We show how local differential methods imply the constraints for related methods using higher derivatives. Experiments show the behavior of these optical flow methods on velocity fields which so not obey the assumptions. Implementation of these methods highlights the importance of numerical differentiation. Numerical approximation of derivatives require care, in two respects: first, it is important that the temporal and spatial derivatives be matched, because of the significant scale differences in space and time, and, second, the derivative estimates improve with larger support.