Spatial filtering in tone reproduction and vision

This thesis is concerned with spatial filtering. What is its utility in tone reproduction? Does it exist in vision, and if so, what constraints does it impose on the nervous system?

Tone reproduction is just the art and science of taking a picture and then displaying it. The sensors available to capture an image have a greater dynamic range than the media that may be used to display it. Conventionally, spatial filtering is used to boost contrast; it ameliorates the loss of contrast that results when the sensor signal range is scaled down to fit the display range. In this thesis, a type of nonlinear spatial filtering is discussed that results in direct range reduction without range scaling. This filtering process is instantiated in a real-time image processor built using analog CMOS VLSI.

Spatial filtering must be applied with care in both artificial and natural vision systems. It is argued that the nervous system does not simply filter linearly across an image. Rather, the way that we see things implies that the nervous system filters nonlinearly. Further, many models for color vision include a high-pass filtering step in which the DC information is lost. A real-time study of filtering in color space leads to the conclusion that the nervous system is not that simple, and that it maintains DC information by referencing to white.

Lattice quantum codes and exotic topological phases of matter

This thesis addresses whether it is possible to build a robust memory device for quantum information. Many schemes for fault-tolerant quantum information processing have been developed so far, one of which, called topological quantum computation, makes use of degrees of freedom that are inherently insensitive to local errors. However, this scheme is not so reliable against thermal errors. Other fault-tolerant schemes achieve better reliability through active error correction, but incur a substantial overhead cost. Thus, it is of practical importance and theoretical interest to design and assess fault-tolerant schemes that work well at finite temperature without active error correction.

In this thesis, a three-dimensional gapped lattice spin model is found which demonstrates for the first time that a reliable quantum memory at finite temperature is possible, at least to some extent. When quantum information is encoded into a highly entangled ground state of this model and subjected to thermal errors, the errors remain easily correctable for a long time without any active intervention, because a macroscopic energy barrier keeps the errors well localized. As a result, stored quantum information can be retrieved faithfully for a memory time which grows exponentially with the square of the inverse temperature. In contrast, for previously known types of topological quantum storage in three or fewer spatial dimensions the memory time scales exponentially with the inverse temperature, rather than its square.

This spin model exhibits a previously unexpected topological quantum order, in which ground states are locally indistinguishable, pointlike excitations are immobile, and the immobility is not affected by small perturbations of the Hamiltonian. The degeneracy of the ground state, though also insensitive to perturbations, is a complicated number-theoretic function of the system size, and the system bifurcates into multiple noninteracting copies of itself under real-space renormalization group transformations. The degeneracy, the excitations, and the renormalization group flow can be analyzed using a framework that exploits the spin model's symmetry and some associated free resolutions of modules over polynomial algebras.

Experimental and theoretical studies of Notch signaling-mediated spatial pattern

Notch signaling acts in many diverse developmental spatial patterning processes. To better understand why this particular pathway is employed where it is and how downstream feedbacks interact with the signaling system to drive patterning, we have pursued three aims: (i) to quantitatively measure the Notch system's signal input/output (I/O) relationship in cell culture, (ii) to use the quantitative I/O relationship to computationally predict patterning outcomes of downstream feedbacks, and (iii) to reconstitute a Notch-mediated lateral induction feedback (in which Notch signaling upregulates the expression of Delta) in cell culture. The quantitative Notch I/O relationship revealed that in addition to the trans-activation between Notch and Delta on neighboring cells there is also a strong, mutual cis-inactivation between Notch and Delta on the same cell. This feature tends to amplify small differences between cells. Incorporating our improved understanding of the signaling system into simulations of different types of downstream feedbacks and boundary conditions lent us several insights into their function. The Notch system converts a shallow gradient of Delta expression into a sharp band of Notch signaling without any sort of feedback at all, in a system motivated by the Drosophila wing vein. It also improves the robustness of lateral inhibition patterning, where signal downregulates ligand expression, by removing the requirement for explicit cooperativity in the feedback and permitting an exceptionally simple mechanism for the pattern. When coupled to a downstream lateral induction feedback, the Notch system supports the propagation of a signaling front across a tissue to convert a large area from one state to another with only a local source of initial stimulation. It is also capable of converting a slowly-varying gradient in parameters into a sharp delineation between high- and low-ligand populations of cells, a pattern reminiscent of smooth muscle specification around artery walls. Finally, by implementing a version of the lateral induction feedback architecture modified with the addition of an autoregulatory positive feedback loop, we were able to generate cells that produce enough cis ligand when stimulated by trans ligand to themselves transmit signal to neighboring cells, which is the hallmark of lateral induction.