Information-theoretic Studies and Capacity Bounds: Group Network Codes and Energy Harvesting Communication Systems

Network information theory and channels with memory are two important but difficult frontiers of information theory. In this two-parted dissertation, we study these two areas, each comprising one part. For the first area we study the so-called entropy vectors via finite group theory, and the network codes constructed from finite groups. In particular, we identify the smallest finite group that violates the Ingleton inequality, an inequality respected by all linear network codes, but not satisfied by all entropy vectors. Based on the analysis of this group we generalize it to several families of Ingleton-violating groups, which may be used to design good network codes. Regarding that aspect, we study the network codes constructed with finite groups, and especially show that linear network codes are embedded in the group network codes constructed with these Ingleton-violating families. Furthermore, such codes are strictly more powerful than linear network codes, as they are able to violate the Ingleton inequality while linear network codes cannot. For the second area, we study the impact of memory to the channel capacity through a novel communication system: the energy harvesting channel. Different from traditional communication systems, the transmitter of an energy harvesting channel is powered by an exogenous energy harvesting device and a finite-sized battery. As a consequence, each time the system can only transmit a symbol whose energy consumption is no more than the energy currently available. This new type of power supply introduces an unprecedented input constraint for the channel, which is random, instantaneous, and has memory. Furthermore, naturally, the energy harvesting process is observed causally at the transmitter, but no such information is provided to the receiver. Both of these features pose great challenges for the analysis of the channel capacity. In this work we use techniques from channels with side information, and finite state channels, to obtain lower and upper bounds of the energy harvesting channel. In particular, we study the stationarity and ergodicity conditions of a surrogate channel to compute and optimize the achievable rates for the original channel. In addition, for practical code design of the system we study the pairwise error probabilities of the input sequences.

From molecules to organs : Microscopy and multi-scale nature of development

Morphogenesis is a phenomenon of intricate balance and dynamic interplay between processes occurring at a wide range of scales (spatial, temporal and energetic). During development, a variety of physical mechanisms are employed by tissues to simultaneously pattern, move, and differentiate based on information exchange between constituent cells, perhaps more than at any other time during an organism's life. To fully understand such events, a combined theoretical and experimental framework is required to assist in deciphering the correlations at both structural and functional levels at scales that include the intracellular and tissue levels as well as organs and organ systems. Microscopy, especially diffraction-limited light microscopy, has emerged as a central tool to capture the spatio-temporal context of life processes. Imaging has the unique advantage of watching biological events as they unfold over time at single-cell resolution in the intact animal. In this work I present a range of problems in morphogenesis, each unique in its requirements for novel quantitative imaging both in terms of the technique and analysis. Understanding the molecular basis for a developmental process involves investigating how genes and their products- mRNA and proteins-function in the context of a cell. Structural information holds the key to insights into mechanisms and imaging fixed specimens paves the first step towards deciphering gene function. The work presented in this thesis starts with the demonstration that the fluorescent signal from the challenging environment of whole-mount imaging, obtained by in situ hybridization chain reaction (HCR), scales linearly with the number of copies of target mRNA to provide quantitative sub-cellular mapping of mRNA expression within intact vertebrate embryos. The work then progresses to address aspects of imaging live embryonic development in a number of species. While processes such as avian cartilage growth require high spatial resolution and lower time resolution, dynamic events during zebrafish somitogenesis require higher time resolution to capture the protein localization as the somites mature. The requirements on imaging are even more stringent in case of the embryonic zebrafish heart that beats with a frequency of ~ 2-2.5 Hz, thereby requiring very fast imaging techniques based on two-photon light sheet microscope to capture its dynamics. In each of the hitherto-mentioned cases, ranging from the level of molecules to organs, an imaging framework is developed, both in terms of technique and analysis to allow quantitative assessment of the process in vivo. Overall the work presented in this thesis combines new quantitative tools with novel microscopy for the precise understanding of processes in embryonic development.

Deep tissue fluorescence imaging with time-reversed light

Advances in optical techniques have enabled many breakthroughs in biology and medicine. However, light scattering by biological tissues remains a great obstacle, restricting the use of optical methods to thin ex vivo sections or superficial layers in vivo. In this thesis, we present two related methods that overcome the optical depth limit—digital time reversal of ultrasound encoded light (digital TRUE) and time reversal of variance-encoded light (TROVE). These two techniques share the same principle of using acousto-optic beacons for time reversal optical focusing within highly scattering media, like biological tissues. Ultrasound, unlike light, is not significantly scattered in soft biological tissues, allowing for ultrasound focusing. In addition, a fraction of the scattered optical wavefront that passes through the ultrasound focus gets frequency-shifted via the acousto-optic effect, essentially creating a virtual source of frequency-shifted light within the tissue. The scattered ultrasound-tagged wavefront can be selectively measured outside the tissue and time-reversed to converge at the location of the ultrasound focus, enabling optical focusing within deep tissues. In digital TRUE, we time reverse ultrasound-tagged light with an optoelectronic time reversal device (the digital optical phase conjugate mirror, DOPC). The use of the DOPC enables high optical gain, allowing for high intensity optical focusing and focal fluorescence imaging in thick tissues at a lateral resolution of 36 µm by 52 µm. The resolution of the TRUE approach is fundamentally limited to that of the wavelength of ultrasound. The ultrasound focus (~ tens of microns wide) usually contains hundreds to thousands of optical modes, such that the scattered wavefront measured is a linear combination of the contributions of all these optical modes. In TROVE, we make use of our ability to digitally record, analyze and manipulate the scattered wavefront to demix the contributions of these spatial modes using variance encoding. In essence, we encode each spatial mode inside the scattering sample with a unique variance, allowing us to computationally derive the time reversal wavefront that corresponds to a single optical mode. In doing so, we uncouple the system resolution from the size of the ultrasound focus, demonstrating optical focusing and imaging between highly diffusing samples at an unprecedented, speckle-scale lateral resolution of ~ 5 µm. Our methods open up the possibility of fully exploiting the prowess and versatility of biomedical optics in deep tissues.