Combinatorial Representations of Block Codes

We look at graphical descriptions of block codes known as trellises, which illustrate connections between algebra and graph theory, and can be used to develop powerful decoding algorithms. Trellis sizes for linear block codes are known to grow exponentially with the code parameters. Of considerable interest to coding theorists therefore, are more compact descriptions called tail-biting trellises which in some cases can be much smaller than any conventional trellis for the same code . We derive some interesting properties of tail-biting trellises and present a new decoding algorithm.

Geometric representations of graphs in low dimension

We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in $1.5 (\Delta + 2) \ln n$ dimensions, where $\Delta$ is the maximum degree of G. We also show that $\boxi(G) \le (\Delta + 2) \ln n$ for any graph G. Our bound is tight up to a factor of $\ln n$. We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree $\Delta$, we show that for almost all graphs on n vertices, its boxicity is upper bound by $c\cdot(d_{av} + 1) \ln n$ where d_{av} is the average degree and c is a small constant. Also, we show that for any graph G, $\boxi(G) \le \sqrt{8 n d_{av} \ln n}$, which is tight up to a factor of $b \sqrt{\ln n}$ for a constant b.

Computational representations for multi state design tasks and enumeration of mechanical device behaviour

The role of a computer emerged from modeling and analyzing concepts (ideas) to generate concepts. Research into methods for supporting conceptual design using automated synthesis had attracted much attention in the past decades. To find out how designers synthesize solution concepts for multi-state mechanical devices, ten experimental studies were conducted. Observations from these empirical studies would be used as the basis to develop knowledge involved in the multi-state design synthesis process. In this paper, we propose a computational representation for expressing the multi-state design task and for enumerating multi-state behaviors of kinematic pairs and mechanisms. This computational representation would be used to formulate computational methods for the synthesis process to develop a system for supporting design synthesis of multiple state mechanical devices by generating a comprehensive variety of solution alternatives.

Use of Exocentric and Egocentric Representations in the Concurrent Planning of Sequential Saccades

The concurrent planning of sequential saccades offers a simple model to study the nature of visuomotor transformations since the second saccade vector needs to be remapped to foveate the second target following the first saccade. Remapping is thought to occur through egocentric mechanisms involving an efference copy of the first saccade that is available around the time of its onset. In contrast, an exocentric representation of the second target relative to the first target, if available, can be used to directly code the second saccade vector. While human volunteers performed a modified double-step task, we examined the role of exocentric encoding in concurrent saccade planning by shifting the first target location well before the efference copy could be used by the oculomotor system. The impact of the first target shift on concurrent processing was tested by examining the end-points of second saccades following a shift of the second target during the first saccade. The frequency of second saccades to the old versus new location of the second target, as well as the propagation of first saccade localization errors, both indices of concurrent processing, were found to be significantly reduced in trials with the first target shift compared to those without it. A similar decrease in concurrent processing was obtained when we shifted the first target but kept constant the second saccade vector. Overall, these results suggest that the brain can use relatively stable visual landmarks, independent of efference copy-based egocentric mechanisms, for concurrent planning of sequential saccades.

Flexible representations of dynamics are used in object manipulation.

To manipulate an object skillfully, the brain must learn its dynamics, specifying the mapping between applied force and motion. A fundamental issue in sensorimotor control is whether such dynamics are represented in an extrinsic frame of reference tied to the object or an intrinsic frame of reference linked to the arm. Although previous studies have suggested that objects are represented in arm-centered coordinates [1-6], all of these studies have used objects with unusual and complex dynamics. Thus, it is not known how objects with natural dynamics are represented. Here we show that objects with simple (or familiar) dynamics and those with complex (or unfamiliar) dynamics are represented in object- and arm-centered coordinates, respectively. We also show that objects with simple dynamics are represented with an intermediate coordinate frame when vision of the object is removed. These results indicate that object dynamics can be flexibly represented in different coordinate frames by the brain. We suggest that with experience, the representation of the dynamics of a manipulated object may shift from a coordinate frame tied to the arm toward one that is linked to the object. The additional complexity required to represent dynamics in object-centered coordinates would be economical for familiar objects because such a representation allows object use regardless of the orientation of the object in hand.

Neural routing circuits for forming invariant representations of visual objects

This thesis presents a biologically plausible model of an attentional mechanism for forming position- and scale-invariant representations of objects in the visual world. The model relies on a set of control neurons to dynamically modify the synaptic strengths of intra-cortical connections so that information from a windowed region of primary visual cortex (Vl) is selectively routed to higher cortical areas. Local spatial relationships (i.e., topography) within the attentional window are preserved as information is routed through the cortex, thus enabling attended objects to be represented in higher cortical areas within an object-centered reference frame that is position and scale invariant. The representation in V1 is modeled as a multiscale stack of sample nodes with progressively lower resolution at higher eccentricities. Large changes in the size of the attentional window are accomplished by switching between different levels of the multiscale stack, while positional shifts and small changes in scale are accomplished by translating and rescaling the window within a single level of the stack. The control signals for setting the position and size of the attentional window are hypothesized to originate from neurons in the pulvinar and in the deep layers of visual cortex. The dynamics of these control neurons are governed by simple differential equations that can be realized by neurobiologically plausible circuits. In pre-attentive mode, the control neurons receive their input from a low-level "saliency map" representing potentially interesting regions of a scene. During the pattern recognition phase, control neurons are driven by the interaction between top-down (memory) and bottom-up (retinal input) sources. The model respects key neurophysiological, neuroanatomical, and psychophysical data relating to attention, and it makes a variety of experimentally testable predictions.