Quantization and erasures in frame representations

Thesis (Sc. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 123-126).Frame representations, which correspond to overcomplete generalizations to basis expansions, are often used in signal processing to provide robustness to errors. In this thesis robustness is provided through the use of projections to compensate for errors in the representation coefficients, with specific focus on quantization and erasure errors. The projections are implemented by modifying the unaffected coefficients using an additive term, which is linear in the error. This low-complexity implementation only assumes linear reconstruction using a pre-determined synthesis frame, and makes no assumption on how the representation coefficients are generated. In the context of quantization, the limits of scalar quantization of frame representations are first examined, assuming the analysis is using inner products with the frame vectors. Bounds on the error and the bit-efficiency are derived, demonstrating that scalar quantization of the coefficients is suboptimal. As an alternative to scalar quantization, a generalization of Sigma-Delta noise shaping to arbitrary frame representations is developed by reformulating noise shaping as a sequence of compensations for the quantization error using projections.(cont.) The total error is quantified using both the additive noise model of quantization, and a deterministic upper bound based on the triangle inequality. It is thus shown that the average and the worst-case error is reduced compared to scalar quantization of the coefficients. The projection principle is also used to provide robustness to erasures. Specifically, the case of a transmitter that is aware of the erasure occurrence is considered, which compensates for the erasure error by projecting it to the subsequent frame vectors. It is further demonstrated that the transmitter can be split to a transmitter/receiver combination that performs the same compensation, but in which only the receiver is aware of the erasure occurrence. Furthermore, an algorithm to puncture dense representations in order to produce sparse approximate ones is introduced. In this algorithm the error due to the puncturing is also projected to the span of the remaining coefficients. The algorithm can be combined with quantization to produce quantized sparse representations approximating the original dense representation.by Petros T. Boufounos.Sc.D

Multiple-Description Coding by Dithered Delta-Sigma Quantization

We address the connection between the multiple-description (MD) problem and Delta-Sigma quantization. The inherent redundancy due to oversampling in Delta-Sigma quantization, and the simple linear-additive noise model resulting from dithered lattice quantization, allow us to construct a symmetric and time-invariant MD coding scheme. We show that the use of a noise shaping filter makes it possible to trade off central distortion for side distortion. Asymptotically as the dimension of the lattice vector quantizer and order of the noise shaping filter approach infinity, the entropy rate of the dithered Delta-Sigma quantization scheme approaches the symmetric two-channel MD rate-distortion function for a memoryless Gaussian source and MSE fidelity criterion, at any side-to-central distortion ratio and any resolution. In the optimal scheme, the infinite-order noise shaping filter must be minimum phase and have a piece-wise flat power spectrum with a single jump discontinuity. An important advantage of the proposed design is that it is symmetric in rate and distortion by construction, so the coding rates of the descriptions are identical and there is therefore no need for source splitting.Comment: Revised, restructured, significantly shortened and minor typos has been fixed. Accepted for publication in the IEEE Transactions on Information Theor

Anthropomorphic Coding of Speech and Audio: A Model Inversion Approach

Auditory modeling is a well-established methodology that provides insight into human perception and that facilitates the extraction of signal features that are most relevant to the listener. The aim of this paper is to provide a tutorial on perceptual speech and audio coding using an invertible auditory model. In this approach, the audio signal is converted into an auditory representation using an invertible auditory model. The auditory representation is quantized and coded. Upon decoding, it is then transformed back into the acoustic domain. This transformation converts a complex distortion criterion into a simple one, thus facilitating quantization with low complexity. We briefly review past work on auditory models and describe in more detail the components of our invertible model and its inversion procedure, that is, the method to reconstruct the signal from the output of the auditory model. We summarize attempts to use the auditory representation for low-bit-rate coding. Our approach also allows the exploitation of the inherent redundancy of the human auditory system for the purpose of multiple description (joint source-channel) coding

Signal Reconstruction from Frame and Sampling Erasures

This dissertation is concerned with the efficient reconstruction of signals from frame coefficient erasures at known locations. Three methods of perfect reconstruction from frame coefficient erasures will be discussed. These reconstructions are more efficient than older methods in the literature because they only require an L × L matrix inversion, where L denotes the cardinality of the erased set of indices. This is a significant improvement over older methods which require an n × n matrix inversion, where n denotes the dimension of the underlying Hilbert space. The first of these methods, called Nilpotent Bridging, uses a small subset of the non-erased coefficients to reconstruct the erased coefficients. This subset is called the bridge set. To perform the reconstruction an equation, known as the bridge equation, must be solved. A proof is given that under a very mild assumption there exists a bridge set of size L for which the bridge equation has a solution. A stronger result is also proven that shows that for a very large class of frames, no bridge set search is required. We call this set of frames the set of full skew-spark frames. Using the Baire Category Theorem and tools from Matrix Theory, the set of full skew-spark frames is shown to be an open, dense subset of the set of all frames in finite dimensions. The second method of reconstruction is called Reduced Direct Inversion because it provides a basis-free, closed-form formula for inverting a particular n × n matrix, which only requires the inversion of an L × L matrix. By inverting this matrix we obtain another efficient reconstruction formula. The final method considered is a continuation of work by Han and Sun. The method utilizes an Erasure Recovery Matrix, which is a matrix that annihilates the range of the analysis operator for a frame. Because of this, the erased coefficients can be reconstructed using a simple pseudo-inverse technique. For each method, a discussion of the stability of our algorithms is presented. In particular, we present numerical experiments to investigate the effects of normally distributed additive channel noise on our reconstruction. For Reduced Direct Inversion and Erasure Recovery Matrices, using the Restricted Isometry Property, we construct classes of frames which are numerically robust to sparse channel noise. For these frames, we provide error bounds for sparse channel noise

The Dynamics of Variability in Introductory Physics Students' Thinking: Examples from Kinematics

Physics education research has long emphasized the need for physics instruction to address students' existing intuitions about the physical world as an integral part of learning physics. Researchers, however, have not reached a consensus-view concerning the nature of this intuitive knowledge or the specific role that it does (or might) play in physics learning. While many early characterizations of student misconceptions cast students' intuitive thinking as largely static, unitary in structure, and counter-productive for the purpose of learning correct physics, much of contemporary research supports a conceptualization of intuitive thought as dynamic, manifold in structure, and generative in the development of expertise. This dissertation contributes to ongoing inquiry into the nature of students' intuitive thought and its role in learning physics through the pursuit of dynamic systems characterizations of student reasoning, with a particular focus on how students settle into and shift among multiple patterns of reasoning about motion. In one thread of this research, simple experimental designs are used to demonstrate how individual students can be predictably biased toward and away from different ways of thinking about the same physical situation when specific parameters of questions posed to students are varied. I qualitatively model students' thinking in terms of the activations and interactions among fine-grained intuitive knowledge and static features of the context. In a second thread of this research, case studies of more dynamic shifts in students' conceptual reasoning are developed from videos of student discussions during collaborative classroom activities. These show multiple local stabilities of students' thinking as well, with evidence of group-level dynamics shifting on the time scale of minutes. This work contributes to existing research paradigms that aim to characterize student thinking in physics education in two important ways: (1) through the use of methods that allow for forms of empirical accountability that connect descriptive models of student thinking to experimental data, and (2) through the theoretical development of explanatory mechanisms that account for patterns in students' reasoning at multiple levels of analysis

Recent Trends in Communication Networks

In recent years there has been many developments in communication technology. This has greatly enhanced the computing power of small handheld resource-constrained mobile devices. Different generations of communication technology have evolved. This had led to new research for communication of large volumes of data in different transmission media and the design of different communication protocols. Another direction of research concerns the secure and error-free communication between the sender and receiver despite the risk of the presence of an eavesdropper. For the communication requirement of a huge amount of multimedia streaming data, a lot of research has been carried out in the design of proper overlay networks. The book addresses new research techniques that have evolved to handle these challenges

Cross-Cutting Narratives of Opioid Use Disorder Among Pregnant Women and Mothers: Implications for Humanistic Care

Opioid-related fatalities in the U.S. have increased drastically. Pregnant women and mothers with opioid use disorders (OUD) are a rapidly growing and vulnerable population. Using a critical narrative approach, this dissertation examines how the syndemic of trauma, substance use, and mental health conditions influences opioid use and treatment trajectories among pregnant women and mothers across the lifecourse. The goal of this dissertation was to examine three discursive resources that shape the social construction of perinatal and maternal opioid use across all strata of social life: macro-level (news media), meso-level (scientific), and micro-level (individual) narratives. Informed by 18-months of ethnographic observation, in-depth interviews, and mixed methods analysis of scientific research and news media coverage of perinatal and maternal OUD, this research brings together the voices of women in recovery, clinicians, social workers, policymakers, and the public. Building from what Sanders (2014) refers to as the “gendered double standard” faced by women with substance use disorders, in this dissertation I characterize the intersecting identities of female, pregnant/mother, and substance user as a gendered triple standard. Throughout this work I argue that being held to this gendered triple standard intensifies the stigma faced by pregnant women and mothers with OUD as they navigate the medical, legal, and social service institutions. Key findings from this research include: (1) a predominating focus on “fetal victimhood” (Knight, 2015), which overlooks the needs of pregnant women and mothers with OUD that run concurrent to ensuring a healthy pregnancy and birth; (2) approaches to addressing the opioid “crisis” that elide key at-risk populations (e.g. People of Color, active substance users, and polysubstance users); (3) “folk” pharmacokinetic knowledge and practices utilized by pregnant women and mothers that function as both facilitators and barriers to treatment engagement; and (4) the traumas associated with institutional policies and procedures specific to the management of opioid use (e.g., practices associated with civil commitment to treatment, loss of custody, and intergenerational family separation via the criminal justice and/or foster care systems). Informed by key findings, this dissertation concludes with five specific recommendations for research and practice