Nguyen T. Thao

Set theoretic compression with an application to image coding

We discuss the concept of set theoretic compression where the input signal is implicitly encoded by the specification of a set which contains it, rather than an estimate which approximates it. This approach assumes the reconstruction of an estimate from the encoded set information only at the decoder side. We explain the motivations of this approach for high signal compression and encoding simplification, and the implication of more complex decoding. We then present the tools to support the approach. We finally show a demonstration of this approach in a particular application of image coding.<<ETX>>

Quantization of overcomplete expansions

We present a method that represents a signal with respect to an overcomplete set of vectors which we call a dictionary. The use of overcomplete sets of vectors (redundant bases or frames) together with quantization is explored as an alternative to transform coding for signal compression. The goal is to retain the computational simplicity of transform coding while adding flexibility like adaptation to signal statistics. We show results using both fixed quantization in frames and greedy quantization using matching pursuit. An MSE slope of -6 dB/octave of frame redundancy is shown for a particular tight frame and is verified experimentally for another frame.

A hybrid fractal-DCT coding scheme for image compression

We introduce a new way to use fractal coding for image compression, based on the parallel use of a fractal encoder and a DCT encoder. The two encoders are given the complementary roles to capture the information of edge and smooth variation, and the information of detail respectively. We show the advantage of using this hybrid coding scheme over the use of a fractal encoder alone, or a DCT encoder alone. This coding scheme is also the occasion to demonstrate a new concept of coding by nonlinear feature separation based on regular and uniform algorithms, suitable for real-time VLSI implementation.

Vector quantization analysis of SD modulation

When considering a class of finite-dimensional input signals such as bandlimited signals in the discrete Fourier sense within a finite time window [O,T] of observation, we show that a single-loop ΣΔ modulator behaves like the encoder of a vector quantizer. The intrinsic behavior of the modulator can be studied by analyzing the partition it generates in the input space. We show that this partition yields a particular structure that we call the “hyperplane wave structure”. As one consequence to this analysis, it can be proved for the considered class of bandlimited signals that the mean squared error (MSE) of reconstruction cannot asymptotically decrease with the oversampling ratio R faster than O(R-4), regardless of the type of reconstruction used. We then generalize this analysis to n-loop ΣΔ modulators and show that the MSE cannot decrease faster than O(R-(2n+2))

Set theoretic inverse halftoning

We give a general method to recover a continuous-tone image from a halftone image in the situation where the halftoning algorithm is known. Using rigorous mathematical tools, the method combines what we call the visual information and the coding information of the halftone data. Although we concentrate on halftoning by error diffusion, the method is applicable to all digital halftoning techniques.

Deterministic analysis of sigma-delta modulation for linear and non-linear signal reconstruction

Deterministic analysis is becoming increasingly popular to study the theoretical mechanisms of ΣΔ modulation, as opposed to the classical noise-shaping approach. This paper gives an overview of its application to the specific problem of signal reconstruction. Once an input signal has been processed through a ΣΔ modulator of given specifications, the fundamental question is to know what deterministic information can be retrieved about this signal from the ΣΔ output. The paper summarizes all results obtained in this research under a unified presentation framework.

Lower bound on the mean squared error in multi-loop /spl Sigma//spl Delta/ modulation with periodic bandlimited signals

Research has been devoted to optimal signal reconstruction by /spl Sigma//spl Delta/ modulation. Using classical linear filtering, the reconstruction mean squared error (MSE) asymptotically decreases with the oversampling ratio R by R/sup -(2n+1)/, where n is the order of the modulator. Using a nonlinear reconstruction scheme, an improvement of the MSE asymptotic behavior by 3 dB per octave of R has been observed for time-varying, periodic and bandlimited input signals. This implies an MSE of /spl Oscr/(R/sup -(2n+2)/). We show for the multi-loop configuration of the /spl Sigma//spl Delta/ modulation and with the same kind of input signals that the asymptotic behavior of the reconstruction MSE cannot be less than /spl Oscr/(R/sup -(2n+2)/), including optimal reconstruction.<<ETX>>

Lower bound on the mean squared error in oversampled quantization of periodic signals

We analyze optimal reconstruction in oversampled A/D conversion, in the case where the input signals are bandlimited and periodic. We show that the reconstruction MSE cannot decrease with the oversampling ratio faster than /spl Oscr/(R/sup (-2)/).<<ETX>>

Local search fractal image compression for fast integrated implementation

The well known drawback of fractal image coding is its heavy computation complexity. In this paper, we show however that only a very small portion of this complexity is needed to encode the contours and the smooth areas of the objects of a certain minimum size. This is simply achieved by a local search of the matching domain blocks without isometry. Therefore, instead of forcing the fractal coder to take care of all types of feature in the image, we limit itself to encoding the features of the type mentioned above. We give the role of capturing the missing detailed features to a DCT coder. This is much more cost effective than using a complete self-sufficient fractal coder. We provide a special method of fractal-DCT output recombination which prevents the introduction of the classic DCT artifacts while preserving the qualities of the local search fractal information. We give experimental results at the given total bit rate of 0.18 bit/pixel.

Analysis framework for the digital inversion of nonlinear deterministic characteristics in /spl Sigma//spl Delta/ modulation

The classical noise-shaping analysis provides a framework to invert any existing distortion in the transfer function of a /spl Sigma//spl Delta/ modulator, provided that the distortion has the form of a time-invariant and linear operation. We establish here a more general algebraic framework for the analysis of /spl Sigma//spl Delta/ modulation, which can explicitly take into account any nonlinear deterministic distortion. We show in this paper that this framework permits the algebraic inversion of nonlinear characteristics in the specific case of static ADC-DAC nonuniformity, which can be seen as a generalization of the linear noise-shaping inversion formula. From this algebraic inversion formula, the classical qualitative knowledge of the behavior of /spl Sigma//spl Delta/ modulation with regard to ADC-DAC nonuniformity can be reexplained in a concise and formal manner, with some new insights and results.