The limitations of commonly used separable extensions of one-dimensional transforms, such as the Fourier and wavelet transforms, in capturing the geometry of image edges are well known. In this paper, we pursue a "true" two-dimensional transform that can capture the intrinsic geometrical structure that is key in visual information. The main challenge in exploring geometry in images comes from the discrete nature of the data. Thus, unlike other approaches, such as curvelets, that first develop a transform in the continuous domain and then discretize for sampled data, our approach starts with a discrete-domain construction and then studies its convergence to an expansion in the continuous domain. Specifically, we construct a discrete-domain multiresolution and multidirection expansion using nonseparable filter banks, in much the same way that wavelets were derived from filter banks. This construction results in a flexible multiresolution, local, and directional image expansion using contour segments, and, thus, it is named the contourlet transform. The discrete contourlet transform has a fast iterated filter bank algorithm that requires an order N operations for N-pixel images. Furthermore, we establish a precise link between the developed filter bank and the associated continuous-domain contourlet expansion via a directional multiresolution analysis framework. We show that with parabolic scaling and sufficient directional vanishing moments, contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves. Finally, we show some numerical experiments demonstrating the potential of contourlets in several image processing applications.

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The contourlet transform: an efficient directional multiresolution image representation

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

1 Introduction to Wavelets 2 A Multiresolution Formulation of Wavelet Systems 3 Filter Banks and the Discrete Wavelet Transform 4 Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Tight Frames, and Unconditional Bases 5 The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients 6 Regularity, Moments, and Wavelet System Design 7 Generalizations of the Basic Multiresolution Wavelet System 8 Filter Banks and Transmultiplexers 9 Calculation of the Discrete Wavelet Transform 10 Wavelet-Based Signal Processing and Applications 11 Summary Overview 12 References Bibliography Appendix A Derivations for Chapter 5 on Scaling Functions Appendix B Derivations for Section on Properties Appendix C Matlab Programs Index

Introduction to Wavelets and Wavelet Transforms: A Primer

In this paper, we develop the nonsubsampled contourlet transform (NSCT) and study its applications. The construction proposed in this paper is based on a nonsubsampled pyramid structure and nonsubsampled directional filter banks. The result is a flexible multiscale, multidirection, and shift-invariant image decomposition that can be efficiently implemented via the a trous algorithm. At the core of the proposed scheme is the nonseparable two-channel nonsubsampled filter bank (NSFB). We exploit the less stringent design condition of the NSFB to design filters that lead to a NSCT with better frequency selectivity and regularity when compared to the contourlet transform. We propose a design framework based on the mapping approach, that allows for a fast implementation based on a lifting or ladder structure, and only uses one-dimensional filtering in some cases. In addition, our design ensures that the corresponding frame elements are regular, symmetric, and the frame is close to a tight one. We assess the performance of the NSCT in image denoising and enhancement applications. In both applications the NSCT compares favorably to other existing methods in the literature

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The Nonsubsampled Contourlet Transform: Theory, Design, and Applications

The outline of this chapter is as follows. Section 2 reviews various types of existing finite impulse response (FIR) and infinite impulse response (IIR) two-channel filter banks. The basic operations of these filter banks are considered and the requirements are stated for alias-free, perfect-reconstruction (PR), and nearly perfect-reconstruction (NPR) filter banks. Also some efficient synthesis techniques are referred to. Furthermore, examples are included to compare various two-channel filter banks with each other. Section 3 concentrates on the design of multi-channel (M-channel) uniform filter banks. The main emphasis is laid on designing these banks using tree-structured filter banks with the aid of two-channel filter banks and on generating the overall bank with the aid of a single prototype filter and a proper cosine-modulation or MDFT technique. In Section 4, it is shown how octave filter banks can be generated using a single two-channel filter bank as the basic building block. Also, the relations between the frequency-selective octave filter banks and discrete-time wavelet banks are briefly discussed. Finally, concluding remarks are given in Section 5.

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Multirate Systems and Filter Banks

Proposes a novel framework for a new class of two-channel biorthogonal filter banks. The framework covers two useful subclasses: i) causal stable IIR filter banks. ii) linear phase FIR filter banks. There exists a very efficient structurally perfect reconstruction implementation for such a class. Filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity. The properties of such a class are discussed in detail. The design of the analysis/synthesis systems reduces to the design of a single transfer function. Very simple design methods are given both for FIR and IIR cases. Zeros of arbitrary multiplicity at aliasing frequency can be easily imposed, for the purpose of generating wavelets with regularity property. In the IIR case, two new classes of IIR maximally flat filters different from Butterworth filters are introduced. The filter coefficients are given in closed form. The wavelet bases corresponding to the biorthogonal systems are generated. the authors also provide a novel mapping of the proposed 1-D framework into 2-D. The mapping preserves the following: i) perfect reconstruction; ii) stability in the IIR case; iii) linear phase in the FIR case; iv) zeros at aliasing frequency; v) frequency characteristic of the filters. >

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A new class of two-channel biorthogonal filter banks and wavelet bases

We consider the minimization of uncoded bit error rate (BER) for the orthogonal frequency division multiplexing (OFDM) system with an orthogonal precoder. We analyze the BER performance of precoded OFDM systems with zero forcing and minimum mean squared error (MMSE) receivers. In the case of MMSE receivers, we show that for quadrature phase shift keying (QPSK), there exists a class of optimal precoders that are channel independent. Examples of this class include the discrete Fourier transform (DFT) matrix and the Hadamard matrix. When the precoder is the DFT matrix, the resulting optimal transceiver becomes the single carrier system with cyclic prefix (SC-CP) system. We also show that the worst solution corresponds to the conventional OFDM system; the conventional OFDM system has the largest BER. In the case of zero forcing receivers, the design of optimal transceiver depends on the signal-to-noise ratio (SNR). For higher SNR, solutions of optimal precoders are the same as those of MMSE receivers.

https://ir.nctu.edu.tw:443/bitstream/11536/27544/1/000184861900011.pdf

BER minimized OFDM systems with channel independent precoders

Presenting the first complete treatment of MIMO transceiver optimization, this self-contained book provides all the mathematical information needed to understand transceiver optimization in a single volume. It begins with a review of digital communication fundamentals, and then moves on to a detailed study of joint transceiver optimization, starting from simple single-input single-output channels all the way to minimum bit error rate transceivers for MIMO channels. Crucial background material is covered, such as Schur convex functions, matrix calculus, and constrained optimization, together with eight appendices providing further background material on topics such as matrix theory, random processes, and sampling theory. A final ninth appendix provides a grand summary of all the optimization results. With 360 illustrations, over 70 worked examples, and numerous summary tables provided to aid understanding of key concepts, this book is ideal for graduate students, practitioners, and researchers in the fields of communications and signal processing.

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Signal Processing and Optimization for Transceiver Systems

Filter Bank Transceivers For Ofdm and Dmt Systems: Preliminaries of digital communications

There has been great interest in the design of filterbank transceivers. Usually, with proper time domain equalization, the channel is modeled as an FIR filter. It is known that for FIR channels, the introduction of certain redundancy allows the receiver to cancel intersymbol interference (ISI) completely, and channel equalization is performed implicitly using FIR transceivers. This scheme allows us to trade bandwidth for ISI cancellation. In this paper, we derive the minimum redundancy required for the existence of FIR transceivers for a given channel. We see that the minimum redundancy is directly related to the zeros of the channel and to the Smith form of an appropriately defined channel matrix.

https://ir.nctu.edu.tw:443/bitstream/11536/28885/1/000174492400008.pdf

See-May Phoong

Papers

A new class of two-channel biorthogonal filter banks and wavelet bases

BER minimized OFDM systems with channel independent precoders

Signal Processing and Optimization for Transceiver Systems

Filter Bank Transceivers For Ofdm and Dmt Systems: Preliminaries of digital communications

Minimum redundancy for ISI free FIR filterbank transceivers