Design of doubly-complementary IIR digital filters using a single complex allpass filter, with multirate applications

It is shown that a large class of real-coefficient doubly-complementary IIR transfer function pairs can be implemented by means of a single complex allpass filter. For a real input sequence, the real part of the output sequence corresponds to the output of one of the transfer functions G(z) (for example, lowpass), whereas the imaginary part of the output sequence corresponds to its "complementary" filter H(z)(for example, highpass). The resulting implementation is structurally lossless, and hence the implementations of G(z) and H(z) have very low passband sensitivity. Numerical design examples are included, and a typical numerical example shows that the new implementation with 4 bits per multiplier is considerably better than a direct form implementation with 9 bits per multiplier. Multirate filter bank applications (quadrature mirror filtering) are outlined

各種の性質を改善した直交DTCWTの設計に関する研究

The Dual tree complex wavelet transforms (DTCWTs) have been found to be successful in many applications of signal and image processing. DTCWTs employ two real wavelet transforms, where one wavelet corresponds to the real part of complex wavelet and the other is the imaginary part. Two wavelet bases are required to be a Hilbert transform pair. Thus, DTCWTs are nearly shift invariant and have a good directional selectivity in two or higher dimensions with limited redundancies. In this dissertation, we propose two new classes of DTCWTs with improved properties. In Chapter 2, we review the Fourier transform at first and then introduce the fundamentals of dual tree complex wavelet transform. The wavelet transform has been proved to be a successful tool to express the signal in time and frequency domain simultaneously. To obtain the wavelet coefficients efficiently, the discrete wavelet transform has been introduced since it can be achieved by a tree of two-channel filter banks. Then, we discuss the design conditions of two-channel filter banks, i.e., the perfect reconstruction and orthonormality. Additionally, some properties of scaling and wavelet functions including orthonormality, symmetry and vanishing moments are also given. Moreover, the structure of DTCWT is introduced, where two wavelet bases are required to form a Hilbert transform pair. Thus, the corresponding scaling lowpass filters must satisfy the half-sample delay condition. Finally, the objective measures of quality are given to evaluate the performance of the complex wavelet. In Chapter 3, we propose a new class of DTCWTs with improved analyticity and frequency selectivity by using general IIR filters with numerator and denominator of different degree. In the common-factor technique proposed by Selesnick, the maximally at allpass filter was used to satisfy the halfsample delay condition, resulting in poor analyticity of complex wavelets. Thus, to improve the analyticity of complex wavelets, we present a method for designing allpass filters with the specified degree of flatness and equiripple phase response in the approximation band. Moreover, to improve the frequency selectivity of scaling lowpass filters, we locate the specified number of zeros at z = -1 and minimize the stopband error. The well-known Remez exchange algorithm has been applied to approximate the equiripple response. Therefore, a set of filter coefficients can be easily obtained by solving the eigenvalue problem. Furthermore, we investigate the performance on the proposed DTCWTs and dedicate how to choose the approximation band and stopband properly. It is shown that the conventional DTCWTs proposed by Selesnick are only the special cases of DTCWTs proposed in this dissertation. In Chapter 4, we propose another class of almost symmetric DTCWTs with arbitrary center of symmetry. We specify the degree of flatness of group delay, and the number of vanishing moments, then apply the Remez exchange algorithm to minimize the difference between two scaling lowpass filters in the frequency domain, in order to improve the analyticity of complex wavelets. Therefore, the equiripple behaviour of the error function can be obtained through a few iterations. Moreover, two scaling lowpass filters can be obtained simultaneously. As a result, the complex wavelets are orthogonal and almost symmetric, and have the improved analyticity. Since the group delay of scaling lowpass filters can be arbitrarily specified, the scaling functions have the arbitrary center of symmetry. Finally, several experiments of signal denoising are carried out to demonstrate the efficiency of the proposed DTCWTs. It is clear that the proposed DTCWTs can achieve better performance on noise reduction.電気通信大学201

Method and apparatus for predicting the direction of movement in machine vision

A computer-simulated cortical network is presented. The network is capable of computing the visibility of shifts in the direction of movement. Additionally, the network can compute the following: (1) the magnitude of the position difference between the test and background patterns; (2) localized contrast differences at different spatial scales analyzed by computing temporal gradients of the difference and sum of the outputs of paired even- and odd-symmetric bandpass filters convolved with the input pattern; and (3) the direction of a test pattern moved relative to a textured background. The direction of movement of an object in the field of view of a robotic vision system is detected in accordance with nonlinear Gabor function algorithms. The movement of objects relative to their background is used to infer the 3-dimensional structure and motion of object surfaces

A New Reduction Scheme for Gaussian Sum Filters

In many signal processing applications it is required to estimate the unobservable state of a dynamic system from its noisy measurements. For linear dynamic systems with Gaussian Mixture (GM) noise distributions, Gaussian Sum Filters (GSF) provide the MMSE state estimate by tracking the GM posterior. However, since the number of the clusters of the GM posterior grows exponentially over time, suitable reduction schemes need to be used to maintain the size of the bank in GSF. In this work we propose a low computational complexity reduction scheme which uses an initial state estimation to find the active noise clusters and removes all the others. Since the performance of our proposed method relies on the accuracy of the initial state estimation, we also propose five methods for finding this estimation. We provide simulation results showing that with suitable choice of the initial state estimation (based on the shape of the noise models), our proposed reduction scheme provides better state estimations both in terms of accuracy and precision when compared with other reduction methods

Efficient algorithm for solving semi-infinite programming problems and their applications to nonuniform filter bank designs

An efficient algorithm for solving semi-infinite programming problems is proposed in this paper. The index set is constructed by adding only one of the most violated points in a refined set of grid points. By applying this algorithm for solving the optimum nonuniform symmetric/antisymmetric linear phase finite-impulse-response (FIR) filter bank design problems, the time required to obtain a globally optimal solution is much reduced compared with that of the previous proposed algorith

Sampling and Reconstruction of Sparse Signals on Circulant Graphs - An Introduction to Graph-FRI

With the objective of employing graphs toward a more generalized theory of signal processing, we present a novel sampling framework for (wavelet-)sparse signals defined on circulant graphs which extends basic properties of Finite Rate of Innovation (FRI) theory to the graph domain, and can be applied to arbitrary graphs via suitable approximation schemes. At its core, the introduced Graph-FRI-framework states that any K-sparse signal on the vertices of a circulant graph can be perfectly reconstructed from its dimensionality-reduced representation in the graph spectral domain, the Graph Fourier Transform (GFT), of minimum size 2K. By leveraging the recently developed theory of e-splines and e-spline wavelets on graphs, one can decompose this graph spectral transformation into the multiresolution low-pass filtering operation with a graph e-spline filter, and subsequent transformation to the spectral graph domain; this allows to infer a distinct sampling pattern, and, ultimately, the structure of an associated coarsened graph, which preserves essential properties of the original, including circularity and, where applicable, the graph generating set.Comment: To appear in Appl. Comput. Harmon. Anal. (2017