Amy E. Bell

A comparison of hardware implementations of the biorthogonal 9/7 DWT: convolution versus lifting

The filter bank approach for computing the discrete wavelet transform (DWT), which we call the convolution method, can employ either a nonpolyphase or polyphase structure. This work compares filter banks with an alternative polyphase structure for calculating the DWT-the lifting method. We look at the traditional lifting structure and a recently proposed flipping structure for implementing lifting. All filter bank structures are implemented on an Altera field-programmable gate array. The quantization of the coefficients (for implementation in fixed-point hardware) plays a crucial role in the performance of all structures, affecting both image compression quality and hardware metrics. We design several quantization methods and compare the best design for each approach: the nonpolyphase filter bank, the polyphase filter bank, the lifting and flipping structures. The results indicate that for the same image compression performance, the flipping structure gives the smallest and fastest, low-power hardware.

Phase retrieval using the wavelet transform

The 1-D phase retrieval problem is to reconstruct a signal given the modulus of its Fourier transform. Conventional solution techniques approximate the solution by discretizing this continuous problem. Our solution obviates this approximation by employing basis functions to better represent a continuous signal. In particular, we use wavelet bases because they provide a structure which permits the use of a fast algorithm and because they incorporate a priori knowledge of the signal. This paper develops our algorithm and illustrates its effectiveness with numerical examples.

Multiplierless filter Bank design: structures that improve both hardware and image compression performance

Design techniques for high-performance, fixed-point, multiplierless filter banks are presented. Image compression using the biorthogonal 9/7 discrete wavelet transform provides a motivating example. Image compression and hardware performance of two commonly used filter structures, direct and cascade, and two known filter bank structures, nonpolyphase and polyphase, are compared. A technique is shown for designing a fixed-point polyphase filter structure, which is highly efficient from a hardware standpoint, such that image-compression quality is not significantly deteriorated by the use of fixed-point mathematics. The result is a polyphase structure with about twice the throughput rate of nonpolyphase structures, and peak signal-to-noise ratio values for lossy compression within 0.2 decibels of those achieved using floating-point filters.

Optimal quantized lifting coefficients for the 9/7 wavelet [image compression applications]

The lifting structure has been shown to be computationally efficient for implementing filter banks. The hardware implementation of a filter bank requires that the lifting coefficients be quantized. The quantization method determines compression performance, hardware size, hardware speed and energy. We investigate the implementation of two lifting coefficient sets, rational and irrational, for the biorthogonal 9/7 wavelet. Six different approaches are used to find optimal quantized lifting coefficients from these sets. We find that the best hardware and PSNR performance is obtained using the rational coefficient set quantized with gain compensation and lumped scaling.

1-D continuous non-minimum phase retrieval using the wavelet transform

The phase retrieval problem arises when a signal must be reconstructed from only the magnitude of its Fourier transform; if the phase information were also available, the signal could simply be synthesized using the inverse Fourier transform. In continuous phase retrieval, most previous solutions rely on discretizing the problem and then employing an iterative algorithm. We avoid this approximation by using wavelet expansions to transform this uncountably infinite problem into a linear system of equations. The wavelet bases permit a solution by incorporating a priori signal information and they provide a structured system of equations which results in a fast algorithm. Our solutions obviate the stagnation problems associated with iterative algorithms, they are computationally simpler and more stable than previous non-iterative algorithms, and they can accommodate noisy Fourier magnitude information. This paper develops our 1-D continuous, non-minimum phase retrieval algorithm and illustrates its effectiveness with numerical examples.

Fast algorithms for systems of equations in wavelet-based solution of integral equations with Toeplitz kernels

The wavelet transform is applied to integral equations with Toeplitz kernels. Such integral equations arise in inverse scattering and linear least-squares estimation. The result is a system of equations with block-slanted-Toeplitz structure. In previous approaches, this linear system was sparsified by neglecting all entries below some threshold. However, in inverse scattering, the Toeplitz kernel may not be a rapidly decreasing function due to reflections from great depths. In this case, neglecting entries below a threshold will not work since the system matrix is ill-conditioned. We use the different approach of exploiting the block-slanted-Toeplitz structure to obtain fast algorithms similar to the multichannel Levinson and Schur algorithms. Since it is exact to within the wavelet-basis approximation, this different approach should prove to be a valuable alternative to the approximate approach of sparsification in cases when the latter does not work.