Wavelet Encoding Research Articles

A novel symmetric cryptography based on the hybrid Haar wavelets encoder and chaotic masking scheme

In this scheme, the authors use the seven chaotic parameters (a,/spl beta/,a,b,x/sub 0/,y/sub 0/,z/sub 0/) and the dimension of Haar wavelet encoder matrix H/sub n/, the combining form of H/sub n/ with subwavelet basis {h/sub 0/,h/sub 1/,...,h/sub n-1/}, the prespecified time span t/sub s/, as well as the coefficients a/sub i/,i=1,...,n, and the prime number k of the collision-free one-way function to serve as the encryption keys. The security property of the proposed cryptosystem results from two main parts: first, the high sensitivity of synchronization versus parameter or initial condition with small variation. It is sensitive enough to make the state trajectories separate from each other at an exponential rate. The second part depends on the Haar wavelets encoder H/sub n/ with its various combining form from the subwavelets basis {h/sub 0/,h/sub 1/,...,h/sub n-1/}; it is composed the same way as the lottery. Furthermore, due to the collision-free one-way function, the chaotic behavior control parameters are always different. Therefore, the system can be secured as long as the first chaotic parameters are kept secret. Moreover, the transmitting error-detection function of the proposed cryptosystem is also addressed.

Method for improved multiband excitation profiles using the Shinnar–Le Roux transform

A method to design multiband RF pulses for magnetic resonance imaging is described. The method is based on the Shinnar–Le Roux transform and involves a phase correction that provides control over the phase of the excited bands. The theory behind the method and this phase correction is outlined. The method is demonstrated with the design of RF pulses for Hadamard encoding and Haar wavelet encoding. Experimentally measured excitation profiles and images for RF pulses designed with the new method are compared to those designed by the conventional method. The conventional method is shown to result in distortion of the excitation profile when the bands are closely spaced. A 78% reduction in this distortion is attributed to the new method. This translates into a 52% reduction of out-of-slice signal in Haar wavelet encoding. Magn Reson Med 42:577–584, 1999. © 1999 Wiley-Liss, Inc.

Method for improved multiband excitation profiles using the Shinnar-Le Roux transform.

A method to design multiband RF pulses for magnetic resonance imaging is described. The method is based on the Shinnar-Le Roux transform and involves a phase correction that provides control over the phase of the excited bands. The theory behind the method and this phase correction is outlined. The method is demonstrated with the design of RF pulses for Hadamard encoding and Haar wavelet encoding. Experimentally measured excitation profiles and images for RF pulses designed with the new method are compared to those designed by the conventional method. The conventional method is shown to result in distortion of the excitation profile when the bands are closely spaced. A 78% reduction in this distortion is attributed to the new method. This translates into a 52% reduction of out-of-slice signal in Haar wavelet encoding. Magn Reson Med 42:577-584, 1999.

A Near-Resonance Solution to the Bloch Equations and Its Application to RF Pulse Design

A near-resonance expansion of the solution to the Bloch equations in the presence of a radiofrequency (RF) pulse is presented in this paper. The first-order approximation explicitly demonstrates the nonlinear nature of the Bloch equations and precisely relates the excitation profile with the RF pulse when the flip angle is less than π/2. As an application of this solution, we present a procedure for designing RF pulses to generate symmetric excitation profiles with arbitrary shapes for new encoding approaches such as wavelet encoding.

Partial wavelet encoding: a new approach for accelerating temporal resolution in contrast-enhanced MR imaging.

We propose a new approach using wavelet encoding to improve temporal resolution in contrast-enhanced magnetic resonance (MR) imaging. Exploiting the unique property of wavelets localized in space and frequency, we construct an efficient encoding scheme to capture signal changes due to contrast agent uptake, which in general is spatially localized with low- and mid-range frequency components. On the basis of space-frequency analysis, we describe mathematical formulations of our method and discuss its theoretical advantages over Fourier-based phase-encoding methods (the keyhole and reduced-encoding imaging by generalized-series reconstruction [RIGR] techniques). The results obtained in computer simulations and a phantom study demonstrate the feasibility and practical advantages of our approach.

Wavelet encoding for improved SNR and retrospective slice thickness adjustment.

A method to encode the slice-select direction in magnetic resonance imaging through the use of a discrete wavelet transform is presented. The paper reports the first implementation of wavelet encoding using the Haar basis. Theoretically, the signal-to-noise ratio (SNR) for two levels of wavelet encoding is expected to be approximately 1.6 times higher than for conventional multislice imaging in which each slice is excited separately. A slightly lower SNR is encountered experimentally because of slice interference. The design of radiofrequency (RF) pulses to excite slices approximating Haar wavelet basis functions is discussed. Wavelet encoding also provides an efficient method for balancing slice thickness and SNR after an image has been acquired. The experiments have been conducted at 1.5 Tesla. Wavelet encoding of slices should be especially suitable for low-field MRI, in which the SNR is lower and the characteristically lower T1 values favor the use of shorter TR, which sometimes forces the slices to be acquired in multiple passes.

Wavelet Encoding for Accelerating Temporal Resolution in Dynamic Contrast-Enhanced MR Imaging

Wavelet Encoding for Accelerating Temporal Resolution in Dynamic Contrast-Enhanced MR Imaging

Wavelet encoding for 3D gradient-echo MR imaging.

An implementation of wavelet encoding to resolve one spatial direction of a three-dimensional gradient-echo magnetic resonance image is described. This is the first report of the use of wavelet encoding with a relatively short repetition time for rapid image acquisition. An important feature of this implementation is that it allows wavelet-encoded images to be reconstructed without phase correction. The images show no discernible degradation compared with images acquired with phase encoding or scaling-function encoding instead of wavelet encoding. In addition to describing the implementation, typical sources of phase errors are investigated both theoretically and experimentally.

Multilevel wavelet-transform encoding in MRI.

This paper explores an alternative to standard Fourier MRI called wavelet-transform encoding. Spatially selective radiofrequency (RF) pulses are used to excite slice profiles in the shape of the wavelet-basis functions of a discrete wavelet transform. Our implementation on a standard commercial whole body MRI system resolves one spatial dimension through wavelet encoding and orthogonal directions through conventional frequency encoding and rectangular-slice selection. The wavelet transform is described as a method for multiresolution analysis, and we show how a high resolution MR image can be constructed from lower resolution images, representing the approximation and detail structures of the object. A multiple level architecture of the wavelet reconstruction is described, thereby allowing wavelet-encoded images to be reconstructed through several modes. Multilevel wavelet-encoded images of a gel phantom are presented that show comparable image quality to Fourier-encoded images of similar signal-to-noise ratio. However, a chemical-shift artifact is identified with this particular implementation.

Theoretical comparison of Fourier and wavelet encoding in magnetic resonance imaging

A theoretical comparison of the wavelet and Fourier encoding methods is made with respect to resolution, sensitivity to artifact, and signal-to-noise ratios (SNR's). A general mathematical description is developed in which magnetic resonance (MR) image encoding is represented by a projection of the function representing the "MR signal density" onto an approximation subspace of the finite energy functions. Characteristics of the subspace are used to define a "generalized" point-spread function for space-variant systems. Using the formal model of MR image encoding it is shown that wavelet encoding approaches the resolution limit defined by Fourier encoding. Artifact is treated according to whether or not the source of the variation in the measured data is stationary. Nonstationary imperfections perturb the projection operation and result in encoding method-dependent effects which can be modeled by a distortion matrix suitable for treating shift variant systems. A framework is developed in which to derive expressions for SNR's applicable to a general class of MR encoding methods.

Signal-to-Noise Ratios and Effective Repetition Times for Wavelet Encoding and Encoding with Wavelet Packet Bases

Wavelet encoding is a method of localizing MR signal. Bands of spins of variable length and position are excited to encode position. Wavelet encoding is a compromise between phase encoding, which excites all the spins with each RF excitation, and line scanning, which excites only one pixel with each RF excitation. Expressions for the SNR are developed. Phase encoding produces optimum SNR, but it has relatively short effective repetition times. Line scanning has poor SNR but has long effective repetition times. Wavelet encoding is generalized to use wavelet packet bases. Wavelet packet bases include a wide class of bases. They include the bases used for Hadamard and phase encoding at one extreme and line scanning at the other. If the appropriate basis is chosen from among the wavelet packet bases, SNR can be freely traded for longer effective repetition times. Several other results are shown; e.g., wavelet encoding is very effective at edge detection in terms of SNR and number of excitations required. In addition, an alternative statistical formulation is introduced that quantifies the bias in the reconstruction. The bias quantifies the sensitivity of each basis to artifacts.

Implementation of wavelet-encoded MR imaging.

Reconstructions of images from wavelet-encoded data are shown. The method of MR wavelet encoding in one dimension was proposed previously by Weaver and Healy. The technique relies on selective excitation with wavelet-shaped profiles generated by special radio-frequency waveforms. The result of the imaging sequence is a set of inner products of the image with orthogonal functions of the wavelet basis. Inversion of the wavelet data is accomplished with an efficient algorithm with processing times comparable with those of a fast Fourier transform. The experiments show that wavelet encoding by selective excitation of wavelet-shaped profiles is feasible. Wavelet-encoded images are compared with phase-encoded images that have a similar signal-to-noise ratio, and there is no discernible degradation in image quality due to the wavelet encoding. Potential benefits of wavelet encoding are briefly discussed.