Fig 1.
Wavelet compression of a time domain pulse.
A 2-band SMS purely amplitude modulated pulse is shown with the corresponding wavelet decomposition. This pulse was selected as the initial condition for the wavelet optimization. The top panel shows the initial condition pulse (blue trace) plotted against the pulse (dashed-dotted red trace) that was wavelet reconstructed exclusively from the 40 wavelet coefficients shown in the bottom panel. The middle panel provides the full wavelet decomposition (using Discrete Meyer wavelet prototypes) exhibiting the sparseness of the wavelet domain. The lower left panel shows the 40 highest energy wavelet coefficients (above the energy threshold of 0.001). The lower right panel plots the root mean squared error (RSME) percentage of the initial condition pulse compared to the wavelet reconstructed pulse after truncation in the wavelet domain to a limited number of coefficients and zero-filling the remainder of the wavelet domain. In this figure, 3200 time domain points were compressed to 40 wavelet domain coefficients representing 98.7% compression of the original time domain pulse.
Fig 2.
Diagram of the pulse optimization workflow.
Wavelet decomposition is performed on the initial condition (input) pulse. Truncation in the wavelet domain selects the desired number of coefficients based on an energy threshold and the remaining coefficients are set to zero. The nonlinear optimization loop iteratively evaluates the cost function until the stopping criteria are satisfied. The final coefficients are wavelet reconstructed to obtain the ideal time domain pulse.
Fig 3.
Graphical display of wavelet coefficients and cost function terms for a 2-band pulse.
Panel (A) at the top shows the 117 wavelet domain coefficients whose amplitudes were optimized. The amplitude modulated time domain pulse with 3200 points shown in panel (B) was generated from the wavelet coefficients displayed in the top panel. Panel (C) shows the transverse magnetization profile at the echo time after Bloch Simulation of excitation and refocusing pulses. Panel (D) displays the phase of the transverse magnetization at the echo time. Panel (E) shows the tip angle for solely the rejection band that was dynamically computed from the net magnetization vector. The lower three plots represent the spin magnetization state from Bloch simulation for a single spin isochromat on resonance at each spatial frequency point.
Fig 4.
Wavelet coefficients for Shinnar-LeRoux (SLR) and wavelet time domain pulses.
A 2-band SMS Shinnar-LeRoux (SLR) purely amplitude modulated pulse is shown (peak B1 of 21.69 μT) along with the corresponding wavelet decomposition on the left column. The right column shows the amplitude modulated wavelet optimized 2-band SMS pulse yielding the same slice bands as the SLR pulse. The wavelet optimization achieved a peak B1 of 19.67 μT (reduction of 10.7% from the comparable SLR pulse). The lower row shows the wavelet coefficients containing the highest energy. In this example, 3200 time domain points were compressed to 75 wavelet domain coefficients.
Table 1.
Comparison of three dual-band pulses.
Fig 5.
Bloch Simulation comparison of three dual-band refocusing pulses.
For the three pulses shown (Fourier pulse, SLR pulse, or Wavelet pulse), Bloch simulation comprised a dual-band excitation pulse followed by one of the three dual-band refocusing pulses plotted in the top panel with 50 spin isochromats and an echo time of 72 ms to simulate a crushed spin echo response. The second panel compares the transverse magnetization magnitude zoomed to each of the two pass bands. The SLR and wavelet pulse exhibit a very similar profile. The bottom panel plots the phase angle in radians of the transverse magnetization for each pulse.
Fig 6.
Reconstructed sagittal brain images obtained with the standard single-band pulse sequence on the General Electric MR950 7.0T scanner are compared to images obtained with the dual-band Fourier, SLR and wavelet-optimized refocusing pulses. All images are shown with the same window level. Single-band images are shown in the top row with standard reconstruction on the scanner. Column A shows the acquired (aliased) images containing two different brain slices superimposed. The two separated slices, after SENSE reconstruction, are shown in column B and column C for the Fourier, SLR and wavelet-optimized refocusing pulses. Other than differing refocusing pulses, all aspects of the scan sequence remained the same.
Fig 7.
Diffusion (DTI) color-coded fractional anisotropy (FA) maps.
The FA maps compare a central sagittal slice for the standard single-band diffusion pulse sequence (panel A) with our SMS sequence using Fourier, SLR and wavelet-optimized refocusing pulses (panels B, C and D). The customized SMS pulse sequence, after reconstruction, yields comparable results the standard sequence (panel A) acquired without slice acceleration. The scan parameters included a 4.0 sec TR, 73.2 ms TE, 30 DTI directions and b = 1000 s/mm2. The color-coding scheme denotes the principal diffusion direction as red for left-right, green for anterior-posterior, and blue for superior-inferior. The level of brightness represents the magnitude of the FA value.