MRIPulses

Module for RF pulse design in MRI.

ab2rf(a, b)

Inverse SLR transform: alpha/beta polynomials to RF pulse.

Arguments

• a::AbstractVector: SLR alpha parameter.
• b::AbstractVector: SLR beta parameter.

Returns

• Vector{ComplexF64}: Designed RF pulse.

b2a(b)

Convert an SLR beta parameter to its minimum-phase alpha parameter.

Arguments

• b::AbstractVector: SLR beta parameter.

Returns

• Vector{ComplexF64}: Minimum-phase SLR alpha parameter.

b2rf(b, cancel_alpha_phs=false)

Convert an SLR beta parameter to an RF pulse.

Arguments

• b::AbstractVector: SLR beta parameter.
• cancel_alpha_phs::Bool: Cancel alpha phase?

Returns

• Vector{ComplexF64}: Designed RF pulse.

calc_ripples(ptype=:st, d1=0.01, d2=0.01)

Calculate effective SLR ripple levels for a specific pulse type.

Arguments

• ptype::Symbol: Pulse type.
• d1::Real: Passband ripple level.
• d2::Real: Stopband ripple level.

Returns

• Tuple{Real, Real, Real}: (bsf, d1, d2).

dinf(d1 = 0.01, d2 = 0.01)

Calculate D_∞ for a linear phase filter. From eqn. (20) in the Pauly reference below, D_∞ = T ⋅ B ⋅ W where

• T = (N-1) Δt is the pulse duration
• B = (Fₚ + Fₛ) / Δt is the filter half-amplitude width
• W = (Fₚ - Fₛ) / (Fₚ + Fₛ) is the fractional transition width.

Arguments

• d1::Real: passband ripple level in M0⁻¹.
• d2::Real: stopband ripple level in M0⁻¹.

Returns

• Real: D infinity.

References

Pauly, Le Roux, Nishimura, Macovski: Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm. IEEE Tr Medical Imaging 1991; 10(1):53-65. https://doi.org/10.1109/42.75611

dz_gslider_b(n=128, g=5, gind=1, tb=4, d1=0.01, d2=0.01, phi=π, shift=32)

Design a gSlider SLR beta parameter.

Arguments

• n::Int: Number of time points.
• g::Int: Number of sub-slices.
• gind::Int: Sub-slice index.
• tb::Real: Time-bandwidth product.
• d1::Real: Passband ripple level.
• d2::Real: Stopband ripple level.
• phi::Real: Sub-slice phase.
• shift::Int: Number of time points shift of pulse.

Returns

• Vector{ComplexF64}: SLR beta parameter.

References

• Setsompop, K. et al. (2018). 'High-resolution in vivo diffusion imaging of the human brain with generalized slice dithered enhanced resolution: Simultaneous multislice (gSlider-SMS)'. Magn. Reson. Med. 79, 141–151. https://doi.org/10.1002/mrm.26653

dz_gslider_rf(n=256, g=5, flip=π/2, phi=π, tb=12, d1=0.01, d2=0.01, cancel_alpha_phs=true)

Design a gSlider RF pulse.

dz_hadamard_b(n=128, g=5, gind=1, tb=4, d1=0.01, d2=0.01, shift=32)

Design a Hadamard-encoded SLR beta parameter.

dz_recursive_rf(; n_seg, tb, n, ...)

Recursive SLR pulse design for variable flip angle or spin-echo sequences.

Arguments

• n_seg::Int: Number of segments designed by recursion.
• tb::Real: Time-bandwidth product.
• n::Int: Pulse length.
• se_seq::Bool: Spin-echo sequence.
• tb_ref::Real: Time-bandwidth product of refocusing pulse.
• z_pad_fact::Real: Zero padding factor.
• win_fact::Real: Applied window factor.
• cancel_alpha_phs::Bool: Absorb alpha phase for flatter phase.
• t1::Real: T1 relaxation time.
• tr_seg::Real: Length of TR segment.
• use_mz::Bool: Design pulses accounting for actual Mz profile.
• d1::Real: Passband ripple level.
• d2::Real: Stopband ripple level.
• d1se::Real: Passband ripple level for spin-echo.
• d2se::Real: Stopband ripple level for spin-echo.

Returns

• Matrix{ComplexF64} or Tuple{Matrix{ComplexF64}, Vector{ComplexF64}}: RF pulse(s).

dzlp(n=64, tb=4, d1=0.01, d2=0.01)

Design a linear-phase FIR filter using the Remez exchange algorithm.

Arguments

• n::Int: Number of time points.
• tb::Real: Time-bandwidth product.
• d1::Real: Passband ripple level.
• d2::Real: Stopband ripple level.

Returns

• Vector{Float64}: Designed filter coefficients.

dzls(n=64, tb=4, d1=0.01, d2=0.01)

Design a linear-phase FIR filter using the least-squares error minimization method.

Arguments

• n::Int: Number of time points.
• tb::Real: Time-bandwidth product.
• d1::Real: Passband ripple level.
• d2::Real: Stopband ripple level.

Returns

• Vector{Float64}: Designed filter coefficients (truncated to n samples).

dzmp(n=64, tb=4, d1=0.01, d2=0.01)

Design a minimum-phase FIR filter.

Arguments

• n::Int: Number of time points.
• tb::Real: Time-bandwidth product.
• d1::Real: Passband ripple level.
• d2::Real: Stopband ripple level.

Returns

• Vector{ComplexF64}: Designed minimum-phase filter coefficients.

dzrf(n=64, tb=4, ptype=:st, ftype=:ls, d1=0.01, d2=0.01, cancel_alpha_phs=false)

Primary function for design of pulses using the Shinnar-Le Roux (SLR) algorithm.

Arguments

• n::Int: Number of time points.
• tb::Real: pulse time bandwidth product.
• ptype::Symbol: pulse type, :st (small-tip), :ex (π/2), :se (spin-echo), :inv (inversion), :sat (saturation).
• ftype::Symbol: type of filter: :ms (sinc), :pm (Parks-McClellan), :min (minphase), :max (maxphase), :ls (least squares).
• d1::Real: Passband ripple level.
• d2::Real: Stopband ripple level.
• cancel_alpha_phs::Bool: For :ex pulses, cancel alpha phase for a flatter profile.

Returns

• Vector{ComplexF64}: Designed RF pulse.

firls(numtaps, bands, desired; weight=ones(length(desired)÷2))

FIR filter design using least-squares error minimization. Translated from SciPy's signal.firls.

Arguments

• numtaps::Int: Number of taps in the FIR filter. Must be odd.
• bands::AbstractVector: Monotonic non-decreasing sequence of band edges.
• desired::AbstractVector: Desired gain at the start and end of each band.
• weight::AbstractVector: Relative weighting for each band.

Returns

• Vector{Float64}: Optimal FIR filter coefficients.

fmp(h)

Convert a linear-phase filter h to its minimum-phase equivalent.

Arguments

• h::AbstractVector: Linear-phase filter coefficients.

Returns

• Vector{ComplexF64}: Minimum-phase filter coefficients.

hadamard(n)

Generate a Hadamard matrix of size n (must be a power of 2).

hankel(c, r = zeros(eltype(c), length(c)))

Construct a Hankel matrix from column c and row r.

leja(r)

Order roots r using Leja ordering for numerical stability.

See Reichel, LAA, 1991: https://doi.org/10.1016/0024-3795(91)90386-B

mag2mp(x)

Convert a magnitude spectrum x to its minimum-phase equivalent.

Arguments

• x::AbstractVector: Magnitude spectrum.

Returns

• Vector{ComplexF64}: Minimum-phase filter.

msinc(n=64, m=1.0)

Generate a Hamming-windowed sinc pulse with m lobes.

Arguments

• n::Int: Number of time points.
• m::Real: Number of lobes.

Returns

• Vector{Float64}: Windowed sinc pulse.

root_flip(b, d1, flip, tb)

Exhaustive root-flip pattern search for minimum peak B1.

Arguments

• b::AbstractVector: SLR beta parameter.
• d1::Real: Passband ripple level.
• flip::Real: Target flip angle.
• tb::Real: pulse time bandwidth product.

Returns

• Tuple{Vector{ComplexF64}, Vector{Float64}}: (rf_out, b_out).

References

• Sharma, A., Lustig, M. and Grissom, W. (2016).

'Root-flipped multiband refocusing pulses'. Magn. Reson. Med. 75(1), 227-237. https://doi.org/10.1002/mrm.25629

sp_fft(x)

SigPy-style centered FFT.

sp_ifft(x, center=true)

SigPy-style centered IFFT.

toeplitz(c, r=c)

Construct a Toeplitz matrix from column c and row r.