Supporting Information Figure S1: Sequence diagrams of (a) scheme 1 using velocity-insensitive control; (b), scheme 2 using velocity-compensated control; (c) scheme 3 using dynamic phase-cycling. The phase cycling schedules of all refocusing pulses for each scheme are provided. Only the first two velocity-encoding steps are shown. 

Supporting Information Figure S2: Diagram of a conventional VSS pulse train or the first step of FT-VSI, including eight blocks: first excitation pulse (E1), first gradient (G1), first refocusing pulse (R1), second gradient (G2), third gradient (G3), second refocusing pulse (R2), fourth gradient (G4), and second excitation pulse (E2). R1, G2, G3 and R2 can be combined as a spatially independent block  for label and velocity-insensitive control.

Supporting Information Figure S3: Simulated Mz-velocity profiles and Mz-position profiles of FT-VSI label and control pulse trains at a B0 range of ±200 Hz and their subtractions with (a,b) scheme 1 using velocity-insensitive control; (c,d) scheme 2 using velocity-compensated control; and (e,f) scheme 3 using dynamic phase-cycling. Compared to schemes 1 and 2, scheme 3 yielded both velocity and spatial responses with much less susceptibility to B0 variations. 

Supporting Information Figure S4: Simulated DC bias with five conventional velocity-selective pulse trains: 1) double refocused composite (DRCP) pulses, (90ºx180ºy90ºx, 1 ms), the same as used in the FT-VSI pulse trains of this study; 2) and 3) double refocused hyperbolic tangent (DRHT) pulses, (3 ms, tanh/tan, a frequency sweep of 35 and 25 KHz) using maximum amplitude of 20 μT and 30 μT respectively; and 4) and 5) BIR-8 with the same HT pulses as in 2) and 3). All sequences were implemented with a total duration of 24 ms and a cut-off velocity of 2 cm/s. For DRCP and DRHT, a 0–180° inner-module phase cycling was used. Only subtracted Mz was shown for (a) scheme 1 using velocity-insensitive control and (b) scheme 3 using dynamic phase-cycling. For BIR-8, dynamic phase-cycling was implemented by adding phases to the second and the fourth adiabatic pulses. White rectangles indicated the typical B0/B1+ range of human brain at 3T (B0: ±200 Hz; B1+ scale: ±0.3). For scheme 1, inadequate refocusing pulses, such as DRCP and even DRHT or BIR-8 with the first HT (20 μT) refocusing pulses, can lead to DC bias in extreme B0/B1+ conditions as explained in Appendix theory. This can be mitigated by employing either more adiabatic refocusing pulses in scheme 1 or the dynamic phase-cycling scheme.