Analytical resolution and noise characteristics of linearly reconstructed magnetic resonance data with arbitraryk‐space sampling

The effects of time‐varying readout gradients and data sampling with variable dwell times in magnetic resonance imaging are examined. General reconstruction formulas are given for linear reconstruction with even k‐space weighting. Closed analytic expressions for estimator variance are given for data sampling during arbitrary gradient waveforms with both uniform k_x step size and nonuniform k_x step size. It is shown that estimator variance increases (the signal‐to‐noise ratio decreases) for nonconstant gradient waveforms. It is also shown that estimator variance is greater for constant k‐space sampling strategies than for constant time sampling at the Nyquist rate. Data collected during a triangular readout gradient waveform, with either constant time or constant k‐space sampling, versus conventional (constant gradient) collection confirms theoretical predictions for estimator variance. The benefits of collecting data while the readout gradient is ramping up from and down to zero are discussed.