= Diffusion MRI gradient sampling
<Diffusion MRI gradient sampling> is the choice of <magnetic field gradient> directions, strengths and repetitions used to acquire <diffusion magnetic resonance imaging> measurements. Under a Gaussian <diffusion tensor> model, different directions probe different values of a <quadratic form>. Acquiring many measurements in an uninformative set of directions cannot determine every tensor coefficient.
For a unit gradient direction $\mathbf g$, a separately measured matched baseline $S_0$ and fixed positive <diffusion weighting b-value> $b$, the ideal signal model is
$$
S_{\mathbf g}=S_0\exp(-b\mathbf g^T\mathbf D\mathbf g).
$$
The <symmetric matrix> $\mathbf D$ has six independent coefficients. Opposite directions $\mathbf g$ and $-\mathbf g$ give the same model prediction. Thus counting orientations without considering their independent information can overstate <diffusion tensor identifiability>.
A 2003 <Monte Carlo simulation> study controlled the observation budget: one baseline plus60 diffusion-weighted observations. Six distinct orientations repeated ten times and30 distinct orientations repeated twice each use60 diffusion-weighted observations. The comparison asks how angular coverage and repetition affect estimated orientation, rather than simply comparing a longer acquisition with a shorter one.<The Effect of Gradient Sampling Scheme on Estimates of Fiber Orientation: Implications for Fiber Tractography by D. K. Jones>[Methods and Table1: observation budget, directions and repetitions]{ref}
The subsequent 2004 study compared direction schemes through estimated orientation, <trace> and <fractional anisotropy> at $b=1000\,\mathrm{s/mm^2}$. These are distinct outputs of the same tensor fit. An acquisition suited to one readout need not minimize uncertainty in another.<The effect of gradient sampling schemes on measures derived from diffusion tensor MRI: A Monte Carlo study by Derek K. Jones>[Primary Abstract: Monte Carlo design and tensor-derived measurements]{ref}
An original design comparison separates two questions. Repeating the three coordinate axes may reduce random variation in their measured attenuations, but it adds no equations for the three off-diagonal tensor entries. Adding suitable mixed directions can supply those equations. Full <matrix rank> establishes uniqueness in the ideal model; practical <measurement uncertainty>, motion and model mismatch still require assessment.
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