 Diffusion tensor identifiability

Diffusion tensor identifiability asks whether the measurements determine a unique diffusion tensor within a stated model. For a constant symmetric matrix $\mathbf{D}$, known matched baseline $S_0$ and known $b_i>0$, defineFor unit directions ${\mathbf{g}}_i$, these equations give a linear system $A\mathbf{d}=\mathbf{y}$ with rowAll six coefficients are uniquely determined in this unrestricted noise-free linear model exactly when $\mathrm{rank}(A)=6$. The condition concerns independent equations, not merely the number of acquired images. Direction design and uncertainty in tensor-derived measurements were examined experimentally through controlled simulations by Derek Jones.^[ref]

For a counterexample, use the six directions $\pm {\mathbf{e}}_x,\pm {\mathbf{e}}_y,\pm {\mathbf{e}}_z$. Each antipodal pair repeats a row, so the matrix rank is three. Only $D_{xx},D_{yy},D_{zz}$ are determined. Even restricting to physically positive tensors does not generally repair this: the identity matrix and a matrix with unit diagonal and $D_{xy}=D_{yx}=\epsilon$ both fit those data for any $|\epsilon |<1$, while having different off-diagonal entries.

Replace the negative axes by $({\mathbf{e}}_x+{\mathbf{e}}_y)/\sqrt{2}$, $({\mathbf{e}}_x+{\mathbf{e}}_z)/\sqrt{2}$ and $({\mathbf{e}}_y+{\mathbf{e}}_z)/\sqrt{2}$. The axes first give the diagonal entries, thenand similarly for $D_{xz}$ and $D_{yz}$. This directly proves uniqueness for the new design. If $D_{xx}=1.8$, $D_{yy}=0.9$ and $y_{({\mathbf{e}}_x+{\mathbf{e}}_y)/\sqrt{2}}=1.55$, all in units of $10^{-3}\mathrm{m}{\mathrm{m}}^2/s$, then $D_{xy}=0.20$ in those units.

Full matrix rank is not a guarantee of a stable estimate in noisy data. If $S_0$ must also be fitted, its logarithm is another unknown; the augmented signal model needs a corresponding independent baseline constraint. The six-coefficient argument above explicitly assumes $S_0$ is separately known. Likewise, tractography requires additional interpretation beyond uniqueness of a local tensor.