Statistical descriptors and binning

Global statistical descriptors

The final solution of the Monte-Carlo inversion algorithm, \(\mathcal{P}(\mathbf{D})\), can be understood as the median of all bootstrap solutions, \(\mathcal{P}_{\mathit{n}_\mathrm{b}}(\mathbf{D})\) with $1\leq \mathit\mathrm \leq \mathit\mathrm$. To quantify the main features of this final solution, one computes for instance the medians over bootstrap solutions of the per-bootstrap means \(\mathrm{Med}_{(\mathit{n}_\mathrm{b})}\,(\mathrm{E}[\chi]_{\mathit{n}_\mathrm{b}})\), variances \(\mathrm{Med}_{(\mathit{n}_\mathrm{b})}\,(\mathrm{V}[\chi]_{\mathit{n}_\mathrm{b}})\) and covariances \(\mathrm{Med}_{(\mathit{n}_\mathrm{b})}\,(\mathrm{C}[\chi,\chi^\prime]_{\mathit{n}_\mathrm{b}})\) of \(\chi=\mathit{D}_\mathrm{iso}, D_\Delta^2\), respectively referred to as the “size” and "shape" of the diffusion tensors building up \(\mathcal{P}(\mathbf{D})\).

Here, the median operator “\(\mathrm{Med}\)” acts across bootstrap solutions, and \(\mathrm{E}[\,\cdot\,]_{\mathit{n}_\mathrm{b}}\), \(\mathrm{V}[\,\cdot\,]_{\mathit{n}_\mathrm{b}}\) and \(\mathrm{C}[\,\cdot,\cdot\,]_{\mathit{n}_\mathrm{b}}\) denote the per-bootstrap average, variance and covariance over bootstrap solution \(\mathit{n}_\mathrm{b}\), respectively. For simplicity, the median operator is implicitly omitted when addressing a statistical descriptor, thereby writing averages, variances and covariances as "\(\mathrm{E}[\chi]\)", “\(\mathrm{V}[\chi]\)” and "\(\mathrm{C}[\chi,\chi^\prime]\)", respectively.

While previous works have relied on means to compute averages across bootstrap solutions [@deAlmeidaMartins_Topgaard:2018 ; @Topgaard:2019], more recent works have employed medians instead, because of their enhanced robustness to statistical outliers [@Reymbaut_accuracy_precision:2020 ; @deAlmeidaMartins:2020].

Bin-specific statistical descriptors

Given that the DTD method builds up \(\mathcal{P}(\mathbf{D})\) as a discrete sum of components, all aforementioned statistical descriptors can be extracted within tissue-specific bins, i.e. subdivisions of \(\mathcal{P}(\mathbf{D})\)'s configuration space.

For instance, the “thin”, "thick" and “big” bins introduced in Refs.[@deAlmeidaMartins:2020 ; @Reymbaut_book_chapter:2020] aim to isolate the signal contributions from white matter, grey matter and cerebrospinal fluid, respectively. The boundaries of these bins, implemented as default bins in dVIEWR, are defined as follows: