2.6. Quantifying uncertainty in the spectra

The jackknife technique (Quenouille, 1949) was used to quantify the uncertainty in the estimated parameters in the following manner: first, the 1D T₁, T₂, and D, estimations were repeated with one data point being left out in each iteration, resulting in 20 spectra for T₂ and D, and 12 for T₁. Then, the 2D spectra, D-T₂, T₁–T₂, and D-T₁, were reconstructed using MADCO with the “leave-one-out” 1D spectra as constraints, resulting in 400 D-T₂ spectra and 240 T₁–T₂ and D-T₁ spectra. The high dimensionality of the spectra prevents direct visualization of the uncertainty, and we therefore adopted a previous approach (Prange and Song, 2009) that maps each spectrum into a single number or numbers whose uncertainty can then be visualized as histograms. We chose to focus on the tissue components apparent volume fractions, which were obtained by integration over the corresponding spectral peak (as described in more detail in section 3.5). Those apparent volume fractions were obtained from each of the leave-one-out spectra, and their standard deviation was used to indicate the uncertainty in their estimation. This process was repeated in each of the image voxels.