Table 1.

WMD-corrected PCA and Variance Mapping.

Initialize an m by n matrix D = [0] For each particle, i – Construct the rotated wedge mask: $$W^{\prime }=C{\mathbf{R}}_{{\theta }_i}\left(W_{v_i}\right)$$ – Apply the wedge mask to the subvolume and average, and band limit the subvolume: $$\begin{array}{c}\hfill t={\mathcal{F}}^{-1}\left(W^{\prime }{X}^{\ast }\right)\hfill \\ \hfill y={\mathcal{F}}^{-1}\left({\mathit{BW}}^{\prime }{X}_i\right)\hfill \end{array}$$ – Form the wedge-masked difference: δ = t − y – Apply an ROI mask, if any: δ = pδ – Centralize or standardize as desired. In most of the work described here, we adjusted δ to be zero mean and unit variance over p, and zero elsewhere. More recently, we have stopped adjusting variance. – Set column i of D to δ: [di] = δ Center the columns of D: ${\mathbf{d}}_i={\mathbf{d}}_i-\frac{1}{n}{\displaystyle {\Sigma }_{i=1}^n}{\mathbf{d}}_i$ Compute the [partial] SVD of the WMDs: USVT = D Cluster using first k rows of SVT as inputs to the chosen algorithm If desired, form the corrected covariance: C = US2UT/(n − 1) and the variance map σ2 = diag(C)