Fig 4. A flowchart of the algorithm for rigid registration of multiple bone images.

The input of the algorithm is N 3D micro-CT images {I ^N} and the index of the root image (I ^Root; 1 ≤ Root ≤ N) to which all other images will be registered. (A) Preprocessing. For each input image I ^N: 1. Zero-out all trabecular regions. 2. Zero-out all background regions. 3. Align bone to axes by applying PCA. 4. Extract cylindrical shape descriptor. (B) Pairwise Registration. For a pair of a source (I ^s) and a target (I ^t) image: 5. For each of the four basic alignments between the bones (with or without proximal-distal inversion × with or without right-left side inversion), calculate the affinity (Ψ) between the extracted shape descriptors of I ^s and I ^t as a function of the rotation angle (Θ) of I ^t about PC1. 6. From each local optimum in the calculated affinity function, perform several volume-based registration steps using NCC as a similarity measure and downhill descent as the optimization method. 7. Identify the path that reached the highest NCC score and optimize it using additional volume-based registration steps until convergence is reached. 8. If NCC <0.7, perform manual validation of the registration. (C) Agglomeration of pairwise to multiple image registration and interpolation. 9. Sort and re-index all bones based on their lengths, from shortest (“1”) to longest (“N”). 10. Register all pairs of bones I ^s, I ^t (s < t) adhering to either one of the following criteria: I ^s and I ^t are lengthwise consecutive: t − s = 1 (subdiagonal entries in D; in blue), or t – s > 1 ∧ Length(I ^t)/Length(I ^s) ≤ 1.2 (non-subdiagonal entries in D; in green). Assign the resulting NCC score in the corresponding cell in D. 11. Calculate the MST of the graph inspired by D, with I ^Root being the root of the tree. 12. Infer all final transformations (A ^N) based on the MST. 13. Transform/interpolate each image I ^N according to its inferred final transformation. The output is N 4 × 4 homogenous transformation matrices $\left\{A_{final}^N\right\}$, each aligns the corresponding input image I ^N to the root image I ^Root, and the N transformed images. For more details, see Materials and Methods.