Fig. 1. Sequence of interpolated slices (and their homotopy trees) obtained using interpolation based on Hausdorff distance with convex mask.

Fig. 2. Sequence of interpolated slices (and their homotopy trees) generated by median set-based interpolation method.

Fig. 3. Sequence of interpolated slices (and their homotopy trees) generated by the interpolation function method. (Note that there are two pores in image (d) and three grains in images (i) and (j).)

Fig. 4. Inclusion property illustrated.

Fig. 5. Inclusion property and homotopy issues.

Fig. 6. General interpolation algorithm. (Functions matching and displace are described later.)

Fig. 7. Separation of outer filled CCs.

Fig. 8. First matching criterion: (a) a slice with one CC A, indicating its MSP and its radius, λ_A; (b) a second slice with CC B; and (c) the proximity zone of A, δλ_A (A) (in gray).

Fig. 9. Matching algorithm.

Fig. 10. Multiple matching and isolated CCs. (a)Slice 1 with (up) and (down); (b) Slice 2 with (up), (bottom-left) and (bottom-right); (c) CCs of slice 1 (in light-gray) with their respective proximity zones (in dark gray) and CCs of slice 2 (in white), and (d) CCs of slice 2 (in light gray) with their respective proximity zones (in dark gray) and CCs in slice 1 (in white).

Fig. 11. Alignment and interpolated set:(a) Input sets A and B and their respective MSP points; and (b) Aligning of both input sets and interpolated set.

Fig. 12. Inner Structures of input sets in Fig. 7. (a) Hole in , (b) Structure inside , (c) Hole in and (d) Structure inside .

Fig. 13. Extreme case in which inclusion is desired.

Fig. 14. Position of holes in interpolated shapes: (a) Shape 1; (b) Shape 2; (c) Aligned outer CCs and interpolated outer grain; (d) Position of holes; (e) Aligned holes; and (f) Interpolated result.

Fig. 15. Selecting new locations for inner structures.

Fig. 16. Displacement algorithm.

Fig. 17. Interpolated results for the extreme example of Fig. 13.

Fig. 18. Simple example: interpolation of a circle and a square. Slices 1 and 2 are images (a) and (i), respectively. Images (b) to (h) are interpolated results. Image (j) represents the homotopy tree for all slices.

Fig. 19. Interpolated results 2 (compare with Fig. 2). Slices 1 and 2 are images (a) and (g), respectively. Images (b) to (f) are interpolated results. Image (h) represents the homotopy tree for all slices.

Fig. 20. Interpolated result 3 (compare with Fig. 3). Slices 1 and 2 are images (a) and (g), respectively. Images (b) to (f) are interpolated results. Image (h) represents the homotopy tree for all slices.

Fig. 21. Interpolation results 4: interpolation of nested CCs. Slices 1 and 2 are images (a) and (g), respectively. Images (b) to (f) are interpolated results. Image (h) represents the homotopy tree for all slices.

Fig. 22. Interpolation results 5: interpolation of slices with different homotopy (different number of grains). Slices 1 and 2 are images (a) and (h), respectively. Images (b) to (g) are interpolated results.

Fig. 23. Interpolation results 6: complex case with different homotopy and nested CCs . Slices 1 and 2 are images (a) and (i), respectively. Images (b) to (h) are interpolated results.

Distance table between CCs of Fig. 10