Figure 1.
Shoot Apical Meristem (SAM): A multilayer cell cluster.
(A) SAM located at the top of the shoot of Arabidopsis, (B) A detailed surface view showing different regions of SAM, (C1–C3) Three consecutive slices of SAM, each m apart, obtained through CLSM technique, (D) A cross sectional side view of SAM, which clearly shows the multiple layers (L1, L2, L3) of tightly packed stem cells and their shapes, (E1–E3) The visible cell walls of individual cells in 3 sparsely sampled consecutive slices of the SAM obtained from the 3D CLSM live imaging dataset.
Figure 2.
A schematic of Voronoi Tessellation and Estimated SAM cell centroids as Voronoi sites.
(A) A Voronoi diagram based on the Euclidean distance metric for twenty one sites in 2D. The figures also show that the Voronoi edges are perpendicular to the line joining any two neighbouring sites. are three of the Voronoi edges and they are the perpendicular bisectors of
respectively. (B) Centroids are estimated for around two hundred cells in a SAM tissue, which are also the sites of Euclidean distance based Voronoi tessellation.
Figure 3.
Generation of the dense point cloud from within the reconstructed SAM surface.
(A) The SAM contours extracted from the confocal image stack using Level-Set segmentation, (B) The SAM surface is reconstructed using linear interpolation on a local neighbourhood of points on the SAM contours, (C) A very dense point cloud is extracted from within the reconstructed SAM surface which is clustered using the proposed reconstruction technique into individual cells.
Figure 4.
Ellipsoidal representation of the AQVT parameters estimated from the sparse data-points.
(A) The Minimum Volume Enclosing Ellipsoids representing the parameter pairs for individual cells are shown in different colors. (B) The same representation viewed from top.
Figure 5.
Sample segmentation and spatio-temporal tracking result.
(A) Raw confocal image slice, (B) Watershed segmented cell edges from the same image in A, (C) Individual cell slices are tracked in z to find correspondence between slices belonging to the same cells. The cells are color coded in the image to show the correspondences. The cells can also be tracked in time (which is shown using the same color code) that can be useful while reconstructing the same cells in consecutive time points to observe the growth of those cells.
Figure 6.
Visualization of the AQVT based 3D reconstruction of SAM cell cluster.
(A) Visualization of the 3D reconstructed structure of a cluster of around 220 closely packed cells using convex polyhedron approximations of the densely clustered data-points for each cell, as obtained from the proposed 3D reconstruction scheme, (B) A subset of cells from the same tissue.
Figure 7.
Reconstruction of a cluster of cells using Euclidean distance based Voronoi tessellation and the proposed AQVT for comparison of the 3D reconstruction accuracy.
(A) Segmented and tracked cell slices for a cluster of fifty two cells from the L1 and L2 layers of SAM. A dense confocal image stack is subsampled at a z-resolution of 1.35 m to mimic the ‘z-sparsity’ observed in a typical Live-Imaging scenario. The slices belonging to the same cell are marked with the same number to show the tracking results. (B) 3D reconstructed structure for a subset of these cells when reconstructed using the Euclidean distance based Voronoi Tessellation. (C) The AQVT based reconstruction result for the same cell cluster.
Figure 8.
Comparison of the 3D reconstruction accuracy for the proposed AQVT based reconstruction against Euclidean distance based Voronoi tessellation.
(A) The cells shown in Figure 7(A) are reconstructed using the Euclidean distance based Voronoi tessellation and the computationally re-sliced cells are compared against the ground truth. (B) The same cells are reconstructed using the adaptive quadratic distance based Voronoi tessellation and then computationally re-sliced along various depths in z at which we also have the ground truth (in terms of the 2D segmentation results of the cell slices), but were not used in generating the reconstruction results. The computationally obtained cell slices are shown in different colors for different cells and they are superimposed by the ground truth segmentation results. (C) The error in reconstruction (similar to the reprojection error) is computed as the Modified Hausdorff Distance (MHD) between the computationally generated cell slices and the segmentation results on the ground truth images of the same cells. The MHD, computed for each of the 52 cells at different depths in the Z-stack are plotted for both the methods to compare the methods against each other. It can be clearly observed from the plots that the reconstruction error is much larger for the Euclidean distance based Voronoi tessellation (VT) than for AQVT, especially at the terminal () slices, between consecutive layers of cells.
Figure 9.
Errors in AQVT estimated cell volumes from their respective ground truth volumes at various levels of sparsity.
A cluster of cells from a 3D confocal image stack with z resolution of 0.225 m is resampled to generate stacks of 5 different levels of sparsity. Each of these resampled stacks is 3D reconstructed using the proposed AQVT and volumes of each of the cells in the cluster are computed. The means and standard deviations of absolute errors in volumes (expressed as a ratio to the ground truth volumes) of all the cells for each sparser stacks are plotted. The average error slowly increases with increased sparsity but is less than 5.3% with a standard deviation of 4% even at 1.35
m/slice (i.e. 3 slices/cell on an average).
Figure 10.
Validation of AQVT on 2D root apex longitudinal cross section data.
(A) Ground truth segmentation of a sample cross sectional slice of root apical meristem tissue (the source images for this tissue can be found in [29]). (B) The zoomed in tissue after segmentation (B-top) and sparser point clouds per cell (in the -
plane) after resampling the tissue at various
-resolutions (zoomed in for clarity). (C) The cells (color-coded) are reconstructed using the proposed AQVT with the resampled point clouds as shown in B to present the change in reconstruction quality with increased sparsity in the sampled point clouds for both the larger and elongated cells towards the outer and upper part and the smaller cells towards the lower central part of the tissue. (D) Quantitative measure of reconstruction errors: the difference between actual and reconstructed cell shapes are computed using modified Hausdorff distance (MHD) and the histograms of MHDs for all the cells at every level of sparsity is plotted.