Fig 1.
Derivatives sum algorithm for 3D nuclear segmentation.
(a) 2D 8-bit gray scale image slice of a 3D stack of dimension 167 × 172 × 39 voxels from the posterior PSM of an 18-somite stage zebrafish embryo. (b) De-noised image after Gaussian blur (σ = 0.5, window size = 5×5 squared pixels). (c) Image smoothened by a non-linear isotropic diffusion filter (κ = 10, n = 4). (d) Magnitude of Gauss gradient (σg = 1.5), (e) Laplacian where positive and (f) Determinant of Hessian where negative, we show the absolute value. (g) De-noised image shown in (c). (h) A tangent hyperbolic masking function (α = β = ε = γ = δ = 1). (i) Masked image obtained by the pair-wise product between the de-noised image in (g) and the masking function in (h). (j) Slices of binary images obtained by Otsu’s thresholding method. (k) Surface rendered 3D binary objects colored with respect to their position along the z-direction. All tunable parameters are highlighted by Greek glyphs in red. See S1 Text for more details.
Fig 2.
Post-processing steps to separate under-segmented objects.
Top panel is a result of the DS algorithm and the bottom panel are the results obtained after post-processing steps for the single stack shown in Fig 1. (a) Stem plot of the volumes of segmented objects obtained after DS algorithm. The black dashed vertical line (at 237 voxels) indicates the empirically calculated ‘Fused object volume’ threshold value above which the objects (with volumes > 160 μm3) were subjected to post-processing steps. Position of the dashed line depends upon the DS algorithm and noise filter parameters and volumes of segmented objects (S1 Text). (b) A 3D rendered object from the left side of the black dashed line in (a) correctly segmented by the DS algorithm as indicated with the centroid position in red. (c) An example of 3D rendering of a fused object, as indicated by the position of the centroid in red, from the right side of the black dashed line in (a). (d) Frequency distribution of voxels in x, y and z direction for the correctly segmented nucleus in (b) obtained after DS algorithm. Note that a single nucleus has a unimodal distribution of voxels in each direction. (e) Frequency distribution of voxels in x, y and z direction for the fused case in (c). The distribution in the z-direction indicates 3 peaks marked by asterisks (*), suggesting three nuclei fused in the z-direction. (f) 3D rendered object shown in (b) is post-processed with K-means (centroid position in blue) and GMM (centroid position in green). All the three methods give the same centroid position. (g) 3D rendered object shown in (c) is post-processed with K-means and GMM that find 3 new centroid positions indicated in blue and green thus segregating fused nuclei. (h) Stem plot after DS with post-processing steps. Blue and green stem plots represent segmented volumes after DS with K-means and DS with GMM, respectively. Post-processing steps tend to reduce under-segmentation. (i) Dependence of the average processing time of post-processing steps GMM and K-means on the maximum number of local peaks mnp. Processing time was measured for each single object with a given mnp value. The error bars indicate standard deviations.
Fig 3.
Assessment of 3D segmentation using synthetic image data.
(a) A stack of slices of a representative synthetic image produced with characteristic SNR and object density. (b) Positions of true centroids of objects (red dots) and centroid positions detected by the Derivatives Sum algorithm (blue dots) in the synthetic image. (c) Dependence of sensitivity and precision of algorithms on the density of objects in synthetic images. SNR = 5. Gaussian filter (σ = 0.5, window size = 5×5 squared pixels) and median filter (window size = 3×3) were used here for de-noising. Parameters in DS are α = β = ε = γ = δ = 1, and σg = 1.1. Scale bars = 10 μm.
Fig 4.
Assessment of 3D segmentation using experimental image data.
(a) Schematic illustration of transplantation experiment. Cells in a donor embryo at blastula stage expressing two histone variants fused to GFP and mCherry (h2Aflv-gfp/ h2Aflv-mCherry) are colored orange. (b) Bright-field image of a 13 somite-stage chimeric embryo. The white box indicates the tailbud. (c) 15 somite-stage chimeric embryo in gfp (left) and mcherry (middle) channels, and merged (right). White box in each image indicates the cropped region (50×50 squared pixels) used for the sensitivity analysis. Inset image is magnification of boxed region. (d) Sensitivity plot over density for five cropped images from four chimeric embryos. Each symbol and error bar indicates the temporal average and standard deviation, respectively, of the sensitivity over 10 time frames for a cropped image. Parameters in DS are α = β = γ = δ = 1, ε = 2, and σg = 1.2. De-noising filters; Gaussian filter (σ = 0.5, window size = 5×5 squared pixels) and median filter (window size = 3×3), Lucy-Richardson deconvolution filter with σ = 0.5, non-linear isotropic diffusion filter (κ = 50, n = 5). Both channels for all five cropped images were processed with same parameter values.