6.4 Active Tessellations
6.4.3 Experiments and validation
We perform three experiments to evaluate the performance of our active tessellation. We first investigate its robustness with respect to noise and dim membranes on synthetic data. Then, we illustrate applications on real data.
Synthetic Data
We compare our approach in term of accuracy against the watershed method [189] imple- mented by I. Arganda-Carreras and D. Legland for the bioimage platform Fiji9. In the experi- ments, we compute the Jaccard index for each cell and take the average. It is this average value that we refer to as Jaccard index in the next sections.
We created a test image that simulates the fluorescence microscopy of a C. elegans embryo with 5 cells (Figure 6.35 (a)). We use the same seed points to initialize the two methods. The initial configuration of the active tessellation is illustrated in Figure 6.35 (a). Its initial similarity with the ground truth corresponds to J = 0.64.
Robustness with Respect to Noise: We corrupted the test image by different levels of additive
Gaussian noise (20 realizations per level of noise, Figure 6.35 (d)). The SNR correspond- ing to a given noise level and Jaccard index were computed. The SNR that we use is the ratio of the mean value of the signal and the standard deviation of the noise. The results are given in Figure 6.36 and illustrated in Figures 6.35 (e) and (f ). We observe that the active tessellation is robust with respect to noise since it is able to give a proper seg- mentation outcome even for low SNRs. On the contrary, the accuracy of the watershed method degrades significantly for a SNR below 2.4.
Chapter 6. Design of Active Contours
(a) Test image. (b) Active tessellation, J = 0.98. (c) Watershed, J = 1.0.
(d) Noisy image. (e) Active tessellation, J = 0.90. (f ) Watershed, J = 0.46.
(g) Image with dimmed mem- branes.
(h) Active tessellation, J = 0.94. (i) Watershed, J = 0.44.
Figure 6.35 – Segmentation outcomes. (a)-(c) Test images. (a) Initial configuration of the active tessellation; (d)-(f ) noisy data with SNR= 0.81; (g)-(i) image with 23.95% of membrane information loss.
Figure 6.36 – Segmentation of noisy data. Evolution of the Jaccard index as a function of the SNR. Filled area: standard deviation across the 20 realizations.
6.4. Active Tessellations
Figure 6.37 – Evolution of the Jaccard index as a function of the dimming percentage.
Robustness with Respect to Dim Membranes: We progressively dimmed the fluorescence sig-
nal on the membranes of the test image (Figure 6.35 (g)). We computed the Jaccard index as a function of the information-loss percentage. This dimming percentage corresponds to the ratio of the mean intensity on the membrane of the test image over the one of the corrupted image. The resulting plot is given in Figure 6.37 and we illustrate results in Figures 6.35 (h) and (i). The active tessellation accurately segments the cells until 49% of information loss while the watershed method can tolerate no more than 15% of information loss, then it quickly decreases. As this model is only based on intensity, it leaks through dim membranes. Due to the structure and smoothness of the active tessellation, the proposed framework does not suffer from leakage.
Real Data
We applied our active tessellation on real biomedical images. These images are challenging because of the presence of noise and gaps in the membranes. For each segmentation, the initial configuration of the active tessellation has 2.6 control points per cell in average. We compute the Jaccard index of each outcome considering a manual segmentation as ground truth. The results obtained are satisfactory in most cases (Figures 6.32 and 6.38).
6.4.4 Conclusions
We have presented a new subdivision-based active contour for the segmentation of cell aggregates. We have modeled the active contour by a smooth tessellation and used the oriented-ridge-based energy term designed in Section 5.3 to efficiently attract the curve toward the membranes. The tessellation structure prevents from overlapping segmentation of the cells and from leakage issues. Moreover, each cell of the segmentation outcome can be
Chapter 6. Design of Active Contours
(a) J = 0.86. (b) J = 0.95.
Figure 6.38 – Cell segmentation of (a) cornea endothelium; (b) C. elegans embryo a in light- sheet fluorescence-microscopy image. Source: R. Jankele and P. Gönczy, EPFL.
easily extracted as a continuous closed curve making possible the computation of cell metrics. We have demonstrated the robustness of our method under noisy conditions and to dim membranes. We have also illustrated its behavior on real bioimages. The main contributions related to this work are:
• The construction of a smooth tessellation to describe an active contour;
• The derivation of an oriented ridge-based energy functional (see (5.3.3) and (5.3.4)); • The implementation of the whole framework.
7
Active Subdivision Surfaces
In this chapter, we extend the 2D multiresolution subdivision snake exposed in Section 6.3 to its 3D counterpart for the extraction of volumetric structures.
Subdivision is widely used in computer graphics for representation and modeling [107, 108]. As it was motivated in Chapter 4, this geometric representation combines the advantages of parametric and mesh-based approaches: The continuously defined limit surface is fully driven by the initial coarse mesh which consists of only few parameters.
The use of subdivision to construct segmentation models was pioneered in 2D by [171] for the DLG-scheme [172]. We then presented a generic framework that is valid for any convergent subdivision schemes in [70]. The extension to 3D models is more challenging. From a computational point of view, the geometry of the surface and the mesh connectivity increase the complexity of the implementation. Shapes are encoded with more control points, with three degrees of freedom for each one, which renders the optimization more complicated and slower. Moreover, it might be necessary to maintain evenly spaced control points to favor a representative sampling of the surface. In the literature, only few works used subdivision to segment volumes. The authors of [170] presented the modelization of left ventricles using Doo-Sabin surfaces [132], while the segmentation of branch vessel structures was performed in [190] using Loop’s subdivision scheme [110].
In this chapter, we present1the generic construction of active subdivision surfaces in the context of any subdivision scheme that operates on triangular meshes. The main contribu- tions related to this work are 1) a new 3D geometrical representation based on subdivision. The subdivision operator confers important properties to the surface such as smoothness, reproduction of desirable shapes and interpolation; 2) the derivation of region- and gradient- based energy functions that are guaranteed to have the proper limit proposed in [52]; 3) the presentation and integration of an algorithmic coarse-to-fine optimization strategy. This speeds up the computations and increases the robustness. We have implemented the method
1This section is based on our work [72], in collaboration with L. Romani and M. Unser. A demo of the corre-
Chapter 7. Active Subdivision Surfaces
as a user-friendly open-source plugin2for the bioimaging platform Icy [11].
Throughout this chapter we use the notations described in Section 4.6. Moreover, we consider orientable closed surfaces, i.e., compact and without boundary, since we want our active surface to segment blob-like objects within 3D images.
The chapter is organized as follows: In Section 7.1, we describe the generic construction of active subdivision surfaces on triangular meshes. Then, in Section 7.2, we provide a coarse- to-fine optimization strategy. In Section 7.3, we perform an extensive validation of active subdivision surfaces on both synthetic and real biological images. In particular, we show that the scheme is robust in the presence of noise and with respect to the initialization. Finally, conclusions are drawn in Section 7.4.
7.1 Framework
We implicitly represent the surface of the deformable model by the continuously defined, orientable, closed limit surfaceσ of a convergent subdivision scheme
σ = lim
k→∞M(k), (7.1.1)
whereM(k)is the triangular mesh at the k-th subdivision step obtained by (4.6.2). Its shape is encoded by the M = N0 control pointsΘ = P(0)=©p(0)[m]ªm∈{0,...,M−1}. The number M
of control points determines the number of degrees of freedom of the model. A small M leads to simple and constrained shapes, while an increase in M brings additional flexibility to approximate arbitrary surfaces. This representation implies that the properties of the active surface depend on the choice of the subdivision scheme (see Section 4.6.3). A mandatory requirement is affine invariance. Moreover, the quality of the segmentation outcome can be influenced as follows: first, the regularity of the surface defines the smoothness of the segmentation result; second, the geometric reproduction properties have to match the shape of interest.
We reduce the energy of the active subdivision surface to an image energy term (see Sec- tion 1.2.2). The smoothness of the surface is ensured by choosing a subdivision scheme that produces at leastC1surfaces. We use a combination of the gradient- and region-based
terms (5.1.8) and (5.2.9) such as
Esnake(P(k)(Θ)) = bEgradSD(P(k)(Θ)) + (1 − b)EintensitySD(P(k)(Θ)), (7.1.2)
whereP(k):= P(k)(Θ), given by (4.6.2), describes the surface and b ∈ [0,1] is a tradeoff parameter
that balances the contribution of the two energies.
2See footnote 1.