A Level Set-Based Model for Image Segmentation under Geometric Constraints and Data Approximation

Abstract

In this paper, we propose a new model for image segmentation under geometric constraints. We define the geometric constraints and we give a minimization problem leading to a variational equation. This new model based on a minimal surface makes it possible to consider many different applications from image segmentation to data approximation.

Keywords: energy minimization; level set methods; numerical analysis.

Figure 1

Left: Dirac distribution $\delta$ (in red). Right: In Gout et al.’s work [19], the authors introduce a regularized function ${\delta }_{\gamma }$ of $\delta$.

Figure 2

In 1D, in [19], the function ${\delta }_{\gamma }(\Phi (t,x,y,z))$ (in red) is equal to 0 outside a neighborhood of $(x,y,z)$ such that $\Phi (t,x,y,z))=0$, i.e., outside the points $r,s\in S=\left\{x,\Phi \left(x\right)=0\right\}$.

Figure 3

Left: the function $\Phi (t,x,y,z)$ (in red) is close to a constant between the points of S. Right: Thus, we have $\nabla \Phi \left(t,x,y,z\right)\cong 0$ (in red) outside the neighborhood of S.

Figure 4

We give 4 examples from the Brats dataset [27] with comparisons between our method and U-Nets. First line: considered images. Second line: initial guess (yellow crosses represent geometrical conditions (set of point(s)), and yellow discus is zero level of initial condition). Third line: ground-truth labels. Fourth line: segmentation obtained using supervised U-Nets [28]. Fifth line: segmentation obtained with our algorithm. In all examples, we considered $\alpha =1$, $\beta =1$, $\delta t=0.3$, and $\delta x=0.1$. The given values represent the Dice score. These results illustrate the efficiency of our proposed approach.

Figure 5

Initial image. The arrow shows the vessel to be segmented (main pulmonary artery).

Figure 6

Studied zone around the main pulmonary artery. We use the model proposed in this paper. Left: the MPA is perfectly segmented on the initial image (we have considered 2 points as geometric conditions). We obtain equivalent results until adding 40% of noise. Middle: after adding 50% of noise on the initial image, the geometric conditions are efficient, but in the right part of the artery, the segmentation contour is (logically) distorted by the noise. Right: after adding 200% of noise, the result is of course worse (except near the geometric conditions).

Figure 7

Left: with the same geometric conditions as in Figure 6 and on the image with the addition of 200% of noise, we test the algorithm given in [12]. We can see that the result is equivalent to the one of our approach near the 2 points to be interpolated but worse than with our algorithm in other zones. Right: the geodesic active contours (without interpolation conditions) do not give a good result (it is well-known that they are sensitive to noise).

Figure 8

Two-dimensional and three-dimensional views of the dataset: Big Island (Hawaii) zone, 8736 points.

Figure 9

Obtained approximation of the Big Island (Hawai’i) zone using a finite element grid of 400 Bogner Fox Schmidt rectangles (of class $C^1$, see [26] for more details). The step $\delta t$ is equal to 0.3. The quadratic error (23) is equal to $6.8\times {10}^{-5}.$ Such quadratic error values are very good in the surface approximation framework, and show that our approach is efficient, even in the case of this rather complex dataset (having large variations). In the global dataset, the maximum error measured is $6\%$, corresponding to a maximal error of 42 m (the location of this maximum error is logically near the steep valleys).

Figure 10

Example of a 3D seismic dataset wherein two continuous reflectors (layer A and layer B) appear.

Figure 11

An example of layers and a vertical fault extracted from the complex 3D dataset of Figure 10. Obtaining such visualization requires for a geologist to directly work on the 3D bloc (almost pixel after pixel); we propose to use a segmentation process with geometric constraints to segment one layer after another.

Figure 12

An example of a studied zone in Normandy (Sainte Marguerite cliffs). From different datasets (including acquisition using drones carrying infrared cameras and photogrammetry). The goal is to precisely reconstruct the topography (credits: Defhy3geo project, with Cerema Normandie).

Figure 13

A studied zone with infrared datasets (Vaches noires cliffs, credits: Defhy3geo project, with Cerema Normandie). Approximation of coastal zones is required for many applications like security concerns (cliffs collapsing), or to study the impact of topography on velocity wind fields (Intertwind project).