Unbiased Finite Element Mesh Delaunay Constrained Triangulation Applied to 2D Images with High Morphological Complexity Using Mathematical Morphology Tools Part 1: Binary Images

Abstract

We propose a method for establishing a Constrained Delaunay Triangulation CDT applied to 2D binary images of high morphological complexity. A prerequisite for CDT is the unbiased definition of the Planar Straight-Line Graph PSLG, which must respect the injective nature of Jordan’s Curve whatever the topology of the image objects. Mathematical morphology provides tools for extracting the image contour, on which points will be judiciously placed at particular points to construct the vector path of the PSLG. Finally, these tools will enable us to implement a judicious pointing process in the image to guarantee the relative equivalence of triangles. The deterministic and rigorous procedure detailed in this article will be generalized in a second article, Part 2, to the case of labeled images for which the definition of the PSLG is more complex to define, since the contour of objects in the image is defined by the set of contours of adjacent objects.

1. Introduction

Two-dimensional imaging is widely used for quantitative image analysis in virtually all fields of research, such as biology, materials science, or calligraphic analyses of texts, to name but a few. Segmentation is a prerequisite for all image analyses. If segmentation results in the division of two unique labels, the image is binary. By convention, the objects in an image all have one pixel, and the 0 value is assigned to the rest of the image. However, although the amount of data pixels is often small, it is often necessary to reduce the size of the data to suit the area of use. Constrained Delaunay triangulation (CDT) is an effective tool for associating a triangular structure with each object in an image. The condition is that the PSLG of the triangulation is faithful to the contour of the objects in the image.

Langer’s method [1], later adopted by Zao [2], involves adapting a coarse triangular grid to an image, which serves as the first mesh. This is refined by successive iterations until the image and mesh converge on an equilibrium structure. It then assigns a unique label value to each mesh element. It establishes a calculation code called the Oriented Finite Element (OFE). Although this technique takes grayscale images as input, this method is classified as a binary-image-meshing method. The algorithm uniquely allows only two groups of elements, the objects and the rest of the image.

Other methods include Marching Techniques. These techniques are very popular, especially in 3D, where the Marching Cube technique [3] is at the heart of the vast majority of mesh codes. In 2D, Marching Square, the 2D version of Marching Cube, extracts the contour and establishes the PSLG of every object in an image, even in a 3D mesh, to establish surface intersections [4].

But the Marching Square Technique, like the Marching Cube Technique, produces configuration ambiguities. While it is possible to resolve these pathological cases in the case of 2D images [5], it is much more difficult to do so in the case of labeled images.

Delaunay triangulation is also widely used in mesh technology. Schewchuk [6,7] proposes a constrained Delaunay triangulation by decimating inappropriate triangles according to angular or shape criteria, starting from a collection of predefined edges that make up the PSLG. This method applies to 2D binary images that are labeled.

In many meshing techniques, PSLG construction is an imperative preamble, as is the case for the frontal advanced method developed by Rassineux specifically in the case of 2D and 3D binary images [8,9]. Like Delaunay triangulation, it requires the PSLG to be defined for all objects in the image. However, if the image contour is sufficiently smooth, it is possible to proceed by parabolic approximation [10,11] or by splines [12].

2. Delaunay Constraint Triangulation of 2D Binary Images

In our study, we will use a constrained Delaunay triangulation (CDT), which is a variant of Delaunay triangulation. It was theoretically defined by Borgers for the non-convex domains [13,14]. To use this, of course, we need to construct the PSLG. In our case, the PSLG is not simply a collection of edges, but a vector path describing a simple closed curve. To ensure that the angles of the triangles are relatively homogeneous, i.e., around 60 degrees, we’ll perform an optimized point process [15]. This step avoids the mesh-optimization step [16,17]. We note that the idea of using a Voronoi diagram, even a centroidal one (CVT), to obtain a mesh with homogeneous triangles is always subject to the risk of cocyclicity if the seeds are collinear or if there are four cocyclic points [18]. These cases have all the more potential as an image is represented by pixels whose coordinates are integer values.

It is very important to specify that we are going to distinguish the case of binary images, where one label is the complement of the other, from the case of labeled images, where each label must be defined by the adjacency of the set of labels that are mytoyen to it, which makes the PSLG more complex to establish. In the binary case, the PSLG is a closed, continuous curve; in the labeled case, the PSLG is a closed, piecewise continuous curve. Image meshing in the case of labeled images will be studied in a forthcoming publication.

Our 2D binary-image-triangulation procedure can be synthesized in several stages:

Figure 1 illustrates the main triangulation steps. Starting from a 2D binary image, Figure 1a, we detect all the points on which the PSLG vectors will be positioned, Figure 1b. Next, the image contour is calculated using the morphological gradient, Figure 1c. Finally, each PSLG vector is defined by iterative geodesic reconstruction along the image gradient. The result is the morphological triangulation of the 2D binary image, Figure 1e.

2.1. Choice of a Binary 2D Image

We chose an X arbitrary complex shape composed of pixels with high morphological complexity, Figure 2, represented by areas circled in red.

These areas indicate that cusps B, thin parts A, and strong concave components B. The set X is not simply connected; it contains holes C, one of which is very close to the outer contour of X, so that X is relatively thin locally. The representation of X in a discrete frame is a binary image matrix composed of a label and its complementary, represented in the matrix by 1 and 0, respectively. Figure 2 shows an example of an image matrix corresponding to a detail of the form X.

2.2. Image Contouring Using Morphological Tools

To establish the image’s PSLG, we first need to define the image contour. Several techniques exist to determine the contour of an image, we have chosen to use the tools of mathematical morphology. This contour is then used to define the vector path of the PSLG.

Determining the contours of objects in an image is a complex process in image analysis, particularly if the image is made up of pixels with a high dynamic range of numerical values. There are, however, a number of methods available for this purpose [19,20,21]. For a binary image composed of pixels, the selection of an object’s edge pixels is trivial, since an image edge pixel (1) must have at least one of its sides connected to a background pixel (0). This simplifies the determination of object contours for simple configurations. The contour of a set is the morphological gradient invented by Beucher. It applies to functions $f\in {\mathbb{R}}^n$ and to sets $X\in {\mathbb{R}}^n$, which is the case for binary images. In this case, the set formulation of the gradient is defined as the subtraction of the dilation of X by a ball B of radius r centered on a point of the image x:

$$\begin{array}{c}\hfill {\delta }_{(B,x,r)}\left(X\right)\end{array}$$

(1)

reduced by the erosion of X by a ball B with the same radius r and the same center x:

$$\begin{array}{c}\hfill {\epsilon }_{(B,x,r)}\left(X\right)\end{array}$$

(2)

Note that there are several formulations of the erode of X: ${\epsilon }_{(B,x,r)}\left(X\right)$, ${\epsilon }_{(B,x)}\left(X\right)$, or ${\epsilon }_{\left(B\right)}\left(X\right)$, respectively of the dilated X: ${\delta }_{(B,x,r)}\left(X\right)$, ${\delta }_{(B,x)}\left(X\right)$ or ${\delta }_{\left(B\right)}\left(X\right)$. We choose the notations ${\epsilon }_{(B,x)}\left(X\right)$ for the eroded X and ${\delta }_{(B,x)}\left(X\right)$ for the dilated X.

In this case, the Beucher gradient is defined by:

$$\begin{array}{c}\hfill \partial \left(X\right)={\delta }_{B\left(r\right)}\left(X\right)\setminus {\epsilon }_B\left(r\right)\left(X\right)\end{array}$$

(3)

In our study, we chose to use the lower or inner or interior gradient ${\partial }^{-}\left(X\right)$, Figure 3c, qui est defined by the set X, Figure 3a, minus its morphological erosion ${\epsilon }_B\left(X\right)$, Figure 3b.

In the lower morphological gradient, this derivation direction allows us to stay within the X set.

Figure 3 illustrates the various steps required to determine the image contour.

In practice, the gradient of X is constructed by the intersection of X with the complementary image of X eroded. These elementary operations modify the shape of a set X using a structuring element, usually a ball B of radius r and centered at a point x of the image $B(x,r)$. Other notations include $B\left(r\right)$ or simply B.

In the case of digital images, the lower ${\partial }^{-}\left(X\right)$ gradient results from the intersection of the image matrix, Figure 3d with detail A, and the complementary of the erosion matrix, Figure 3e. The result of the morphological gradient is a binary image matrix, Figure 3f, whose values at one correspond to the contour defined by the Beucher gradient.

Note that this calculation follows the intuitive idea of a classical derivative by considering the constant label areas of the image as zero. It gives a value at the boundary of X. For a binary image, we assign one to all non-zero values of the gradient. This approach, applied to the entire X image, is used to determine the global contour, Figure 3c.

We also note that Beucher also proved that the morphological gradient and the gradient defined in the continuous case ∇ are equal.

2.3. Sample Points on the Morphological Gradient Image

The points sampled along the image of the X gradient are used to construct the PSLG. The random sampling of points in the gradient image matrix does not guarantee controlled point spacing. Furthermore, to best describe the shape of the set X, the extremities of the vectors must be placed on remarkable points of the contour: points or concavities. For real shapes that are sometimes complex, these points cannot always be located analytically using splines or polynomials on bends or inflections in the contour.

2.4. Detection of Remarkable Points on the Image Contour

There are three types of remarkable points. They are located on convex parts in red, Figure 4, while cusps or concavities in the image are shown in blue and green, Figure 4.

2.4.1. Remarkable Points on the Convex Parts of X

The remarkable points on the convexities of X are located on the $S_q\left(X\right)$ skeleton of the X Endoskeleton [22].

2.4.2. Skeletonization of X

Skeletonization is a well-known procedure in mathematical morphology. It is defined by morphological openings based on two fundamental operations, dilation and erosion.

Let $\stackrel{\circ }{B}(x,r)$ be an open ball on B with center x and radius r with $\stackrel{\circ }{B}(x,r)\subset B(x,r)$. $S_r\left(X\right)$ is the ensemblist subtraction of this morphological erosion of X reduced by the unit opening on this erode:

$$\begin{array}{c}\hfill \begin{array}{c}{S}_r\left(X\right)={\mathcal{E}}_{\stackrel{\circ }{B}(x,r)}\left(X\right)\setminus {\gamma }_{B(x,\eta )}\left({\mathcal{E}}_{\stackrel{\circ }{B}(x,r)}\left(X\right)\right)\hfill \end{array}\end{array}$$

(4)

In detail A of Figure 5a, we first erode X, Figure 5b.

We realize the morphological opening of this erosion, i.e., we erode and then dilate this erode. The result is that only the thin end of X, Figure 5c, disappears. X dilated by this opening means that only this point remains, Figure 5d.

The skeleton of X will be the union of the centers of these maximal balls:

$$\begin{array}{c}\hfill \begin{array}{c}{S}_q\left(X\right)=\underset{r>0}{\cup }\underset{\eta >0}{\cap }{\mathcal{E}}_{\stackrel{\circ }{B}(x,r)}\left(X\right)\setminus {\gamma }_{B(x,\eta )}\left({\mathcal{E}}_{\stackrel{\circ }{B}(x,r)}\left(X\right)\right)\hfill \end{array}\end{array}$$

(5)

The first point of the skeleton $S_r\left(X\right)$, Figure 5d, corresponds to the center of a circle tangent to only two points on the contour, Figure 6b, of X.

Figure 6a represents its previously calculated inner gradient.

The skeleton is an iterative process, and each point of the skeleton corresponds to the center of a ball B of a maximal circle of radius r, Figure 6b.

The Lantuéjoul Skeleton Formulation is also known as the maximal ball skeleton. The skeleton is thinly homotopic because it retains the same number of connected components of X, Figure 6c,d.

2.4.3. Detection of Simple Points on $S_q\left(X\right)$

The digital representation of the skeleton is an image whose thickness is at most one pixel; this is known as the line of a skeleton, Figure 7a.

The points of the samples are the extremity or skeleton end points.

We carry out a hit-and-miss operation to extract the simple points of the skeleton, which are the remarkable points of the convex parts of X, Figure 7d.

Figure 7b shows that only the conformal configurations of the simple points are considered; the result of $HMT\left(X\right)$ leads to a sub-matrix $\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)$ located exactly at the position of the single point. A non-conformal configuration leads to a zero sub-matrix.

2.4.4. Remarkable Points on the Concave Parts of X

Remarkable points located on the concavities and cusps of X are detected on the skeleton of X’s complement Exoskeleton. We therefore perform exactly the same operations as before on $X^c$, Figure 8.

The various remarkable points of the PSLG are brought together in an image, Figure 9c, by joining the points extracted from the convexities, Figure 9a, and concavities, Figure 9b, of X.

2.5. Construction of the Contour Vectors of the PSLG

2.5.1. Position of the Problem—Remark on Euclidean Metrics

The PSLG cannot be constructed by calculating the nearest neighbor from a sampling point.

Let $P_0$ be the nearest Euclidean neighbor of $P_1$, Figure 10a, and the vector $\overrightarrow{P_0{P}_1}$ is established.

On the other hand, $P_n$ is closer to $P_1$ than $P_2$, so establishing the vector $\overrightarrow{P_1{P}_n}$ is a break in the vector definition of the contour, Figure 10b.

2.5.2. Sampling of PSLG Points by Successive Geodesic Morphological Dilation

Between two successive remarkable points, we calculate ${\delta }_G^{\left(n\right)}\left(P_i\right)$, the unitary iterative geodesic dilation of the first remarkable point $P_i$, which we call the geodesic seed, on the ${\partial }^{-}\left(X\right)$ contour up to the second remarkable point. At each iteration $i=\lambda$, with $\lambda =cte$ for the demonstration, a new point $P_{i\lambda }$ is established, Equation (6):

$$\begin{array}{c}{\delta }_G^{\left(\lambda \right)}\left(P_i\right)=\underset{\lambda times}{\underbrace{\delta (\dots \delta (\delta }}\left(P_i\right)\cap G)\cap G\dots )\cap G\hfill \\ orifweadmitthat\hfill \\ {\delta }_G\left(P_i\right)\stackrel{def}{=}(P_i\oplus B_{(x,r)})\cap G\\ so\hfill \\ {\delta }_G^{\lambda }\left(P_i\right)=\underset{\lambda times}{\underbrace{{\delta }_G\left(P_i\right)\circ ...\circ {\delta }_G\left(P_i\right)}}\hfill \end{array}$$

(6)

The dilatation is isotropic in ${\mathbb{R}}^n$ so that if $P_i$ is the germ, each ${\delta }_G^{\lambda }\left(P_i\right)$ leads to two possible directions of propagation for the first point, $i=0$, and ${\delta }_G^{\left(\lambda \right)}\left(P_0\right)$ leads to two possible positions of $P_1$, Figure 11a.

To construct the PSLG, we choose an arbitrary direction, Figure 11b.

A new point $P_{i+1}$ is placed, and we therefore define a vector $\overrightarrow{P_i{P}_{i+1}}$. In Figure 11b, the vectors $\overrightarrow{P_0{P}_1}$ and $\overrightarrow{P_1{P}_2}$ are established.

The process of constructing vectors is continued by iterative geodesic dilation until idempotence is reached, thus closing the contour of each object, Equation (7):

$$\begin{array}{c}\hfill \delta ({\delta }_G^{\left(n\right)}\left(P_i\right))={\delta }_G^{\left(n\right)}\left(P_i\right)\end{array}$$

(7)

At idempotency, we say that we achieved a geodesic reconstruction of G by successive geodesic dilations of the unit dimensions, Equation (8):

$$\begin{array}{c}\hfill {\gamma }_G\left(P_i\right)=R_G\left(P_i\right)={\delta }_G^{+\infty }\left(P_i\right)\end{array}$$

(8)

Figure 11b shows that geodesic idempotency is achieved at the construction of $\overrightarrow{P_n{P}_0}$.

These vectors, constructed by geodesic expansion, are known as PSLG geodesic vectors.

2.6. Adaptive Sampling of Triangulation Points

To ensure a good definition of the contour, locating the remarkable points is a necessary but not sufficient condition. It is also necessary for the vector polygonal curve thus established to remain injective; it must not intersect itself, thus respecting the principle of the Jordan curve [23] illustrated in Figure 12.

For example, for any $\Gamma$ Jordan’s curve in ${\mathbb{R}}^2$, injective by definition, the polygonal curve induced by discretization points that are too far apart creates a vector, in red in Figure 12, which breaks the injective character of the curve.

Figure 12 illustrates a case of the non-injectivity of Jordan’s curve by establishing the vector $\overrightarrow{P_8{P}_9}$, which intersects the vectors $\overrightarrow{P_4{P}_5}$ and $\overrightarrow{P_5{P}_6}$.

This singularity is detected when the ratio k between the Euclidean distance and the geodesic distance is too high. In practice, it should not exceed three times the Euclidean distance:

$$\begin{array}{c}\hfill d_X\left(x,y\right)=kd\left(x,y\right)where1\le k\le 3\end{array}$$

(9)

In the morphological sense, the fine parts of the contour of X are detected by the intersection of the contour of X and the complementary of X minus its opening of size r:

$$\begin{array}{c}\hfill P\left(X\right)={\partial }^{-}\left(X\right)\cap (X\cap {\left(Xo{B}_r\right)}^c)\end{array}$$

(10)

The radius r of the ball B makes it possible to locate fine zones, Figure 13a, on which we carry out a densification ${\lambda }_2=2\lambda$ so that the polygonal curve retains its injective character, Figure 13b.

2.7. Location of Points in X

The second stage of Delaunay’s constrained triangulation consists of distributing points uniformly throughout the X set in order to guarantee optimum relative equality of the triangles.

In our case where X is not included in a parallelepipedic field, the Poisson point process can be established simply [22]. Lantuéjoul redefines a process of uniform point implantation inside any set [15]. The hit-and-run procedure in 2D, illustrated in Figure 14a, begins by implanting a point $x_i$ when the intersection of a straight line $L_i$ passes through this point when the contour of the image ${\partial }^{-}\left(X\right)$ is non-empty: $L_i\cap {\partial }^{-}\left(X\right)\ne \left\{\varnothing \right\}$.

To ensure that the points are evenly spaced, we introduce a repulsion distance r around each of the $x_i$ points. This distribution and spacing allow us to guarantee a homogeneous triangulation of our image.

The probability of a point x being contained in a mask X is

$$\begin{array}{c}\hfill P(x,X)=\underset{Z}{\int }f(x,y)dyx\in X\end{array}$$

(11)

with

$$\begin{array}{c}\hfill f(x,y)={\displaystyle \frac{2}{d{w}_d}}{\displaystyle \frac{1}{l(x,y,X){\left|x-y\right|}^{d-1}}}\end{array}$$

(12)

where $x\in X$; $x\ne y$; $d,\omega$ = constant; and $l(x,y,X)$ is the length of the segment $(x,y)\in X$.

This iterative process is continued until the mesh density, Figure 14b, is reached. To establish triangles with a shape close to equilateral, we impose a repulsion distance $\lambda$ on the points $x_i$. Note that the introduction of this distance breaks the uniformity property of the distribution but allows us to establish a homogeneous distribution of the points included in X:

$$\begin{array}{c}\hfill d(x_i,x_{i+1})\le \lambda \end{array}$$

(13)

All the triangulation points are combined with the remarkable points. The set of points required for the constrained Delaunay triangulation is shown in Figure 15.

2.8. Results of Delaunay Constraint Morphological Triangulation

Figure 16 illustrates the automatic meshing of the first label, label 1, in the image.

The triangulation faithfully respects the shape constraint by positioning the remarkable points. It should also be noted that with this method, there is no Staircase effect due to the discretization of the image into pixels.

2.9. Triangulation of the Set’s Complementary X

The Delaunay constrained triangulation of the 0 label is performed on the complementary image of the 1 label. It is produced using the same process as described above. The triangulation of this label must absolutely respect the common contour between the two labels while ensuring the continuity of the triangulation, Figure 17.

The final triangulation result demonstrates that coupling mathematical morphology with Delaunay constrained triangulation allows us to define a robust, automatic, and optimized procedure for the method proposed in this study.

2.10. Controlled Densification of the Triangulation

2.10.1. Morphological Densification

An important improvement to this technique concerns the automatic and controlled densification of a mesh density in fine areas of the image. To solve this problem, differentiated densification is proposed in these critical areas.

The densification concerns the fine zones of X, which implies a redensification of the local contour of these zones.

2.10.2. Local Densification of X

In mathematical morphology, the ball B of radius r, $B(r,x)$, can be used as an indicator of these fine areas. We therefore define the fine morphological parts of X, with $P_X$ being the set intersection between X and the complementary of its open by a ball B of a radius slightly larger than the size of the fine zones, Equation (14):

$$\begin{array}{c}\hfill P_X=X\cap {\left(XoB(r,x)\right)}^c\end{array}$$

(14)

In practice, r can be calculated using an opening granulometry operation [22].

In our case, Figure 18 illustrates the zones considered to be fine in the one label of the image.

The location of these critical zones in blue, Figure 19, allows for the use of a different local mesh density ${\theta }_2$.

For example, a double density of points ${\theta }_2=2{\theta }_1$ can be defined in these new morphologically localized zones in blue, Figure 19b.

2.10.3. Local Densification of the Contour X

To remain consistent with ${\theta }_2$, it is important to redefine a new ${\lambda }_2=2\lambda$ locally on the contour in this new area of the image. The areas of the contour concerned by the local densification of X, noted $P_g$, are deduced from $P_X$. In fact, they are identified as being the intersection between the X contour and $P_X$, Figure 20:

$$\begin{array}{c}\hfill \left[H\right]P_g=P_X\cap {\partial }^{-}\left(X\right)\end{array}$$

(15)

Figure 20 illustrates the contour zones adjoining the X zones identified previously. The location of these critical zones enables a different local mesh density ${\lambda }_2$ to be used, Figure 19b. For example, a double density of points ${\lambda }_2=2{\lambda }_1$ can be defined in these new contour zones in green, Figure 19b, while a ${\lambda }_1$ density is implemented on the rest of the contour in black, Figure 19b.

2.10.4. Union of Densities and Triangulation

The different densities ${\lambda }_1$, ${\lambda }_2$, ${\theta }_1$, and ${\theta }_2$ are grouped together, and the triangulation process is carried out continuously on this anisotropic density field, Figure 19c. It is important to note that the different densities are mutually constrained; when a point in a process is generated, it is defined with a repulsion zone in which the implantation of a new point is forbidden.

2.10.5. Internal De-Densification Control $\theta$

At constant ${\lambda }_1$ and ${\lambda }_2$, the PSLG defined by the contour vector constraint and established by the geodesic vectors whose remarkable points are one of the morphological constraints makes it possible to establish different mesh densities ${\theta }_1$ and ${\theta }_2$ while maintaining a Hausdorff distance of less than one pixel. The mesh shown in Figure 21 has a mesh density of ${\theta }_2={\displaystyle \frac{1}{2}}{\theta }_1$, Figure 21b, and ${\theta }_3={\displaystyle \frac{1}{5}}{\theta }_1$, Figure 21c.

Figure 21c illustrates the robustness of the method, which forces the triangulation to be included in X. The triangulation remains coherent despite a relatively low $\theta$ density. However, there is a flattening of certain triangulations, which is not necessarily inappropriate for image simplification or visualization applications that only require a reduction in the data (pixels).

The number of elements is directly linked to the different densities $\theta$ and $\lambda$ set by the user: these are two triangulations input data. Compared with Marching-Type Techniques, where the mesh density is at least equal to the number of pixels, it is often necessary to simplify and then optimize in order to avoid the Staircase effect of the pixels.

2.10.6. Hausdorff Distance

The positioning of the triangles on the image contour is checked by measuring any discrepancy between the real image and the mesh. This must be as small as possible. The Hausdorff distance is used to measure this gap. Huttenlocher applies this formulation to digital screening [24], and Aspert measures the deviation of a triangular mesh from a surface [25].

In our case, we need to evaluate the distance separating the contour established by the Beucher gradient ${\partial }^{-}\left(X\right)$, Figure 22b, from the vectors of the triangulation contour, Figure 22a.

This distance is defined by

$$\begin{array}{c}\hfill d({\partial }^{-}\left(X\right),L\left(X\right))=max\left\{\underset{x\in {\partial }^{-}\left(X\right)}{sup}d(x,L\left(X\right)),\underset{x\in L\left(X\right)}{sup}d(x,{\partial }^{-}\left(X\right))\right\}\end{array}$$

(16)

in the Euclidean formulation and

$$\begin{array}{c}\hfill d({\partial }^{-}\left(X\right),L\left(X\right))=inf{\displaystyle \{}\epsilon ,{\partial }^{-}\left(X\right)\subset {\delta }_{B\left(\epsilon \right)}\left(L\left(X\right)\right),L\left(X\right)\subset {\delta }_{B\left(\epsilon \right)}\left({\partial }^{-}\left(X\right)\right)\}\end{array}$$

(17)

in the morphological formulation, where $B\left(\epsilon \right)$ is the closed ball of radius $\epsilon$ and $d(x,{\partial }^{-}\left(X\right))={inf}_{y\in {\partial }^{-}\left(X\right)}d(x,y)$, the Euclidean distance from x to ${\partial }^{-}\left(X\right)$.

In our approach, the measurement of this distance is less than the pixel $d({\partial }^{-}\left(X\right),L\left(X\right))<1$, Figure 22c, so we consider that the two contours are superimposable.

We check that our method preserves the integrity of the contour of each of the objects in the image.

2.10.7. Comparison with the Marching Square Technique

The aim of this section is to position our approach in relation to current techniques. In the field of Marching Square, the list of calculation codes is far from exhaustive, but we chose Avizo Fire (Edition 8.0.0) for comparison. The aim is to highlight the problems associated with this method.

With the Marching Square Technique, the mesh density is maximal, and the number of elements is at least equal to the number of pixels, Figure 23a.

As Marching Square faithfully follows a shape composed of pixels, we observe a Staircase discretization of the mesh at the boundary of the object, Figure 23b,c and Figure 24b.

However, there are other smoothing methods, such as the one established by [26] or [27], for example.

It is important to note that a drastic reduction in the number of mesh elements can lead to breaks in the continuity of the mesh, Figure 24c.

3. Application to Microstructure Images from Scientific Fields

We have explored many scientific fields, such as medicine and biology, but it is in the field of metallurgy that we have found images of materials with shapes whose morphology can be extremely complex, and whose contours can be difficult to approximate with a polynomial approximation such as a spline.

The chosen example is a cross-section of an agglomerate of silver particles mixed with particles of oxides (tin and bismuth), Figure 25.

These powders, known as STOB, silver tin oxide with bismuth addition, are produced chemically using a traditional powder metallurgy technique of agglomerate granulometry adjusted by thermal granulation. This ensures that the porosity and oxide additions are evenly distributed. Tin oxide, $S_n{O}_2$, reinforces the matrix, while bismuth oxide, $B{i}_2{O}_3$, and copper oxide, $CuO$, improve the wetting of the tin oxide by the silver.

The morphological complexity of the image is obvious, as it contains numerous cusps and holes. Establishing the PSLG on this image is tricky using traditional methods. The image presented in Figure 25 is the result of a segmentation of a grayscale image acquired using an electronic microscope.

The morphological Delaunay constraint triangulation procedure is performed on the entire image, Figure 26.

All the morphological complexities were ideally managed by our algorithm, and the Hausdorff distance calculation is well below one. This precision is all the more apparent in the details, Figure 27a, where our meshing method enables precise triangulation, Figure 27c, for complex morphology, Figure 27b or even Figure 27e, when the thickness of the objects in the image is one pixel, Figure 27d.

4. Conclusions

Delaunay constrained morphological triangulation applied to a binary image can be considered an automatic and universal procedure, as it is unbiased both in the definition of the PSLG and in the number of labels to be considered, which is not limited, unlike the Marching Technique.

What is also notable about our meshing method is that the number of meshing elements is optimized and much lower than the number of pixels in the image. It is therefore not necessary to supplement the procedure with remeshing techniques. This can sometimes be detrimental to the final mesh, i.e., remeshing techniques, which are often based on the principle of element decimation, which can lead to the local deletion of fine components of the image in our meshing procedure. Our method is therefore particularly well suited to experimental binary images, as it is capable of accurately describing all the real morphologies of microstructures that are often difficult to model using polynomials. A Hausdorff distance is always less than the size of a pixel, which makes our method both precise and accurate. In light of these promising results, we will be proposing in the future the generalization of the method to 2D labeled images.

5. Future Work: Labeled Case (Part 2)

The challenges associated with labeled images are particularly pronounced in many fields. In medical imaging, the complexity lies in the fact that the contour of an anatomical structure is often defined by the contours of adjacent structures. For example, the contour of an organ may be partially defined by neighboring organs. Similarly, in the case of EBSD images in materials science, the contours of crystalline grains are determined by adjacent grains, complicating the precise definition of boundaries. Managing junction points where several grains meet and share edges between grains requires high precision to avoid topological errors. In both fields, these complex interactions require advanced techniques and robust algorithms to faithfully represent the structures and their interactions in a Planar Straight-Line Graph (PSLG).