2.1.

Opening and Closing by Reconstruction

In mathematical morphology (MM), the basic transformations are the erosion ${\epsilon }_{\mu B}(f)(x)$ and dilation ${\delta }_{\mu B}(f)(x)$, where $B$ represents the three-dimensional (3-D) structuring element which has its origin in the center. Figure 1 illustrates the shape of the structuring element used in this paper. $\breve{B}$ denotes the transposed set of $B$ with respect its origin, $\breve{B}=\{-x\text{:}x\in B\}$, $\mu$ is a size parameter, $f\text{:}{Z}^3\to Z$ is the input image, and $x$ is a point on the definition domain.

Fig. 1

Three-dimensional (3-D) structuring element used in this paper.

The next equations represent the morphological erosion ${\epsilon }_{\mu B}(f)(x)$ and dilation ${\delta }_{\mu B}(f)(x)$:⁴¹

In addition, the opening (closing) by reconstruction has the characteristic of modifying the regional maxima (minima) without affecting the remaining components to a large extent. These operators use the geodesic transformations.^32,42 The geodesic dilation ${\delta }_f^1(g)$ and erosion ${\epsilon }_f^1(g)$ are expressed as ${\delta }_f^1(g)=f\wedge {\delta }_B(g)$ with $g\le f$, and ${\epsilon }_f^1(g)=f\vee {\epsilon }_B(g)$ considering $g\ge f$, respectively. When the function $g$ is equal to the morphological erosion or dilation, the opening ${\tilde{\gamma }}_{\mu B}(f)(x)$ or closing ${\tilde{\phi }}_{\mu B}(f)(x)$ by reconstruction is obtained. Formally, the next expressions represent them

2.2.

Viscous Opening ${\tilde{\gamma }}_{\lambda ,\mu }$ and Viscous Difference

The viscous opening ${\tilde{\gamma }}_{\lambda ,\mu }$ and closing ${\tilde{\phi }}_{\lambda ,\mu }$ defined in Ref. 29 allow one to deal with overlapped or chained components. These transformations are denoted by

Equation (2) uses three operators, the morphological erosion ${\epsilon }_{\lambda }$, the opening by reconstruction ${\tilde{\gamma }}_{\mu -\lambda }$, and the morphological dilation ${\delta }_{\lambda }$. The morphological erosion ${\epsilon }_{\lambda }(f)$ allows one to discover and disconnect the $\lambda$-components (all components where the structuring element can go from one place to another by a continuous path made of squares and whose centers move along this path). Then, the opening by reconstruction ${\tilde{\gamma }}_{\mu -\lambda }{\epsilon }_{\lambda }(f)$ removes all regions less than $\mu -\lambda$ around the $\lambda$-components. Finally, because the viscous opening is defined on the lattice of dilatation, the ${\delta }_{\lambda }$ must be obtained on ${\tilde{\gamma }}_{\mu -\lambda }{\epsilon }_{\lambda }(f)$. An example of Eq. (2) is given in Fig. 2. The original image is exhibited in Fig. 2(a). Figure 2(b) shows the morphological erosion ${\epsilon }_{\lambda =6}(f)$. Notice that there are several components around the brain. The image in Fig. 2(c) corresponds to the transformation ${\tilde{\gamma }}_{\mu =16-\lambda =10}{\epsilon }_{\lambda =6}(f)$. In this image, several components have been eliminated by the process of opening by reconstruction. Figure 2(d) displays the result of the transformation ${\delta }_{\lambda =6}$ ${\tilde{\gamma }}_{\mu =10}{\epsilon }_{\lambda =6}(f)$.

Fig. 2

Viscous opening illustration: (a) original volume $f$, (b) ${\epsilon }_{\lambda =6}(f)$, (c) ${\tilde{\gamma }}_{\mu =16-\lambda =10}{\epsilon }_{\lambda =6}(f)$, and (d) ${\delta }_{\lambda =6}$ ${\tilde{\gamma }}_{\mu =10}{\epsilon }_{\lambda =6}(f)$.

Viscous openings permit the sieving of the image through the viscous difference.²⁹ This is defined in

According to the explanation given above, the erosion ${\epsilon }_{\lambda }$ discovers the $\lambda$-components and the difference ${\tilde{\gamma }}_{{\mu }_1-\lambda }({\epsilon }_{\lambda })-{\tilde{\gamma }}_{{\mu }_2-\lambda }({\epsilon }_{\lambda })$ with $\lambda \le {\mu }_1\le {\mu }_2$ sieving the image, whereas ${\delta }_{\lambda }$ is necessary to obtain the viscous component. The viscous difference gives the information of all discovered disconnected components of a certain size $\lambda$ when $\mu$ is increased.

2.3.

Lower Leveling

The lower leveling transformation is presented as follows:³⁰

On the other hand, the selection of the marker $g$ is very important. In Ref. 30, for example, the following marker was used to segment the brain:

The parameter $\alpha$ helps to control the reconstruction of the marker $g$ into $f$. An example to illustrate the performance of Eq. (5) is given in the next section.

3.1.

Composition of Morphological Connected Transformations

As previously stated, viscous opening and lower leveling allow separating the chained components, and it is natural to think of combining both operators to get one transformation capable of having better control on the reconstruction process. Following this idea, an option is to use the viscous opening as a marker of the lower leveling to eliminate a great portion of the skull; posteriorly, the resulting image is again processed with a similar filter to eliminate the remaining regions around the brain. Such a procedure represents a sequential application of the combined transformations considering different parameters in order to have increasing control in the reconstruction process. The purpose is to eliminate the skull softly in two steps. The following equation permits the disconnection of the chained components and comes from the combination of Eqs. (2) and (5):

Nevertheless, Eq. (7) produces an unsatisfactory performance. Figure 4 shows an example of the transformation ${\eta }_{\mu =1,{\alpha }_2=20,{\lambda }_2=8,{\mu }_2=10,{\alpha }_1=10,{\lambda }_1=10,{\mu }_1=12}(f)$. Some slices of the original volume can be seen in Fig. 4(a) where the segmentation results in the creation of holes on the brain, as is illustrated in Fig. 4(b). The viscous opening causes this behavior, since all components not supporting the morphological erosion of size $\lambda$ will merge with the background because the 3-D structuring element produces stronger changes as the size of the structuring element is increased. One way to get better segmentations consists of computing the operator in Eq. (8), where $f$ is the input image, ${\overline{h}}_{\rho }$ represents the mean filter of size $\rho$, and $T_a$ expresses a threshold between $[a,255]$ sections. The mean filter ${\overline{h}}_{\rho }$ partially closes the holes, $T_a$ and permits the selection of certain regions of interest, and ${\xi }_{\rho ,a}$ helps to obtain a portion of the original image

Fig. 4

Segmentation of a volume [taken from the database Internet Brain Segmentation Repository (IBSR) with 20 subjects] using Eq. (7). (a) Slices of the original volume and (b) slices corresponding to ${\eta }_{\mu =1,{\alpha }_2=20,{\lambda }_2=8,{\mu }_2=10,{\alpha }_1=10,{\lambda }_1=10,{\mu }_1=12}(f)$.

The combination of Eqs. (7) and (8) gives the next operator as a result

To apply Eq. (9), the reasoning below needs to be considered.

3.2.

Parameters $\alpha$, $\lambda$, and $\mu$

3.2.1.

Parameter $\alpha$

The following analysis corresponds to Eq. (5) since Eq. (9) uses it. Large $\alpha$ values produce (i) a time reduction in reaching the final result and (ii) a larger control in the reconstruction process.

The quantification of the time when Eq. (5) is applied on a volume of 60 slides—using as a marker ${\tilde{\gamma }}_{\lambda =10,\mu =12}$ and the lower leveling ${\mathrm{\Psi }}_{\mu =1,\alpha }^{\infty }$ with $\alpha =1$, 3, 6, 9, 12—is presented in Fig. 5. This figure displays several slices belonging to the output volumes obtained for different $\alpha$ values.

Fig. 5

Execution time of Eq. (5) and some output slices corresponding to several processed volumes. (a) Time spent to compute Eq. (5) considering 60 slices, the marker corresponds to the viscous opening with $\lambda =10$, $\mu =12$. The leveling is applied considering $\alpha =1$, 3, 6, 9, 12; (b) graph corresponding to the data presented in (a); (c) slice of the original volume; (d) brain section taking from the viscous opening and used as marker; (e)–(i) set of slices taken from the volumes processed with $\alpha =1$, 3, 6, 9, 12, as is illustrated in Fig. 4.

3.2.2.

Parameter $\lambda$

The adequate election of the parameter $\lambda$ will bring, as a consequence, the disconnection between the skull and the brain. Such a parameter is computed from a granulometric analysis applying Eq. (10)³⁷

Eq. (10)

$$\upsilon =\frac{\mathrm{vol}[{\gamma }_{\lambda }(f)]-\mathrm{vol}[{\gamma }_{\lambda +1}(f)]}{\mathrm{vol}[f]},$$

where vol stands for the volume, i.e., the sum of all gray levels in the image, and ${\gamma }_{\lambda }(f)$ represents the morphological opening size $\lambda$. The graph in Fig. 6(a) corresponds to the application of Eq. (10) taking $\lambda \in [1,30]$. This graph has three important intervals. The interval where $\lambda \in [1,6]$ shows the elimination of an important part of the skull. Hence, in order to detect the brain, $\lambda$ will take values greater than 6, i.e., $\lambda \ge 6$.

Fig. 6

Granulometric curves computed on certain volume taken from the database IBSR with 20 subjects. (a) Granulometry computed from Eq. (10) and (b) viscous granulometry obtained from Eq. (11).

3.3.

Simplification of Eq. (9)

The fact of sequentially applying two transformations along with a mask to obtain Eq. (9) brings as a consequence the use of a large number of parameters. This problem is considered and a simplification is proposed as follows:

According to Eq. (12), the marker ${\xi }_{{\rho }_1,a_1}{\tilde{\gamma }}_{{\lambda }_1,{\mu }_1}$ is obtained from the viscous opening, and it is propagated by the lower leveling transformation ${\mathrm{\Psi }}_{\mu ,{\alpha }_1}^{\infty }$. Equation (12) presents the following benefits when compared with respect to Eq. (9): (1) the use of fewer parameters and (2) a reduction of the execution time. The performances of Eqs. (9) and (12) are illustrated in Sec. 4.

4.1.

Parameters Involved in Eqs. (9) and (12)

Tables 1Table 2Table 3–4 contain the parameters used in Eqs. (9) and (12) to segment the volumes belonging to the IBSR1 and IBSR2 datasets. In the volumes of IBSR1, the neck was cropped to obtain similar images to those of IBSR2. Differences among the volumes with respect to the intensity, size, and connectivity, originate the parameters’ variation. The guidelines for the parameter selection are given below:

Fig. 7

Some brain slices corresponding to the segmentation of the volume IBSR1_100 using Eq. (9) with the parameters defined in Table 1.

Fig. 8

Some brain slices corresponding to the segmentation of the volume IBSR1_100 using Eq. (12) with the parameters defined in Table 2.

Table 1

Parameters corresponding to Eq. (9). The processed dataset was IBSR1.

Table 2

Parameters corresponding to Eq. (12). The processed dataset was IBSR1.

Table 3

Parameters corresponding to Eq. (9). The processed dataset was IBSR2.

Table 4

Parameters corresponding to Eq. (12). The processed dataset was IBSR2.

4.2.

Comparison Results

Figure 9 illustrates the resulting segmentation corresponding to the volume IBSR1_16 under the application of Eqs. (9), (12), and by BET (default parameters) implemented in the MRIcro software.⁴³ Figure 9(a) shows the original slices. Figure 9(b) displays the respective manual segmentations. Figure 9(c) presents the application of Eq. (9) with the parameters given in Table 1. Figure 9(d) presents the application of Eq. (12) with the parameters given in Table 2. Figure 9(e) shows the brain extraction using the BET algorithm. The parameters selected for the set of 20 brains are intensity $\text{threshold}=0.50$ and $\text{vertical gradient}=0.0$. In order to compare the segmentations, the Jaccard and the Dice coefficients are computed. Table 5 contains the indices corresponding to BET and Eqs. (12) and (9) for the IBSR1.

Fig. 9

Images illustrating the segmentation of volume IBSR1_016 through several methods. (a) Brain sections corresponding to the original volume IBSR1_016; (b) manual segmentations provided by the IBSR website; (c) application of Eq. (9) with the parameters defined in Table 1; (d) application of Eq. (12) with the parameters defined in Table 2; and (e) slices obtained from BET using a $\text{fractional intensity}=0.5$ and $\text{vertical gradient}=0.0$.

Table 5

Jaccard and Dice indexes corresponding to IBSR1 dataset and segmented with BET, Eqs. (9) and (12) considering the parameters presented in Tables 1 and 2.

A similar procedure is applied to the IBSR2 database. Figure 10 presents the segmentations corresponding to the IBSR2_04 volume. Figure 10(a) shows the original slices. Figure 10(b) displays the respective manual segmentations. Figure 10(c) presents the application of Eq. (9) with the parameters given in Table 3. Figure 9(d) presents the application of Eq. (12) with the parameters given in Table 4. Figure 9(e) shows the brain extraction using the BET algorithm with the default parameters. Table 6 contains the Jaccard and Dice indices corresponding to BET and Eqs. (12) and (9) for the IBSR2.

Fig. 10

Images illustrating the segmentation of volume IBSR2_04 through several methods. (a) Brain sections corresponding to the original volume IBSR2_04; (b) manual segmentations provided by the IBSR website; (c) application of Eq. (9) with the parameters defined in Table 3; (d) application of Eq. (12) with the parameters defined in Table 4; and (e) slices obtained from BET using a $\text{fractional intensity}=0.5$ and $\text{vertical gradient}=0.0$.

Table 6

Jaccard and Dice indexes corresponding to IBSR2 dataset and segmented with BET, Eqs. (9) and (12) considering the parameters presented in Tables 3 and 4.

Table 7 contains the mean values of the indices presented in Tables 5 and 6 together with the mean values of the indices reported in Refs. 11, 12, and 40.

Table 7

Jaccard and Dice mean indexes corresponding to IBSR1 and IBSR2 datasets and some reported in the literature.

4.3.

Discussion

Some commentaries on the segmented volumes are presented as follows:

Fig. 11

Time performance of Eq. (9). (a) Consumed time by Eq. (9) when the number of slices increases. (b) Graph of the information presented in (a).