and Israel M. Santillán-Méndez<sup>1</sup>

<sup>1</sup>*División de Investigación y Posgrado de la Facultad de Ingeniería, Universidad Autónoma de Querétaro, Centro Universitario S/N, CP. 76010, Querétaro* <sup>2</sup>*Posgrado de la Facultad de Medicina, Departamento de Investigación, Universidad Autónoma de Querétaro, Clavel # 200, Fraccionamiento Prados de la Capilla, Santiago de Querétaro* <sup>3</sup>*Universidad Politécnica de Querétaro, Carretera Estatal 420 S/N, El Rosario, El Marqués, CP. 76240 Querétaro México*

#### **1. Introduction**

#### **Notation**

170 Advances in Brain Imaging

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On the other hand, congenital strabismus affects 3% of the world-wide population. Nevertheless, it is not known with precision, the anatomical site or microstructural damage underlying the origin of the disease and the zones or nuclei involved. This situation arises from the difficulty of applying morphometric studies in vivo, which would provide an accurate identification of the involved zones and nuclei and would give a better description of diseases such as dissociated strabismus, which is a form of congenital strabismus (6)(7)(8). Therefore, evidence related to the etiology of this disease has been collected from

<sup>173</sup> Comparison of Granulometric Studies of Brain Slices

from Normal and Dissociated Strabismus Subjects Through Morphological Transformations

Some forms of dissociated strabismus, besides presenting alterations in the digitized brain mapping (DBM), show ocular movements with angles of variable presentation. When movement is directed in ward, the condition is called SSAV, when it is directed outward it is known as DHD. This last visual alteration is generally symmetrical and concomitant with posterior electroencephalographic alterations, while SSAV tends to present more anterior and

Cortical areas in dissociated strabismus can be observed and differentiated with high accuracy by means of techniques of optical dissector as those employed in (10). These authors show, in rodents, changes in cortical cellular density related to visual activity using optical dissector techniques or in vitro methods such as cytochrome oxidase (11)(12)(13). As a result of these studies, the authors demonstrate the presence of structural changes in monkey's cerebral cortex with strabismus and amblyopia; however, given the nature of these methods, it is not

Up to the present moment, no morphological alterations have been described in any of the varieties of strabismus. But based on the experience drawn from neurofunctional studies of patients with dissociated strabismus, using DBM (6)(7)(8), single photon emission computerized tomography (SPECT) (14)(15) and nuclear proton magnetic resonance spectroscopy (H-NMRS) (9) it is expected that forthcoming evidence on this subject will soon

Moreover, for the first time, by means of DBM (7)(8), the participation of the cerebral cortex in dissociated strabismus was demostrated. This finding was lately verified by means of SPECT studies. This technique enabled the localization of cortical areas involved in dissociated strabismus and in an epilepsy case. SPECT results (14)(15) showed neuroadaptive changes consistent with the improvement of glucose consumption in the cerebral cortex of patients

The combined application of DBM and H-NMRS methods has demonstrated that cortical regions in dissociated strabismus patients present neuroelectric and biochemical alterations compatible with epileptogenous disease (9). The difficulty lies in identifying, by means of neuroimage studies applied to live subjects, the microstructural alterations related to the origin of dissociated strabismus in children; these micro-alterations cannot be determined with conventional methods of imagenology such as CT or MRI. So far, one alternative for identifying structural changes in cortical regions would be by means of a granulometric

Granulometric analysis is a methodology developed within mathematical morphology. Mathematical morphology is a technique widely used in image processing. This technique

neurofunctional studies (6)(7)(8)(9).

asymmetric alterations (6)(7)(8)(9).

possible to apply them to strabismic children.

allow the identification of microstructural changes.

analysis of the diverse regions under study.

with strabismus treated with the botulinic toxin or surgery.

(*ψλ*)*λ*∈*Z*<sup>+</sup> a family of transformations depending on an unique positive parameter *<sup>λ</sup>*


*SSAV*1, *SSAV*2 subject 1 and subject 2 with strabismus syndrome of angular variability

*DHD*1, *DHD*2 subject 1 and subject 2 with dissociated horizontal deviation

*DBM* Digitized brain mapping

*CT* Computed tomography

Occipital lobe is located in the posterior portion of brain, contiguous to parietal and temporal lobes. Brain lobes present at their most superficial portions, pyramidal as well as starred neurons forming nuclei conforming GM. These neurons are ordered in six laminae on the brain surface to form the brain cortex. The cellular groups in this cortical regions are disposed in an orderly fashion forming columns; their axons extend deeper into the WM (1).

WM is composed by neuronal axons interconnecting cortical zones with other nuclei, for example, magnocellular (great size) and parvocellular (small size) cells, whose projections go to the geniculate lateral bodies toward the cerebral cortex of the occipital lobes (1).

GM, as well as WM, are part of the visual pathway, but only the most superficial layer of the brain that contains cellular nuclei conforming the GM is known as brain cortex; whereas the WM is basically constituted by the axons of these cells (2)(3).

The cerebral cortex is traversed by circumvolutions (gyrus and sulcus) of varied aspects, making it difficult to identify with precision where a brain specific cortical area begins and ends, when it is first observed by means of conventional neuroimaging studies, as is the case of MRI or computed tomography (CT) (1)(2)(3). In spite of this difficulty, it has been possible to obtain important advances in cerebral cortex morphometric studies by means of voxel-based morphometry (VBM). In (4), the authors find a GM diminution in the calcarine sulcus containing the primary visual cortex; they also demonstrate reductions in parieto-occipital areas and in the ventral temporal cortex in children and adult patients with amblyopia. This condition is characterized by dimness of vision with no-improvement in spite of using the best optical correction. In the same way, but using a MR-based morphometric technique, a study on adult patients with exotropia (5), reports a GM increase in motor cortical regions, in conjunction with a diminution in visual cortex, suggesting morphometric changes related to cerebral plasticity. In spite of these morphometric advances, no studies are available in which alterations in the WM and GM in child cerebral cortex with congenital strabismus are compared with that of healthy children of the same age.

2 Will-be-set-by-IN-TECH

*χWM* normalized volumetric measure of granulometric residues of clear structures

*ζGM* normalized volumetric measure of granulometric residues of dark structures

*SSAV*1, *SSAV*2 subject 1 and subject 2 with strabismus syndrome of angular variability

Occipital lobe is located in the posterior portion of brain, contiguous to parietal and temporal lobes. Brain lobes present at their most superficial portions, pyramidal as well as starred neurons forming nuclei conforming GM. These neurons are ordered in six laminae on the brain surface to form the brain cortex. The cellular groups in this cortical regions are disposed

WM is composed by neuronal axons interconnecting cortical zones with other nuclei, for example, magnocellular (great size) and parvocellular (small size) cells, whose projections

GM, as well as WM, are part of the visual pathway, but only the most superficial layer of the brain that contains cellular nuclei conforming the GM is known as brain cortex; whereas the

The cerebral cortex is traversed by circumvolutions (gyrus and sulcus) of varied aspects, making it difficult to identify with precision where a brain specific cortical area begins and ends, when it is first observed by means of conventional neuroimaging studies, as is the case of MRI or computed tomography (CT) (1)(2)(3). In spite of this difficulty, it has been possible to obtain important advances in cerebral cortex morphometric studies by means of voxel-based morphometry (VBM). In (4), the authors find a GM diminution in the calcarine sulcus containing the primary visual cortex; they also demonstrate reductions in parieto-occipital areas and in the ventral temporal cortex in children and adult patients with amblyopia. This condition is characterized by dimness of vision with no-improvement in spite of using the best optical correction. In the same way, but using a MR-based morphometric technique, a study on adult patients with exotropia (5), reports a GM increase in motor cortical regions, in conjunction with a diminution in visual cortex, suggesting morphometric changes related to cerebral plasticity. In spite of these morphometric advances, no studies are available in which alterations in the WM and GM in child cerebral cortex with congenital strabismus are

in an orderly fashion forming columns; their axons extend deeper into the WM (1).

go to the geniculate lateral bodies toward the cerebral cortex of the occipital lobes (1).

WM is basically constituted by the axons of these cells (2)(3).

compared with that of healthy children of the same age.

*DHD*1, *DHD*2 subject 1 and subject 2 with dissociated horizontal deviation

(*ψλ*)*λ*∈*Z*<sup>+</sup> a family of transformations depending on an unique positive parameter *<sup>λ</sup>*

conforming the image

conforming the image

*SS* subjects with dissociated strabismus

*DHD* Dissociated horizontal deviation

*DBM* Digitized brain mapping *CT* Computed tomography

*SSAV* Strabismus syndrome of angular variability *WM* , *GM* white and grey matter, respectively.

*CS* Control subject

On the other hand, congenital strabismus affects 3% of the world-wide population. Nevertheless, it is not known with precision, the anatomical site or microstructural damage underlying the origin of the disease and the zones or nuclei involved. This situation arises from the difficulty of applying morphometric studies in vivo, which would provide an accurate identification of the involved zones and nuclei and would give a better description of diseases such as dissociated strabismus, which is a form of congenital strabismus (6)(7)(8). Therefore, evidence related to the etiology of this disease has been collected from neurofunctional studies (6)(7)(8)(9).

Some forms of dissociated strabismus, besides presenting alterations in the digitized brain mapping (DBM), show ocular movements with angles of variable presentation. When movement is directed in ward, the condition is called SSAV, when it is directed outward it is known as DHD. This last visual alteration is generally symmetrical and concomitant with posterior electroencephalographic alterations, while SSAV tends to present more anterior and asymmetric alterations (6)(7)(8)(9).

Cortical areas in dissociated strabismus can be observed and differentiated with high accuracy by means of techniques of optical dissector as those employed in (10). These authors show, in rodents, changes in cortical cellular density related to visual activity using optical dissector techniques or in vitro methods such as cytochrome oxidase (11)(12)(13). As a result of these studies, the authors demonstrate the presence of structural changes in monkey's cerebral cortex with strabismus and amblyopia; however, given the nature of these methods, it is not possible to apply them to strabismic children.

Up to the present moment, no morphological alterations have been described in any of the varieties of strabismus. But based on the experience drawn from neurofunctional studies of patients with dissociated strabismus, using DBM (6)(7)(8), single photon emission computerized tomography (SPECT) (14)(15) and nuclear proton magnetic resonance spectroscopy (H-NMRS) (9) it is expected that forthcoming evidence on this subject will soon allow the identification of microstructural changes.

Moreover, for the first time, by means of DBM (7)(8), the participation of the cerebral cortex in dissociated strabismus was demostrated. This finding was lately verified by means of SPECT studies. This technique enabled the localization of cortical areas involved in dissociated strabismus and in an epilepsy case. SPECT results (14)(15) showed neuroadaptive changes consistent with the improvement of glucose consumption in the cerebral cortex of patients with strabismus treated with the botulinic toxin or surgery.

The combined application of DBM and H-NMRS methods has demonstrated that cortical regions in dissociated strabismus patients present neuroelectric and biochemical alterations compatible with epileptogenous disease (9). The difficulty lies in identifying, by means of neuroimage studies applied to live subjects, the microstructural alterations related to the origin of dissociated strabismus in children; these micro-alterations cannot be determined with conventional methods of imagenology such as CT or MRI. So far, one alternative for identifying structural changes in cortical regions would be by means of a granulometric analysis of the diverse regions under study.

Granulometric analysis is a methodology developed within mathematical morphology. Mathematical morphology is a technique widely used in image processing. This technique

Formally, the morphological opening *γμB*(*f*)(*x*) and closing *ϕμB*(*f*)(*x*) are expressed as

<sup>175</sup> Comparison of Granulometric Studies of Brain Slices

from Normal and Dissociated Strabismus Subjects Through Morphological Transformations

where the morphological erosion *εμB*(*f*)(*x*) and the morphological dilation *δμB*(*f*)(*x*) are *εμB*(*f*)(*x*) = ∧{ *<sup>f</sup>*(*y*) : *<sup>y</sup>* <sup>∈</sup> *<sup>μ</sup>B*ˇ*x*} and *δμB*(*f*)(*x*) = ∨{ *<sup>f</sup>*(*y*) : *<sup>y</sup>* <sup>∈</sup> *<sup>μ</sup>B*ˇ*x*}. Here, <sup>∧</sup> is the inf

The morphological opening and closing can be interpreted in the following way, both morphological transformations allow the elimination of components that can not be contained by the structuring element. Morphological opening works in the interior of the function, while

On the other hand, throughout the paper, we will use size 1, or size *μ* of the structuring element. Size 1 means a square of 3 × 3 pixels, while size *μ* means a square of (2*μ* + 1)(2*μ* + 1) pixels. For example, if the structuring element is size 3, then the square will be 7 × 7 pixels, i.e, 49 neighbors are analyzed. In any size of the structuring element the origin is located at its

In Fig.1 the erosion, dilation, opening and closing are illustrated by using a size 5 structuring

(a) (b) (c) (d)

(e)

Fig. 1. Images illustrating several morphological operators using a structuring element size 5.

a) Input image, b) erosion, c) dilation, d) opening, and e) closing.

*γμB*(*f*)(*x*) = *δμB*ˇ(*εμB*(*f*))(*x*) *and ϕμB*(*f*)(*x*) = *εμB*ˇ(*δμB*(*f*))(*x*) (1)

follows:

center.

element.

operator and ∨ is the sup operator.

the morphological closing in the complement of the function.

was initially used to solve a real problem applied to the study of porous means in materials science (16). Currently, one of its multiple applications is the processing of medical images.

In this paper we present a methodology to segment brain MRI <sup>1</sup> by using the morphological opening by reconstruction (17; 18). In particular, the segmentation of regions included in the occipital lobe and areas nearest to this brain structure is performed. Subsequently, two granulometric studies are carried out in a similar way to that followed in granulometric density studies (19). The first study, consists in analyzing clear and dark structures in the segmented deskulling brain for SS and CS groups. Subsequently, a similar procedure is done but on the WM and GM for the subject under study. In each of the granulometric studies, mean patterns are obtained and compared against the granulometric patterns belonging to the CS group. This comparison enables the establishment of volumetric differences, and the introduction of an index, which is useful to understand the behavior of the structures detected in the WM and GM for CS and SS.

It is important to mention not only that the age of the six participants in this study is seven years old; but also that, the GM and WM are segmented by using the methodology followed in (20). From the latter procedure, only some output images are presented to illustrate such segmentations.

This paper is organized as follows. In section 2, a background of the different transformations and concepts related to mathematical morphology are presented. The morphological opening and closing are defined in subsection 2.1. While the opening and closing by reconstruction are presented in subsection 2.2. In subsection 2.3, the granulometry notion, as well as the equations that work in a way similar to those in granulometric density are given. On the other hand, a methodology to segment regions of interest from MRI by using morphological transformations is provided in section 3. In subsection 3.1 several patterns describing the granulometric density of clear structures, dark regions, WM and GM are introduced. The intervals considered for the different structure sizes and a volumetric analysis based on a given index are reported in section 4. Finally, conclusions are presented in section 5.

#### **2. Background on morphological transformations**

As follows a background on morphological transformations employed for the treatment of MRIs is presented.

#### **2.1 Definitions of some morphological transformations**

In mathematical morphology increasing and idempotent transformations are frequently used. Morphological transformations fulfilling these properties are known as morphological filters (16; 21; 22). The basic morphological filters are the morphological opening *γμB*(*f*)(*x*) and closing *ϕμB*(*f*)(*x*) using a given structural element. In this paper, a square structuring element is employed, where *B* represents the structuring element of size 3 × 3 pixels, which contains its origin. While *<sup>B</sup>*<sup>ˇ</sup> is the transposed set (*B*ˇ= {−*<sup>x</sup>* : *<sup>x</sup>* <sup>∈</sup> *<sup>B</sup>*}) and *<sup>μ</sup>* is a homothetic parameter.

<sup>1</sup> The MRIs used in this paper were obtained from an equipment Philips Intera of 1.5 T (Philips Medical Systems Best Netherlands), using a sequence fast feel echo (FFE), with echo time TE = 6.9 ms, repetition time TR = 25 ms, deviation angle FA = 30 degrees, excitation number NSA = 1, vision field FOV = 230 mm and slice number = 120.

Formally, the morphological opening *γμB*(*f*)(*x*) and closing *ϕμB*(*f*)(*x*) are expressed as follows:

$$
\gamma\_{\mu B}(f)(\mathbf{x}) = \delta\_{\mu \underline{\mathcal{B}}}(\varepsilon\_{\mu B}(f))(\mathbf{x}) \quad \text{and} \quad \varrho\_{\mu \underline{\mathcal{B}}}(f)(\mathbf{x}) = \varepsilon\_{\mu \underline{\mathcal{B}}}(\delta\_{\mu B}(f))(\mathbf{x}) \tag{1}
$$

where the morphological erosion *εμB*(*f*)(*x*) and the morphological dilation *δμB*(*f*)(*x*) are *εμB*(*f*)(*x*) = ∧{ *<sup>f</sup>*(*y*) : *<sup>y</sup>* <sup>∈</sup> *<sup>μ</sup>B*ˇ*x*} and *δμB*(*f*)(*x*) = ∨{ *<sup>f</sup>*(*y*) : *<sup>y</sup>* <sup>∈</sup> *<sup>μ</sup>B*ˇ*x*}. Here, <sup>∧</sup> is the inf operator and ∨ is the sup operator.

The morphological opening and closing can be interpreted in the following way, both morphological transformations allow the elimination of components that can not be contained by the structuring element. Morphological opening works in the interior of the function, while the morphological closing in the complement of the function.

On the other hand, throughout the paper, we will use size 1, or size *μ* of the structuring element. Size 1 means a square of 3 × 3 pixels, while size *μ* means a square of (2*μ* + 1)(2*μ* + 1) pixels. For example, if the structuring element is size 3, then the square will be 7 × 7 pixels, i.e, 49 neighbors are analyzed. In any size of the structuring element the origin is located at its center.

In Fig.1 the erosion, dilation, opening and closing are illustrated by using a size 5 structuring element.

(a) (b) (c) (d)

4 Will-be-set-by-IN-TECH

was initially used to solve a real problem applied to the study of porous means in materials science (16). Currently, one of its multiple applications is the processing of medical images. In this paper we present a methodology to segment brain MRI <sup>1</sup> by using the morphological opening by reconstruction (17; 18). In particular, the segmentation of regions included in the occipital lobe and areas nearest to this brain structure is performed. Subsequently, two granulometric studies are carried out in a similar way to that followed in granulometric density studies (19). The first study, consists in analyzing clear and dark structures in the segmented deskulling brain for SS and CS groups. Subsequently, a similar procedure is done but on the WM and GM for the subject under study. In each of the granulometric studies, mean patterns are obtained and compared against the granulometric patterns belonging to the CS group. This comparison enables the establishment of volumetric differences, and the introduction of an index, which is useful to understand the behavior of the structures detected

It is important to mention not only that the age of the six participants in this study is seven years old; but also that, the GM and WM are segmented by using the methodology followed in (20). From the latter procedure, only some output images are presented to illustrate such

This paper is organized as follows. In section 2, a background of the different transformations and concepts related to mathematical morphology are presented. The morphological opening and closing are defined in subsection 2.1. While the opening and closing by reconstruction are presented in subsection 2.2. In subsection 2.3, the granulometry notion, as well as the equations that work in a way similar to those in granulometric density are given. On the other hand, a methodology to segment regions of interest from MRI by using morphological transformations is provided in section 3. In subsection 3.1 several patterns describing the granulometric density of clear structures, dark regions, WM and GM are introduced. The intervals considered for the different structure sizes and a volumetric analysis based on a

given index are reported in section 4. Finally, conclusions are presented in section 5.

As follows a background on morphological transformations employed for the treatment of

In mathematical morphology increasing and idempotent transformations are frequently used. Morphological transformations fulfilling these properties are known as morphological filters (16; 21; 22). The basic morphological filters are the morphological opening *γμB*(*f*)(*x*) and closing *ϕμB*(*f*)(*x*) using a given structural element. In this paper, a square structuring element is employed, where *B* represents the structuring element of size 3 × 3 pixels, which contains its origin. While *<sup>B</sup>*<sup>ˇ</sup> is the transposed set (*B*ˇ= {−*<sup>x</sup>* : *<sup>x</sup>* <sup>∈</sup> *<sup>B</sup>*}) and *<sup>μ</sup>* is a homothetic parameter.

<sup>1</sup> The MRIs used in this paper were obtained from an equipment Philips Intera of 1.5 T (Philips Medical Systems Best Netherlands), using a sequence fast feel echo (FFE), with echo time TE = 6.9 ms, repetition time TR = 25 ms, deviation angle FA = 30 degrees, excitation number NSA = 1, vision field FOV = 230

**2. Background on morphological transformations**

**2.1 Definitions of some morphological transformations**

in the WM and GM for CS and SS.

segmentations.

MRIs is presented.

mm and slice number = 120.

(e)

Fig. 1. Images illustrating several morphological operators using a structuring element size 5. a) Input image, b) erosion, c) dilation, d) opening, and e) closing.

*f* 

*f* 

*x* 

Fig. 2. a) Original image *f* and the marker *g* = *ε*(*f*), b) Original image f and the marker *g* = *δ*(*f*), c) Opening by reconstruction, which uses erosion as marker, d) Closing by

In Fig. 3 the procedure to find *χClear* on one slice is illustrated; this procedure is the same in the case of equation 4. On the other hand, in the last image of Fig. 3, the arithmetic difference is calculated between openings of sizes 5 and 6. This last image illustrates well some components that are not visible at first sight; however, they are detected by a granulometric

In Fig. 4 we present a set of brain MRI slices belonging to a patient with dissociated strabismus classified as SSAV (in the following, in order to simplify the notation we will use for example,

)( ~γ*f*

(a) (b)

(c) (d)

**3. MRI processing through morphological transformations**

reconstruction, which uses dilation as marker.

process.

*x* 

<sup>177</sup> Comparison of Granulometric Studies of Brain Slices

*f* 

*f* 

*x* 

)( ~ ϕ*f*

*x* 

δ*f* )(

ε*f* )(

from Normal and Dissociated Strabismus Subjects Through Morphological Transformations

μB

#### **2.2 Opening and closing by reconstruction**

The reconstruction transformation notion is a useful concept introduced by mathematical morphology. These transformations allow the elimination of undesirable regions without considerably affecting the remaining structures of the image. This characteristic is due to the fact that these transformations are built by means of geodesic transformations (19). The geodesic dilation *δ*<sup>1</sup> *<sup>f</sup>*(*g*)(*x*) and the geodesic erosion *<sup>ε</sup>*<sup>1</sup> *<sup>f</sup>*(*g*)(*x*) of size one are given by *δ*1 *<sup>f</sup>*(*g*)(*x*)=*f*(*x*) <sup>∧</sup> *<sup>δ</sup>*(*g*)(*x*) with *<sup>g</sup>*(*x*) <sup>≤</sup> *<sup>f</sup>*(*x*) and *<sup>ε</sup>*<sup>1</sup> *<sup>f</sup>*(*g*)(*x*)=*f*(*x*) ∨ *ε*(*g*)(*x*) with *g*(*x*) ≥ *f*(*x*), respectively. When the function *g*(*x*) is equal to the erosion or the dilation of the original function, we obtain the opening *γ*˜*μB*(*f*)(*x*) and the closing *ϕ*˜*μB*(*f*)(*x*) by reconstruction (17; 18; 23):

$$\tilde{\gamma}\_{\mu B}(f)(\mathbf{x}) = \lim\_{\mathbf{n} \to \infty} \delta\_f^{\mathbb{II}}(\varepsilon\_{\mu B}(f))(\mathbf{x}) \quad \text{and} \quad \tilde{\varphi}\_{\mu B}(f)(\mathbf{x}) = \lim\_{\mathbf{n} \to \infty} \varepsilon\_f^{\mathbb{II}}(\delta\_{\mu B}(f))(\mathbf{x}) \tag{2}$$

In Fig. 2 the performance of the opening and closing by reconstruction is illustrated. Note in Figs. 2(c) and 2(d) that some components have been eliminated, while the remaining are maintained equal to those in the original image.

#### **2.3 Granulometry**

Granulometry is the distribution by sizes of particles that constitute an aggregate; it is employed in diverse areas to describe the qualities of size and shape of individual grains within a product. The concept of granulometry was introduced by G. Matheron at the end of the sixties and is presented as follows (24).

**Definition 1** (Granulometry)**.** *Let* (*ψλ*≥0)*λ*∈*Z*<sup>+</sup> *be a family of transformations depending on an unique positive parameter λ. This family constitutes a granulometry if and only if the next three properties are verified: (i)* ∀ *positive λ*, *ψλ is increasing; (ii)* ∀ *positive λ*, *ψλ is antiextensive, and (iii)* ∀ *positive λ and μ*, *ψλψμ* = *ψμψλ* = *ψmax*(*λ*,*μ*)

The family of morphological openings and closings for the numerical case {*γμB*}, {*ϕμB*} with *μ* = {1, .., *n*} fulfils this last definition.

In this paper, we try to detect some characteristics of the different structures conforming the image by means of plots; which are obtained in a way similar to a granulometric density (19). Equations 3 and 4 are used in this article to deduce the granulometric plots. Such equations enable us to obtain a normalized volumetric measure of the granulometric residues of clear (*χClear*) and dark (*ζDark*) structures conforming the image (25; 26). Note that the structures of different dimensions are detected from the diverse *μ* sizes of the structuring element.

$$\chi\_{\text{Celer}} = \frac{vol\left(\gamma\_{(\mu-1)B}(f)(\mathbf{x})\right) - vol(\gamma\_{\mu B}(f)(\mathbf{x}))}{vol(f(\mathbf{x})) + 1} \tag{3}$$

$$\mathcal{L}\_{Dark} = \frac{vol(\varphi\_{\mu B}(f)(\mathbf{x})) - vol(\varphi\_{(\mu - 1)B}(f)(\mathbf{x}))}{vol(f(\mathbf{x})) + 1} \tag{4}$$

Where *vol* represents the volume; the unit has been added in the denominator of equations 3 and 4 to avoid any indetermination.

6 Will-be-set-by-IN-TECH

The reconstruction transformation notion is a useful concept introduced by mathematical morphology. These transformations allow the elimination of undesirable regions without considerably affecting the remaining structures of the image. This characteristic is due to the fact that these transformations are built by means of geodesic transformations (19).

respectively. When the function *g*(*x*) is equal to the erosion or the dilation of the original function, we obtain the opening *γ*˜*μB*(*f*)(*x*) and the closing *ϕ*˜*μB*(*f*)(*x*) by reconstruction

In Fig. 2 the performance of the opening and closing by reconstruction is illustrated. Note in Figs. 2(c) and 2(d) that some components have been eliminated, while the remaining are

Granulometry is the distribution by sizes of particles that constitute an aggregate; it is employed in diverse areas to describe the qualities of size and shape of individual grains within a product. The concept of granulometry was introduced by G. Matheron at the end of

**Definition 1** (Granulometry)**.** *Let* (*ψλ*≥0)*λ*∈*Z*<sup>+</sup> *be a family of transformations depending on an unique positive parameter λ. This family constitutes a granulometry if and only if the next three properties are verified: (i)* ∀ *positive λ*, *ψλ is increasing; (ii)* ∀ *positive λ*, *ψλ is antiextensive, and*

The family of morphological openings and closings for the numerical case {*γμB*}, {*ϕμB*} with

In this paper, we try to detect some characteristics of the different structures conforming the image by means of plots; which are obtained in a way similar to a granulometric density (19). Equations 3 and 4 are used in this article to deduce the granulometric plots. Such equations enable us to obtain a normalized volumetric measure of the granulometric residues of clear (*χClear*) and dark (*ζDark*) structures conforming the image (25; 26). Note that the structures of

*<sup>χ</sup>Clear* <sup>=</sup> *vol*(*γ*(*μ*−1)*B*(*f*)(*x*)) <sup>−</sup> *vol*(*γμB*(*f*)(*x*))

*<sup>ζ</sup>Dark* <sup>=</sup> *vol*(*ϕμB*(*f*)(*x*)) <sup>−</sup> *vol*(*ϕ*(*μ*−1)*B*(*f*)(*x*))

Where *vol* represents the volume; the unit has been added in the denominator of equations 3

different dimensions are detected from the diverse *μ* sizes of the structuring element.

*<sup>f</sup>* (*εμB*(*f*))(*x*) *and <sup>ϕ</sup>*˜*μB*(*f*)(*x*) = lim*n*→<sup>∞</sup> *<sup>ε</sup><sup>n</sup>*

*<sup>f</sup>*(*g*)(*x*) of size one are given by

*<sup>f</sup>*(*δμB*(*f*))(*x*) (2)

*<sup>f</sup>*(*g*)(*x*)=*f*(*x*) ∨ *ε*(*g*)(*x*) with *g*(*x*) ≥ *f*(*x*),

*vol*(*f*(*x*)) + <sup>1</sup> (3)

*vol*(*f*(*x*)) + <sup>1</sup> (4)

*<sup>f</sup>*(*g*)(*x*) and the geodesic erosion *<sup>ε</sup>*<sup>1</sup>

**2.2 Opening and closing by reconstruction**

*<sup>γ</sup>*˜*μB*(*f*)(*x*) = lim*n*→<sup>∞</sup> *<sup>δ</sup><sup>n</sup>*

maintained equal to those in the original image.

the sixties and is presented as follows (24).

*μ* = {1, .., *n*} fulfils this last definition.

and 4 to avoid any indetermination.

*(iii)* ∀ *positive λ and μ*, *ψλψμ* = *ψμψλ* = *ψmax*(*λ*,*μ*)

*<sup>f</sup>*(*g*)(*x*)=*f*(*x*) <sup>∧</sup> *<sup>δ</sup>*(*g*)(*x*) with *<sup>g</sup>*(*x*) <sup>≤</sup> *<sup>f</sup>*(*x*) and *<sup>ε</sup>*<sup>1</sup>

The geodesic dilation *δ*<sup>1</sup>

*δ*1

(17; 18; 23):

**2.3 Granulometry**

Fig. 2. a) Original image *f* and the marker *g* = *ε*(*f*), b) Original image f and the marker *g* = *δ*(*f*), c) Opening by reconstruction, which uses erosion as marker, d) Closing by reconstruction, which uses dilation as marker.

In Fig. 3 the procedure to find *χClear* on one slice is illustrated; this procedure is the same in the case of equation 4. On the other hand, in the last image of Fig. 3, the arithmetic difference is calculated between openings of sizes 5 and 6. This last image illustrates well some components that are not visible at first sight; however, they are detected by a granulometric process.

#### **3. MRI processing through morphological transformations**

In Fig. 4 we present a set of brain MRI slices belonging to a patient with dissociated strabismus classified as SSAV (in the following, in order to simplify the notation we will use for example,

SS).

As follows, we introduce the algorithm to achieve the deskulling step for the regions under study. This algorithm was applied to images acquired from the subjects in this study ( CS and

<sup>179</sup> Comparison of Granulometric Studies of Brain Slices

from Normal and Dissociated Strabismus Subjects Through Morphological Transformations

(a) (b) Fig. 5. DBMs showing the electrical behavior of two strabismic children. Darker regions

indicate greater electrical activity. (a) Subject with DHD, (b) Subject with SSAV.

Fig. 4. High resolution axial slices taken from SSAV1

$$\Upsilon\_{\text{Clear}} = 0.00268$$

γȝ=5 - γȝ=6

Fig. 3. Exemplification of the procedure followed to obtain the *χClear* value of one slice. Bottom image illustrates the arithmetic difference between openings.

the acronym SSAV1 to denote subject 1 with dissociated strabismus classified as SSAV; and the same is extended for DHD subjects.)

The interest in analyzing the regions nearest and within the occipital lobe comes from the finding that in patients with dissociated strabismus, using DBM, an important increment in the electrical activity in the alpha rhythm has been observed (7; 8; 14; 15). This rhythm has been associated with an occipital distribution. An example of these maps is provided in Fig. 5, in which we can note, in black color, an increment in the electrical activity in the alpha rhythm, in accordance with the scale situated at the right and top of the images. In this way, ten axial slices are considered for analysis. The slices are parallel and within the visual via.

On the other hand, images in Fig. 4 are useful to present examples of output images, after the algorithm to carry out the deskulling step is applied.

As follows, we introduce the algorithm to achieve the deskulling step for the regions under study. This algorithm was applied to images acquired from the subjects in this study ( CS and SS).

Fig. 4. High resolution axial slices taken from SSAV1

8 Will-be-set-by-IN-TECH

*vol*(γȝ=6) = 16094 pixels

*vo*l(γȝ=5) = 16138 pixels

χClear= 0.00268

*vol*(f) = 16393 pixels

γȝ=5 - γȝ=6

Fig. 3. Exemplification of the procedure followed to obtain the *χClear* value of one slice.

slices are considered for analysis. The slices are parallel and within the visual via.

the acronym SSAV1 to denote subject 1 with dissociated strabismus classified as SSAV; and

The interest in analyzing the regions nearest and within the occipital lobe comes from the finding that in patients with dissociated strabismus, using DBM, an important increment in the electrical activity in the alpha rhythm has been observed (7; 8; 14; 15). This rhythm has been associated with an occipital distribution. An example of these maps is provided in Fig. 5, in which we can note, in black color, an increment in the electrical activity in the alpha rhythm, in accordance with the scale situated at the right and top of the images. In this way, ten axial

On the other hand, images in Fig. 4 are useful to present examples of output images, after the

Bottom image illustrates the arithmetic difference between openings.

the same is extended for DHD subjects.)

algorithm to carry out the deskulling step is applied.

Fig. 5. DBMs showing the electrical behavior of two strabismic children. Darker regions indicate greater electrical activity. (a) Subject with DHD, (b) Subject with SSAV.

(a)

(b)

(c)

<sup>181</sup> Comparison of Granulometric Studies of Brain Slices

from Normal and Dissociated Strabismus Subjects Through Morphological Transformations

(d)

(e)

(f) Fig. 6. Images illustrating the procedure followed for skull elimination. (a) Cropped images

taken from SSAV1; (b) Opening by reconstruction; (c) Threshold; (d) Closing by

reconstruction; (e) Mask; (f) Manual elimination.

*Algorithm ( deskulling step ):*


With respect to step ii) of the mentioned algorithm, an opening by reconstruction size 6 is applied. This size was elected from the graphic in Fig. 7(a). The plot was obtained by computing the volume on the image *<sup>γ</sup><sup>μ</sup>* <sup>−</sup> *<sup>γ</sup><sup>μ</sup>*+1. Such graphic shows the contained structures (clear regions) in the image of size *μ*. Observe that an important structure of white regions is found between values 1-3, where the skull information is located. To verify this, in Fig. 7(b) we have the original image, on the right we have the eroded image size 3; and this eroded image is used as marker to obtain the opening by reconstruction (last image on the right). However, given that the skull is surrounded by regions with lower intensity levels, a marker given by the morphological erosion of size *μ* = 6 does not allow the skull reconstruction with its original intensity levels. In Fig. 7(c), the erosion size 6 for several input images is presented; these images are used as markers to obtain the opening by reconstruction. If smaller sizes of the structuring element are selected, for example *μ* = 1, then the skull is not completely attenuated and the proposed algorithm fails in step iii). For this reason it is crucial to have this situation in mind . In Fig. 8 we illustrate the application of the opening by reconstruction size *μ* = 1 to the images located in Fig. 6(a). Note that pixel intensity levels in the skull are hardly attenuated.

#### **3.1 Granulometric patterns**

Once the skull has been separated and the undesirable regions on the images have been suppressed, a granulometric study is carried out to determine the *χClear* and *ζDark* values on the segmented slices. *χClear* and *ζDark* quantities(see equations 3 and 4) provide information about the distribution density of the clear and dark structures within the image. In Fig. 9 we present in a general way the procedure to obtain the granulometric patterns for dark regions for both CS cases, and one SS. A similar method is applied in the case of clear structures. In i) Given that the interest zones are located around and within the occipital lobe, the original images located in Fig. 4 are cropped at section 160 to separate the regions of interest. This

ii) Once the images are cropped, we proceed to the deskulling step. For this, the opening by reconstruction size 6 is employed (see equation (2)). The opening by reconstruction has the property of avoiding the generation of new structures. In Fig. 6(b), a set of output images

iii) Thresholding of the images in Fig. 6(b) between sections 80-255. Here the skull is eliminated and several pores appear in the binary image as a result of the thresholding

iv) Subsequently, a closing by reconstruction size 6 is applied with the purpose of closing the pores originated by the thresholding. The output images are displayed in Fig. 6(d). v) A mask is obtained with respect to the original image (see Fig. 4), i.e. every pixel in the images in Fig. 6(d) takes the corresponding grey level in the original image. This situation

vi) A manual segmentation is carried out by a specialist in strabismus. Several undesired regions in the images of Fig. 6(e) are eliminated. The suppressed regions correspond

With respect to step ii) of the mentioned algorithm, an opening by reconstruction size 6 is applied. This size was elected from the graphic in Fig. 7(a). The plot was obtained by computing the volume on the image *<sup>γ</sup><sup>μ</sup>* <sup>−</sup> *<sup>γ</sup><sup>μ</sup>*+1. Such graphic shows the contained structures (clear regions) in the image of size *μ*. Observe that an important structure of white regions is found between values 1-3, where the skull information is located. To verify this, in Fig. 7(b) we have the original image, on the right we have the eroded image size 3; and this eroded image is used as marker to obtain the opening by reconstruction (last image on the right). However, given that the skull is surrounded by regions with lower intensity levels, a marker given by the morphological erosion of size *μ* = 6 does not allow the skull reconstruction with its original intensity levels. In Fig. 7(c), the erosion size 6 for several input images is presented; these images are used as markers to obtain the opening by reconstruction. If smaller sizes of the structuring element are selected, for example *μ* = 1, then the skull is not completely attenuated and the proposed algorithm fails in step iii). For this reason it is crucial to have this situation in mind . In Fig. 8 we illustrate the application of the opening by reconstruction size *μ* = 1 to the images located in Fig. 6(a). Note that pixel intensity levels in the skull are

Once the skull has been separated and the undesirable regions on the images have been suppressed, a granulometric study is carried out to determine the *χClear* and *ζDark* values on the segmented slices. *χClear* and *ζDark* quantities(see equations 3 and 4) provide information about the distribution density of the clear and dark structures within the image. In Fig. 9 we present in a general way the procedure to obtain the granulometric patterns for dark regions for both CS cases, and one SS. A similar method is applied in the case of clear structures. In

mainly to dura mater and cerebellum; this can be appreciated in Fig. 6(f).

procedure. The processed images are shown in Fig. 6(c).

is illustrated in Fig. 6(a).

*Algorithm ( deskulling step ):*

is presented in Fig. 6(e).

is presented.

hardly attenuated.

**3.1 Granulometric patterns**

Fig. 6. Images illustrating the procedure followed for skull elimination. (a) Cropped images taken from SSAV1; (b) Opening by reconstruction; (c) Threshold; (d) Closing by reconstruction; (e) Mask; (f) Manual elimination.

following the procedure depicted in Fig. 9. Notice that the first two graphs of mean volume correspond to the DHD subjects and CS group; while the last two to the SSAV subjects and CS

Fig. 10, we present the granulometric patterns for the SS and the mean pattern for the CS

<sup>183</sup> Comparison of Granulometric Studies of Brain Slices

from Normal and Dissociated Strabismus Subjects Through Morphological Transformations

The granulometric curves in Fig. 10 provide general information for the clear and dark regions detected in the whole deskulling brain. These curves have the disadvantage of being derived from non-autodual transformations, i.e. clear and dark structures are not treated in individual form. As a consequence, it is necessary to separate WM and GM to obtain the granulometric patterns of these regions. In this work, GM and WM were segmented by applying the methodology used in (20). Some output images illustrating the segmentation of such regions are presented in Fig. 11, and the granulometric patterns of WM and GM for the SS and SC are presented in Fig. 12. These graphics were obtained under the procedure depicted in Fig. 9.

In particular, we consider three main groups of clear and dark structures. These groups take into account the size of the structuring element (the structuring element used in this paper is

*Group 1(Small structures).-*In this group the structures within the sizes 1 and 2 of the structuring element are comprised. *Group 2(Medium structures).-* This group contains the structures located in the size interval 3 to 6 of the structuring element. *Group 3(Large structures).-*Finally,

In both clear and dark regions, a great variation in the curves of DHD subjects is observed with respect to the CS, mainly in medium and large sizes. This indicates the lack of smooth

*ii) Clear and dark regions analysis from graphics in Fig. 10 for SSAV case.-* Main changes in clear structures are observed in large size structures; while dark structures vary for all sizes. Curves in Fig. 10 indicate the existence of important variations in clear and dark structures in the DHD and SSAV subjects with respect to the CS. However, clear and dark structures are mixed, and results difficult to infer, since it is not possible to determine whether the predominance or the absence of some structures sizes is due to clear or dark components. This situation occurs because WM and GM have different intensities, and the morphological transformations used to build equations 3 and 4 are not autoduals, i.e, they do not try clear and dark structures in a separate form; this causes that, some clear and dark components are

Trying to avoid this inconvenience, WM and GM will be analyzed in separate ways with the

From graphs in Fig. 12, for WM and GM in the SS case, we observe the following: a) a lack of small components with respect to CS; b) the existent small components are thin; and c)

this group comprises structure sizes 7-17 of the structuring element.

*i) Clear and dark regions analysis from graphics in Fig. 10 for DHD case.-*

purpose of finding some morphometric differences between CS and SS.

*iii) WM and GM analysis from graphics in Fig. 12 for DHD case.-*

medium and large structures predominate in WM and GM.

group.

**4. Results**

a square, see subsection 2.1).

mixed during the processing.

transitions between the analyzed structures.

Fig. 7. Size election of structuring element to attenuate the skull. a) Graph of structures size vs volume on the image *<sup>γ</sup><sup>μ</sup>* <sup>−</sup> *<sup>γ</sup><sup>μ</sup>*+1, b) Original image , eroded size 3 and opening by reconstruction size 3; c) Original images, eroded size 6, and opening by reconstruction size 6

(c)

Fig. 8. Opening by reconstruction size *μ* = 1 applied to images in Fig. 6(a).

following the procedure depicted in Fig. 9. Notice that the first two graphs of mean volume correspond to the DHD subjects and CS group; while the last two to the SSAV subjects and CS group. Fig. 10, we present the granulometric patterns for the SS and the mean pattern for the CS

The granulometric curves in Fig. 10 provide general information for the clear and dark regions detected in the whole deskulling brain. These curves have the disadvantage of being derived from non-autodual transformations, i.e. clear and dark structures are not treated in individual form. As a consequence, it is necessary to separate WM and GM to obtain the granulometric patterns of these regions. In this work, GM and WM were segmented by applying the methodology used in (20). Some output images illustrating the segmentation of such regions are presented in Fig. 11, and the granulometric patterns of WM and GM for the SS and SC are presented in Fig. 12. These graphics were obtained under the procedure depicted in Fig. 9.
