Shape Bottlenecks and Conservative Flow Systems - CiteSeerX

Shape Bottlenecks and Conservative Flow Systems Jean-Fran¸cois Mangin Service Hospitalier Fr´ed´eric Joliot, CEA 4, place du g´en´eral Leclerc 91401 Orsay Cedex, France [email protected]

Jean R´egis Service de Neurochirurgie Fonctionnelle et St´er´eotaxique C.H.U. La Timone 254 rue Saint Pierre 13005 Marseille, France

Abstract This paper proposes an alternative to mathematical morphology to analyze complex shapes. This approach aims mainly at the detection of shape bottlenecks which are often of interest in medical imaging because of their anatomical meaning. The detection idea consists in simulating the steady state of an information transmission process between two parts of a complex object in order to highlight bottlenecks as areas of high information flow. This information transmission process is supposed to have a conservative flow which leads to the well-known Dirichlet-Neumann problem. This problem is solved using finite differences, over-relaxation and a raw to fine implementation. The method is applied to the detection of main bottlenecks of brain white matter network, namely corpus callosum, anterior commissure and brain stem.

1. Introduction





Bottlenecks and mathematical morphology

Shape bottlenecks have been widely used in mathematical morphology to decompose complex objects in simpler parts. The usual approach is the following. First, the whole object is eroded in order to get disconnection of its different parts at the bottleneck level (see Fig. 1 and 2). Then parts of interest are reconstructed in order to recover their original shapes. In fact, the reconstruction process itself can be performed according to two slightly different points of view. When only specific object parts are of interest, the connected components corresponding to these parts are selected as seeds using problem dedicated procedures. Then a dilation of limited size (usually the size of the previous erosion) is applied conditionally to the initial object. This kind of approach is used today by a number of teams in order to

Vincent Frouin Service Hospitalier Fr´ed´eric Joliot, CEA 4, place du g´en´eral Leclerc 91401 Orsay Cedex, France [email protected]

segment the brain in 3D T1-weighted MR images [14]. For other applications, the reconstruction process is applied until convergence, which means that the final result is simply a generalized Vorono¨ı diagram constructed for the set of selected seeds. Such an approach can be applied for instance to segment brain white matter in four main parts corresponding to brain and cerebellum hemispheres (see Fig. 1 and 3). With a modern point of view, this kind of morphological deconnection schemes could be related to the decomposition obtained using the reaction/diffusion space [12]. Indeed morphological erosion can be related to an evolutionary partial differential equation corresponding to pure reaction [3]. During these morphological segmentation processes, few attention is given to shape bottlenecks for themselves. However, in medical imaging, shape bottlenecks have often an important anatomical meaning which justifies their detection. For instance, the corpus callosum, which is morphologically the bottleneck which links both brain hemispheres, has been intensively studied for morphometric purpose (male/female differences...) and is a subject of increasing interest in image processing [7, 19]. Therefore usual morphological deconnection schemes are sometimes not sufficient to analyze complex anatomical shapes. A decomposition in parts and bottlenecks is then required. Extending the morphological approach with this purpose is possible. For instance junction points of the Vorono¨ı diagram are a first kind of representation of bottlenecks but it appears difficult to master their localization. Indeed, the diagram is highly dependent on the choice of the discrete distance used to construct it, on the kind of structuring element used for erosion and on the shape of the bottleneck. Moreover, junction point sets do not really represent bottlenecks because they do not share the initial object dimension (3D objects give usually 2D junctions). This problem could be addressed through the study of a new object constituted by the difference between the initial object and a morphological opening of the initial object (the result of a limited re-

Axial slices

Coronal slices Figure 2. 3D renderings of the four seeds obtained from a 7mm erosion of brain white matter (erosion computed using 3D chamfer distances taking into account voxel anisotropy [13]) Sagittal slices Figure 1. A few slices of a T1 weighted MR image acquired with a Signa GE 1.5 T. (124 slices, 1mm slice thickness, 0.9mm pixel size)

construction of seeds as described in first paragraph). Indeed this new object should include bottlenecks which could be detected using the Vorono¨ı diagram junctions mentioned above as seeds. Unfortunately difficulties could occur with convoluted objects because bottlenecks could be connected to other fine parts of the initial object removed during the opening.






Axial slices

Coronal slices

Sagittal slices Figure 3. A few slices of a generalized Vorono¨ı diagram of brain white matter computed for 4 seeds obtained from a 7mm erosion. Erosion and Vorono¨ı diagrams have been computed using anisotropic chamfer distances taking into account voxel anisotropy. Brain white matter has been simply segmented by thresholding a presegmented brain obtained by standard mathematical morphology [14].

White matter network of the brain

In this paper we propose a different approach to address shape bottleneck detection. This approach is inspired by the anatomical nature of brain white matter bottlenecks. White matter of brain hemispheres is mostly made up by association and commissural fibers which interconnect different cortical areas [15]. Commissural fibers interconnecting cortical regions of both hemispheres cross mainly in the corpus callosum and the anterior commissure (see Fig. 4). Other commissures generally interconnect internal nuclei. They are usually difficult to identify because of MR image resolution. Several reasons call for an important development of studies of brain white matter network in the future. There is a relative lack of knowledge of this network. In the human brain, long association systems are mainly known from gross dissection. Almost nothing is known on their inter-individual variability. Nevertheless, experimental literature gives a fairly complete account of these connections in primate. Information on the fiber tracts should become of crucial importance for the interpretation of functional images originating from positron emission tomography (PET), magnetic resonance imaging (fMRI) and especially magneto-encephalography (MEG) and electroencephalography (EEG), because of their accurate temporal resolution (ms). Moreover, the organization of callosal connections gives an insight into myelo- and cytoarchitecture, which results in a direct delimitation of functional areas [6, 18]. Detection of long association fibers could also be of fair importance for neurosurgical planning and morphometric studies. Lastly, new imaging methods allow the detection of fiber tract orientation. Indeed magnetic resonance diffusion tensor imaging shows the anisotropic diffusion of water in anisotropic tissues which is related to fiber tract direction [4].






Conservative flow systems

Usual T1 or T2-weighted MR images do not allow the fiber tract tracking but we will propose a way of highlighting some white matter network bottlenecks. Let us consider for instance information transmission between left hemisphere cortex and right hemisphere cortex. All information has to cross using commissural fibers. Hence, information flow

Corpus Callosum

Anterior Commissure

Hence the detection of white matter bottlenecks between both cerebral hemispheres could be inferred from the solution of the Laplace equation with appropriate boundary conditions. In the following we address the discretization of this equation for a domain of arbitrary shape and the boundary condition choice. Then we describe the resolution by a raw to fine process and over-relaxation. Lastly we present some experiments with real data aiming at the detection of corpus callosum, anterior commissure and brain stem.

2. Discrete formulation of the Laplace equation

Figure 4. Commissural connections of the teleencephalon as seen from the basal side of the brain [15].

should be higher in commissures than anywhere else. In order to apply this idea we have to simulate a steady state of information transmission in order to get an image of information flow in white matter. This phenomenon can be modelized using partial differential equations, such a model being clearly a huge simplification of reality. Since information sources are only located outside white matter, the information transmission process has a conservative flow inside white matter. Let 
   denote the information potential located at point   . Whatever the domain  included in white matter, the following result is true:  

grad 







    





div
grad

"

 

 



#





Finite differences

(1)

 and denotes the normal oriented towards where  exterior. The usual transformation of Eq.1 obtained from Green formula gives: !

The Laplace equation is one of the fundamental equations of physics, mechanics and applied mathematics. Indeed, it is the prototype of elliptic linear homogeneous equations. Therefore its resolution has been intensively dealt with in literature [8]. Three kinds of problem rely on Laplace equation according to boundary conditions. If value is imposed on the whole domain boundary, we get a Dirichlet problem. If  ' ( value is imposed on the whole boundary, we get a Neumann problem. Lastly, let  denote the domain bound.ary with )* (subsets with non null areas). If

,+ value is imposed on  and  ' ( value is imposed on

., we get a Dirichlet-Neumann problem. The idea presented in introduction will lead to Dirichlet-Neumann problems. Indeed this idea consists in the simulation of information transmission from a boundary subset /10  towards another subset 20  . With this purpose, we set 43 on 65 / , on 2 , and  "' 786 on :9
/ + 2 , where 3 and 5  are constant values with 3?@9
"/ 2A . Existence and unicity of the solution + of continuous Dirichlet-Neumann problems is a well-known result (with a sufficient regularity of boundary conditions). We will now address the discrete formulation of a DirichletNeumann problem with a domain of arbitrary shape.



(2)

Assuming the regularity of the solution, partial derivatives are classically approximated using finite differences. Indeed such an approach is sufficient for our purpose and much simpler than finite elements with a domain of arbitrary shape. From the discretization of Laplacian operator, we get for  interior the usual consistent second order discrete Laplace equation: B

Since Eq.2 is true for any  we get the well-known Laplace equation (assuming regularity of ): $

% 

(3)

3 C 

 B

3 M 

 B J

&

We use here the term “information potential” with a very arbitrary meaning... J

B 9

  

9

9 3 O 

 

B

  B 

9D 9D 9D

E
"FGHG IKJE
"LJ

B

GHG I

E
"FGHG IKJE
"FGNJ E
"FGHG IKJE
"FGHG NJ

B

(4)

G I B





 

where 3 C  3 M  3 O correspond to voxel sizes. It should be noted that this numerical scheme has to be viewed as a network of 6-connected points of which values are related by Eq. 5. Indeed, since we deal with domains of arbitrary shape, a real discretization should involve a discretization step largely smaller than voxel sizes. In the following, for the sake of simplicity, we assume 3 C  3 M )3 O . Then Eq. 5 amounts to the simple relation: 
  

where
"FGHG I






I
"FGHG I


  E
. C M O B



(5)

denotes 6-neighborhood of point

.

The discrete version of Dirichlet boundary condition is straightforward:



8/

 82 E




3

E


65

(6)

The discrete version of Neumann boundary condition is far more problematic, especially for a domain defined by a binary image, because a lot of points do not have a clear normal partial derivative. The usual approach consists in substituting normal second order partial derivatives by tangential second order partial derivatives using the Laplace equation [8]. Let be a boundary point, namely a point of which 6-neigborhood is not included in  . Let  be the “local dimension” of  at that point, namely the number of grid directions for which at least one 6-neighbor of belongs to  . Lastly let (respectively ) be the set of directions with only one 6-neighbor in  (respectively two 6-neighbors  - ). A solution to deal with difficult sit B uations where cardinal( ) ; consists in imposing a null first partial derivative in each direction of (simple cases B where cardinal( )  are naturally processed). Then for any point we can write





 















-

-

"3 ! - 
KJ 
. where   is the direction of  . Summing over  we get $#&%(')+*-, '/.0*1%('23*547698 $# ;98 % '2>*54@?A' 69B * card : ;9< 8= ' *C, '/.D*1%E'23*GF 6 8 HI; 8 % '2>*K4L?A' 69B * Laplace card : ;9< 8J , '/.0*1%('23* card F@698 MH N '%E'PO & *QF : %('2>*K4L%E'PO *1*54@?A' 6 8 *SR : 8 68 

E
8





E
KJ

3

D







UT

-

T



ZY

 (









9


" 


/



B

\[





2A +



E


B J

DI

$]



\





KJ

L 5X

T5VWV

\ -




 

(7)

Successive over relaxation

The Dirichlet-Neumann problem reduces now to a huge linear system through Eq. 5, 6 and 7. We solve this system using the usual iterative scheme corresponding to successive over relaxation. The process is initialized by



Dirichlet-Neumann boundary conditions



T

3 because 
.J 3  J 
 9 3  9 D
"( #3
7J
 (  is a direction of , card( ) denotes cardinal). Finally we get the following second order discrete version of Neumann condition:


 

9


"/ +

2E

(^
( 

3 J

5  D

(8)

a`b

Then the iterative process can be written  (9) _
" 9
"/ + 2  

dcfe 
 c c   
B 9+g 
 J g A  d 
(
   ih where 
 is a coefficient given by Eq. 5 or 7 or zero if   ' h  , and Bkj g j D . Convergence of this kind

of process for domains with simple shapes is well-known [8]. Convergence in the case of domains defined by complex binary images is questionable even if in practice we did not run into problems. Indeed, as mentioned above, our discretization step seems too large to hope a good approximation of the continuous case. Moreover, convergence speed is very difficult to assess because with convoluted domains large wavelength oscillations could require a large number of iterations to disappear. Nevertheless this last problem can be forgotten because we do not need an exact solution for our purpose. A 2D illustration of the result of successive over relaxation is proposed in Fig. 5. High and low potential Dirichlet conditions have been imposed on parts of left and right B  I hemisphere white matter surface ( 3% I    5> ). These parts have been simply defined by a geodesic distance to the left most and right most points. Successive B  over-relaxation has been applied with  and 1000 iterations. The corpus callosum is clearly highlighted in the information flow image. The information potential image shows that the iterative process has not fully converged in some very fine brain convolutions, which has no consequence on bottleneck detection. This image shows also that isopotential lines are orthogonal to white matter surface which tends to show that our Neumann boundary conditions have been respected.

ml n

l






Diffusive process

In order to interpret our iterative process independently of the continuous problem, another point of view related to diffusion on a regular network seems preferable. In fact iterative process (9) turns out to almost correspond to the well-known integration scheme of the diffusion equation [17, 11, 3]. Indeed Eq. 9 can be written

/cfe 
( g J

c 




 

" c
8 9 c
E (10) Af  /  h B (in all cases    d  
  ). Hence similarity with diffusive processes ish straightforward: e
8 
(8  J  

(8   9
(  . (11) A  / 
 M where
(     would be a conduction coefficient. In fact some differences remain for “corner points”, namely points B with cardinal( ) ; , where some 
(8    
  . h h of This observation led us to question our discretization 

raw data and domain of flow





High pot.

Low pot.

















potentiel definition and information flow

Neumann boundary condition. Therefore we tried a slightly different scheme which could be interpreted as “geodesic isotropic diffusion”:



  g J






9


"/ +

Af  / 

! 

information potential Figure 5. 2D illustration of the bottleneck detection method. High and low potential Dirichlet conditions have been fixed on parts of left and right hemisphere white matter surface. The corpus callosum is clearly highlighted in the information flow image.

/ce 
(  c
( `b B c 9 c 

"
(8
(   2E



(12)

Up to now, we have observed no significative differences between the results of schemes 9 and 12, but further work has to be done on this subject. It should be noted that an important difference remains with standard diffusion. Indeed thanks to boundary conditions and since we are only interested in the steady state, we can use a larger integration constant  than usual bounds for stability [11]. Stationary boundary conditions drive then the evolutionary diffusive process toward a steady state.

3. Applications We will now propose a few applications of ideas developped above for the detection of main white matter network bottlenecks. In all cases, the domain corresponding to white matter has been defined by thresholding a presegmentation of the brain in a T1-weighted MR image using standard mathematical morphology [14].

Corpus Callosum

Ant. commi.

Figure 6. Anterior commissure in atlas of Nieuwenhuys [15] and in real MR slice





Raw to fine resolution

In order to get reasonable computation times, the iterative process defined above has been implemented using a raw to fine approach. Indeed, the over relaxation scheme requires about 1000 iterations before reasonable convergence with highly convoluted objects, which costs more than one hour of computation on a conventional workstation for a 3D binary object including several million points. Hence, most iterations are performed at highest levels of a resolution pyramid which allows us to obtain convergence in less than 10mn on conventional Sun Sparc stations. It should be noted that the iterative process could be implemented easily on massively parallel architectures, like usual diffusive schemes. The resolution pyramid is computed classically. Resolution of the 3D binary image is reduced level by level (by a factor 2) using median filtering which preserves as far as possible initial shape. All experiments presented further have been obtained using a 4 level resolution pyramid. The number of iterations performed at each level, from the lowest resolution to the finest, are 1000, 800, 200, 100 (these choices have been fixed empirically, a serious study of these parameter influence is far beyond the scope of the paper). The highest level is initialized according to Eq. 6. Other levels are initialized by the result of respective previous level. 






Corpus callosum and anterior commissures

The first experiment aimed at the detection of the corpus callosum and the anterior commissure, which is an important landmark in the field of functional imaging, because it allows transformation towards the widely used Talairach referential [9, 10, 20]. Fig. 4, 6 and 7 give a good idea of the anatomical nature of the two structures of interest. These structures interconnect both cortical hemispheres. An automatic approach has been recently proposed in order to

Corpus Callosum

Figure 7. Corpus callosum in atlas of Nieuwenhuys [15] and in real MR slice

detect intersection between anterior commissure and interhemispheric plane using 2D scene analysis [2]. Two different boundary conditions have been tried for this experiment. The first one consists in the definition of two sub-parts of hemispheric white matter surfaces using a threshold (60 grid bonds) on a geodesic distance to left most and right most points (see Fig. 5). The second one consists in using seeds of both hemispheres obtained by erosion (cf. introduction and Fig. 2). This second approach is less satisfying because it does not correspond any more to the simulation of information transmission between both cortical hemispheres. Nevertheless, it is interesting because of its generic aspect. Indeed, such an approach can be combined with the usual morphological approaches in any case where bottleneck detection is of interest. For both boundary condition choices, high and low potential values have been fixed to 5000 and -5000. Fig. 8 and 9 propose a glimpse of the result for the first choice. The anterior commissure and the corpus callosum are clearly highlighted in information flow images. A third domain of high flow is located in the brain stem. This domain includes posterior commissure and other fibers which actually inter-connect the tele-encephale (brain hemispheres) and the cerebellum. These three main areas can be sorted according to mean value, highest value and volume after segmentation by hysteresis thresholding. Another way of using this result could stemmed from the intersection with inter-hemispheric plane which can be detected independently [2, 5] (this could turn out to be a better approach for morphometric studies [7] because the 3D volume of commissures is dependent on the chosen threshold). Lastly small areas of high flow can be observed near Dirichlet boundary conditions. These areas could be removed by postprocessing according to their highest values and distance to boundary. Fig. 10 proposes the result for the second choice, which presents less parasitic high flow areas

Figure 8. Three orthogonal slices of raw data information potential near boundaries. This could stem from the fact that seed boundaries are less convoluted than white matter surface. Lastly, Fig. 11 proposes a 3D rendering of the corpus callosum detected as the largest connected component over a high threshold in the first choice information flow image. 






Brain stem

Another experiment has been done in order to detect brain stem as bottleneck between teleencephalon and cerebellum. Low and high potential Dirichlet boundary conditions have been simply applied respectively to the 20 top most and the 20 bottom most slices. Fig. 12 shows that the situation is more complex than for corpus callosum because of the presence of a “bulb” called pons in the middle of brain stem.

4. Conclusion In this paper we have presented an alternative to mathematical morphology to analyze complex shapes. This approach aims mainly at the detection of shape bottlenecks which are often of interest in medical imaging because of their anatomical meaning. The detection idea consists in simulating the steady state of an information transmission process between two parts of a complex object in order to highlight bottlenecks as areas of high information flow. This information transmission process is supposed to have a conservative flow which leads to the well-known DirichletNeumann problem. Hence, a new application of partial differential equations (PDE) to image processing has been found. Like some other applications of PDE to this field, the idea consists in defining a problem which steady state is

information flow Figure 9. Detection of corpus callosum and anterior commissure: High and low potential Dirichlet boundary conditions have been fixed on parts of the left and right hemisphere white matter surfaces. Three orthogonal slices of information potential and information flow during the steady state are displayed. The anterior commissure and the corpus callosum are clearly highlighted in information flow images.

information flow Figure 10. Same results as in Fig.9 with different boundary conditions. High and low potential Dirichlet boundary condition have been fixed on both seeds of brain hemispheres given by an initial erosion of 7mm (cf. Fig. 3). This result presents less parasitic high flow areas.

raw data

information flow

Figure 11. 3D rendering of corpus callosum detected as the largest connected component over a high threshold in the information flow image. The rendering relies on an initial triangulation according to the method described in [1]

Figure 12. Detection of brain stem as bottleneck between teleencephalon and cerebellum (3 orthogonal slices). Low and high potential Dirichlet boundary conditions have been simply applied respectively to the 20 top most and the 20 bottom most slices. The result shows that the situation is more complex than for corpus callosum because of the presence of a “bulb” (namely the pons) in the middle of brain stem.

of interest for the segmentation purpose. A similar idea has been proposed for instance in the case of anisotropic diffusion through the addition of a term expressing the deviation between original and filtered image to the diffusion scheme [16]. Applications have been proposed dedicated to the detection of brain white matter network bottlenecks. The result are promising but further work has to be done in order to study robustness of the method.

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