Functional Maps for Brain Classification on Spectral Domain

  • Simone Melzi
  • Alessandro Mella
  • Letizia Squarcina
  • Marcella Bellani
  • Cinzia Perlini
  • Mirella Ruggeri
  • Carlo Alfredo Altamura
  • Paolo Brambilla
  • Umberto Castellani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10126)

Abstract

In this paper we exploit the Functional maps approach for brain classification. The functional representation of brain shapes, or their subparts, enables us to improve the detection of morphological abnormalities associated with the analyzed disease. The proposed method is based on the spectral shape paradigm that is largely used for generic geometric processing but still few exploited in the medical context. The key aspect of the Functional maps framework is that it moves the estimation of correspondences from the shape space to the functional space enhancing the potential of spectral analysis. Moreover, we propose a new kernel, called the Functional maps kernel (FM-kernel) for the Support Vector Machine (SVM) classification that is specifically designed to work on the functional space. The obtained results for bipolar disorder detection on the putamen regions are promising in comparison with other spectral-based approaches.

Keywords

Spectral shape analysis Functional maps Brain classification Diseases and disorders detection 

1 Introduction

Automatically detection of abnormal anatomical shapes derived from diseased subjects is a fundamental goal in medical imaging. This task is typically formulated as a two-class classification problem, assigning to each shape a healthy or diseased label [30, 31]. In particular, thanks to the increased amount of data available, the attention of researchers is often focused on advanced learning-by-example methods [2, 4, 6, 14, 15, 29]. These tools require good shape representation and measure that encodes the relationship between the shapes. The desired representation should be informative, concise and efficient in computational terms. In order to capture possible brain deformations due to the disease, it is convenient to exploit geometry and topology properties of the anatomical parts as shape representation [12, 13, 18]. To this aim, new spectral shape descriptors and methods have been adopted in this area [5, 26], aiming at investigating advanced shape analysis approaches for the characterization of brain structures.

In this work we propose a new method for shape classification based on the Functional maps framework [22]. The main idea of Functional maps consists of defining a functional space for each surface and therefore representing relations between surfaces as linear maps between these functional spaces. In this fashion, the correspondences between pair of shapes is carried out on the functional representation rather than the physical space in a more flexible and easy to compute way. The characterizations of the shapes are based on point descriptors and parts derived from a shape segmentation procedure that can be encoded as functions defined on the surfaces. These corresponding functions give rise to linear constraints on the linear map between the two spaces. The solution can be computed by solving an optimization problem. Finally, choosing a proper basis for each functional space, the desired Functional maps can be carried out by applying standard linear algebraic techniques.

The contribution of the proposed method is two-fold:
  • Firstly we extend the use of Functional map to the medical domain, to improve the encoding of morphological relations between pairs of brain-shapes.

  • Secondly we propose a new dissimilarity measure properly designed for the functional space. In particular, from this dissimilarity measure we derived a well defined new kernel, namely the Functional maps kernel (FM-Kernel) that is effective and theoretically founded.

We evaluated our method for the characterization of brain abnormalities in the context of mental health research. In particular, we propose a brain classification study on a dataset of patients affected by bipolar disorder and healthy controls. We focused on the putamen region, which is a deep gray matter brain structure, part of the basal ganglia, a functional and anatomical heterogeneous region which is thought to be affected, particularly in shape, by bipolar disorder [17]. In order to check the actual effectiveness of the proposed method and the richness added by the Functional maps framework in this context, we compared our method with more classical shape analysis methods based on a spectral approach.

Roadmap. The rest of the paper is organized as follows. In Sect. 2 we give a brief overview of the related works, highlighting connections with our method. Section 3 summarizes the background on the Functional maps framework. The proposed method and the derived FM-Kernel are presented in Sect. 4. Then, the experimental Sect. 5 shows the results of our approach in comparison with other spectral-based methods for the brain classification on the putamen regions. Finally, in Sect. 6, some conclusions are drawn and future works are envisaged.

2 Related Work

In literature there are plenty of methods for identifying and detecting alterations in anatomical shapes. For brevity here we focus on the approaches that characterize the shapes by adopting a spectral shape analysis strategy. A first method based on spectral properties was proposed in [11], where spherical harmonic descriptors (SPHARM) are computed on brain surfaces after a shapes registration step. In [26] Reuter et al. introduced a spectral global descriptor, namely Shape-DNA. This signature is defined as the increasing ordered sequence of the first Laplace-Beltrami operator (LBO) eigenvalues. The Shape-DNA is invariant to the isometric deformations and by neglecting higher frequencies of the shape it is also robust to noise. This descriptor is proposed for two different versions: the external surfaces and the entire volume. The two surface-based and volume-based versions are also introduced by Castellani et al. in [5] where a well known point signature, the Heat Kernel Signature (HKS) [10, 28], has been extended to describe the entire shape by leading to the so called Global Heat Kernel Signature (GHKS). Differently from Shape-DNA this approach is based on a point signature that encodes local information. Furthermore the GHKS allows a multi-scale analysis that enhances the discriminative properties of the signature. Note that both the approaches [5, 26] do not require an explicit registration phase for shape comparison. In [20], a collection of three well known spectral descriptors, the previously cited HKS, the Wave Kernel Signature WKS [1] and the Scale Invariant Heat Kernel Signature SI-HKS, [3] are computed at every vertex of the mesh and then used in a Bag of features framework for spectral shape analysis of brain structures in order to detect the Alzheimers Disease. The multiscale analysis is instead the basic idea of [32]. This approach encodes the volumetric geometry information starting from the volumetric LBO and obtaining a multi-scale volumetric morphology signature which describes the transition probability by random walk between the point pairs and depends on heat transmission time.

Finally, starting again from the LBO eigendecomposition an interesting technique is recently presented by Rabiei et al. in [24]. In this work the Graph Windowed Fourier[27] is exploited to encode the geometric properties of the brain cortex. More specifically, a Gyrification Index is introduced to represents at every point how much the surface is folded.

Differently from all these methods we propose to move the comparison between shapes from the descriptors spaces to the functional spaces defined on the surfaces. Shifting the focus on functional spaces can be effective and productive as for example in [19]. This work proposes a spectral framework namely Brain Transfer to transfer functions between different shapes, in order to explore the shape and functional variability of retinotopy. Conversely, to obtain and analyze this representation we propose the use of the Functional maps framework defined by Ovsjanikov et al. in [22]. This construction is founded on the LBO eigendecomposition and involves diffusion spectral descriptors and their desired properties.

3 Background

In this section we briefly introduce the Functional map framework presented in [22]. In order to achieve comparison and classification among a family of similar surfaces it can be useful to recover a point-to-point map T between every pair X and Y of smooth surfaces embedded in \(\mathbb {R}^{3}\), defined as:
$$\begin{aligned} T: X \longrightarrow Y, \end{aligned}$$
(1)
such that for every fixed point \(x \in X \), \(y = T(x) \in Y \) is the corresponding point of x. We can consider \(\mathcal {F}(X,\mathbb {R})\) and \(\mathcal {F}(Y,\mathbb {R})\) the spaces of integrable real valued functions defined on the surfaces. T naturally induces a map between the functional spaces, namely the functional map. The functional map for the pair of surfaces YX is a map between their two functional spaces:
$$\begin{aligned} C: \mathcal {F}(Y,\mathbb {R}) \longrightarrow \mathcal {F}(X,\mathbb {R}). \end{aligned}$$
(2)
Indeed for every function \(f \in \mathcal {F}(Y,\mathbb {R}) \) defined on Y the functional map C is defined by the composition with T as \( C(f) = f \circ T \), as reported in the following commutative diagram:
In the discrete setting, given a couple of basis for the functional spaces, C can be represented as a matrix. Fixing a pair of basis for the functional spaces, as for example the eigenfunctions of the Laplace Beltrami operator, the functional map can be represented in this basis reducing the size and the computational cost of its computation. If T is given than C can be easily computed. Otherwise as suggested in [22] it is possible to approximate this functional map, adopting a set of linear constraint optimizations. In our implementation the imposed constraints are two-fold. The first related to a set of pairs of functions that are stable with respect to deformations. The second is based on commutativity with pairs of corresponding operators. In general, the function constraints do not resolve the symmetry ambiguity, in fact the selected functions are usually symmetric. For this reason, in addition to the function constraints, in [22] was proposed to add the commutativity constraint.
$$\begin{aligned} C = \underset{Q}{{\text {argmin}}} \sum _{i \in I} \vert \vert Q f_{i} - g_{i} \vert \vert _{F}^{2} + \alpha \sum _{j\in J} \vert \vert R_{j}Q - QS_{j} \vert \vert _{F}^{2}. \end{aligned}$$
(3)
where \(\left\{ f_{i} \right\} _{i \in I}\) is a collection of functions defined on the surface Y, and \(\left\{ g_{i} \right\} _{i \in I}\) is the set of corresponding functions to those selected on Y. \(\alpha \in \left[ 0, 1 \right] \) is a real parameter that allows us to choose how much to give importance to the second constraint. In this way the first part of the optimization function is minimized when \(\forall f \in \mathcal {F}(Y,\mathbb {R}) \), that is selected as stable function, C(f) is as equal as possible to \(g \in \mathcal {F}(X,\mathbb {R}) \) the selected stable function defined on X that matches f. These stable functions can be selected in several way. They can be chosen between the point descriptors that are invariant to isometric deformations. If these functions are known they can be selected as landmark point correspondences or segment correspondences. The second part of the optimization function is called the Operator Commutativity constraint. Here for every pair of corresponding operator \((S_{j}, R_{j})\) belonging to \(\left\{ S_{j} \right\} _{j \in J}\) operators of Y and \(\left\{ R_{j} \right\} _{j \in J}\) operators of X this minimization force the following diagram to commute \(\forall j \in J\):

Fixing a proper basis for each functional space, the computation of the functional map can be efficiently done adopting some linear algebraic techniques at the same time also reducing the dimensionality of the problem. For a deeper analysis of the properties and uniqueness of the definition of such functional map refer to [22]. Even though the original method is quite general, in this work we will consider only brain shapes or subparts of brain represented in the discrete setting as a triangle mesh. Concluding this section, we desire to point out that the selection of functions and operators thanks to which we can obtain the functional map is a fundamental step for our work. The choices and the reasons for these choices will be presented in more detail in the next section.

4 Proposed Method

In this section we show the main contributions of our paper that are: (i) the design of a Functional maps framework on the spectral domain for brain comparison, and (ii) the customized Functional maps kernel for brain classification.

4.1 Computing Functional Maps

As mentioned in Sect. 3, we can approximate the Functional map C taking into account two sets of linear constraints. In particular, such constraints are defined by pairs of corresponding functions and by operators that satisfy the commutativity property with respect to C.

In this work we assume that in the absence of disease and disorders the brain surfaces are closer to isometric shapes with respect to the variations caused by the presence of disturbances. Therefore, for shapes belonging to the same class it is possible to find a map T that can be approximated by an isometry. Thus, a good approximation of the Functional maps, in order to detect disorders, can be computed starting from isometry invariant descriptors and operators. This is the motivation that has driven our choices of function and operator constraints.

For the operator commutativity we consider the Laplace-Beltrami operator (LBO), a positive semidefinite differential operator, defined on the smooth manifold. The LBO is fully described in terms of the Riemannian metric and therefore it is invariant to isometric deformations of the surface [25]. In the discrete setting the LBO can be computed using the classical cotangent formula [21, 23]. As functions constraint we take two spectral point descriptors, which are also selected as probe functions in the original functional maps framework. These descriptors namely HKS and WKS are known to be stable and invariant to isometries.

The first one is the Heat Kernel Signature (HKS) [10, 28] given by
$$\begin{aligned} h(x,t)=\sum _{i=1}^n e^{-\lambda _it}\phi _i^2(x) \end{aligned}$$
(4)
where \(\lambda _i\), \(\phi _i\) are eigenvalues and eigenfunctions of the LBO eigendecomposition and n is the number of selected eigenfunctions.
In the same way we can define the second descriptor, namely the Wave Kernel Signature (WKS) [1], as
$$\begin{aligned} w(E,x)=\sum _{i=0}^n \phi _i(x)^2f_E(\lambda _i)^2 \end{aligned}$$
(5)
where E is an approximation of the energy expected value, and \(f_E^2\) is an energy probability distribution.

As extensively argued in [1, 10, 28] we chose these two spectral signatures because they have a lot of interesting property. The HKS ensures the so called informative theorem which states that if X and Y are two compact manifold and the eigenvalues of the respective Laplace-Beltrami operators are not repeated, then the heat HKS is preserved for every isometry T between two manifold X and Y, i.e. \(h_X(x,t)=h_Y(T(x),t)\). Although it is possible for some shapes to have some eigenvalues that are very close each others by leading to a switch in the order, in the practical experience the HKS descriptors are quite robust with respect to this non optimal situation.

The WKS is also intrinsic and informative, i.e. once again for every isometry T between X and Y, we have that \(w_X(E,x)=w_Y(E,T(x))\), for every \(x\in X\) and for every \(E\in \mathbb {R}\) (see Fig. 1). So these descriptors could better represent small variations among near isometric shapes belonging to the same class, allowing a better realization of the functional map.
Fig. 1.

Distribution of WKS values for several shapes of putamen region and energy values. From left to right \(E=10,140,180\). Lines 1 and 2 subjects with bipolar disorder, lines 3 and 4 normal controls.

4.2 Functional Maps Kernel

As shown in Sect. 3, we can estimate the map C for every pair of surfaces (XY). For the sake of clarity we denote with \(C_{X,Y}\) the map between X and Y. Now, we need a specific kernel based on this map to perform our classification task. Given the pair (XY), we compute two maps: (i) \(C_{X,Y}\) defined from \(\mathcal {F}(Y,\mathbb {R})\) to \(\mathcal {F}(Y,\mathbb {R})\) and, (ii) the inverse \(C_{Y,X}\). Clearly the exact Functional map from a functional space \(\mathcal {F}(X,\mathbb {R})\) and itself is the identity map \(Id_{X}\).

If the estimated maps are correct we can draw the following commutative diagram:
This diagram shows that a function should remain the same when it is moved from shape X to Y, and than put it back to X again.
In order to quantify how well the maps \(C_{X,Y}\) and \(C_{Y,X}\) have been calculated, we can define the following measure:
$$\begin{aligned} \vert \vert C_{Y,X}C_{X,Y} - Id_{X} \vert \vert _{F} , \end{aligned}$$
(6)
which tells us how much the previous diagram is actually commutative. Now we infer that if two surfaces are in the same class, i.e. they do not differ sensibly, we can compute \(C_{X,Y}\) and \(C_{Y,X}\) in a sufficiently exact way, such that \(C_{X,Y}C_{Y,X}\approx Id_{X}\). Thus for surfaces that belongs to the same class we obtain small value in the Eq. 6, conversely these score will be higher if the surfaces come from different classes. At this point we can advisedly define the following distance function:
$$\begin{aligned} d(X,Y) = \frac{1}{2} ( \vert \vert C_{Y,X}C_{X,Y} - Id_{X} \vert \vert _{F} + \vert \vert C_{X,Y}C_{Y,X} - Id_{Y} \vert \vert _{F} ). \end{aligned}$$
(7)
This distance function has the following nice properties:
  • Symmetry:\(d(X,Y) = d(Y, X)\), \(\forall X, Y\).

  • Zero diagonal:\(d(X,X) = 0\), \(\forall X\).

  • Nonnegativity:\(d(X,Y) \ge 0\), \(\forall X, Y\).

Thanks to this properties and referring to [16] we can define a distance substitution kernel on the distance d, that we will call Functional maps kernel (FM-kernel). Given a collection of surfaces \(\left\{ X_{i} \right\} _{i \in I} \) we define the FM-kernel as:
$$\begin{aligned} K(i,j)=e^{-\gamma d(X_i,X_j)^{2}}, \forall i, j \in I. \end{aligned}$$
(8)
As shown in [16] the obtained kernel can be successfully applied in SVM for classification.

5 Results

In this section we show how the Functional maps framework together with our new FM-kernel improve the brain classification performance on the spectral domain. With this aims we explore the comparison with all the spectral methods that are more related to our framework. We also report the results obtained using different classifiers, namely the Support Vector Machines (SVM) and the Nearest Neighbour (NN) classifier.

5.1 Materials

We analyze a dataset of patients affected by bipolar disorder and healthy control subjects. More precisely, 34 control subjects (22 males, \(29 \pm 5\) years old (y.o.)), 34 patients affected by bipolar disorder (15 males, \(45 \pm 13\) y.o.) underwent an MRI session. MRI data were obtained using a Siemens 3.0 T Magnetom Allegra MRI scanner (Siemens Ag). The following parameters were used for T1-weighted images: 256 256 256 voxels, \(1 \times 1 \times 1\,{\text {mm}}^3\), TR 2060 ms, TE 3.93 ms, flip angle \(15^{\circ }\). Cortical and subcortical surfaces were obtained using FreeSurfer version 4.3.11 [9]. First, non-brain tissues were excluded, then images were segmented into white and gray matter (WM and GM respectively), and then, meshes of the boundaries between WM and GM and between GM and CSF were estimated. We focused on the putamen, a deep gray matter brain structure, which is thought to be modified in the shape in subjects that are affected by bipolar disorder [17]. The process encoded by the functional map framework is shown in Fig. 2. The function defined on the first shape is represented by the WKS descriptor. Such function is map to the second shape by using C by showing that the transported WKS values are very similar to the original one.
Fig. 2.

A couple of putamen surfaces and two WKS descriptors computed on them. Through the C map calculated using the Functional maps framework, we transport the function defined on the first shape to the second one and as shown we achieve a good approximation of the desired function on the second shape.

5.2 Comparison with Other Methods

We compare our method with the state of the art spectral methods. In order to establish how much the Functional maps framework and the proposed FM-kernel improve the classification results, we select methods that encode the same information used in our Functional maps construction. We consider the Shape-DNA (S-DNA) descriptor [26], i.e., the increasing ordered sequence of the first LBO eigenvalues. We evaluate also the so called Global Heat Kernel Signature (GHKS) [5], a multiscale histogram representation of the pointwise HKS. Similarly, we define the Global Wave Kernel Signature (GWKS) based on the WKS. Finally, since in our method the information coming from LBO, HKS, and WKS is integrated into the same framework we carried out a further evaluation with descriptors obtained by the concatenation of GHKS and GWKS (GHKS + GWKS), or GHKS, WKS and S-DNA (ALL3desc).

5.3 Comparison with Different Classifiers

We show the results obtained by different choices of classifiers. Specifically, we adopt a K-Nearest Neighbor (KNN) classifier, with \(k=6\) and the standard SVM classifier using LIBSVM [7]. A cross-validation scheme is introduced to estimate the SVM parameters as suggested in [7].
Table 1.

Results in classification for the bipolar disorder on the putamen shapes. The evaluated methods are SVM and KNN classifiers

Classification results

Method

Ours

S-DNA

GHKS

GWKS

GHKS + GWKS

ALL3desc

SVM

72.06

70.59

67.65

69.12

69.12

70.59

KNN

64.71

60.29

63.24

61.76

60.29

63.24

Table 1 shows the results. Our proposed approach outperforms all the other methods, also in their joined version. This confirm our claim that performing the classification on the functional space improves the results. Nevertheless, our method performed at best for both SVM and KNN showing its independence from the choice of classifier. Since the proposed FM-kernel is designed specifically for the SVM classifier it does not surprise that the best performance was obtained with this classifier.

6 Conclusion

In this paper the Functional maps approach for brain classification in the spectral domain is proposed. We introduced a specific kernel for SVM classification, namely the FM-kernel, based on the integration among different spectral shape analysis operators and descriptors. We evaluated our new classification method for bipolar disorder detection on the putamen regions by showing very promising results in comparison with other spectral-based approaches. As future works we consider to learn more suitable spectral descriptors for specific tasks as suggested in [8]. In particular, we will focus on the reduction of the importance of the isometry constraint between shapes that is difficult to justify from the clinical point of view even if it is working well in practice. We could also include further information related to the anatomical structure as additional constraints for the Functional maps framework such as the parts of a prior available shape segmentation procedure. Moreover, we will consider that the major variability is for the non-healthy subjects and therefore a new classifier based on the training of a single class (the healthy one) will be considered. Finally, a more exhaustive clinical evaluation will be carried out by exploring other brain regions and by enlarging the cohort of available subjects.

Footnotes

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Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Simone Melzi
    • 1
  • Alessandro Mella
    • 1
  • Letizia Squarcina
    • 2
  • Marcella Bellani
    • 3
  • Cinzia Perlini
    • 4
  • Mirella Ruggeri
    • 3
  • Carlo Alfredo Altamura
    • 5
  • Paolo Brambilla
    • 5
    • 6
  • Umberto Castellani
    • 1
  1. 1.Computer ScienceUniversity of VeronaVeronaItaly
  2. 2.Scientific Institute IRCCS E. MedeaBosisio PariniItaly
  3. 3.Section of PsychiatryAOUI VeronaVeronaItaly
  4. 4.Section of Clinical Psychology, Department of Neuroscience, Biomedicine and Movement SciencesUniversity of VeronaVeronaItaly
  5. 5.Department of Neurosciences and Mental Health, Fondazione IRCCS Ca Granda Ospedale Maggiore PoliclinicoUniversity of MilanMilanItaly
  6. 6.Department of Psychiatry and Behavioural SciencesUniversity of Texas Health Science CenterHoustonUSA

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