Roberto Ugolotti

Pablo Mesejo
email: pmesejo@ce.unipr.it

Youssef S G Nashed
email: nashed@ce.unipr.it

Stefano Cagnoni
email: cagnoni@ce.unipr.it

GPU-based Automatic Configuration of Differential Evolution: a case study

Keywords: Automatic Algorithm Configuration, Global Continuous Optimization, Particle Swarm Optimization, Differential Evolution, GPGPU

The performance of an evolutionary algorithm strongly depends on the choice of the parameters which regulate its behavior. In this paper, two evolutionary algorithms (Particle Swarm Optimization and Differential Evolution) are used to find the optimal configuration of parameters for Differential Evolution. We tested our approach on four benchmark functions, and the comparison with an exhaustive search demonstrated its effectiveness. Then, the same method was used to tune the parameters of Differential Evolution in solving a real-world problem: the automatic localization of the hippocampus in histological brain images. The results obtained consistently outperformed the ones achieved using manually-tuned parameters. Thanks to a GPU-based implementation, our tuner is up to 8 times faster than the corresponding sequential version.

Introduction

In this paper we consider the problem of automatizing the configuration of an Evolutionary Algorithm (EA) [START_REF] Eiben | Introduction to Evolutionary Computing[END_REF]. This problem is particularly important because EAs are usually strongly influenced by the choice of the parameters which regulate their behavior [START_REF] Eiben | Parameter tuning for configuring and analyzing evolutionary algorithms[END_REF].

Regarding automatic algorithm configuration, it is important to distinguish between parameter tuning and parameter control. In the former, parameter values are fixed in the initialization stage and do not change while the EA is running; in the latter, parameter values are adapted as evolution proceeds. This work is focused only on choosing parameter values, either numerical (integer or real) or nominal (representing different design options for the algorithm), which will then be kept constant during evolution.

Virtually, all EAs have free parameters that are partly set by the programmer and/or by the user. In this scenario, automatic parameter configuration is desirable, since manual parameter tuning is time consuming, error-prone, and may introduce a bias when comparing a new algorithm with a reference, caused by better knowledge of one with respect to the other or to possible differences in the amount of time spent tuning each of them. Nevertheless, it is important to take the computational burden of these tuners into account, since they usually need to perform many tests to find the best parameter configuration for the algorithm under consideration.

In this regard, GPU implementations have arisen as a very promising way to speed-up EAs. Modern graphics hardware has gained an important role in parallel computing, since it has been used to accelerate general computations (General Purpose Graphics Processing Unit programming), as well as in computer graphics. In particular, CUDA TM (Compute Unified Distributed Architecture) is a parallel computing environment by nVIDIA TM , which exploits the massively parallel computation capabilities of its GPUs. CUDA-C [START_REF][END_REF] is an extension of the C language that allows development of GPU routines (termed kernels), that can be executed in parallel by several different CUDA TM threads. We use GPU implementations of real-valued population-based optimization techniques (Particle Swarm Optimization (PSO) [START_REF] Kennedy | Particle Swarm Optimization[END_REF] and Differential Evolution (DE) [START_REF] Storn | Differential Evolution -a simple and efficient adaptive scheme for global optimization over continuous spaces[END_REF]) to automatically configure DE parameters. The implementations of both metaheuristics on GPU are publicly available in a general-purpose library1 . DE has been chosen as the method to be tuned mainly because it is the optimizer initially proposed and employed in the real-world problem under consideration [START_REF] Mesejo | Automatic hippocampus localization in histological images using differential evolution-based deformable models[END_REF], while PSO was chosen as tuner because of its easy parallelization and its ability to reach good results using a limited number of fitness evaluations.

The choice of using Evolutionary Algorithms or Swarm Intelligence [START_REF] Bonabeau | Swarm Intelligence: From Natural to Artificial Systems[END_REF] algorithms as tuners derives from the large number of attractive features they offer: robust and reliable performance, global search capability, virtually no need of specific information about the problem to solve, easy implementation, and, above all, implicit parallelism, which allows one to obtain a parallel implementation starting from a design which is also conceptually parallel.

We first demonstrate that our method works properly by optimizing DE parameters for some well-known benchmark functions and compare these results with the ones obtained by a systematic search of the parameter space. Then, we apply the same approach to the real-world problem of localizing the hippocampus in histological brain images, obtaining a significant improvement with respect to the manually-tuned algorithm. Finally, we prove that our GPU-based implementation significantly reduces computation time, compared to a sequential implementation.

The remainder of this paper is structured as follows: in Section 2 we provide the theoretical foundations necessary to understand our work, including the metaheuristics and the problem used to test the system. In Section 3, a general overview of the method is presented, providing details about its GPU implementation. Finally, Section 4 presents the results of our approach followed, in Section 5, by some final remarks and a discussion about future developments.

In this section, we will briefly describe the metaheuristics used in this work (DE and PSO) and the hippocampus localization task used as test, and make a short review of the approaches in which metaheuristics have been used to tune EAs.

Particle Swarm Optimization

Particle Swarm Optimization is a well-established stochastic optimization algorithm which simulates the behavior of bird flocks. A set of N particles moves through a "fitness" function domain seeking the function optimum. Each particle's motion is described by two simple discrete-time equations which regulate the particles' velocity and position:

v n (t) = w • v n (t -1) + c 1 • rand() • (BLP n -P n (t -1)) + c 2 • rand() • (BN P n -P n (t -1)), P n (t) = P n (t -1) + v n (t),
where P n (t) and v n (t) represent the n th particle's position and velocity at time t; c 1 , c 2 are positive constants which represent how strongly cognitive and social information affects the behavior of the particle; w is a positive inertia factor (possibly depending on time t) used to balance PSO's global and local search abilities; rand() returns a vector of random values uniformly distributed in [0, 1]; BLP n and BN P n are, respectively, the best-fitness location visited so far by particle n and by any particle in its neighborhood; this may include a limited set of particles (lbest PSO) or coincide with the whole population (gbest PSO). One of the most commonly used PSO versions is based on a "ring topology", i.e. each particle's neighborhood includes the particle itself, the K particles preceding it, and the K particles which immediately follow it.

Differential Evolution

Differential Evolution is one of the most successful EAs for global continuous optimization [START_REF] Das | Differential Evolution: A Survey of the State-of-the-Art[END_REF]. DE perturbs the current population members with the scaled difference of distinct individuals. At every generation, every individual X i of the current population acts as a parent vector, for which a donor vector D i is generated. In the original version of DE, the donor vector for the i th parent (X i ) is created by combining three distinct randomly-selected elements X r1 , X r2 and X r3 . The donor vector is then computed as:

D i = X r1 + F • (X r2 -X r3 ),
where the scale factor F (usually F ∈ [0, 1.5]) is a parameter that strongly influences DE's performances. This mutation strategy is called random, but other mutation strategies have been proposed. Two of the most successful ones are best (equation 1) and target-to-best (TTB) (equation 2):

D i = X best + F • (X r1 -X r2 ),
(1)

D i = X i + F • (X best -X i ) + F • (X r1 -X r2 ), (2)
where X best represents the best-fitness individual in the current population.

After mutation, every parent-donor pair generates one child (called trial vector, T i ) by means of a crossover operation. Two kinds of crossover are typically used: binomial (uniform) and exponential ; both are regulated by a parameter called crossover rate (Cr ∈ [0, 1]). In the former, a real number r ∈ [0, 1] and an integer M ∈ [1, S], where S is the size of the search space, are randomly generated for each element of the vector. The trial vector is then defined as follows:

T i,j = D i,j if r ≤ Cr or j = M X i,j otherwise , j = 1 . . . S
The exponential crossover, instead, generates an integer L according to this pseudo-code:

L = 0 do L = L + 1 r = rand() while ((r ≤ Cr) and (L ≤ S))
This value is then used to generate the trial vector:

T i,j = D i,j for j = M S . . . M + L -1 S X i,j otherwise , j = 1 . . . S
where • S represents the modulo S function.

After the crossover operation, the trial vector is evaluated and its fitness is compared to its parent's: only the one having the best fitness survives in the next generation, while the other is discarded from the population.

Hippocampus Localization in Histological Images

The hippocampus is a structure located in the mammalian brain that has long been known for its crucial role in learning and memory processes, as well as an early biomarker for Alzheimer disease and epilepsy. Thus, its automatic, robust and fast localization is of great interest for the scientific community.

As a real-valued benchmark for our method, we consider the problem of localizing the hippocampus in histological images (see Figure 1) extracted from the Allen Brain Atlas (ABA), a publicly available image database [1] which contains a genome-scale set of histological images (cellular-resolution gene-expression profiles) obtained by In Situ Hybridization of serial sections of mouse brains.

In particular, a localization method based on Active Shape Models has been recently presented to tackle this kind of images [START_REF] Mesejo | Automatic hippocampus localization in histological images using differential evolution-based deformable models[END_REF]. It uses a medial shape-based representation of the hippocampus in polar coordinates, with the objective of creating simple models that can be managed easily and efficiently. Two parametric models (see Figure 1b) representing the two parts that compose the hippocampus (called SP and SG, see Figure 1a) are moved and deformed by DE according to an intensity-based similarity function between the model and the object itself.

Each model comprises two sets of points: the goal is to overlap the first one (Inner Set, I) to the part of the hippocampus to be located, while placing the other one (Outer Set, O), obtained by rigidly shifting the first one, immediately outside it. This model is subjected to external forces (driven by the image features) and internal forces (driven by the model itself). The target function H to be maximized (see [START_REF] Mesejo | Automatic hippocampus localization in histological images using differential evolution-based deformable models[END_REF] for a more detailed description) has three components: the external energy EE, the internal energy IE, and the contraction factor C.

H = EE -(IE + C)
In turn, EE can be divided in two components: P E, that depends on the control points of the model (denoted by black dots in Figure 1b) and CE, that is computed considering the points which belong to the segments connecting them:

EE = γ P • P E + γ C • CE P E = n i=1 [T (N 3 (I i )) -T (N 3 (O i ))] CE = n i=2 p j=1 T (I i-1 + j p + 1 (I i -I i-1 )) - n i=2 p j=1 T (O i-1 + j p + 1 (O i -O i-1 ))
where I i and O i represent the points of the two sets, γ P and γ C are positive values that weigh the two components, N 3 (P ) is a 3 × 3 neighborhood centered in P , T (P ) is the intensity of P if P is a point, or the average intensity if P is a neighborhood, and p is the number of points sampled in each segment.

The internal energy IE is computed as:

IE = ξ ρ • n i=2 (ρ i -ρ mi ) 2 + ξ ϑ • n i=2 (ϑ i -ϑ mi ) 2
where ξ ρ and ξ ϑ are two positive weights that regulate the deformability of the model. (ρ i , θ i ) represents a point of the model in polar coordinates, while (ρ mi , θ mi ) represents a point of a reference model derived empirically from a set of "training" images.

Finally, the contraction factor C also regulates the model's deformability to avoid unfeasible situations that are not allowed by the nature of the hippocampus and is defined as follows:

C = ξ c • I n -I 1 (a) (b) (c)
Fig. 1: In the upper row, an anatomical atlas representation of the hippocampal region that highlights the two parts to be recognized (SP and SG) and an example of the parametric models used to localize the hippocampus; below, four examples of hippocampus images taken from the ABA.

Automatic Configuration of EAs

Parameter tuning for configuring and analyzing EAs is a topic frequently dealt with in the last years [START_REF] Eiben | Parameter tuning for configuring and analyzing evolutionary algorithms[END_REF][START_REF] Smit | Comparing parameter tuning methods for evolutionary algorithms[END_REF]. In fact, the idea of a meta-EA was already introduced in 1978 by Mercer and Sampson [START_REF] Mercer | Adaptive search using a reproductive metaplan[END_REF], while Grefenstette, in 1986 [START_REF] Grefenstette | Optimization of control parameters for genetic algorithms[END_REF], conducted more extensive experiments with a meta-GA and showed its effectiveness. However, the problem of tuning EAs by means of meta-EAs implies tackling two challenges: the noise in (meta)-fitness values and the very expensive (meta)-evaluations. Our GPU-based implementation tries to reduce the impact of the latter problem. Recently, REVAC, a specific type of meta-EA where the population approximates the probability density function of the most promising areas of the utility-landscape (similar to Iterative F-RACE), was shown to be able to find much more robust parameter values than one could set by hand, outperforming the winner of the competition on the CEC 2005 test-suite [START_REF] Smit | Beating the 'world champion' evolutionary algorithm via REVAC tuning[END_REF].

An attempt to find the optimal parameters for DE using a metaheuristic called Local Unimodal Sampling is described in [START_REF] Pedersen | Tuning and Simplifying Heuristical Optimization[END_REF]. As well, PSO parameter tuning has already been proposed based on other overlaying optimizers, like in [START_REF] Meissner | Optimized Particle Swarm Optimization (OPSO) and its application to artificial neural network training[END_REF],

where PSO itself is used to this end.

3 Parameter Configuration using Metaheuristics

Parallel Implementation

The GPU-based implementation of the metaheuristics employed in this paper has been presented in [START_REF] Nashed | libCudaOptimize: an open source library of GPU-based metaheuristics[END_REF].

The first implementations of PSO and DE based on nVIDIA CUDA TM were developed in 2009 and 2010, respectively [START_REF] De Veronese | Swarm's flight: Accelerating the particles using C-CUDA[END_REF][START_REF] De Veronese | Differential evolution algorithm on the GPU with C-CUDA[END_REF]. After that, other implementations of DE have been developed [START_REF] Krömer | Many-threaded implementation of differential evolution for the CUDA platform[END_REF], and fast versions of PSO have been implemented by removing the constraint of synchronicity between particles [START_REF] Mussi | GPU-based asynchronous particle swarm optimization[END_REF]. The early GPU PSO implementations suffered from a coarse-grained parallelization (one thread per particle), that did not offer the opportunity to compute the fitness function, usually the most time-consuming process, in parallel over the problem dimensions. This aspect has been improved by letting each thread manage a single dimension of each particle, adding a further level of parallelism [START_REF] Mussi | Evaluation of parallel Particle Swarm Optimization algorithms within the CUDA architecture[END_REF]. Similar inefficiencies characterized the early implementations of DE. These problems were subsequently addressed by [START_REF] Krömer | Many-threaded implementation of differential evolution for the CUDA platform[END_REF] and [START_REF] Nashed | libCudaOptimize: an open source library of GPU-based metaheuristics[END_REF].

Our GPU-based implementations of PSO and DE share a similar kernel configuration. Both are implemented as three distinct kernels: (i) the first kernel generates the solutions to be evaluated, (ii) the second kernel implements the fitness function, and (iii) the last kernel updates the population.

General Design

A tuner (or meta-optimizer) is implemented exactly in the same way as any optimizer used to solve any other problem. The only significant difference consists in the evaluation of the fitness function F . In this case, each particle X i = {x 1 , . . . , x n } of a tuner represents an optimizer (or better, a set of m ≤ n parameters that describes an optimizer). An optimizer O i (X i ) is instantiated using the parameters X i and the entire optimization process is repeated on the test function T times to compute fitness f i as the average fitness over the T runs. This optimizer, used to tune DE, has been introduced in [START_REF] Ugolotti | Algorithm Configuration using GPU-based Metaheuristics[END_REF].

PSO and DE are real-valued optimization methods, so the problem of representing nominal parameters needs to be addressed. We chose to represent each nominal value by a vector which associates a number to each option available; the option associated with the highest number is then selected. Figure 2 shows the encoding of a particle representing a DE instance. Table 1 shows the range and possible values of the DE parameters in the representation we have used.

n n i=1 xi 2 + n i=1 0.5 • i • xi 2 2 + n i=1 0.5 • i • xi 2 4 U S Schwefel 1.2 [-100, 100] n n i=1 i j=1 xj 2 U NS Rastrigin [-5.12, 5.12] n n i=1 x 2 i -10 • cos(2πxi) + 10 M S Rosenbrock [-100, 100] n n i=2 100(xi -x 2 i-1 ) 2 + (1 -xi-1) 2 M NS

Experiments

In this section, we will describe the two sets of tests performed on benchmark functions and on the hippocampus localization problem. Tests were run on a 64-bit Intel Core i7 CPU running at 3.40GHz using CUDA v. 4.2 on an nVidia GeForce GTX680 graphics card with compute capability 3.0.

Benchmark Functions

Four classical benchmark functions (see Table 2) having different nature (unimodal/multimodal and separable/non separable) have been selected to test our method. For each function, we systematically sampled (with the sampling steps shown in Table 1) the parameter search space and ran 10 independent repetitions for each of the 13728 parameter combinations generated. This way, we created a ground truth used to assess the results of the meta-optimization. The termination criteria was set after 1000 generations and the search space size was set to 16.

Parameter tuning has been repeated 10 times (5 using DE as meta-optimizater, 5 using PSO) on each function. Table 3 shows the parameters used by the tuners based on DE and PSO. In order to choose them, we performed the tuning procedure described here over 8 benchmark functions using "standard" parameters for PSO and DE (see [START_REF] Ugolotti | Algorithm Configuration using GPU-based Metaheuristics[END_REF]). The best parameter configuration found, based on the number of times in which it obtained the best fitness value, was used in our tests. Notice that, with this parameter configuration, a maximum of 1536 sets of parameters (64 generations × 24 individuals) are evaluated in each run.

Table 4 shows, for each function, the ranges of the parameters for the best 10 settings found in the systematic search and for the 10 combinations obtained by the meta-optimization process. As can be observed, all parameter ranges virtually overlap, which proves that the meta-optimizer operates correctly. For every set of parameters, we performed 100 independent runs on the corresponding function. All optimizers were able to obtain a median value of 0 over the corresponding function. The Wilcoxon signed-rank test confirmed the absence of statistically significant differences between the results of the systematic search and the tuning, since the p-values obtained were always larger than 0.01.

To assess the speedup of the GPU implementation, we performed the same experiments adapting a sequential C++ DE implementation 2 . We compared the execution time of our method and of a sequential version of the tuner, using DE as meta-optimizer on Rastrigin function. Using the same parameters previously shown, the GPU version takes an average of 164.4 s for the entire meta-optimization process, while the sequential version takes 1007.5 s (6.1 times slower). Moreover, if we increase the problem dimension from 16 to 64, the two Table 5: The ten sets of parameters generated by the meta-optimization processes. The three columns for SP and SG (see Figure 1a) indicate the average fitnesses and standard deviations and if there is a significant improvement against the original parameters (+), no difference (=) or a worse result (-). This is a maximization problem: a higher value represents a better solution. versions take 458.2 s and 3683.7 s, respectively and, consequently, the speedup increases to 8. Please notice that these speedups are only indicative of the problem, since they depend on a very high number of parameters, like population size, number of evaluations, degree of parallelism and complexity of the fitness function and, obviously, on the hardware on which the optimizer is run.

Parameters

Hippocampus Localization

After checking the correctness of our approach, we repeated the same procedure to solve the hippocampus localization task described in Section 2.3. The parameters of the meta-optimizer are the same used in the previous sections.

Table 5 shows the parameters obtained by our meta-optimizer and by the manually-tuned ones used in [START_REF] Mesejo | Automatic hippocampus localization in histological images using differential evolution-based deformable models[END_REF]. To exclude any possible bias in favor of the former, we used a slightly different (and better working) setting for the reference, increasing its population to the maximum allowed, 150. This choice has been made because the implementation in [START_REF] Mesejo | Automatic hippocampus localization in histological images using differential evolution-based deformable models[END_REF] was sequential and this led us to use a lower number of particles for speed issues, while in our GPU implementation population size has no impact on computation time.

Each row in the table presents the method used in the meta-optimization process and the parameters found. Each set of parameters have been tested 10 times over 320 different hippocampus images.

In order to check the statistical significance of the results obtained, a Kruskal-Wallis test was performed with a level of confidence of 0.01. Since the normality and homoscedasticity assumptions were not met, as checked through the application of Kolmogorov-Smirnov and Bartlett's tests, non-parametric tests were used. The p-value was close to zero, suggesting that at least one sample median is significantly different from the others. The columns termed "Cmp" indicate the results of the Wilcoxon signed-rank test used to assess the statistical significance of the difference between the generated optimizer and the reference manually-tuned one (Original) for the two parts that compose the model (SP and SG). This value is a "+" if the results of the automatically tuned DE are significantly better than the original ones (significance of p = 0.01, using the Bonferroni-Holm correction), a "-" when the original method is better, and a "=" when there are no statistically significant differences.

As can be seen, meta-optimization always leads to similar results, and the automatically selected parameters always lead to higher mean and lower standard deviation than the ones set after a time-consuming manual tuning.

Discussion

In this paper we proposed a method to find the best parameters for an Evolutionary Algorithm (in this case, Differential Evolution) to solve a real-world problem. This method uses an evolutionary process to search the space of the algorithm parameters to find the best possible configuration for it.

We tested this method on four benchmark functions and the comparison with a systematic search proved that our method is actual effective. Afterwards, we repeated the same procedure optimizing the DE algorithm parameters in a realworld application, where Differential Evolution is used to localize hippocampi in brain histological images. The results obtained by the automatically-tuned optimizers outperform the manually-tuned ones almost systematically.

As future work, we will try to increase the degree of parallelization in order to take even more advantage of the parallel nature of the algorithms.

We evaluate the goodness of our tuning procedure by the quality of the solution obtained by the tuned optimizer. In the same way, we could add another step to empirically find a good set of parameters for the meta-optimizer. Of course, this could lead to an infinite repetition of the same procedure and to an exponential increase in optimization time. However, our practical aim is just to improve the performance of manually-tuned optimizers without increasing complexity too much, when dealing with problems in which the optimizer's performance is particularly sensitive to the parameters.

Fig. 2 :

 2 Fig. 2: Representation of a set of parameters used in the tuner. The first three elements represent numerical parameters (integer and real-valued numbers), while the last five encode the nominal parameters, i.e. crossover type and mutation strategy. In this case n = 8 values are needed to represent m = 5 parameters.

Table 1 :

 1 Range of DE parameters allowed by the tuning algorithms. The last column shows the sampling step used in the systematic search used as reference.

	Parameter	Values	Step
	Crossover Rate	[0.0, 1.0]	0.1
	Scale Factor (F )	[0.0, 1.5]	0.1
	Population Size	[30,150]	10
	Crossover	{binomial, exponential}	-
	Mutation	{random, best, target-to-best} -

Table 2 :

 2 Benchmark functions. For every function, the table displays name, search space, formula, modality (multimodal, unimodal) and separability (separable, non separable). Optimum fitness values are 0 for all functions.

	Name	Range	Formula	Modality Separability
	Zakharov	[-10, 10]

Table 3 :

 3 Parameters of DE/PSO tuners.

	DE	PSO
	TTB Mutation	C1 = 1.525
	Exponential Crossover	C2 = 1.881
	F = 0.52	w = 0.443
	Cr = 0.91	Ring Topology (K = 1)
	Population Size = 24, Generation = 64

Table 4 :

 4 Range of the parameters found by systematic search and metaoptimization. P is population, M ut is mutation strategy, Cross is Crossover Type. M ut ∈ {Best, T T B} M ut ∈ {Best, T T B} M ut ∈ {Rand, Best} M ut ∈ {Best, T T B} Cross ∈ {Bin, Exp} Cross ∈ {Bin, Exp} Cross ∈ {Bin, Exp} Cross ∈ {Bin, Exp} M ut ∈ {Best, T T B} M ut ∈ {Best, T T B} M ut ∈ {Rand, Best} M ut ∈ {Best, T T B} Cross ∈ {Bin, Exp} Cross ∈ {Bin, Exp} Cross ∈ {Bin, Exp} Cross ∈ {Bin, Exp}

	Zakharov	Schewfel 1.2	Rastrigin	Rosenbrock
		Systematic Search
	Cr ∈ [0.8, 0.9]	Cr ∈ [0.9, 1.0]	Cr ∈ [0.0, 0.4]	Cr ∈ [0.8, 0.9]
	F ∈ [0.5, 0.6]	F ∈ [0.5, 0.7]	F ∈ [0.1, 0.6]	F ∈ [0.6, 0.7]
	P ∈ [100, 150]	P ∈ [120, 150]	P ∈ [90, 150]	P ∈ [90, 150]
		Meta-Optimization
	Cr ∈ [0.799, 1.0]	Cr ∈ [0.935, 1.0]	Cr ∈ [0.0, 0.344]	Cr ∈ [0.740, 0.939]
	F ∈ [0.433, 0.714]	F ∈ [0.535, 0.667]	F ∈ [0.207, 0.596]	F ∈ [0.646, 0.686]
	P ∈ [111, 146]	P ∈ [80, 150]	P ∈ [79, 149]	P ∈ [83, 150]

The code can be downloaded from http://sourceforge.net/p/libcudaoptimize[START_REF] Nashed | libCudaOptimize: an open source library of GPU-based metaheuristics[END_REF]

http://www1.icsi.berkeley.edu/˜storn/code.html

Acknowledgments

Roberto Ugolotti is funded by Fondazione Cariparma. Pablo Mesejo and Youssef S. G. Nashed are funded by the European Commission (Marie Curie ITN MIBISOC, FP7 PEOPLE-ITN-2008, GA n. 238819).


Abigaël Pesses

Pesses Abigaël, « Localizing women's role in a globalized

Based on ethnographic data collected among Karen communities and indigenous associations of Chiang Mai province within the past 15 years, this presentation aims at questioning the opportunities and limits of women's empowerment through the strategies driven by development actors into the local arena.

By integrating gender issues in the development discourse, the role and abilities assigned to indigenous women has been embedded into national and global political frames: from state integration policies that emphasize women's role as a pillar of the household economy to the contest of world environmental policies and the promotion of women's indigenous skills in the safeguarding of biodiversity.

As observed, this empowerment is continuously negotiated within an institutionalized process which categorizes gender field and people. This historical and localized process activates new dynamics in social organization paving the way to a range of actors' strategies in development policies. This paper will question how the promotion of the "locality" became one of the main human and ideological resource, by mobilizing together women and development actors in civil society debate and economic local arena.

Introduction

This presentation aims at questioning the role assigned to tribal women in the development policies made in Thailand. On the basis of ethnographic data collected in Karen communities and indigenous associations of the Chiang Mai province within the past 15 years, I would like to compare development strategies directed to women groups and discuss the opportunities and limits of women's political and economic empowerment.

State's gender technologies

In Thailand, the adoption of the international rhetoric on women's rights has been officially brought to national consciousness with the concept of "popular participation", including women, promulgated by the World Bank in 1978 [START_REF] Papanek | Women and National Development -The complexities of Change[END_REF]. The objective of the socalled "popular participation" was to encourage village communities to govern themselves and to increase the people's ability in local "self-development".

The practical application of this message came with a national process of administrative decentralisation which led to the creation of a village committee, articulated with the creation of 3 groups based on gender and age categories. These groups are namely:

The group of "household fathers", (klum pho baan); the group of "household mothers", (klum mae baan) and the "youth group", (klum jauwachon). These three groups became the main apparatus to receive the state macro policies and the various development projects coming from the outside.

These state macro policies directed to hill tribe groups like the Karen were following the lines of a larger integration policy started in the 60'ees and mainly focussed on national security problems: namely opium cultivation, communist insurgencies and deforestation.

The policy aimed at integrating the hill tribe villages into the thai administrative system, to create a sense of national consciousness, and to replace shifting methods of cultivation, accused to be responsible of deforestation, by sedentary wet-rice and cash-crop cultivation. Thus :

The group of Houselhold fathers Men's groups were set up to articulate the relation between villagers and local administrative authorities and to capture the agricultural policies oriented toward market. In order to replace "slash and burn" techniques of cultivation, men could receive trainings on alternative cash-crop cultivation as a mean to provide income. How to use fertilizers and how to plant different species adapted to the highland temperate climate and the need of the market: salad, passion fruit, coffee.

Youth's groups for developing prevention policies towards drugs traffic, issues on national security and the protection of the environment.

The group of Household mothers

As deforestation was also linked to population growth, household mothers' groups were set up in order to drain family planning policies to reduce birth rate and to promote health care.

Apart from this boby control, initiatives were implemented to help women to participate in the household economy.

The advent of a women group in a village systematically implied the creation of a common women bank to encourage savings and loans at a very low rate of interests and to capture external development subvention.

Tribal women groups thus received training on how to organise a group, supervise the bank and manage the family budget.

In order to get alternative income, local administrative authorities also encouraged them to collectively produce local goods and handicraft so that they could be more competitive in the market.

As we can observe, the institutional policies directed toward local villagers are addressed through strict functionalistic strategies, which act as technologies of powers [START_REF] Foucault | Histoire de la sexualité (Tome 1)-La volonté de savoir[END_REF]) linked to the production of gender development groups (Lauretis, 1987). In that manner, economic development issues are systemically thought in terms function and status based on the stereotypes of what should be the standard men and women activities (Pesses 2007):

For men: agricultural practices linked to the shift toward a market economy and leading roles of mediation with the outside powers.

For women: taking care of the children, fructifying the family budget, and taking part to a more global economy through the promotion of the local handicraft and the national cultural heritage.

Despite the integration of gender issues in the development discourse, supposed to taking into account the asymmetric relations between men and women in the process of production and decision-making, state development practices focus on women only, and avoid the question of power relations.

The woman is promoted as an individual equal to man. But this recognition is realized through the naturalization process of her place and role in the social space. Assigned to the social category of "mother", its economic and political role are, by nature, confined to the domestic sphere. A separation which contributes to exclude them from the entire social process of production, which they are part of as producers [START_REF] Polanyi | The great transformation: The political and economic origins of our time[END_REF][START_REF] Godelier | La production des grands hommes[END_REF].

This process, coming from above, questions the way local development actors interact more concretely with the reality of the Karen social organization and accompanied its mutation through a greater involvement in the capitalist economy (Keyes, 1979;Kunstadter and al. 1978).

Into a moving social organization

I will now develop two contrastive examples of non-states development actors that interfere in the "local arena" (De Sardan 2005) -Catholic missionaries and environmentalist NGOsto show how they apprehends women's empowerment, and both compose with the local social organization and the state policies.

Catholic missionaries: the men's frontier and the local assignation

First it is important to notice that the Christian missionaries of Bétharram, a French catholic organization, were the first development actors to enter the Chiang Mai Karen local scene after the Second World War. They were then followed in 1969 by Buddhist missionary monks who were sent by the government in the mountains in order to facilitate the integration of highlanders into the Thai nation-state.

The conversion to Buddhism or Christianity was mostly achieved through the renunciation of the ancestor cult [START_REF] Yoshimatsu | The Karen World: The Cosmological and Ritual Belief System of the Sgaw karen in Northwestern Chiang Mai Province, Final Research Report presented to the National Research Council of Thailand[END_REF]. This ritual, performed in case of the illness of a family member, consists in the sacrifice of a pig and chicken to appease ancestral spirits. The ritual sacrifice is consumed in the women spatial domain, the hearth of the kitchen, and is linked to domestic women activities, rearing child and animals.

Villagers, both men and women, justified the break of the ancestral cult as a possibility to increase their economic capital by saving the livestock usually dedicated to expensive curing sacrifices. Moreover, as the ritual performance implied the presence of all family members, they could also extend their sphere of mobility, either to search for seasonal wage works or to send their children to the lowland schools.

As the ancestor cult is usually referred as a women's rite, men, on the other hand, are in charge of the sacrifice to negotiate the good relationships with the external tutelary spirits of the territory. In the same time, both Buddhist and catholic missionaries provide villagers with alternative rituals to relatives the territorial taboos that limited their access to land.

The religious change were thus accompanying other socio-economic changes linked to a sedentarisation process based on rice field subsistence mixed with complementary swidden cultivation and cash crop income.

By that way, the traditional prestige system based on rituals, marital status, age, wisdom and number of child, was progressively challenged by new factors of empowerment in the village society: the control over land, the accumulation of cash and the ability to deal with external development or religious agents. If, in Karen social organization, resources such as land and livestock are owned by women and men, men gained the opportunity to enhance their status in society by seeking economic or political opportunities in the lowland surroundings, while women were confined in the village.

On that point, Christian missionaries, from the beginning of their arrival, strongly included women into development strategies by emphasizing their role in the maintenance of a subsistence economy and the research on village self-sufficiency from external support.

According to the saying of father of the Bétharram mission, when missionaries entered the Chiang Mai region after the Second World War, many men were mortgaging their land and family livestock in order to buy opium to the Hmong. In the eyes of the missionaries, compared to men, women were considered less individualistic and more trusted agents for managing the family budget.

In order to fight against the chronic indebtedness of the peasants, missionaries started to look for women with a strong character able to lead women associations and supervise rice-bank, cow bank, and women saving banks. As Christian Karen do not practices sacrifice anymore, the benefits of the women savings banks could be partly reinvested in a new prestige system based on the accumulation of material wealth, investment in child's education, and redistribution to the poor.

From what I could observe in the field, the Christian women groups were playing a leading local role in propagating their own economical initiatives of funding and solidarity in the surrounding villages while participating in the extension of the Christian community network beyond the village level.

The women's deprivation of political function was thus counterbalanced by their ability to contribute to the maintenance of a subsistence economy and the creation of inter-village women group networks.

At the same time, teenage girls could benefit the network of missionaries to pursue professional training in the catholic school of Chiang Mai city. The minority of girls who had such opportunity to study today exercise valued professions of nurse or teacher in mountain schools and clinics. Thanks to their ability to read and write thai, and to gain cash income, they gained more mobility, and feel more and more like equal partners to men in their relation to a Thai dominant environment.

The fact that Christian missionaries strategically relied on women to support their propositions of development have opened them a pass to a greater access to education and entrepreneurship in the local arena. But to do so, missionaries had first to pass the frontier of men to drive their empowerment strategies toward women. And they did it without contesting the traditional gender role in the local the structure of power relations, nor the national macro policies agenda.

The separation between the two spatial spheres of rituals characterising traditional Karen organazationthe inside family rites of the women and the outside territorial rites of the men [START_REF] Hayami | Ritual and Religious Transformation among Sgaw Karen of Northern Thailand[END_REF])has been reproduced and adapted to the social changes introduced by the engagement in a capitalist economy:

Women priority exercise their authority on household budget and participate to the preservation of a routed subsistence economy and actions of solidarity inside a local arena, whereas men are in charge of negotiating the political and economical relations with external agents.

Environmentalist NGOs: The step to civil society for sustainabilities

I now would like deepen this process with a comparative example, showing how in contrast, environmentalist NGOs, used the issue of women's empowerment to contest state policies. Since the early nineties, faced by the radicalisation of governmental policies concerning the protection of environment, a network of Thai environmentalist NGOs, has started to play a very active role in the local and national scene in order to avoid the relocation of highland communities away from "national parks" (Hirsch, 1997).

In that context, some Karen activist have constructed a discourse which defends their rotational agriculture system (rai mun wien), condemned by the law and progressively abandoned to the profit of commercial cultivation [START_REF] Laungaramsri Pinkeaw | Redefining Nature: Karen Ecological Knowledge and the Challenge to the Modern Conservation Paradigm[END_REF]. Rai mun wien is now valued by the Karen activists as a fundamental aspect of their culture, strongly connected to all the different aspects of their livelihood: subsistence production, rituals, handicraft, medicine, and gender relations.

They thus tend to condemn the destructive impacts of the governmental cash crop policy on the environment and the subsequent degradation of the women's status. The main argument of the activists consists in promoting the women's knowledge in the complementary division of gender labour tasks which prevailed under swidden cultivation.

On the one hand, men had to domesticate the land through appropriate rituals and endorsed the physical work to clear the new fields. On the other hand, women were in charge of sowing the fields, harvesting the plants, and preserving a large array of seed grain for the following year. They thus disposed of specific knowledge on plants and a significant role in the maintenance of bio-diversity. But, this women ability has been challenged by the state and foreign development programs which favour men involvement in cash cropping or monocropping without considering the traditional labour division between the sexes.

New technical knowledge and responsibilities accessible to men in many domains, like agriculture, trade, and communication with external agents thus contributed in the reducing of the women's role to be vassals of the men. Their lack of knowledge in these new domains, added to the smaller opportunities to speak Thai as the men, excludes them greatly from decision-making about development.

In this context, NGO activists consider alternative solutions of "sustainable" development that integrate the cultural dimension of gender labour division which prevailed in a subsistence economy. In order to empower the women, it proposes to revitalise locally based women traditional knowledge and activities: that is to say the promotion of weaving, rearing animals, and biological diversified agriculture through the perpetuation of rotational farming.

Paradoxically, this discourse which underlines the asymmetric relations between men and women in a process of socio-economic mutation, relay on development strategies that reproduce and reinforce the traditional division of the genders through the prescription of typical female roles.

But while trying to value the women's knowledge and role in swidden cultivation, NGOs have given a new echo to their voice in the political struggle of minority groups for claiming community rights to manage forest resources [START_REF] Ganjanapan | The Politics of Environment in Northern Thailand: Ethnicity and Highland Development Programmes[END_REF][START_REF] Laungaramsri Pinkeaw | Redefining Nature: Karen Ecological Knowledge and the Challenge to the Modern Conservation Paradigm[END_REF][START_REF] Pesses | Horizons d'une population frontière. Mise en scène de l'indigénisme et écologie en Thaïlande[END_REF]).

During my field work in 2000, I was able to observe many tribal women of all ages joining a demonstration at the Chiang Mai city hall to claim for the recognition of land community rights. This participation gave birth to a tribal women association that advocates issues concerning access to rights, education, justice, and land resources. By that way, women could become actors involved in the civil society debate and in a global movement for the preservation of indigenous patrimonies.

Conclusion : Beyond the "locality"

To conclude, I would like, on the basis of these field observations, questions the more general limits of women economical and political empowerment in the context of development.

Whereas the global discourse propagated by development actors defends the equality between men and women in all domainseducation, economics, and politicsmost the practical strategies used to reach that aim are linked to the revitalization of specific female abilities. Many development terms are still associated with feminine roles in the sphere of subsistence, family, solidarity, tradition, locality, preservation or sustainability.

If empowerment there is, as it has been shown, it passes through these words. That is to say, women's empowerment is negotiated within an institutionalized process of categorizing gender abilities, the dynamics of a local social organization, and the variations of the development actors' agenda. As observed, the promotion of the "locality" became one of the main human and ideological resource that mobilize together women and development actors in economic local arena or in the civil society debate.

Then, the question is: Do the development projects have or can go beyond the local genders roles?

On the one hand, the revitalization of women's traditional role and attributes give them a status of importance in the local society.

But on the other hand, if these attempts of preservation seem to protect from a process of global socio-economic mutation, it stop from reflecting on the growth of the women's social mobility and their capacity of adaptation outside the local frontiers.


Roberto Ugolotti

Youssef S G Nashed
email: nashed@ce.unipr.it

Pablo Mesejo
email: pmesejo@ce.unipr.it

Stefano Cagnoni
email: cagnoni@ce.unipr.it

Algorithm Configuration using GPU-based Metaheuristics

Keywords: I.2.8 [Artificial Intelligence]: Problem Solving, Control Methods, and Search-Heuristic methods Parallel Computing, GPGPU programming, Automatic Parameter Configuration, Evolutionary Algorithm, Swarm Intelligence

In this paper, a GPU-based implementation of Differential Evolution (DE) and Particle Swarm Optimization (PSO) in CUDA is used to automatically tune the parameters of PSO.

The parameters were tuned over a set of 8 problems and then tested over 20 problems to assess the generalization ability of the tuners. We compare the results obtained using such parameters with the 'standard' ones proposed in the literature and the ones obtained by state-of-the-art tuning methods (irace). The results are comparable to the ones suggested for the standard version of PSO (SPSO), and the ones obtained by irace, while the GPU implementation makes tuning time acceptable.

To the best of our knowledge, this is the first time that a general purpose library of GPU-based metaheuristics is used to solve this problem, as well as being one of the few cases where DE and PSO are both used as tuners.

INTRODUCTION

One of the main problems to be tackled when applying Evolutionary Algorithms (EAs) and Swarm Intelligence (SI) techniques is the strong influence of parameter selection on their performance.

In this paper, we use the GPU implementation of two real-valued population-based optimization techniques (Particle Swarm Optimization (PSO) [START_REF] Kennedy | Particle Swarm Optimization[END_REF] and Differential Evolution (DE) [START_REF] Storn | Differential Evolution -a simple and efficient adaptive scheme for global optimization over continuous spaces[END_REF]) to automatically configure another metaheuristic (PSO). This SI method is tuned using a benchmark of 8 different optimization problems (also implemented on GPU).

The main contribution of this work is three-fold: (i) the first documented use, to the best of our knowledge, of GPU-based metaheuristics devoted to the automatic configuration of algorithms, (ii) a comparison with the results obtained using the parameters selected by the irace software package [START_REF] López-Ibáñez | The irace package, iterated race for automatic algorithm configuration[END_REF], and using a classically and well-established set of manually selected parameters [1]), and (iii) the use of metaheuristics not only to tune numerical parameters but also to make some qualitative decision about the design of the . algorithm to tune, by choosing categorical/nominal parameters (as, for instance, the PSO topology).

PROPOSED METHOD

The implementation of the metaheuristics employed in this paper has been presented in [START_REF] Nashed | A comparative study of three GPU-based metaheuristics[END_REF]. The code used in this research is available at http://sourceforge.net/p/libcudaoptimize [START_REF] Nashed | libCudaOptimize: an open source library of GPU-based metaheuristics[END_REF].

When DE or PSO are used as tuner, they work as any other optimizer. The only difference lies in fitness evaluation and how fitness values are stored and used. Each particle in a tuner represents a set of parameters of the optimizer to tune (in this case PSO). This optimizer is tested T times on a set of F functions. The resulting fitness is then computed based on the average and standard deviation of the best fitness obtained over all functions; that is a 2 × F array of values. A set of parameters is better than another if it obtains better results in the majority of the F functions. In case of a tie, the winner is selected comparing the sum of Welch's t test values over the N functions under consideration:

R = N i=1 (X i a -X i b ) s i a 2 +s i b 2 T
where a and b are the two sets of parameters (or the two optimizers), X i and s i are the mean and standard deviation obtained by each optimizer on function i. If R < 0, the optimizer a is better (since we are looking for the minima of the functions), otherwise b wins. During the tuners' execution, optimizers are compared pairwise while, at the end of the optimization process, the surviving particles are compared with one another in a full tournament to choose the winner. PSO and DE are real-valued optimization methods, so the problem of representing nominal parameters needs to be addressed. Our choice has been to represent each nominal value by a vector with as many values as the options available and choose the parameter corresponding to the element with the highest value. For instance, one of the parameters we want to optimize in PSO is topology. We can Table 1: Results (average and standard deviation) obtained on the complete dataset of 20 functions (see [START_REF] Nashed | A comparative study of three GPU-based metaheuristics[END_REF] for the complete definition) and using the best configuration obtained by all tuners. All functions were optimized using 30 dimensions and 30 runs per function. Due to lack of space, only the functions that showed statistically significant differences are included. choose between ring, star and "global-best" topology. This means that three values for each particle in the PSO/DE tuner are reserved to this choice and the largest one in the best particle is chosen as winner (see Figure 1). The range of the PSO parameters allowed by the tuning algorithm are C1, C2 ∈ [0, 4], w ∈ [0, 1.5], Pop. size ∈ [32, 150], and topology ∈ Gbest, Ring, Star.

RESULTS AND CONCLUSIONS

We performed parameter tuning on a training set of 8 functions (numbers 0, 5, 6, 8, 9, 10, 12, 13 in [START_REF] Nashed | A comparative study of three GPU-based metaheuristics[END_REF]) and a validation set of 20. Our goal was to check (i) whether our method was able to get results competitive with other more complex and/or computationally expensive state-of-the-art tuners (irace), (ii) whether we could find repeatable patterns in the parameter sets obtained by different techniques, and (iii) the generalization ability of every tuner.

The comparison was performed by running 10 times our tuners and irace, and searching the best population size and topology with the parameter values for C1, C2, w suggested in [1]. We then selected the best parameter configuration found by each method, based on the Friedman test and the visual examination of the results using boxplots. The resulting configurations, shown in Table 2, are similar: all of them use global best topology, a small population size, an inertia value between 0.35 and 0.65, while extreme values are rejected. Moreover, all of them have C2 > C1, pointing out that, given a global topology, a good configuration relies on social information to reach good results. The standard parameters seem to need a larger population to balance the smaller importance of the "collaboration between particles" factor: one could say that there is less cooperation between particles, but this fact is attenuated by the larger swarm size.

Table 1 reports the results obtained by the best configuration found by each tuner (best set of SPSO parameters) for the functions where there are statistically significant differences among the optimizers. The first column is the function under consideration. The following ones are divided into four blocks, each of which corresponds to either a tuner or SPSO. Within each block, the average best fitness and the standard deviation over all runs are reported. Results reported on a gray background highlight those cases in which the median over 30 runs obtained by one method is significantly better than the one yielded by the other methods, according to the Friedman test, with a level of significance of 0.01. As posthoc procedure, the Dunn-Sidak correction was applied to study the existence of pairwise statistical differences.

As expected, the results are very similar, since the resulting classifiers are similar to each other: they all use global topology, a similar population size while the setting of parameters C1, C2 and w lie within very small ranges. It should be noticed that the three tuned optimizers reach good results also in functions that were not included in the training phase, showing good generalization.

In conclusion, we can say that simple EA-based parameter tuning is able to get comparable results to a well-established state-ofthe art method in parameter tuning, like irace. Moreover, in our case, the parallel implementation makes the tuning process computationally efficient, which allows our tuners to perform their task in a much more reasonable time with respect to sequential implementations.

Figure 1 :

 1 Figure 1: Representation of a set of PSO parameters inside the tuner. The first four elements represent numerical parameters (either integer or real-valued numbers), while the last three contribute to the decision about a nominal parameter (in this case topology).

Table 2 :

 2 Best automatically-tuned parameter values found by DE, PSO, and irace, in 10 runs of every tuner, and 10 of PSO with standard parameters. See[START_REF] Nashed | A comparative study of three GPU-based metaheuristics[END_REF] for a definition of the symbols.

			DE		PSO		irace	Standard
		Avg		Std	Avg	Std	Avg	Std	Avg	Std
	f 4	2.57e-045	1.43e-045	0.00e+000	0.00e+000	0.00e+000	0.00e+000	9.34e-046	8.50e-046
	f 7	5.86e+000	3.56e-001	6.35e+000	4.42e-001	6.31e+000	4.30e-001	6.17e+000	4.08e-001
	f 10	4.43e+003	1.63e+003	4.07e+003	2.09e+003	3.98e+003	1.52e+003 9.27e+003 3.10e+003
	f 13	2.11e+001	1.50e+000	1.38e+001	8.51e-001	1.55e+001	2.64e+000	2.10e+001 4.45e+000
	f 19	1.46e-001	6.37e-001	4.25e-001	6.54e-001	4.71e-001	7.74e-001	5.34e-001	9.68e-001
	Param.	DE	PSO	irace SPSO [1]
	C 1	0.953 1.525	1.243	1.496
	C 2	2.429 1.881	2.222	1.496
	w	0.636 0.443	0.351	0.730
	Pop. Size	37	34	35	60
	Topology gbest	gbest	gbest	gbest

Acknowledgments

Roberto Ugolotti is funded by Compagnia di S.Paolo and Fondazione Cariparma. Youssef S. G. Nashed, and Pablo Mesejo are funded by the European Commission (Marie Curie ITN MIBISOC, FP7 PEOPLE-ITN-2008, GA n. 238819). The authors want to thank Marco Grossi and Gian Maria Marconi for their precious help.


Pablo Mesejo
email: pmesejo@ce.unipr.it

Stefano Cagnoni
email: cagnoni@ce.unipr.it

Alessandro Costalunga
email: alessandro.costalunga@studenti.unipr.it

Davide Valeriani
email: davide.valeriani@studenti.unipr.it

Segmentation of Histological Images using a Metaheuristic-based Level Set Approach

Keywords: I.5.4 [Pattern Recognition]: Applications-Computer Vision; I.2.10 [Artificial Intelligence]: Vision and Scene Understanding Algorithms, Experimentation Hippocampus, Differential Evolution, Real-Coded Genetic Algorithms, Particle Swarm Optimization, Histological Images, Level Set method, Image Segmentation

This paper presents a two-phase method to segment the hippocampus in histological images. The first phase represents a training stage where, from a training set of manually labelled images, the hippocampus representative shape and texture are derived. The second one, the proper segmentation, uses a metaheuristic to evolve the contour of a geometric deformable model using region and texture information.

Three different metaheuristics (real-coded GA, Particle Swarm Optimization and Differential Evolution) and two classical segmentation algorithms (Chan & Vese model and Geodesic Active Contours) were compared over a test set of 10 histological images. The best results were attained by the real-coded GA, achieving an average and median Dice Coefficient of 0.72 and 0.77, respectively.

INTRODUCTION

Image segmentation is defined as the partitioning of an image into non-overlapping regions that are homogeneous with respect to some visual feature such as intensity, color or texture [START_REF] Pham | Current Methods in Medical Image Segmentation[END_REF]. In particular, image segmentation plays a crucial role in many biomedical imaging applications by automating and facilitating the delineation of anatomical structures and different regions of interest (ROIs). As one can easily check taking a look at the literature [START_REF] He | A comparative study of deformable contour methods on medical image segmentation[END_REF][START_REF] Heimann | Statistical shape models for 3D medical image segmentation: a review[END_REF][START_REF] Petitjean | A review of segmentation methods in short axis cardiac MR images[END_REF], most of the segmentation algorithms developed so far have been applied to MRI and CT images, not paying too much attention to other imaging modalities like ultrasound or microscopy. Even when dealing with histological images, it is surprising to check that most literature about their processing and analysis is focused on registration and 3D reconstruction of the whole brain with very few papers about segmentation. This paper presents a method aimed at segmenting anatomical structures in histological images using deformable models and metaheuristics.

Among the different anatomical structures which make up the mammalian brain, the hippocampus is particularly interesting due to its crucial role in learning and memory processes [START_REF] Norman | How hippocampus and cortex contribute to recognition memory: revisiting the complementary learning systems model[END_REF]. Moreover, it has recently been demonstrated that the volume of the hippocampus is an early biomarker for Alzheimer's disease. Therefore, there is great interest in understanding the cellular and molecular events that take place in this structure, under both normal and pathological conditions.

In particular, we consider the problem of segmenting the hippocampus in histological images extracted from the Allen Brain Atlas (ABA): a huge, publicly available image database, which has recently provided scientists with a gene-expression map for future study and investigation. The ABA contains a genome-scale collection of histological images (cellular resolution gene-expression profiles) obtained by In Situ Hybridization (ISH) of serial sections [START_REF]Allen Reference Atlases[END_REF] of mouse brains.

As in most medical imaging applications, the main problems to be tackled are related with the characteristics of the images involved in the segmentation:

• natural variability of brain structure shapes in different subjects;

• fuzziness of the hippocampus boundary;

• grained patterns with many irregularities, which hamper the classification of individual pixels in the anatomical structures under consideration;

• rotation or displacement of the imaged structures on the slice with respect to a "standard" alignment;

• variable brightness of the images within the same set;

• presence of artefacts: tears, scraps, bubbles, streaks in tissues, partial cut-off of regions; and

• large image size (the typical resolution of ABA images is about 15,000 × 7,000 pixels, and the ROIs about 2,500 × 2,000 pixels).

A procedure based on Active Shape Models and Random Forests (ASM+RF) [START_REF] Mesejo | Automatic Segmentation of Hippocampus in Histological Images of Mouse Brains using Deformable Models and Random Forest[END_REF][START_REF] Mesejo | Automatic hippocampus localization in histological images using Differential Evolution-based deformable models[END_REF] was recently developed to accurately segment these images. The results obtained were very promising in comparison with many other segmentation techniques [START_REF] Mesejo | An experimental study on the automatic segmentation of in situ hybridization-derived images[END_REF] but, despite good performance, they also highlighted some drawbacks:

• impossibility to deal with topological changes in a natural manner;

• ad hoc nature, since the procedure needs a training set of shapes to manually create the parametric template and its possible deformations, as well as a training set of textural patterns for the ensemble classifier used to refine the results of the segmentation; and

• long and complex pipeline of four independent stages (initialization of the deformable model, localization of the anatomical structure of interest, segmentation, and refinement of the segmentation using classifiers). Therefore, a more integrated and compact approach would be desiderable.

The main idea of this paper is to overcome some of these problems using another kind of deformable model (the level set method) that can: a) easily handle topological changes of the contours, b) be adapted to solve large dimensional problems without great effort, and c) easily determine the areas inside and outside an active contour.

The work developed here is based on the one by Ghosh et al in [START_REF] Ghosh | Segmentation of medical images using a genetic Proc. Conference[END_REF][START_REF] Ghosh | A genetic algorithm-based level set curve evolution for prostate segmentation on pelvic ct and mri images[END_REF] which, in turn, was inspired by [START_REF] Tsai | A Shape-Based Approach to the Segmentation of Medical Imagery Using Level Sets[END_REF]. It is important to underline that this research represents one of the very few cases in which metaheuristics have been used to optimize a geometric deformable model, let alone their application to histological images, and that the current work also presents significant differences with respect to the papers mentioned above:

• an intensity-based term from Chan & Vese approach [START_REF] Chan | Active contours without edges[END_REF] has been included in the fitness function, trying to combine a region-based approach with prior knowledge about texture and shape;

• in order to adapt this approach to the particular nature of histological images, the textural part has also been critically modified: we do not use Laws' textural measures or Gabor wavelet transform-based features, like in the original paper, but textural features extracted from the Gray Level Co-occurrence Matrix, as explained in [START_REF] Mesejo | Automatic Segmentation of Hippocampus in Histological Images of Mouse Brains using Deformable Models and Random Forest[END_REF];

• in relation to the genetic operators used, the singlepoint crossover used in [START_REF] Ghosh | Segmentation of medical images using a genetic Proc. Conference[END_REF] has been replaced by a realcoded one like the BLX-α, due to the nature of the GA chromosomes; and The paper is structured as follows: in section 2 we provide the theoretical foundations of our work, as well as an overview of previous related work. In section 3, a general overview of the method is presented, providing details about the segmentation of the hippocampus. Finally, section 4 presents results on 10 real images with appropriate statistical tests, followed, in section 5, by some final remarks and a discussion about possible future developments.

BACKGROUND

This section briefly introduces some of the fundamental computer vision and soft computing techniques applied in this paper, as well as the main contributions to the field found in literature.

The classical gradient search techniques are effective when the problem to optimize satisfies tight constraints. However, when the search space is discontinuous, noisy, highdimensional, or multimodal, then metaheuristics offer a number of attractive features: no requirement for a differentiable or continuous objective function, global search capability, or virtually no need of specific information about the problem to solve.

Particle Swarm Optimization

PSO [START_REF] Kennedy | Particle Swarm Optimization[END_REF] is a bio-inspired optimization algorithm based on the simulation of the social behavior of bird flocks. In the last fifteen years PSO has been applied to a very large variety of problems and numerous variants of the algorithm have been presented.

During the execution of PSO, a set of particles moves within a (fitness) function domain searching for its optimum (best fitness value). The motion of each particle is driven by the best positions visited so far by the particle itself and by the best positions found by the entire swarm (gbest PSO) or by particles in some pre-defined neighborhood (lbest PSO). Consequently, each particle relies both on "individual" and on "swarm" intelligence, and its motion can be described by the following two simple equations which update the particles' position and velocity:

Pn(t) = Pn(t -1) + vn(t) vn(t) = w • vn(t -1) + c1 • rand() • (BPn -Pn(t -1)) + c2 • rand() • (BLPn -Pn(t -1))
where Pn(t) and vn(t) are the position and velocity of the n th particle in iteration t; c1, c2 and w (inertia factor) are positive constants; rand() returns random values uniformly distributed in [0, 1]; BPn is the best-fitness location visited so far by the particle and BLPn is the best-fitness location visited so far by any particle of a neighborhood which may be as large as the current swarm: in this case, this position corresponds to the global best.

In particular, we have used a version of PSO already presented in [START_REF] Ugolotti | Automatic hippocampus localization in histological images using PSO-based deformable models[END_REF], where the inertia factor w adapts its value to the fitness function of each particle and a particle is reinitialized in case of stagnation. [START_REF] Storn | Differential Evolution-a simple and efficient adaptive scheme for global optimization over continuous spaces[END_REF], has recently been shown to be one of the most successful Evolutionary Algorithms (EAs) applied to global continuous optimization. Unlike traditional EAs, DE perturbs the current population members with the scaled differences of randomly selected distinct individuals. In the first iterations, the elements are widely scattered in the search space and have great exploration ability while, as optimization proceeds, the individuals tend to concentrate in the regions of the search space with better values, such that the search automatically focuses on the most promising areas. In DE, every element acts as a parent vector and, for each of them, a donor vector is created. In the original version of DE, the donor vector for the i th parent (Xi) is generated by combining three random distinct elements Xr1, Xr2 and Xr3. The donor vector Vi is calculated as follows:

Differential Evolution

DE, first introduced by Storn and Price

Vi = Xr1 + F • (Xr2 -Xr3)
where F (scale factor) is a parameter that strongly influences DE's performances and typically lies in the interval [0.4, 1]. After mutation, every parent-donor pair generates an offspring (the so-called trial vector) by means of a crossover operation. Two kinds of crossover are typically used: binomial (also called uniform) and exponential. Besides F , the crossover rate Cr is another parameter which regulates the searching behavior of DE.

Real-Coded Genetic Algorithms

Genetic Algorithms (GAs) are stochastic, parallel search algorithms based on the mechanics of natural selection [START_REF] Goldberg | Genetic Algorithms in Search, Optimization and Machine Learning[END_REF]. GAs were designed to efficiently search large, non-linear, poorly-understood search spaces where expert knowledge is scarce or difficult to encode and where traditional optimization techniques fail. They are flexible, robust, and exhibit the adaptiveness of biological systems.

GAs process a population of solutions by three operations: selection, crossover and mutation. In the initial formulation, the solutions were binary encoded; however, other encoding types have been taken into account for the representation. In particular, real encoding seems particularly natural when optimizing parameters in continuous domains. In that case, a chromosome is a vector of real numbers, each of which is one of the parameters to be optimized. GAs based on realnumber representations are called real-coded GAs (RCGAs).

There is a compelling evidence indicating that classical discrete crossover operators (DCOs), i.e., all the crossover operators used for binary encoding which are directly applicable to real coding (like the simple, two-point and uniform crossover operators), are ineffective for RCGAs [START_REF] Herrera | A taxonomy for the crossover operator for real-coded Genetic Algorithms: An experimental study[END_REF][START_REF] Herrera | Tackling real-coded genetic algorithms: Operators and tools for behavioural analysis[END_REF]. Since the crossover operators that exploit the numerical nature of the real coding (aggregation-based operators, like arithmetic or geometric, and neighborhood-based crossover operators, like BLX-α or SBX) consistently outperform the classical DCOs, these kinds of operators have been considered in this work.

Deformable Models

Deformable Models (DMs) [START_REF] Terzopoulos | Deformable models[END_REF] are curves or surfaces, defined within the image domain, that are deformed under the influence of "internal" forces, related with the curve features, and "external" forces, related with the image regions surrounding the curve. Internal forces enforce regularity constraints and keep the model smooth during deformation, while external forces are defined such that the model is attracted toward an object or other features of interest within the image.

Basically DMs can be parametric or geometric. The former represent curves and surfaces explicitly in their parametric forms during deformation, leading to a fast interaction with the model and a compact representation of the object to segment. On the other hand, geometric DMs can handle topological changes naturally, since they are based on the theory of curve evolution and represent curves implicitly as the level set of a higher-dimensional scalar function [START_REF] Osher | Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations[END_REF]. Among the parametric DMs we may mention: Active Shape Models [START_REF] Cootes | Active shape models-their training and application[END_REF], Deformable Templates [START_REF] Zhong | Object tracking using deformable templates[END_REF], Topological Active Nets [START_REF] Ibáñez | Genetic approaches for topological active nets optimization[END_REF] and Active Contour Models ("snakes") [START_REF] Kass | Snakes: Active contour models[END_REF]. Examples of geometric DMs are Geodesic Active Contours (GAC) [START_REF] Caselles | Geodesic active contours[END_REF] and the Chan&Vese model (CV) [START_REF] Chan | Active contours without edges[END_REF].

Although originally developed for computer vision applications to natural scenes and computer graphics problems, the potential of deformable models in medical image analysis has already been proven [START_REF] He | A comparative study of deformable contour methods on medical image segmentation[END_REF][START_REF] Heimann | Statistical shape models for 3D medical image segmentation: a review[END_REF].

Related Work

We shortly review some of the most relevant and recent approaches which combine geometric deformable models and evolutionary computation techniques. In [START_REF] Ghosh | A genetic algorithm-based level set curve evolution for prostate segmentation on pelvic ct and mri images[END_REF], a GA is used to perform level set curve evolution using texture and shape information to automatically segment the prostate in CT and MRI pelvic images. In [START_REF] Mesejo | An experimental study on the automatic segmentation of in situ hybridization-derived images[END_REF], an Active Shape Model optimized by DE is used as the initial contour for two wellknown geometric DMs (CV and GAC). In [START_REF] Feltell | 3d level set image segmentation refined by intelligent agent swarm[END_REF], the initial segmentation based on the level set method is refined using swarms of intelligent agents. Finally, in [START_REF] Heydarian | Optimizing the Level Set Algorithm for Detecting Object Edges in MR and CT Images[END_REF], a GA is used to find an optimal set of parameters for a level set method used to segment MRI and CT.

OVERVIEW OF THE SYSTEM

As explained in the introduction, our work essentially adapts the ideas introduced by Ghosh and Mitchell in [START_REF] Ghosh | Segmentation of medical images using a genetic Proc. Conference[END_REF] to the segmentation of the hippocampus in histological images. Such an adaptation solves some of the problems detected in the original proposal and takes into account the specificities of the case under study.

On the one hand, we introduce a real-value crossover operator conforming to the nature of our chromosomes (see Section 2.3). On the other hand, we use textural features that differ from the initially proposed ones, since it has been shown that the Grey-level Co-occurrence Matrix (GLCM) obtained very good results with the same image modality [START_REF] Mesejo | Automatic Segmentation of Hippocampus in Histological Images of Mouse Brains using Deformable Models and Random Forest[END_REF]. Due to the huge size of these images, the textural segmentation is only applied for a limited number of iterations, to guide the search to promising regions. Finally, the structure orientation (pose) is not taken into account during optimization because of two main reasons: it increases the temporal and computational cost of the optimization, adding a burden that is not justified by the results obtained.

The method consists of two phases: training and proper segmentation (see Figure 1). In the first one, from a set of manually segmented training images we compute the average shape, the main modes of variation, and the median texture, trying to characterize an "ideal" hippocampus. An implicit representation of the segmenting curve is created by applying principal component analysis to a collection of signed distance representations of the training data. During segmentation, the metaheuristic searches for the best weights to linearly combine the mean shape and shape variabilities, and the texture enclosed by the evolving contour is compared with the "representative" texture.

Training Phase

Shape

First of all, the shape priors are derived from the training set. To do so, each contour from the training data is represented as the zero level set of the signed distance function ψi(x,y), i = 1 . . . n, where (x,y) are the pixel coordinates, and n is the number of training contours used to assess shape variability. The signed distance function is the shape representation: the boundaries of each shape are embedded as the zero level set of a signed distance function with negative distances assigned to the inside and positive distances assigned to the outside of the object.

The mean level set function is defined as: Φ(x, y) = 1 n n i=1 ψi(x, y) Mean offset functions are then derived by subtracting the mean from the signed distance representations of the training contours: ψi = ψi -Φ Assume that the images are of size N = N1 × N2. Given the huge size of the images under consideration, all images have been resized to N1 × N2 = 500 × 500 pixels. The columns of the mean offset functions (|N | = N1 × N2) are then serially stacked to form one column vector βi of size 1 × N . The shape variability matrix S (of size N × n) is obtained from these n column vectors: S = [β1, β2, β3, . . . , βn] The variance in shape is then computed by an eigenvalue decomposition on this shape variability matrix:

1 n SS T = U ΣU T
where U is an N × n matrix whose columns represent n orthogonal modes of shape variation, Σ is an n × n diagonal matrix of eigenvalues, and the columns of U = [ui] are the corresponding eigenvectors. In this paper, instead of computing the eigenvectors of such a large matrix, we have considered the smaller one, 1 n S T S, because it is more computationally efficient [START_REF] Leventon | Statistical Models for Medical Image Analysis[END_REF] to obtain the n different eigenshapes {Φ1, Φ2, Φ3, . . . , Φn}.

After this procedure, introduced in [29], the mean shape and shape variability derived as described are used to define a level set function that implicitly represents the segmenting curve:

Φ[w] = Φ + k i=1 wjΦj Thus, the task of the metaheuristic will be to find the values of w that minimize a fitness function to be defined in the test phase.

Texture

A texture is a set of visual elements occurring in some regular or repeated pattern. GLCM was introduced by Haralick [START_REF] Haralick | Textural Features for Image Classification[END_REF] and is a feature-based method that characterizes a texture as a homogeneous distribution of feature values. A co-occurrence matrix describes how often a gray level appears in a specified spatial relationship to another gray level. The entry at (i, j) of the GLCM indicates the number of occurrences of the pair of gray levels i and j which are a distance d apart along a given direction θ. The values of d and θ are parameters for constructing the GLCM.

The textural priors used here are the same as in [START_REF] Mesejo | Automatic Segmentation of Hippocampus in Histological Images of Mouse Brains using Deformable Models and Random Forest[END_REF], which yielded very good performance. The training patterns, as well as the test values calculated during the segmentation, were encoded as a vector of 11 textural features:

• first order measures: mean, standard deviation, skewness, kurtosis, entropy, coefficient of variation and energy;

• second order measures: contrast, correlation, energy and homogeneity from the GLCM, using (1,1) as spatial relationship (i.e. θ = 315 • and d=1 pixel), employing a window of size 30×30 pixels.

The main idea is to capture the textural essence of the hippocampus and compare, during the segmentation phase, the texture enclosed by the evolving surface of our DM with the "ideal" texture of the training set. To create this median texture, p points per image in the training set were randomly selected inside the hippocampus, and t textural patterns were extracted using those points as pixels-of-interest. This procedure creates a (t•p)×n matrix, n being the number of training images, and t • p the number of textural features multiplied by the number of selected points. In this case, t = 11 and p = 100. From this matrix, the median is calculated to obtain a "general representation" of the texture in the hippocampus: all points which are closer to these values are assumed to belong to the hippocampus.

Test Phase

The test phase corresponds with the proper segmentation of the object of interest. The implicit representation of the contour is deformed guided by a metaheuristic which tries to fit the boundaries of the hippocampus. To do so, such a metaheuristic stochastically generates weights to combine the mean and the variability to create new shapes.

The fitness function combines region-and texture-based terms. The former represents the Chan&Vese model [START_REF] Chan | Active contours without edges[END_REF], while the latter is the euclidean distance between the texture enclosed by our contour and the median texture found in our training set.

The energy functional to minimize is the following:

F (i1, i2, C) = α • (µ • Length(C) + ν • Area(inside(C)) +λ1 inside(C) |u0(x, y) -i1| 2 dxdy +λ2 outside(C) |u0(x, y) -i2| 2 dxdy) +β • (d(t(C), T ))
where u0 is the image to segment, formed by two regions (internal and external), C is the evolving curve, i1 and i2 are constants depending on C (pixel intensity averages of u0 inside and outside C, respectively), and d(t(C), T ) the euclidean distance between the median texture extracted from the training set (T ) and the actual texture enclosed by the evolving contour (t(C)). Using this functional, our model simultaneously takes into account intensity and texture criteria. In this case, both terms have been weighted equally (α = β = 1).

Finally, a fast refinement step is applied. This refinement sequentially applies 50 iterations of a local implementation of Chan&Vese [START_REF] Wang | An efficient local Chan-Vese model for image segmentation[END_REF], that takes into account only a neighbourhood of the boundary, and the removal of connected components whose area is smaller than a threshold (in this case 500 pixels).

EXPERIMENTAL RESULTS

To check the performance of the new method, 25 and 10 manually segmented images were used as training and test set, respectively. The 10 images selected as test are representative of the problem and include different scenarios (see Figure 2). Every segmentation method was run 15 times per image (excluding C&V and GAC because they are deterministic methods and always produce the same result). The performance of GA (LS-GA), PSO (LS-PSO) and DE (LS-DE) was compared with the one obtained by two classical geometric DMs: a region-based method (CV) and an edge-based one (GAC). With respect to the population and the number of iterations used in the metaheuristic, the same configuration as the original paper was used (50 individuals, 100 iterations), and the images were resized to 500 × 500 pixels.

Standard segmentation metrics were used to evaluate performance and the results were analyzed using proper statistical tests. The Jaccard similarity index (JI) and the Dice coefficient (DSC) measure set agreement: a value of 0 indicates no overlap with the ground truth, and a value of 1 indicates perfect agreement. In turn, the Hausdorff distance (HD) represents a measure of the spatial distance between two sets of points: it is the largest of all the distances from any point in the resulting segmentation to the closest point in the ground truth.

In order to perform a fair comparison with the deterministic/classical methods, the latter have been intensively tuned using an exhaustive search among the most commonly used parameter values over the complete dataset (training and test sets). Also, for these two methods, the input images were pre-processed using a median filter (5 × 5 in CV and 10×10 in GAC) and, as post-processing, the removal of connected components smaller than 500 pixels was also applied.

Since the normality and homoscedasticity assumptions were not accomplished, as checked through the application of Kolmogorov-Smirnov and Bartlett's tests, non-parametric tests were used. To check the statistical significance of the results obtained, a Friedman test was performed with a level of confidence of 0.01 for the null hypothesis that all samples are drawn from distributions with the same median. Since the p-value was near zero, the statistical test suggested that, at least, one sample median is significantly different from the others. Pairwise statistical differences were studied using Friedman test with the Tukey-Kramer correction, and real-coded GA was found to be the best method among the ones in the comparison. Some conclusions can be derived from the numerical data in Table 2 and the visual information contained in Figure 3. The first conclusion could be the difficulty of tackling satisfactorily these images, since established and well-known segmentation methods, like C&V and GAC, did not obtain as good results as one could expect. In particular, GAC presents problems when the difference, in terms of gradient values, between the hippocampus and the background is small. In fact, between the two classical methods, the regionbased one (CV) obtained better results than the one relying on edge information. This can be justified by the fuzzy boundaries of the hippocampus that may have hampered the performance of GAC. Also, the dissatisfactory results obtained by CV can be explained by the complex background of these images, since it is well-known that the performance of CV is better when dealing with images where foreground and background determine two regions with clearly different average intensity levels.

Considering the stochastic approaches, the best results were obtained by LS-GA, which appeared to be the best in all metrics. Real-coded GA obtained the best results in all images but in one, while the worst results were obtained by DE. Obviously, this is only a preliminary study, so the convergence problems of DE and PSO, which in some cases are not able to obtain good results, should be investigated more deeply. For example, the PSO topology used here has been the global best; tests with other topologies should be performed to check if there is some improvement in the results. Also, in the case of DE, tests with other crossover operators could be run to study the influence of this factor in the general performance of the segmentation algorithm. Finally, experiments with a larger population size or a larger number of iterations should be tested, considering the huge search space these techniques have to explore.

From the values of HD, interesting conclusions can also be drawn. It is interesting to check how our shape-based approach has a smaller (best) average and median HD with respect to C&V and GAC. This could be justified by the absence in the latter of any kind of shape restriction, that impose that the pixels are segmented only taking into account intensity and boundary information, respectively.

CONCLUSIONS AND FUTURE DEVELOPMENTS

This paper investigates hybridizations between metaheuristics and the level set method, and represents a first approach to the segmentation of histological images using eigenshapes, whose linear combination is optimized using PSO, DE and real-coded GA. On the one hand, metaheuristics can optimize the energy function of the deformable model or find the most suitable parameters for such a model. On the other hand, the level set method represents an elegant solution for the main drawbacks of parametric deformable models, like the possibility of managing topological changes in a natural manner. With these concepts in mind, this paper studies ways of using a training set of shapes and textures to solve a difficult problem, like segmenting the hippocampus in histological images, and compare different global search optimization techniques. Finally, this work represents one of the very few cases in which PSO and DE have been used to optimize the level set method, and also one of the few examples of application of geometric DMs to the segmentation of histological images.

It is important to remark that the work presented here is a first approach that should be refined. First, it is slow compared to classic approaches based on only one feature (intensity, edge)1 . Obviously, if someone is looking for a fast method this would not be the best option: what this method can offer is accuracy. For instance, segmentations obtained by LS-GA have a median Dice coefficient which is three times better than GAC. Also, a more sophisticated use of the textures could be taken into account (for example, differentiating the two structures of the hippocampus), or including more prior information and restrictions (for instance, the relative position of the hippocampus with respect to other organs). In any case, this initial approximation has shown good performance using three different standard metrics and the results obtained have been better than the ones obtained by very well-established segmentation algorithms.

Figure 1 :

 1 Figure 1: General Overview of the Hippocampus Segmentation System

Figure 2 :

 2 Figure 2: Test set used in experiments and boxplots of the DSC results obtained per image by the stochastic methods under study. Each plot refers to the image having the corresponding position in the upper row.

Table 1 :

 1 Parameters used in testing the different algorithms. The values for LS-GA, LS-DE and LS-PSO were based on the literature and on a brief empirical study about the suitability of different combinations of parameters. 01 (length term) β = -1 (expansion weight) Uniform Crossover c1 = 2.05 M ut = 0.09 ν = 0 (area term) α = 3 (contour weight) DE/target-to-best/1 c2 = 1.75 T ournament(4) λ1 = λ2 = 1

Figure 3 :

 3 Figure 3: Box-plots representing the global DSC results obtained by the five methods compared over the whole dataset.

Table 2 :

 2 Segmentation results using three different metrics: Dice Similarity Coefficient (DSC), Jaccard Index (JI), and Hausdorff Distance (HD).

					Methods
			LS-GA	LS-DE	LS-PSO	C&V	GAC
		Average	0.7204	0.4145	0.5383	0.5044	0.3966
		Median	0.7695	0.3446	0.6622	0.5294	0.2758
	DSC	Std	0.1672	0.2842	0.2614	0.1546	0.2151
		Best	0.8806	0.8716	0.8711	0.6648	0.7950
		Worst	0.2363	0.0059	0.0015	0.1107	0.2404
		Average	0.5871	0.3058	0.4120	0.3485	0.2710
		Median	0.6271	0.2081	0.4950	0.3600	0.1601
	JI	Std	0.1745	0.2515	0.2378	0.1212	0.1962
		Best	0.7867	0.7725	0.7717	0.4979	0.6597
		Worst	0.1340	0.0029	0.0007	0.0586	0.1366
		Average	420.8674	617.6873	549.0358	762.4849	622.4544
		Median	176.0064	372.8288	338.5550	553.3118	507.1295
	HD	Std	518.0279	549.3174	541.2218	354.1919	242.5857
		Best	28.3456	45.0627	45.0627	403.6087	395.1266
		Worst	1.7878e+003 2.1191e+003 1.7196e+003 1.4290e+003 1.0341e+003

In order to give some reference in this regard, the average time to perform one run with our metaheuristic-based approach on a MacBook Pro i7 dual core

@ 2,7GHz with 8GB of RAM, and using MATLAB as programming environment, it is around 10 times slower than C&V and GAC, that take approximately 5 minutes.

ACKNOWLEDGMENTS

Pablo Mesejo is funded by the European Commission through the MIBISOC project ("Medical Imaging using Bio-Inspired and Soft Computing", MIBISOC Marie Curie Initial Training Network, FP7 PEOPLE-ITN-2008, GA n. 238819).