Genetic Adaptation of Level Sets Parameters for Medical Imaging Segmentation

Transcription

1 Genetic Adaptation of Level Sets Parameters for Medical Imaging Segmentation Dário A. B. Oliveira & Raul Q. Feitosa Catholic University of Rio de Janeiro, Brazil Mauro M. Correia Department of Surgery, School of Medicine, Unigranrio and National Cancer Institute- INCA, Brazil ABSTRACT This chapter presents a method based on level sets to segment organs using Computer Tomography (CT) medical images. Initially, the organ boundary is manually set in one slice as an initial solution, and then the method automatically segments the organ in all other slices, sequentially. In each step of iteration it fits a Gaussian curve to the organ s slice histogram to model the speed image in which the level sets propagate. The parameters of our method are estimated using Genetic Algorithms (GA) and a database of reference segmentations. The method was tested to segment the liver using 20 different exams and five different measures of performance, and the results obtained confirm the potential of the method. The cases in which the method presented a poor performance are also discussed in order to instigate further research. KEYWORDS Medical Imaging, Level Sets, Genetic Algorithms, Segmentation Parameters Estimation, Computer Tomography. INTRODUCTION In medical imaging analysis, image-guided surgery and organs visualization, segmentation is a crucial step. The segmentation process is particularly arduous in abdominal computer tomography (CT) images because different organs lie within overlapping intensity value ranges and are often near to each other anatomically. Therefore, usually it is not possible to define accurately the boundaries of organs and lesions using simple threshold based segmentation. On the other hand more complex algorithms involve more parameters that must be properly adjusted in order to generate good results for a wide set of exams.

2 Numerous techniques have been proposed in the literature for extraction of organs contours in abdominal CT scans. They can be roughly divided in two main groups: model driven and data driven approaches (Masutani et al, 2005). Model driven techniques (e.g. Lamecker et al, 2004; Soler et al, 2000) use pre-defined models to segment the desired object from the available images. In this kind of technique a model describing the organ to be segmented is defined in terms of object characteristics such as position, texture and spatial relation to other objects, and the algorithm searches the images for instances that fit the given model. Data driven techniques (e.g. Fujimoto et al, 2001; Kim et al, 2000) try to emulate the human capacity of identifying objects using some similarity information present on image data, automatically detecting and classifying objects and features in images. Many of them use known techniques such as region growing and thresholds, combined with some prior knowledge about the object being analyzed. Level set methods (Osher & Sethian, 1998) are model driven methods that rely on partial differential equations to model deforming isosurfaces. These methods have been used successfully in medical image processing but usually require human intervention to set an initial solution and indicate explicitly when the model should stop expanding. In semi automatic level sets based methods, the user is minimally required to manually set parameters that segment correctly the desired object, which is a time consuming trial and error task. The manual definition of parameter values on level sets methods is a complex task, because their relation with the final result is unclear and there is no guarantee that the optimal set of values will be determined by an trial and error approach. Such parameters in the implementation of traditional level sets are related to the curves mean curvature, propagation rate and advection of the curve to certain characteristics of the image. Therefore, there is a demand for automatic methods to define such parameters automatically. These methods can use important information provided by a dataset or a specialist to built or adjust a given model. In this way the process learns subjective information from the data, adding this knowledge to the model, making it more robust to deal with problems similar to the ones presented in the training dataset. Genetic Algorithms (GA) can be used to find optimal parameters in different applications. GAs are a computational search technique to find approximate solutions to optimization problems, based in the biological evolution of species as presented by Charles Darwin (Darwin 1859). In this chapter we propose a model driven method based on level sets to segment the liver using CT images. From an initial user-defined liver segment in one slice, the method segments the liver through all other slices, using a Gaussian fit to define the speed image where the level sets propagates. The initial solution at each slice is defined as the region previously segmented on an adjacent slice. We propose also the use of an evolutionary approach to determine appropriate parameter values based on a set of reference segmentation results. Experiments using five exams as training set and other 15 exams for validation indicated the good performance of the proposed method. The cases with poor performance are discussed to instigate further research. The subsequent text is organized in the following way. Firstly the theoretical fundamentals of level sets and genetic algorithms are presented. Then, we present the proposed segmentation method in details and the parameters estimation experiments. Finally, we report some results, and present the main conclusions generated.

3 THEORETICAL FUNDAMENTALS In this section we introduce theoretical fundamentals related to the level sets method and present an overall description of genetic algorithms. Level Sets The level set methods theory was originally developed by Sethian and Osher (Osher and Sethian, 1998; Sethian, 1999) for analyzing and computing moving fronts in a variety of different settings. These methods were firstly introduced in medical imaging by Malladi et al. (Malladi et al, 1995). Level set is a numerical technique to model and compute continuous deformable models using an implicit representation. It can be applied to different areas, such as computer graphics, computer vision and image processing. In image processing it was successfully applied to segment structures with boundaries not clearly defined, that usually cannot be segmented using simple data driven methods, based on thresholds and line extraction. The level set approach takes into account other properties such as mean curvature to model the segmented object, in such a way that it is possible to avoid undesirable leaks toward neighboring structures, preserving the boundaries of the segmented object even when they are not clearly defined through the entire object. The main idea of level sets is to embed the deformable model in a d+1 dimensional space, in order to segment iteratively an object in a d dimensional space, using partial differential equations. The main advantage of level sets is that it allows changes of surface topology implicitly. As it embeds the evolving surface, also called interface, in a higher dimensional function, the interface can split into several connected components or merge from different connected components naturally. It happens because the embedding level sets function, which is used to evolve the surface, remains continuous. Considering ψ(x,t) the level sets function, x the position vector and t the time step of the level set evolution, the evolving surface is represented as the zero level set of ψ(x,t)=0. This time step controls the evolution of the interface in time. The interface ψ(x,t+1)=0 is computed taking into account the interface of the previous time step ψ(x,t)=0, using partial differential equations. The segmentation result is achieved when the RMS difference between ψ(x,t)=0 and ψ(x,t-1)=0 is less than a pre-defined minimum RMS value. The level sets function is usually a smooth well behaved function, for numerical stability reasons during partial differential equations computation. This work uses the signed distance function that computes the distance of each voxel to its closest voxel in the interface. This distance is negative inside the interface, and positive outside, in such a way that the position of a given point in relation to the interface can be easily defined by analyzing the signal of the level set function in the given point. Using an initial surface S0, which is consequently ψ(x,0)=0, the level set function is evolved under the control of the differential equation 1, that defines the displacement of the interface in a time step. dψ/dt = - αa(x) ψ βp(x) ψ + γz(x) К ψ (1)

4 Note that the gradient ψ (or its module) of ψ(x,t) appears in all terms of the equation. As ψ(x,t) was defined as the signed distance function, the gradient of ψ(x,t) points from inside to outside considering the interface ψ(x,t)=0, and ψ = 1, by definition. It defines, together with other parameters, the direction and magnitude of interface evolution. A, P and Z, are usually calculated from the input image. A is a vector field responsible for the advection term. It attracts the evolving surface to determinate features, usually related to boundaries of objects, and pre-defined barriers. It is weighted by the constant advection weight α, and multiplied by the gradient of ψ(x,t). P is a propagation image, also called speed image, which controls how the level sets propagates through the coordinate space. This image normally has high values in regions where the interface can expand quickly, and values close to zero in regions where it should move slowly or stop (normally close to important features). It is weighted by the constant propagation weight β and multiplied by the module of the gradient of ψ(x,t). К is the mean curvature of the interface, and is defined as the divergence of the normal to the interface. It is usually calculated using first and second derivatives of the interface, based on finite differences. In this way, К > 0 for convex regions, and К < 0 for concave regions. Z is a spatial modifier for the mean curvature К, and modifies the value of К in a determinate spatial position. In this work P was defined in such a way that the curvature has less importance when close to important features. It is weighted by the constant mean curvature weight γ and multiplied by the module of the gradient of ψ(x,t). The basic level sets model has these three terms but a segmentation algorithm based on level sets may add new different terms, or omit one or more terms, depending on the complexity of the problem. Many of the parameters mentioned until now must be properly adjusted for the level sets method to produce accurate results. The determination of appropriate parameter values usually cannot be done using heuristics mainly due to the complexity of the target application. It is generally also not possible to determine the optimum values using non-linear optimization methods for parameter adjustment, due to multiple local minima. An alternative to estimate segmentation parameters of complex functions, with many local minima, are Genetic Algorithms. Genetic Algorithms Genetic Algorithms are a computational search technique to find approximate solutions to optimization problems. They are based in the biological evolution of species as presented by Charles Darwin (Darwin, 1859). The main principle of the Darwin s Theory of Evolution is that individual characteristics are transmitted from parents to children over generations, and individuals more adapted to the environment have greater chances to survive and pass on particular characteristics to their offspring.

5 Figure 1: Basic GA processes. In evolutionary computing terms an individual represents a potential solution for a given problem, and its relevant characteristics with respect to the problem are called genes. In this work, as we are applying GAs to find optimal segmentation parameters, the genes are a set of the proposed segmentation method parameters. A population is a set of individuals in a particular generation, and individuals in a population are graded as to their capacity to solve the problem. That capacity is determined by a fitness function, which indicates numerically how good an individual is as a solution to the problem (Michalewicz, 1994). GAs propose an evolutionary process to search for solutions that maximize or minimize a fitness function. This search is performed iteratively over generations of individuals. For each generation the less fitted individuals are discarded and new individuals are generated by the reproduction of the fittest. The creation of the new individuals is done by the use of genetic operators. A genetic operator represents a rule for the generation of new individuals. The classical genetic operators are crossover and mutation. Mutation changes gene values in a random fashion, respecting the genes search spaces. Mutation is important to introduce a random component in the search of a solution in order to avoid convergence to local minima, as it may lead the search to unvisited search space regions. Crossover operators act by mixing genes between two individuals to create new ones that inherit characteristics of the original individuals. The general idea is that an individual s fitness is a function of its characteristics, and the exchange of good genes may produce better fitted individuals depending on the genes inherited from their parents. Although less fitted individuals can also be generated by this process, they will have a lower chance of being selected for reproduction. These two kinds of genetic operators have complementary behaviors. Mutation tries to lead the search to unvisited search regions, changing significantly the population genes in order to find the global optimal solution. On the other hand, crossover operations tend to indicate the optimal local solution, and usually are responsible for the convergence of population genes during the evolutionary process.

6 Other genetic operators can be found in the literature (Michalewicz, 1994). Most of them are variants of crossover and mutation, adapted for specific types of problems. SEGMENTATION METHOD The proposed method relies on two heuristics: the liver parenchyma is roughly homogeneous, and liver veins are mainly inside the liver, as well as liver nodules. The impact of these heuristics on cases where peripheral nodules and veins are present is discussed in details in the section on experimental results. The input images are not pre-processed. Pre-processing images can possibly change important information of boundaries. The segmentation method is able to deal with discontinuities on the border using the curvature term, and possibly it can achieve more accurate results (that means, more precise boundaries) by not changing, for example with a smoothing procedure, the input images. Firstly an initial solution is defined in just one slice, to be used as a sample of the liver. This initial solution is expected to contain great part of the liver, but does not need to be accurate, as it will be later deformed too. Then an iterative process takes place, both upwards and downwards through the whole exam, in which the liver is segmented at each slice sequentially. In this process the initial solution of a new slice to be processed is defined as the previously segmented liver result of the adjacent slice. The same procedure is applied in each slice. It receives as input data, the image slice and an initial solution defined as the result obtained in the adjacent slice. This initial solution is then deformed towards the liver boundaries using an approach based on level sets, with some of the parameters being adjusted at each slice. In this work, the advection term was not used, because it is closely related to boundary features and the boundaries of the liver cannot be robustly detected as the liver usually share close intensity values with some of its anatomical neighboring structures. Considering that the liver s parenchyma is roughly homogeneous it is possible to define two thresholds TL and TH that define a gray level range where most of parenchyma voxels lie on (see Figure 3). To define the propagation term one must define the speed image and the propagation weight β. We define the speed image as a function based on the two automatically defined thresholds TL and TH (TL<TH) and the input image g(x). This function is expressed in equation 2, and depicted in Figure 2: P(x)= g(x) TL if g(x) < (TH+TL)/2 TH g(x) otherwise (2)

7 Figure 2: P(x) definition. The propagation P(x) in equation above is positive when the pixel intensity lies inside the range [TL,TH] and negative when it is outside the range. It implies that the surface expands in regions where pixel values are inside the range, and shrinks otherwise. It can be further observed in equation 2 that the surface s evolving speed decreases or even becomes zero, in regions where pixel values are close to TL and TH. In this sense, considering that TL and TH are values close to the ones observed on the liver boundaries, the surface evolution stops when achieves the liver boundaries. Therefore, the choice of proper values for TL and TH is crucial. We propose a simple algorithm to set up TL and TH, relying on the assumption that the voxel intensities inside the liver follow a Gaussian shaped distribution. Firstly the intensity histogram of the region inside the given initial solution at the respective iteration step is computed. Then a Gaussian curve is fitted to the histogram, using the well known Levenberg-Marquadt non-linear minimization estimator (Levenberg, 1944; Marquadt 1963), and two thresholds TL and TH are computed as the values where the Gaussian curve achieves two pre-defined values GL and GH, as depicted in Figure 3. In this model, the grey level range [TL,TH] is meant to represent the liver parenchyma. The values of GL and GH are parameters that must be properly defined during the adaptation of parameters. Figure 3: Gaussian thresholds estimation.

8 The mean curvature term is defined by the spatial modifier Z, the mean curvature К and the weight γ. The spatial modifier Z was defined like P, in such a way that the importance of the mean curvature К decreases when close to the boundaries. This avoids that boundary details are missed by smoothing too much the surface on boundaries of the segmented object. Once the values for GL, GH, β and γ are selected, the level sets process can start at a given slice, deforming the initial solution towards the liver boundaries until the convergence criterion is achieved (minimum RMS). The level sets segmentation procedure confine the voxels inside the liver parenchyma, since their intensities meet the estimated Gaussian distribution, but nodules and veins, which normally appear respectively as darker and brighter regions are not captured. This problem is partially eliminated by the use of a fill-holes morphologic algorithm (Soille, 2003), that merges to the result of the preceding step veins and nodules that are totally inside the liver. However the problem persists when a nodule or vein is at the periphery, when they usually do not appear in the final result. This algorithm is applied sequentially for each slice in the exam, and ends when it achieves the first and last slices, or when an initial segment is vanished by the level sets algorithm in a given slice, indicating a limit of the liver. PARAMETERS ADAPTATION Processing Scheme There are different methods for tuning parameters and most of them are based on pre-defined heuristics. These heuristics usually include the definition of some rules and values based on trial and error, experimentally. Other methods of optimization are based on non-linear/linear minimization algorithms, and whenever using these methods one must be careful with problems of local minima. GA is an automatic method for optimization, well known by their capacity of dealing with local minima. In the context of machine learning, GAs use information delivered by a specialist (in this work the specialist gives the first segment as sample), to learn by example part of the model, using the learnt information to deal with similar situations. For these reasons GAs were chosen for estimation of parameters in this work. The use of GAs to estimate optimal parameters for different methods, whenever the complexity of the function does not allow a heuristic based definition, is increasing. Some approaches for different image segmentation methods were already published (e.g. Costa et al, 2008) and the results are encouraging. In the devised GA each individual consists of a set of segmentation parameter values; each parameter is represented by a gene. The fitness of each solution (individual) is calculated by comparing the segmentation produced by the solution with the reference segmentation, using five different measures of performance (Heimann et al, 2007) that will be further presented. As described in the previous section, the segmentation method has a set of five parameters to be optimized: 1. Minimum RMS: the convergence criterion of level sets function, defined in terms of the root mean squared (RMS) change in the level set function. 2. GL: Gaussian low factor 3. GH: Gaussian high factor

9 4. β: level set propagation weight 5. γ: level set mean curvature weight Initially a set of solutions (initial population) is generated by computing each parameter value (genes) randomly, in given ranges. As the evolutionary process advances, the best solutions (fittest individuals) are selected and new solutions (generations) are created from them (reproduction). Figure 4: Fitness evaluation. The selection of individuals for reproduction takes the fitness values into consideration, so that the fittest individuals have a larger probability of being selected. Furthermore, the best individuals from one generation are kept in the next generation. The evolutionary process stops after a fixed number of generations, and the gene values of the fittest individual are taken as the final (adapted) segmentation parameter values. Reproduction procedure As stated before, the initial population is created by setting random values for the genes of each individual. After fitness evaluation a new population is created by replacing the Q worst individuals of the prior population, being Q a positive integer value smaller than the population size. The new individuals are created by genetic operations over selected individuals of the prior population. The selection of individuals is done by a roulette mechanism, which takes into consideration normalized fitness values (Davis, 1990). The following genetic operators were used (Davis, 1990; Michalewicz, 1994). One point crossover: two individuals exchange genes; arithmetic crossover: a linear combination of a set of genes of two individuals is performed; mutation: the value of a gene is substituted by a random value.

10 The selection of the reproduction operation is also done by a roulette mechanism, considering a predefined probability value for each operator. To help preventing convergence to local minima, the operators application probabilities are interpolated during the evolution process (Davis, 1990), decreasing crossover probability while increasing mutation probabilities. The mutation and crossover rates are changed through the evolution process linearly, using the initial and final rates and the number of generations provided. In each generation the best individuals are kept to the next generation, according to a steady state rate. This rate specifies the amount of individuals that will be kept to the next generation, guiding the generation to good solutions regions in the search space. It is also possible to make more than one experiment in sequel. In that configuration, the best individuals of one experiment are kept to the next experiment, where they are used as seeds, guiding the following experiment to good solutions. In this work, the GA was configured experimentally as the following: Number of generations = 30; Population Size = 30; Initial crossover rate = 0.8; Final crossover rate = 0.65; Initial mutation rate = 0.1; Final mutation rate = 0.8; Initial Steady State rate = 0.8; Final Steady State rate = 0.2; Number of sequenced experiments = 2; Rate of seed from the first experiment to the second = 0.1; Fitness evaluation The fitness function must indicate the similarity between the segmentation result and the reference segmentation. In mathematical terms, given a set of reference segments M and a parameter vector N a fitness function F(M,N) that appropriately expresses the goodness of a segmentation outcome must be defined. Once the fitness function F is chosen, the task of the GA is to search for the parameter vector Wopt, for which the value of F is minimum: W opt =arg w (min[f(m,w)]) (3) The fitness function devised in this work is defined as the mean of five scores derived from five disparity metrics between the result and the reference surfaces. These metrics try to numerically compare two different segmentations, represented by two different surfaces. In this way, they are measures of spatial differences between the two surfaces provided. The scores computed over these metrics take into account the uncertainty of a human specialist, in such a way that the score gives a measurement of success of the method if compared with a human specialist. The five disparity metrics used in this work are provided by SLiver07 conference (Heimann et al, 2007) and a deep discussion with the reasons why these five metrics were chosen is also found in Heimann et al (2007):

11 1. Volumetric overlap (VOE): is the number of voxels in the intersection of segmentation and reference, divided by the number of voxels in the union of segmentation and reference. 2. Relative absolute volume difference, in percent (RVD): 1 minus the total volume of the segmentation divided by the total volume of the reference 3. Average symmetric absolute surface distance, in millimeters (ASD): the border voxels of segmentation and reference are determined and for each voxel in these sets, the closest voxel in the other set is determined (using Euclidean distance). All these distances are stored, for border voxels from both reference and segmentation. The average of all these distances gives the averages symmetric absolute surface distance. 4. Symmetric RMS surface distance, in millimeters (RMSSD): is similar to the previous measure, but stores the squared distances between the two sets of border voxels. After averaging the squared values, the root is extracted and gives the symmetric RMS surface distance. 5. Maximum symmetric absolute surface distance, in millimeters (MSD): is similar to the previous two, but only the maximum of all voxel distances is taken instead of the average. The scores are calculated from these disparity metrics as follows. Initially, several manual segmentations are taken from different specialists for each exam. Then, the mean surface from the specialists is taken as the reference, and for each metric the mean absolute differences among the specialist results to that reference are computed. The values of these computed mean differences are taken as acceptable errors, in such a way that an algorithm with differences exactly equal to these values receives score of 75 points. If the segmentation is exactly equal to the reference, then the score is 100, and the score decreases linearly as the difference increases. These scores were defined taking into account the human variability of identifying the liver, in terms of these evaluation metrics, in such a way that high-scored results can be considerate as precise as a human specialist result. EXPERIMENTAL EVALUATION In order to evaluate the performance of the proposed method a software prototype was developed in C++/C#. The prototype uses the library that implements the GA, and the ITK library (Yoo et al, 2002) which implements the level sets framework used in the segmentation method. It is also possible to visualize the results in different slice directions, as well as the 3D organs models. It was also developed an environment where the user can specify different GA configurations, enable and disable search of parameters, and define their default values and search ranges. It is possible to define the training set, and fitness aggregation type (mean of fitness values, or minimum fitness value). In our experiment a set of five exams was used for estimation of segmentation parameter values. Then, using the optimal segmentation parameters found, the other 15 exams available on the dataset were evaluated. Both input and reference data were provided by SLiver07 conference (Heimann et al, 2007), and the parameter values found by the GA in our experiment were: 1. Minimum RMS: GL: GH: β: γ:

12 The results obtained by the presented approach are shown in Table 1, where both the metric value and its correspondent score are shown in each cell for the different evaluation metrics. We compiled our results, showing the best, worst, and mean results for all test set. It is possible to notice that some metrics such as RMSSD and MSD are more rigorous than other, but the algorithm achieved good results. According to (Heimann et al, 2007), the maximum score is equal to 100 and corresponds to a perfect match with the reference. A score equal to 75 corresponds roughly to a segmentation done manually by a specialist. For scores higher than 60 disparities from the reference segmentation are hardly detectable visually. Evaluation Best Worst Mean VOE (Score) 5.45 (78.70) (52.82) 7.35 (71.29) RVD (Score) (96.63) 8.12 (56.80) (82.27) ASD (Score) 0.76 (80.85) 3.57 (10.70) 1.35 (66.25) RMSSD (Score) 1.69 (76.46) 8.22 (0) 3.05 (58.58) MSD (Score) (77.59) (27.21) (64.72) Total Score Table 1: Liver segmentation results. The method attained a good performance in 17 of the 20 exams, in which the overall score is above 65. Through all these exams it was possible to visually attest that only small errors were present, when detectable. Figure 5 shows the result obtained in the best result, with overall score of It is possible to notice that the liver boundaries are accurately defined. Figure 5: Best result obtained: axial view; coronal view; sagittal view. In 3 exams, though, the results contain some significant errors that can be verified visually. The exam with the lowest score among all tested exams is depicted in Figure 6, with an overall score of In this exam the liver has a large nodule, which causes a leak of the segmented region towards adjacent darker structures. This can be easily explained by the adopted heuristic that fits a Gaussian to characterize the liver parenchyma. When a nodule with size comparable to the liver parenchyma is present, naturally this approach is susceptible to imperfections, as the histogram will not be Gaussian shaped. However, considering the size of the nodule, the result is

13 acceptable, and the final contour defines properly great part of the liver. This robust behavior can be related to the way TL and TH are computed, which is performed at each slice, having their values adapted to different regions of the liver. Figure 6: Leak caused by a big nodule: axial view; coronal view; sagittal view. The result of another exam with low score (46.18) has a single major error, caused by a peripheral nodule not classified as liver. This is again due to the adopted heuristic that considers a single Gaussian curve to model liver tissue. Once the nodule in this exam appears much darker than the liver parenchyma, its voxel intensities lie outside the range [TL,TH] defined by the Gaussian fit. As the nodule is at the periphery, the morphological fill holes filter was not able to correct the error, and therefore the nodule region was not included in the final result. Therefore, important metrics such as RMSSD and MSD, which take into account the maximum differences between the reference and the resulting surfaces, generated low scores. In the last exam with poor performance, we see a leak towards the mediastinum, with overall score of This leak towards the mediastinum is commonly observed in many different approaches, mainly because the mediastinum share some intensity values with the liver and is anatomically close to the liver in a wide extent. This makes it hard to define properly the boundary between them in some cases. Some post processing can be done to avoid it, once the mediastinum has a known relative position to the liver. Another possibility would be the inclusion of the advection term to create a barrier between the liver and the mediastinum. This could be manually defined by the operator using a proper user interface. Our results can be easily compared with many other approaches, since the data and evaluation metrics were obtained from the website of the liver segmentation competition held in the Sliver07 conference, and the results of other approaches are also available there. If compared with other automatic and semi-automatic methods presented in the conference s website, all of them using the same dataset used in this work, our method had a good performance, being ranked as the 6 th best method among 26, as seen in Table 1. In this table it is possible to observe that many methods use deformable models approaches, and the estimation of parameters is mainly defined experimentally, or not approached. Some of the methods seem to be repeated, but actually they are versions of the same basic algorithm with small modifications. Authors Score Segmentation Method Estimation of Parameters Kainmueller et al Combination of a constrained free-form and statistical deformable model Analysis of histogram

14 Dawant et al Level sets Not available Lee et al Level sets Defined experimentally Wimmer et al Level sets Not available Wimmer et al Level sets Not available Oliveira et al Level sets Evolutionary Algorithm Freiman et al Bayesian classification and Active Not available contour Heimann et al Statistical shape model Evolutionary Algorithm Saddi et al Statistical shape model Not available Schmidt et al Explicit knowledge model Not available Chi et al Gradient vector field based active contour Defined experimentally Seghers et al Active Shape Model Defined experimentally Livseg D active contours plus manually Not available defined constraints Livseg D active contours Not available Campadelli et al Fast marching algorithm Defined experimentally Furukawa et al Level sets Not available Schmidt et al Explicit knowledge model Not available Rikxoort et al K nearest neighbor Defined experimentally Slagmolen et al Atlas plus non-rigid registration Quasi Newton optimizer and atlas based definition Beichel et al Graph cut Defined experimentally Campadelli et al fast marching algorithm Defined experimentally Susomboon et al Region growing plus texture based training Expectation Maximization Susomboon et al Region growing plus texture based training Expectation Maximization Susomboon et al Region growing plus texture based training Expectation Maximization Susomboon et al Region growing plus texture based training Expectation Maximization Susomboon et al Region growing plus texture based training Expectation Maximization Table 1: Comparison of different methods for liver segmentation - data available online on

15 The performance of the method in terms of computational processing time is also reasonable. The estimation of parameters took a long time (about one week), but need to be performed only once. Once the parameters were estimated, the method is quite fast. The computation time depends on the exam size: a 210 slices exam, takes 204 seconds to segment the liver, while a 110 slices exam takes 105 seconds. CONCLUSIONS We have presented a method to segment the liver based on level sets, using an evolutionary method to estimate its parameters. The use of level sets, a consolidate approach to segment medical images, achieved good a performance in the tested exams, and the evolutionary approach based on GAs to estimate segmentation parameters, has been shown robust in our experiments. The use of GA provided an automatic method for learning important information from the specialist, by adjusting the model according to the examples presented during the evolution process. This permits the provided methodology to deal with the inherent subjectivity present in the process of segmenting organs in computer tomography. The method performed well for most CT exams of the available dataset. The evidence that confirms the hypothesis that the liver parenchyma histogram is usually Gaussian shaped can be provided by the final results achieved. Yet, it some limitations became apparent, especially when there are nodules or veins close to the liver s periphery, and also in the presence of nodules with volume similar to the liver parenchyma. These cases indicate directions for the continuation of this research aiming at a further refinement of the proposed method. FUTURE RESEARCH DIRECTIONS For further developments, it is possible to mention that the method can be applied to segment other organs beside the liver, especially assuming that the organ is roughly homogeneous (which means that the organ voxels intensity value histogram is possibly Gaussian shaped) and continuous in the slice direction. The evolutionary approach can be also used to estimate segmentation parameters for other organs, if a proper training set with some reference segmentations of the organ is available. Other suggestions for further research would be a better modeling to build the speed image considering also the information of liver internal structures, such as vessels and nodules. Another possibility would be use the advection term to suppress or reinforce some specific barriers, which could be used to avoid leaking and also enable the inclusion of peripheral nodules and veins in the final result. REFERENCES Costa, G.A.O.P., Feitosa, R.Q., Cazes T.B., & Feijó, B. (2008). Genetic Adaptation of Segmentation Parameters. In T.Blaschke, S. Lang, & G. Hay (Eds.) Object-Based Image Analysis: Spatial concepts for knowledge-driven remote sensing applications ( ). New York: Springer-Verlag. Davis L. (1990). Handbook of Genetic Algorithms. New York: Van Nostrand Reinhold Company.

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