Entropy-Based Automatic Segmentation and Extraction of Tumors from ...

Entropy-Based Automatic Segmentation and Extraction of Tumors from Brain MRI Images Maria De Marsico1(B) , Michele Nappi2 , and Daniel Riccio3 1

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Sapienza University of Rome, Rome, Italy [email protected] 2 University of Salerno, Fisciano, Italy [email protected] University of Naples Federico II, Naples, Italy [email protected]

Abstract. We present a method for automatic segmentation and tumor extraction for brain MRI images. The method does not require preliminary training, and uses an extended concept of image entropy. The latter is computed over gray levels (which are in fixed number) instead of single pixels. The obtained measure can be assumed as a measure of homogeneity/similarity of regions, and therefore be the base for image segmentation. Being independent from the number of pixels in the image, its computation is highly scalable with respect to image resolution. As a matter of fact, it is always carried out over the fixed number of 256 gray levels. Moreover, being region-based rather than pixel-based, the measure is also more robust to slight differences in orientation.

Keywords: Entropy

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· MRI segmentation · Tumor extraction

Introduction

This work proposes an automatic approach to brain tumor segmentation in Magnetic Resonance Images (MRI). Most present methods require supervised training. An example can be found in [12] which exploits SVM approach. On the contrary, we propose an unsupervised technique, along the same line of [9]. The lack of a training phase can be considered as an advantage: methods with this characteristic are generally robust to some extent to changes in capture technologies, possibly causing consequent changes in score distributions. Of course, medical imaging can be an aid, but cannot substitute in any case the humanin-the-loop approach . What can be reasonably avoided is to engage physicians with cases which are negative with very high confidence. Luckily enough, these are the majority of cases. This relieves humans from the most burden of work, allowing one to concentrate on cases marked as ambiguous as well as positive by the automatic system, therefore increasing the overall efficiency. Among brain tumors, gliomas are the most widespread kind of primary malignant brain tumor (not deriving from a metastasis process) among adults (about c Springer International Publishing Switzerland 2015  G. Azzopardi and N. Petkov (Eds.): CAIP 2015, Part II, LNCS 9257, pp. 195–206, 2015. DOI: 10.1007/978-3-319-23117-4 17

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70%). They can be classified in four World Health Organization (WHO) grades. Grades I and II (Low-Grade) can be considered semi-malignant tumors that carry a better prognosis, whereas grades III and IV tumors (High-Grade) are malignant tumors that almost certainly lead to a death of the subject [4]. We have two possible classes for input images: with a possible tumor, and with no tumor. The first one also includes images with multi-focal tumors, i.e., tumors located in two or more different regions in the image. The properties to take into account for a binary classification are those clearly distinguishing the two classes under consideration. In the case of brain, we can distinguish two main sources for prior knowledge. The first accredited one is the so called probability brain atlas, reporting the computed probability of tissues locations. The goal of such an atlas is to detect and quantify distributed patterns of deviation from normal anatomy, in a 3-D brain image from any given subject. Related algorithms analyze a sufficiently large reference population of normal scans and automatically generate color-coded probability maps of the anatomy of new subjects. A first work dealing with the design and implementation of a technique for creating a comprehensive probabilistic atlas of the human brain can be found in [16]. A version especially devoted to the lobe region is created in [10]. Examples of works using this source for prior knowledge can be found in [13] and [3]. The second widely used information is related the general sagittal symmetry of healthy brain. The underlying assumption of methods using this kind of information is that areas that break this symmetry are highly suspect as they are most likely parts of a tumor or represent anyway a kind of pathology. Examples of works based on this assumption can be found in [15], [2], and [9]. Our work develops following this approach. It is worth underlining that shape analysis is not a feasible choice in this problem, since tumors can get different shapes according to the way they spread along the fibers of the white matter. The contribution of this work is to propose the use of a revisited notion of entropy to compute a measure of the homogeneity of set of objects. This measure can underlie unsupervised clustering, and in particular, when either pixels or image regions are the objects of interest, it can provide accurate image segmentation.

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MRI Brain Images

A very short presentation of the different MRI modes seems worth, since we will see that images captured with different techniques provide different information. MRI scanners use strong magnetic fields and radio waves to form images of the body. Contrast in MR imaging can be manipulated to a much greater extent than in other imaging techniques, by varying the amount of excitation exercised by the magnetic field and the number of repetitions. Nevertheless, certain diagnostic questions require even sharper regions and this can be obtained by the application of contrast agents, each achieving a specific result. A separate sequence is obtained from each of them, and often more sequences are captured for better diagnosis accuracy (multi-contrast MRI). Present clinical practice deals with T1-weighted (T1 for short), T1-weighted contrast enhanced

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(T1c for short), T2-weighted (T2 for short), and a kind of T2-weighted denoted as Fluid-attenuated inversion recovery (FLAIR), where T1 stands for the longitudinal relaxation time, and T2 for transverse relaxation time. Obtaining T1-weighted contrast enhanced images requires the contrast agent (usually gadolinium) to be injected into the patient blood, making this kind of magnetic resonance quite invasive, while the others just require oral administration. Further details on the use of contrast agents are out of the scope of this paper, but the interested reader can find them in [14]. BRATS [11] is a very popular public database for experiments with 3D volumes, with a complete series of images captured by such contrast agents for each slice. For our experiments, we used BRATS-1 dataset with ground truth from the 2012 edition (available at http://challenge.kitware.com/midas/folder/102). Figure 4(a) shows examples of images for a single slice of a single volume from BRATS with variations obtained with different contrast agents. More details on the database can be found in the Section 6.1 below. T1 is the most commonly used sequence for structural analysis and for annotation of healthy tissues. In T1c, tumor edges appear lighter because the contrast agent concentrates there due to the perturbations of barriers of encephalic blood in the proliferation of tumor region. In this sequence, it is easy to distinguish a necrotic region and the active part of the tumor. In T2, the edema region around the tumor appears as lighter. FLAIR helps separating the edema region from the cerebro-spinal fluid (CSF). The regions identified by the different techniques are those relevant to the determination of the tumor approximate size and position. These characteristics explain why more sequences are used. It is to notice that BRATS sequences are quite well aligned, and in any case our approach is region-based rather than pixel-based, and therefore it is more robust to slight differences in orientation.

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Entropy as a Homogeneity Measure

The classical way to exploit entropy for image analysis is to evaluate the degree of randomness of image pixels. Each pixel x in an image I is considered as a symbol in the alphabet emitted by a source S. In the case of a gray scale image, the alphabet is represented by the set of 8-bit integers in the range [0 − 255]. The image histogram represents the frequency table of all such symbols. Once its values have been normalized in the range [0, 1], and according to the total number of pixels in the image, each of them represents the probability of occurrence of the corresponding symbol in I. Entropy H(I) can be therefore defined as:  255 (1) H(I) = − k=0 p(k)log2 p(k) It is trivial to notice that Equation 1 can be generalized to express in general the amount of homogeneity in a set of any kind of objects, given that the suited abstractions are devised. From this we can implement an accurate clustering algorithm which, when the objects of interest are image regions, underlies image segmentation. For readers’ sake, we first summarize the appropriate notation.

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More details associated with a number of possible applications of the same concepts to different problems can be found in [5], [6]. We consider a gallery G of objects/elements/observations (for short, objects from here on), and any similarity measure d, which associates a real scalar value to any pair of feature vectors (template) extracted from the objects of interest according to the chosen set of discriminating characteristics. In order to get preliminary definitions, we assume to compare a probe template v (extracted from a new object to classify) with a gallery template gi . We get s(v, gi ) and denote it as si . In particular, after a possible score normalization, si is a real value in the interval [0, 1]. We can then assume a probability distribution over the gallery G such that the score si can be interpreted as the probability that template v conforms to gi that is: si,v = p(v ≈ gi )

(2)

In order to compute a value for the entropy of the gallery we can take each element of the class in turn to play the role of v, and compute all intra-class similarities. After denoting as Q the number of pairs qi , qj  in G such that si,j > 0 we can write: H(G) = −

 1 qi,j ∈Q si,j log2 (si,j ) log2 (|Q|)

(3)

The above equations can be used for clustering unclassified objects of interest, either as in [6] or as in [7]. We report here the approach essentials, while further details can be found in the cited paper. The value of H(G) represents a measure of heterogeneity for a set of objects G. As such, it can be used to order all the objects in the overall gallery according to their informative power. Given G, the proposed procedure computes an all-against-all similarity matrix M and, using its elements, the value for H(G). For each object gi ∈ G, M is then used to compute the value of H(G gi ) that would be obtained by ignoring gi . The object gi , achieving the minimum difference f (G, gi ) = H(G) − H(G gi ), is selected; the matrix M is updated by deleting the i − th row and column, and the process is repeated, until all elements of G have been selected. In practice, we first select the most representative samples, i.e. those causing the lower entropy decrease. This approach progressively reduces the inhomogeneity of a set of samples. In this work we perform clustering using the ordering of objects induced by the iterative application of f (·, ·), in a way similar to the method proposed in [7]. We describe it in the following section.

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Entropy-Based Clustering to Segment MRI Images

In the specific context of this work, namely the segmentation of MRI images of the brain district with possible presence of a tumor, we use the above specification of entropy to obtain a smart and adaptive quantization of gray tones in a gray scale image. In other words, given a picture where gray levels occur, we

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want to identify representative subsets of these, so as to facilitate the partitioning of the image into regions, i.e., clusters of gray levels with similar information content. As a consequence, the objects we are dealing with are the single image pixels and their gray levels in the range [0, ..., 255]. It is important to preliminarily notice that a similarity measure achieving our goal, once defined, must be computed on a per image basis. In other words, to serve as a firm basis for clustering, the difference of any two gray levels cannot be fixed once and for all, but depends on the content of the single image I at hand. This is to be taken into consideration to interpret the following discussion. As a first option, we might trivially consider a similarity measure s inversely proportional to the absolute difference between two different gray levels. However, in our tests this produced one single cluster. The main reason is that this measure does not take into account the actual presence in the image I at hand of pixels with a certain gray level. This can be obtained by weighting each pixel with the probability that it is present in I, before computing the difference. The latter can be easily derived from the normalized histogram computed on I. However, even this solution produces anomalies (black and white with respective probabilities larger than 0 and 0 go in the same cluster). This is because now the actual difference between levels has been smoothed too much. Therefore, both difference and weighted difference must be considered in some way. A last factor to take into account is the actual topological distribution, or sparsity, of pixels with the same gray level in I. This measure can be identified as the standard deviation σ(g) of the coordinates of the pixels with a certain gray level g. Since we are computing a similarity between two gray levels, we have to derive a kind of joint sparsity index, which, given two gray levels gi and gj , can be defined as: σ(gi , gj ) =

min{σ(gi ), σ(gj )} max{σ(gi ), σ(gj )}

(4)

The final similarity can be computed as: s(gi , gj ) = 1 − ψ(|p(gi )gi − p(gj )gj |) + log(|gi − gj | + 1) + σ(gi , gj )

(5)

where ψ is a normalization function to have the weighted difference in the range [0,1], the logarithm balances the contribution of the difference, and the addition of 1 avoids the presence of a term log(0). Notice that for simplicity of notation the mention of the single image I is omitted from all equations, but it is implicit. Applying the similarity measure s(gi , gj ) to a gray scale MRI image, the plot of function f (G, gi ) assumes a typical pattern where the values associated to the gray tones, in the order they are selected by the function, are arranged along a curve with a typical coarse parabolic behavior, and where at a finer detail we can identify peaks and downslopes. We can observe that gray tones with the same degree of representativeness for image I tend to arrange themselves along a descending portion of the curve, while the peaks can be considered as the starting element of a new cluster. Figure 1 shows the iteration of the function on the x axis, and the value of f on the y axis. Each point in the plot can be labeled with the gray level that achieves the maximum of f (value on y) at the iteration

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(on the x). The Figure displays a zoomed portion to show labels. The clustering algorithm then proceeds as follows: a) create the first cluster and enter the first element; b) if the current element is on a descending portion of the curve, include it in the current cluster, otherwise create a new cluster and add that element; c) if there are more elements to cluster, return to b). After clustering procedure is complete, we compute the centroid for each cluster, and substitute it for all the cluster elements, to produce an image Ic . In certain cases, the algorithm produces a over-segmentation, for which there are clusters that could be possibly fused together. For this reason, after the clustering process, we perform a merging of clusters by taking into account consistency among neighbors. Given a cluster, we check if there are other clusters within a certain threshold t, with which to carry out a merge operation. For a cluster whose representative is the gray level gk , the algorithm tries to merge it with the clusters represented by gray levels in the interval [gk+1 , gk+t ]. If this is done for gray level gk+h , the latter becomes the starting point for the search of other clusters that can be melted, checking the interval [gk+h+1 , gk+h+t ]. For this work, we experimentally set t = 4. When two clusters respectively represented by gray levels gi and gj are merged, all the gray levels in the image I represented by gj are replaced with gi . We denote the final image as Icm . The procedure ends when no cluster can be further fused. Notice that, though the similarity measure we adopt takes topology into account, clusters are created over gray levels, and not over regions. This entails that groups of pixels of the same cluster may be located in different regions of the image. Figure 2 shows an histogram with representative gray levels after clustering step (blu/dark lines) and final levels after merging (red/light lines). Figure 3 shows in pseudo-colors an example of a MRI slice and of the obtained clustering of gray levels. Our clustering algorithm has several advantages:

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- With the use of entropy it is not necessary to specify in advance the number of clusters (as in a normal algorithm based on fuzzy techniques). - In fuzzy clustering, assignment of elements to individual clusters is made pixel by pixel, while by the method based on entropy it is made on the basis of gray levels, which are in a fixed number. The higher the resolution of the image, the more significant the saving we obtain in terms of computational cost. It is worth noticing that no knowledge about the training data is hard engineered in the final algorithm and its associated parameters. Even if the concrete instantiation of the similarity measure and the obtained values depend on the content of each single image, its definition holds in general and is is automatically derived from the image. As usual, some parameters may depend on the kind of images, but not on the specific dataset. The most important consequence is that the method can be used without major changes for the segmentation of any other kind of gray scale images. A different approach to entropy-based segmentation, applied to satellite images, can be found in [1]. However, since entropy is computed on single-pixels basis, it is not equally scalable to resolution.

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Fig. 2. An histogram with representative gray levels after clustering step (blu/dark lines) and final levels after merging (red/light lines)

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MRI Image Segmentation and Extraction of Tumor Region

In general, a tumor may not be evident in all MRI slices. As an example, in Figure 4(a) it is only visible in T1c, T2 and FLAIR images. The tumor mass is composed of a nucleus, which in most MRI images tends to the white color, and an edema region around, which instead in most MRI images tends to a gray color with lighter shade than the mass of the healthy brain. In the extraction phase of the tumor mass, the algorithm we propose checks if the original gray level image I image contains pixels with light gray level; if this is the case, it analyzes

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Fig. 3. A brain MRI slice with the result of entropy-based clustering of gray levels

the possibly different clusters in Icm obtained in the previous step that contains those gray levels: for each such cluster the algorithm checks if it is dense (its pixels are concentrated in a small space) or scattered (its pixels are distributed in a large space). Since groups of pixels of the same cluster can be far from each other in the image, one can exploit the variance of pixel coordinates to determine if a cluster is dense (small variance) or scattered (high variance). The search for the nucleus is carried out by analyzing the FLAIR, T1c and T2 images. In particular, the algorithm performs a product pixel by pixel between FLAIR and T1c images, the result of which is is compared to a threshold. Similarly we proceed with the pair of images FLAIR and T2. The results of these two products undergo a logical AND operation. As for edema, we analyze the FLAIR image, from which we extracted the whitish gray levels . From this set we eliminate all the pixels previously classified as as nucleus. The full mass is given by the union of the nucleus and edema. Figure 4 (c) shows the ground truth of a slice (left) and the result of the algorithm for the extraction of the tumor mass (right). The time needed for the segmentation of a single slice is in the order of milliseconds.

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Experimental Results

In this section we report the achieved results, after introducing the dataset used for the experiments. 6.1

BRATS-1 Dataset

The first step of the experimentation was to choose a dataset of MRI volumes that is representative of the problem and widely accepted, so as to make comparisons with the results obtained by other methods in literature. We decided to use the MRI volumes provided by Kitware / MIDAS, available at http://challenge.kitware.com/midas/folder/102. The dataset contains multi-

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contrast MRI images from 30 patients with gliomas (both low-grade, and highgrade, and both with or without resection). In addition, the images were manually annotated by experts, to provide a ground truth (or gold standard) for three different classes of targets: core, edema and complete mass . For each patient, images are available with different contrast types, namely T1, T2, FLAIR and post-gadolinium T1 (T1c). All volumes were co-registered to the T1 contrast image and interpolated to 1 mm isotropic resolution. On the contrary, the images have not been co-registered in order to place the volumes of all patients in a single reference space. The manual segmentations (file names ending in ” truth.mha”) have only three levels of intensity: 1 for edema, two for active tumor, and 0 for the rest. The dataset contains also images in which the tumor mass was simulated for 25 low-grade and 25 high-grade tumors. These simulated images faithfully follow the conventions used for real data. 6.2

Compared Method

As for comparison, we chose one of the few methods in literature that does not require a training phase, namely the one proposed by Dvorak and Bartusek [9], which uses the detection of asymmetries in MRI images and is one of the latest proposals in literature. A further set of candidates would include the methods presented at MICCAI 2012 Challenge, whose results are reported in [11]. However, the challenge was focused on segmentation of tumor and edema separately, so that the results reported in this paper cannot be compared to those described in [11]. Furthermore, our work is fully automatic and does not require any training phase, as all methods proposed in the proceedings. Training causes a twofold limitation. On the one hand, if it is run on the overall dataset at hand, it requires some inter-volume normalization that may introduce inaccuracies if the conditions are not perfectly similar. On the other hand, if it is patient-specific it introduces a further complication in the diagnostic process and makes it longer. In the method proposed in [9], the mid-sagittal plane must be detected first, to correctly align the head. Assuming that the head has been already aligned and the skull is approximately symmetric, the symmetry plane divides the volume of the detected brain into left and right halves of the same size. In the aligned volume, the method locates the asymmetric parts. The algorithm scans both halves symmetrically by a cubic block, whose size is computed from the size of the image. Normalized histograms with the same range are computed for corresponding cubic regions, left and right, and they are compared by Bhattacharya coefficient. Since this coefficient expresses similarity, its complement is used to evaluate asymmetry. Since regions overlap during block sliding, the average asymmetry is computed for each pixel. The whole cycle is repeated at different resolutions. Each cycle outputs an asymmetry map. The product of values corresponding to a particular pixel in the different maps creates a multiresolution asymmetry map. This computation is performed for each contrast volume separately. For each multi-resolution asymmetry map a threshold is set to extract pathological (asymmetric) regions. As it can be noticed, though being region-based, this method relies on a number of computations which depend on

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the number of pixels in the image and on its original resolution. Therefore, it might not scale well with respect to high resolution settings. 6.3

Evaluation of the Algorithm Based on Gray Level Entropy

The accuracy of results of segmentation and tumor extraction is measured using Dice coefficient [8]: 2|A ∩ B| DC = (6) |A| + |B| where A and B denote the ground truth and the result from the tested method respectively. DC is in the range [0, 1], with 1 identifying a perfect segmentation and 0 a completely wrong one. From the dataset described in the previous section, we report here an example of processing steps of a slice from the first volume BRATS HG0001, namely slice 100: Figure 4 shows the original multi-contrast images, the images after entropy-based quantization, and the comparison between extracted tumor mass and ground truth. The achieved DC values for nucleus, edema and total tumor mass were respectively 0.94, 0.82, and 0.86. We ran the method on the complete dataset (all volumes and all images, both real and simulated data). Table 1 reports the cumulative results, compared with those achieved by the approach by Dvorak and Bartusek discussed above. In the table, Symmetry denotes the method in [9] and Entropy denotes our method. Moreover, HG stands for highgrade and LG for low-grade cases, while cells report the average DC achieved.

Fig. 4. Slice 100 from BRATS HG0001: (a) (b) from left to right and top to bottom original images and quantized ones for respectively T1, T1c, T2, and FLAIR; (c) ground truth on the left and algorithm results on the right for tumor extraction

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Table 1. Comparison with results achieved by [9]

Real Data Simulated data

Simmetry HG Simmetry LG Entropy HG Entropy LG 0.67 ± 0.22 0.78 ± 0.10 0.68 ± 0.17 0.63 ± 0.17 0.80 ± 0.10 0.72 ± 0.05 0.73 ± 0.18 0.71 ± 0.19

According to [17], a value of DC > 0.7 indicates an excellent similarity. This statement was met for both high and low grade gliomas in both real and simulated data. Even if the results achieved by our method are sometimes lower than the compared one, though comparable, we want to underline some points. The first consideration is that our method can be further improved. As an example, we obtained some very preliminary results suggesting that a smarter merging strategy can improve the final quantization of image gray levels, e.g., when the merging phase after clustering takes into account the topology of clusters in image space to decide their fusion. The second consideration is that our method is independent from the image resolution, i.e., from the number of pixels, since clustering is performed on a gray level basis, and these are in a fixed number. Last but not least, since we do not rely on any kind of specific geometric feature of the analyzed region, e.g., symmetry, our method might be effective with other anatomical districts as well, where symmetry cannot be assumed. The authors of [9] state that their method would not work for highly rotated volumes, even though the perfect alignment is not necessary. Our method is robust to rotations, since it does not assume any reference axis.

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Conclusions and Future Work

In this work we presented a completely automatic method for tumor mass extraction from brain MRI images. It exploits a revised concept of entropy to perform gray level quantization. The advantages of this method are manifold. It is completely automatic and does not require any kind of training. The main component of its cost is related with the number of gray levels (fixed at 256) in the image rather than to the number of pixels (dramatically increasing with resolution). It is region-based, therefore it is robust to orientation. It does rely on any pre-defined anatomical feature, e.g., symmetry, therefore can be quite straightforwardly extended to any anatomical district. Further improvements are planned in the near future to make the merging process, following the gray level clustering operation, more accurate, by taking into account topological features of the regions that are going to be merged. Finally, the similarity measure between gray levels will be further investigated.

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