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Use of chaos concept in medical image segmentation Sarada Prasad Dakua
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a
Qatar Science & Technology Park, Qatar Robotic Surgery Center, Qatar Foundation, Education City, P.B. 210000, Doha, Qatar
To cite this article: Sarada Prasad Dakua (2013): Use of chaos concept in medical image segmentation, Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 1:1, 28-36 To link to this article: http://dx.doi.org/10.1080/21681163.2013.765709
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Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2013 Vol. 1, No. 1, 28–36, http://dx.doi.org/10.1080/21681163.2013.765709
Use of chaos concept in medical image segmentation Sarada Prasad Dakua* Qatar Science & Technology Park, Qatar Robotic Surgery Center, Qatar Foundation, Education City, P.B. 210000, Doha, Qatar Despite its long track record, segmentation in medical image computing still remains an active field of research, largely due to the complexities of in-vivo anatomical structures, cross-subject and cross-modality variations. Clinically, it has many benefits for effective patient management, both in terms of pre-operative planning and post-operative assessment of the efficacy of therapeutic procedures. Research efforts are focused on novel, clinician friendly, robust and fast segmentation methodologies. In this paper, we present a novel algorithm for efficient segmentation based on Chaotic theory; the preliminary results show the potential of the proposed technique.
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Keywords: computed tomography; magnetic resonance; image segmentation; chaos theory
1.
Introduction
The study of shape and motion of body organs, say heart or liver, is important because most of the diseases associated with these organs are believed to be strongly correlated with their shape and motion. The image segmentation is known to provide adequate information about the shape and size of an object. However, if we consider manual segmentation, it is not only a tedious and time-consuming process but also an inaccurate one, it is therefore desirable to use algorithms that are accurate and require little user interaction. Cardiac magnetic resonance (CMR) imaging is a leading modality for imaging the heart and clinical diagnosis due to some of its remarkable advantages (Frangi et al. 2001). Segmentation of left ventricle (LV) in CMR images, obtained from patients having serious ischaemia, is difficult. In this type of images, LV is nearly indistinguishable from the surrounding muscles to the naked eye. The epicardium is comparatively difficult than endocardium to extract due to small variation in intensity in the region of myocardium. Furthermore, the liver images with big tumours from computed tomography (CT) are the other potential examples of this category. In some cases, the low contrast and presence of big tumours in liver images make the segmentation process even more difficult due to the non-homogeneity nature of the object. A rich tradition of work (Dakua and Sahambi 2009) in image segmentation has focused on the establishment of appropriate image (object) models. Because of the space constraint, we restrict ourselves to only a few of them. Although level set methods gained tremendous popularity, still some problems like computational complexity and re-initialisation (Paragios 2003; Li et al. 2005) of the zero level set
*Email:
[email protected] q 2013 Taylor & Francis
exist. In the early level set methods, the computation is carried out on the entire domain making the computation slow. Narrow band level set methods (Adalsteinsson and Sethian 1995) restrict the computation to a narrow band around the zero level set, but it does not reduce the computational cost to a reasonable limit (Gonzalez et al. 2004). Active contours without edges (Chan and Vese 2001) are useful in detecting the object edge irrespective of the initial contour placement. At the same time, mean intensities of all the regions must be different for a successful segmentation. Atlas-guided approaches have been applied mainly in MR brain imaging; transfer of labels with segmentation is one of the major advantages of these methods. They also provide a standard system for studying morphometric properties (Davatzikos et al. 1996). Even with non-linear registration methods, accurate segmentation of complex structures is still difficult due to anatomical variability. Moreover, atlas-based segmentation has been of limited use in presence of large spaceoccupying lesions (Cuadraa et al. 2006). The watershed algorithm uses mathematical morphology to partition images into homogeneous regions (Vincent and Soille 1991). This method can suffer from over-segmentation, which occurs when the image is segmented into an unnecessarily large number of regions. Thus, watershed algorithms in medical imaging are usually followed by a post-processing step to merge separate regions that belong to the same structure (Sijbers et al. 1997). In the literature, there are also many model-based segmentation techniques (Frangi et al. 2001). One substantial limitation of such methods is that their accuracy is sometimes restricted either
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Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization by the uncertainties in image content or by the intrinsic properties of the model itself, such as prior shape constraints (Qazi et al. 2010). A fast and semi-automatic algorithm proposed in Grady (2006) is based on Random Walk. It does not suffer from the ‘small cut’ problem and extends naturally to an arbitrary number of labels; however, it is sensitive to initial seeds placement. In this way, the list of methods, their merits and demerits continue. Without doubt, graph-based methods have advanced our understanding of image segmentation and have successfully been employed since sometime without heavy reliance on explicitly learned/encoded priors. To have a fair idea of the literature, we include here some of the standard methods. Intelligent scissors is a boundarybased interactive method that computes minimum-cost path between user-specified boundary points (Mortensen and Barrett 1998). It treats each pixel as a graph node and uses shortest path graph algorithms for boundary calculation. However, this is unable to integrate any regional bias naturally, which is overcome by the Graph Cut method as followed. Graph Cut (Boykov and Jolly 2001) is a combinatorial optimisation technique; for the case of two labels [i.e. object and background (BG)], the globally optimal pixel labelling (with respect to defined cost function) can be efficiently computed by maxflow/ min-cut algorithms. Given user-specified object and BG seed pixels, the rest of the pixels are labelled automatically. It minimises an energy function consisting of a data term [computed using colour likelihoods of foreground (FG) and BG] and a spatial coherency term. The latter term is the length of the boundary modulated with the contrast in the image; therefore, minimising the energy with this term has a bias towards shorter boundaries. In particular, it is hard for the Graph Cut approach to segment thin elongated structures. GrabCut (GC) (Rother et al. 2004) extends Graph Cut by introducing iterative segmentation scheme that uses graph cut for intermediate steps. The user draws rectangle around the object of interest; this gives the first approximation of the final object/BG labelling. Then, each iteration step gathers colour statistics according to current segmentation, re-weights the image graph and applies graph cut to compute new refined segmentation. In this method, the colour of the FG and/or BG is known a priori to be described by some parametric distribution (a mixture of the Gaussians). If the parameters of these distributions are allowed to change, such a non-local prior depending on the segment as a whole becomes very hard to express with the local edge weights. There are also other graph-based image segmentation methods such as Random Walk (as described in the above paragraph) in the literature and the list continues. However, most of the graph-based methods do exhibit certain limitations; the bottom line to get a generic method for medical image segmentation is still difficult. This is because medical images have their own unique properties. In many cases, the objects to be segmented are very different
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in their structure and appearance from the objects that are common in photo editing. Probably that is why much research effort is being applied for developing specific and efficient segmentation methods for the medical images domain. In this work, we present a Chaotic theory-based 3D semi-automatic segmentation algorithm for medical images such as liver CT that is typically based on Graph theory concept; the preliminary results show the potential of the proposed technique.
2.
Method and materials
A French mathematician Henri Poincare first developed this chaotic model (CM) (David 1995) by observing a significant deviation in the output if the input is varied even slightly. In his ‘three body system’ the object follows a different path at a slight change in initial condition. Sixty years later, this kind of divergent behaviour was revisited by Edward Lorenz with a set of three equations as follows: dx ¼ X_ ¼ s ðy 2 xÞ; dt dz _ ¼ Z ¼ xy 2 bz; dt
dy _ ¼ Y ¼ ðrx 2 y 2 xzÞ; dt ð1Þ
where t (x, y, z), s, r and b denote time, dependent variables, Prandtl number, Rayleigh number and width – height ratio, respectively. Lorentz has found the values of s and b as 10 and 8=3, respectively as the best representation in his experiment; we also persist with his defined values in this work.
2.1 Exposition of chaos concept for image segmentation We know that any set of pixels which is not separated by boundaries is connected and each maximal region of connected pixels is called a connected component, then the set of connected components partitions an image into segments. In this context, let ›s be a neighbourhood system (here it is 4) and cðsÞ be the set of neighbours that is connected to the point s, for all s and k, the set cðsÞ must have the properties cðsÞ , ›s and k [ cðsÞ , s [ cðkÞ. This correspondence presents a graph-based method for image segmentation where the candidate image is treated as a graph (G) or network; all graph edges are assigned some nonnegative weight (cost) by a Gaussian weighting function 2 we ¼ e2jðxi 2xj Þ (Zhu et al. 2003), where xi is the pixel intensity at node i. The graph is decomposed into maximal strongly connected components; the components are considered as vertices of the graph (Brualdi and Ryser 1991). We look at the strongly connected components corresponding to vertices in the graph and call them leading strong connected components (LSCCs). Since there are usually more than one such LSCC due to the in-homogeneity
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S.P. Dakua
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(b)
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(f)
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Figure 1. Demonstration of chaos-based image segmentation: (a) seed placement, (b) probability map due to BG seed, (c) probability map due to FG seed, (d) label map, (e) map after gradient operation and (f) segmented image.
nature of a typical medical image, it is not possible to synchronise the entire network. Here, the individual systems meaning different objects present in an image are considered chaotic. Therefore, in order to make a specific individual system stable (since we are interested in one object, say liver or heart), it is required to impose some kind of constraints (by placing seed points on the image as shown in Figure 1), called boundary conditions. Here comes the Chaotic theory that can be applied in image segmentation ensuring deterministic convergence by keeping initial conditions constant. The scenario is analogous to ‘iron particles moving randomly in a cell and a strong magnet is suddenly placed on its center’. Our objective is to find the probabilities of the particles reaching the magnet.
The eigenvalues of the above matrix is obtained by solving the following matrix:
2.2
A point is stable when its eigenvalues are all negative and all three eigenvalues g1 ; g2 ; g3 will be negative when
Stable points
The stable points define the stability nature of the system. In order to find these points, its equilibria need to be _ obtained (Boccaletti et al. 2000). For this, we need to set X, _Y and Z_ to zero. The equilibrium points can be obtained as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0; 0; 0Þ, S1 ¼ffi ð2 bðr 2ffi 1Þ; 2 bðr 2 1Þ; r 2 1Þ and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2 ¼ ð bðr 2 1Þ; bðr 2 1Þ; r 2 1Þ. Since s and b are fixed, r is the only term of concern. Some cases may be considered to examine the suitable value of r to have a deterministic system, for example 0 , r , 1, r ¼ 1 and r . 1. It is observed that when 0 , r , 1 and r ¼ 1 there is no chaos. When r . 1, the stable points S1 and S2 are checked by linearising the Lorentz equation and finding its eigenvalues. Linearising the system at a point, say Y; ZÞ, using the Jacobian matrix we get, ðX; 2 3 2 2s X_ 6 7 6 6 Y_ 7 ¼ 6 r 2 Z 4 5 4 Y Z_
s 21 X
32 3 0 X 76 7 6 7 X 7 54 Y 5: 2b Y
ð2Þ
2
2s 2 l
6 det6 4 r2Z Y
s
0
21 2 l
X
X
2b 2 l
3 7 7 ¼ 0: 5
ð3Þ
The eigenvalues are found ffi to be l1 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 ¼ ð2s 2 1 ^ ðs þ 1Þ2 þ 4sðr 2 1ÞÞ=2; l3 ¼ 2b. Y; ZÞ to be either S1 or S2 and plug them If we consider ðX; into (3), the following eigenvalues can be obtained:
g 3 þ ðb þ s þ 1Þg 2 þ ðs þ rÞbg þ ð1 2 rÞ2sb ¼ 0: ð4Þ
r,
s ðs þ b þ 3Þ ¼ rc : s2b21
ð5Þ
Substituting the pre-defined values of s and b, one can get r < 24:63; these values are used to empirically determine j of we by following j ¼ k þ ð0:67r 0:25 =½1 þ 1:48s 0:56 4=9 Þ (Incropera and DeWitt 2000; Sijbers and Dekker 2004), k being a constant. In the context with computation, the points S1 and S2 are now stable, the process is periodic and easy to solve. From (2), the Lorentz system can be formulated as
1 d{x 2 ðtÞ þ y 2 ðtÞ 2 ðr þ sÞ2 } 2 dt ( ) ð6Þ rþs 2 rþs 2 2 2 ¼ 2 sx ð t Þ þ y ð t Þ þ þb : 2 2
Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization It defines the initial paths of all the particles. The probability of a particle to reach the magnet due to its magnetic force depends on its position, resistance along the trajectory and other constraints. In context with image, where only one variable (grey-scale/pixel intensity value) is being considered, upon substituting the corresponding values of the parameters such as s, r and b, the above equation can be generalised in Dirichlet format (Courant and Hilbert 1989) as ð 2 D½v ¼ j7vj dV; ð7Þ
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V
where v is the field (twice differentiable boundary condition on dV), and V is the region (domain V of Rn ). The solution to (7) has to satisfy the boundary conditions imposed by the initial seeds for image segmentation (described in Section 2.4). On the other hand, (6) is a noncompact group and needs a special function approach (like Laplace transform) rather than a distribution valued concept to get solved. However, the classical Laplace transform still rests essentially upon the integral relation between Legendre functions (Mopsik 1994) ð1
Pj ðxÞQl ðxÞ dx ¼ ½ðl 2 jÞðl þ j þ 1Þ21 ;
ð8Þ
1
where l; j [ Z; ðl 2 jÞ . 0 and ðl þ jÞ . 21; its generalisations (Pl is the Legendre polynomial) become the representation functions of the Lorentz group and the Ql the so-called second-kind functions. There are some limitations to this approach viz., its asymptotic behaviour and basis dependency. The classical Laplace transformation on the real line converges for all exponentially bounded functions; and it gains a large measure of its usefulness from the fact that when the contour of integration in the inversion formula is pulled back to the left half plane, its contribution can ultimately be ignored to the integral when compared with those of the singularities it has crossed. Such is not the case in case of Legendre Pl functions since these are ill behaved as l ! ^1. Therefore, the Legendre transform pair should be arranged to have a convergent formula. The canonical method of decomposing the quasi-regular representation of Lorentz group into its irreducible components has been given by Gelfand and Graev (Meinolfa 1991). It consists of passing to the transform Ð ^ ¼ f ðxÞdðux 2 1Þ du, where du is the space functions fðxÞ usual invariant measure upon the two sheeted hyperboloid, and then carrying out a Fourier transform parallel to the generator of the cone: ~ lÞ ¼ fðx;
ð1 21
31
cone by the following transforms ~ þ ; lÞ ¼ fðx
ð1 0
~ 2 ; lÞ ¼ fðx
^ a xÞe2la da; fðe
ð1
ð10Þ ^ 2a xÞe2ðlþ1Þa da: fðe
0
Both of these integrals converge for l sufficiently positive defining analytic functions of l. In the context with segmentation, image pixels are treated as iron particles and we have to find the probabilities with which a pixel reaches the initial seed point. The strong correlation among object pixels by various elements is described in Section 2.3.
2.3 Theoretical elements for strong correlation Synchronisation is an alternative measure of a system showing chaotic behaviour, where algebraic connectivity is a key measure of network’s connectedness (Elson et al. 1998). A connected graph should be more responsible towards synchronisation since there is more influence among nodes (Wu 2006). For a graph G with algebraic connectivity a, if a . 0 , G is strongly connected. The algebraic connectivity of the graph is defined as minx–0;Pxi ¼0 ðx T Lx=x T xÞ, where L is the Laplacian matrix; the higher the algebraic connectivity of the underlying network the easier it is to synchronise the network. In other words, the strong correlation among nodes of the graph is largely based on the characteristics of Laplacian matrix (Wu and Chua 1995). These facts establish the role and importance of Laplacian matrix in chaos-based image segmentation. An attempt has been made to demonstrate the current method for image segmentation through Figure 1, there are two seed points (one for FG and one for BG) resulting two sets of probabilities. The method is based on Graph theory and the details are given in Section 2.4.
2.4
Image segmentation
The input image I is represented as a graph, G ¼ ðV; EÞ; where V and E represent the set of vertices and edges, respectively; pixels are the nodes of the graph. A weight is associated with each edge based on some property of the pixels that it connects, such as their image intensities. . Input: the seed points are set on different labels in
the image. ^ a xÞe2la da; fðe
ð9Þ
where a is a variable. If f ðxÞ is integrable (which is the case here), it is natural to introduce the Laplace transform on the
. Laplacian matrix is built based on the computed
edge weights. The Laplacian matrix of the graph is defined as a zero row sum matrix L such that for i ¼ j and i – j, Lij ¼ 2wij and degree of the vertex, respectively; otherwise Lij ¼ 0. It can alternatively
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S.P. Dakua
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(b)
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Figure 2. (a –d) Segmented liver images, (e– h) segmented images superimposed on the ground truth images, (j) set of liver images of a subject, (j) lower part of the liver, (k) upper part of the liver and (l) 3D liver.
be defined as L ¼ D 2 A, where A is the adjacency matrix and D is the diagonal matrix. . The Laplacian matrix is partitioned, ! Lm B L¼ . Subscript m and u represent B T Lu
likelihood for marked (seeded) and unmarked (unseeded) pixels, respectively. . The linear system is set up as, Lu X u ¼ 2B T X m where the variable X u represents the set of probabilities corresponding to unseeded nodes, X m is the set of probabilities corresponding to seeded
Table I. The employed metrics for quantitative evaluation. Measure
Definition
Hausdorff distance (HD) Dice coefficient (DC) False positive ratio (FPR) False negative ratio (FNR) Sensitivity Specificity Precision Accuracy Mean error rate (MER) Intra-region (I h ) uniformity (Nazif and Levine 1984) Pratt’s figure of merit (FOM) (Pratt et al. 1978)
Minimum distance between two sets of points Quantity of overlapping of two contours Fragment of pixels incorrectly segmented Fragment of pixels incorrectly rejected from segmentation True positive/(true positive/false negative) True negative/(true negative þ false positive) True positive/(true positive þ false positive) (True positive þ true negative)/total samples (False positive þ false negative)/total samples £ 100 Index of homogeneity inside a region Segmentation accuracy
Ih
0.87 0.85 0.89 0.90 0.95 4.24 4.29 3.99 4.06 3.89 0.8490 0.8232 0.9134 0.8869 0.9516 0.8534 0.8419 0.9012 0.8879 0.9350
MER Accuracy Precision
0.8487 0.8829 0.7813 0.8321 0.7755 0.8367 0.8217 0.8921 0.8791 0.9475
Specificity Sensitivity FNR
0.0084 0.0092 0.0078 0.0079 0.0073 0.0234 0.0256 0.0193 0.0211 0.0182
FPR DC
0.8356 0.8147 0.8790 0.8643 0.9188 3.6452 3.9421 3.1589 3.4401 3.0479
HD Method
NCut (Jianbo and Malik 2000) K-Means clustering (Tong et al. 2006) Growcut (Vezhnevets and Konouchine 2005) GC (Shoudong et al. 2009) Our method
Segmentation comparison with different methods. Table 2.
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0.8009 0.8670 0.6937 0.7156 0.6244
FOM
Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization
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nodes. Lu , B and Lm correspond to the matrix decomposition of L. . The linear system is to be solved to get X u . In this way, one can get a set of probabilities for each seed point. . The most probable pixels are then marked with label numbers. For example in Figure 1, there are only two labels, i.e. blue!1 and red!2. So the image becomes a mask of ones and twos. . Finally, gradient of the image is computed, resulting nonzero values at the intensity transition. The locations of these nonzero (marked as nz in the figure) values are the object boundary pixel indices. 2.5 Method summary The image is treated as a graph; the BG constant regions are subtracted by an empirical threshold value to enhance the computation. The methodological operation is explained in Figure 1; a seed point is individually placed on both FG and BG. The probability values at each node are calculated when the seed point is placed on the FG and BG separately. The label map is built by considering the maximum of two probabilities at a node. Finally, gradient operation on the label map determines the coordinates that carry nonzero value as the desired contour coordinates.
3. Results The liver database (20 subjects) used in this experiment is obtained from Medical Image Computing and Computer Assisted Intervention (MICCAI) 2007 Grand Challenge workshop (MICCAI Grand Challenge 2007). The pixel spacing varies between 0.55 and 0.8 mm, the inter-slice distance from 1 to 3 mm. The resolution is 512 £ 512 voxels in-plane. We have qualitatively and quantitatively tested the proposed method on each subject of the above dataset. The qualitative results are shown in Figure 2. The segmented images are superimposed on the ground truth images; a complete match in each case may be observed reflecting the method accuracy. The volumetric liver (as shown in Figure 1) is constructed by properly accompanying the 2D contours of all slices. The quantitative measures used in this experiment are provided in Table 1 and the values are given graphically in Figure 3. Furthermore, we have used the MICCAI standard measures to examine the current method’s precision and achieved the results as follows. Our method has achieved a mean of overlap error (%), volumetric difference (mm), average surface distance (mm), root mean square surface distance (mm) of 9.6, 2 4.7, 2.2 and 3.7, respectively. As reference, one of the best methods of MICCAI 2007 challenge achieved a mean of respective measures as 7, 2 3.6, 1.1 and 2.3, respectively which
Figure 3.
0
0
5
Spec.(GC) Prec.(GC) Spec.(CM) Prec.(CM)
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DC (GC) Sens.(GC) DC(CM) Sens.(CM)
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Values of (a) HD, (b) DC and sensitivity, (c) FPR, (d) FNR, (e) specificity and precision, (f) accuracy and FOM, (g) MER and (h) I h .
0.7
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Value
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Value
15
15
20
20
34 S.P. Dakua
Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization shows potential of the method. In addition to these, a comparison with a few standard graph-based methods is given in Table 2 that also supplements the merit of the proposed method over others. For a single image, it takes nearly 3 s to complete the operation on a system with 2GB RAM core2duo processor.
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4.
Conclusions
The radiologists usually prefer simple and efficient segmentation methods that provide clear shape description of the object. There are many 3D segmentation techniques available to date; but if, at some instance, it is required to move to a specific slice based on the segmentation output, most of these methods would turn futile. Volumetric object reconstruction from segmented slices is an ameliorate option for this purpose. In this scenario, this paper has presented a 3D semi-automatic graph-based segmentation algorithm based on Chaotic theory which is deterministic and time efficient. In future, we intend to explore its behaviour on various subjects and different modalities.
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