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A novel combined level set model for automatic MR image segmentation

A novel combined level set model for automatic MR image segmentation DE GRUYTER Current Directions in Biomedical Engineering 2020;6(3): 20203006 Jianzhang Li*, Sven Nebelung, Björn Rath, Markus Tingart and Jörg Eschweiler A novel combined level set model for automatic MR image segmentation Abstract: Medical image processing comes along with object main classes represent the typical LSM, which are edge-based segmentation, which is one of the most important tasks in that [2, 3] and region-based models [4–6]. field. Nevertheless, noise and intensity inhomogeneity in Edge-based models use edge information to detect object magnetic resonance images challenge the segmentation boundaries. This type of model has been found very sensitive procedure. The level set method has been widely used in object to image noise [1]. Region-based models detect the object by detection. The flexible integration of energy terms affords the calculating intensity in the foreground and background on the level set method to deal with variable difficulties. In this paper, image domain. In magnetic resonance (MR) images, the we introduce a novel combined level set model that mainly intensity inhomogeneity occurs from a non-uniform magnetic cooperates with an edge detector and a local region intensity field due to a variety of reasons [7]. Particularly, any intensity descriptor. The noise and intensity inhomogeneities are inhomogeneity may lead to erroneous segmentation outcomes eliminated by the local region intensity descriptor. The edge when the object has a similar intensity as the background. Two detector helps the level set model to locate the object boundaries more precisely. The proposed model was validated similar local region descriptors were proposed to overcome on synthesized images and magnetic resonance images of in such problems [5][6]. Pixels inside the selected region are vivo wrist bones. Comparing with the ground truth, the calculated based on local intensity similarity to avoid proposed method reached a Dice similarity coefficient of > inhomogeneity on global image domain. 0.99 on all image tests, while the compared segmentation By adding different energy functionals, the LSM can deal approaches failed the segmentations. The presented combined with variable image processing scenes. In this paper, we level set model can be used for the object segmentation in propose a combined level set model for object segmentation in magnetic resonance images. MR images of in vivo wrist bones. Our model consists of an Keywords: level set method, MRI, segmentation, intensity edge detector and a local intensity descriptor for the inhomogeneity segmentation purpose, and a regularization term to maintain evolving stability. Experimental results on synthesized images https://doi.org/10.1515/cdbme-2020-3006 and MR images are presented to demonstrate the power and opportunities of the proposed combined level set model. Our study aimed to provide an automatic MR image segmentation 1 Introduction approach with a high segmentation accuracy compared to existing methods. In the medical field, decomposition of an image is challenging due to images’ poor quality, e.g. occlusion, low signal, and contrast, or noises. The Level Set Method (LSM) 2 Material and Method has become a popular technique in recent years [1, 2]. Two 2.1 LSM ______ *Corresponding author: Jianzhang Li: Department of Given a moving curve 𝒞 , the core of the LSM is to Orthopaedic Surgery, RWTH Aachen University Clinic, implicitly describe 𝒞 by the zero level of a higher dimensional Pauwelsstraße 30, 52074, Aachen, Germany. E-mail: function 𝜙: 𝛺 → ℜ as: 𝒞(𝑡) = {𝑥 ∈ 𝛺 | 𝜙 (𝑥, 𝑡 ) = 0} . The jli@ukaachen.de evolving equation can be expressed in the following partial Sven Nebelung: Institute of Diagnostic and Interventional Radiology, University Hospital Düsseldorf, Düsseldorf, Germany. differential equation: Björn Rath: Department of Orthopaedic Surgery, Klinikum Wels- Grieskirchen, Wels, Austria. 𝝏𝝓 = 𝓕 |𝜵𝝓 |, (1) 𝝏𝒕 Markus Tingart, Jörg Eschweiler: Department of Orthopaedic Surgery, RWTH Aachen University Clinic, Aachen, Germany. Open Access. © 2020 Jianzhang Li* et. al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License. Jianzhang Li et. al., A novel combined level set model for automatic MR image segmentation — 2 where ∇ is the gradient operator, and ℱ the speed function that 𝓔 (𝝓 ) = 𝜶 (|𝜵𝝓 (𝒙 )| − 𝟏 ) 𝒅𝒙 , (6) 𝑹𝒆𝒈 controls the motion of the contour. By choosing the function where 𝛼 > 0 is the coefficient. ℱ, the LSM can reach different segmentation goals. A typical choice is to define the ℱ as a boundary detector [3] or a global intensity descriptor [4]. 2.5 Combined level set model and numerical implementation 2.2 Edge detection energy term As described above, the proposed combined level set Edge information is widely applied for segmentation model in this work is defined as: methods [2, 3, 8]. Here, we utilize the edge detection energy 𝓕 = 𝓔 + 𝓔 + 𝓔 . (7) 𝑪𝒐𝒎𝒃𝒊𝒏𝒆𝒅 𝑹𝒆𝒈 𝒅𝑬𝒈𝒆 𝑳𝒂𝒄𝒍𝒐 term to guide the evolving curve. The image 𝐼: 𝛺 → ℜ is a After replacing energy terms, we can obtain the speed given grey level image on the domain 𝛺 , we define a function energy functional ℱ as: 𝑔 as an edge detector by: (| ( )| ) 𝓕 = 𝜶 ∫ 𝜵𝝓 𝒙 − 𝟏 𝒅𝒙 + 𝑪𝒐𝒎𝒃𝒊𝒏𝒆𝒅 𝒈 = , (2) | | 𝟏 𝜵𝑮 ∗𝑰 ( )| | ∑ ( )| ( ) 𝝁 ∫ 𝒈𝜹 𝝓 𝜵𝝓 𝒅𝒙 + 𝝀 ∫ ∫ 𝑲 𝒙 − 𝒚 𝑰 𝒚 − 𝒊 𝝃 𝒊𝟏 𝜴 𝜴 𝜴 where 𝐺 ∗ 𝐼 is the convolved 𝐼 by a Gaussian kernel 𝐺 with ( )| ( ) 𝒇 𝒙 𝑳 𝝓 𝒅𝒚 𝒅𝒙 . (8) 𝒊 𝒊 a standard deviation 𝜎 . For a level set function 𝜙: 𝛺 → ℜ , we define the edge detection energy term ℰ (𝜙 ) by: In practical implementation, the Heaviside function 𝐻 and the Dirac delta function 𝛿 are approximated by following | | 𝓔 (𝝓 ) = 𝝁 ∫ 𝒈𝜹 (𝝓 ) 𝜵𝝓 𝒅𝒙 , (3) 𝒅𝑬𝒈𝒆 𝜴 smoothed functions 𝐻 and 𝛿 respectively, defined as: 𝟏 𝒙 𝟏 𝝅𝒙 | | 𝟏 + + 𝒔𝒊𝒏 , 𝒙 ≤ 𝝐 𝟐 𝝐 𝝅 𝝐 where 𝜇 > 0 , 𝛿(𝑥) is the Dirac delta function. 𝑯 = , (9) 𝟏, 𝒙 > 𝝐 𝟎, 𝒙 < −𝝐 2.3 The local intensity energy term and 𝟏 𝝅𝒙 | | 𝟏 + 𝒄𝒐𝒔( ) , 𝒙 ≤ 𝝐 𝝐𝟐 𝝐 𝜹 = , (10) To enhance the segmentation against image noise and 𝝐 𝟎, |𝒙 | > 𝝐 , intensity inhomogeneity in MR images, a Gaussian distribution kernel works as the local region descriptor. where 𝜖 is the coefficient and usually set to the value of 1.5. Comparing with [5], the Gaussian distribution kernel has such The minimization of ℱ concerning to 𝜙 can be property that weights the intensities around a small interested attained using standard gradient descent method: 𝝏𝝓 𝝏𝓕 area. For a given point 𝑥 ∈ 𝛺 , we define the local intensity = − , (11) 𝝏𝒕 𝝏𝝓 energy term as: 𝓔 ,𝝓 𝒇 (𝒙 ), 𝒇 (𝒙 ) = 𝝀 ∫ ∫ 𝑲 (𝒙 − 𝑳𝒂𝒄𝒍𝒐 𝟏 𝟐 𝒊𝟏 𝒊 𝝃 where 𝜕ℱ ⁄ 𝜕𝜙 is the Gâteaux derivative of the ℱ. Hence, the 𝜴 𝜴 𝒚) |𝑰 (𝒚 ) − 𝒇 (𝒙 )| 𝑳 𝝓 (𝒚 ) 𝒅𝒚 𝒅𝒙 , (4) corresponding gradient flow equation is expressed as: 𝒊 𝒊 𝝏𝝓 𝜵𝝓 𝜵𝝓 = −𝜶 𝜵 𝝓 − 𝒅𝒊𝒗 − 𝝁𝜹 (𝝓 )𝒅𝒊𝒗 𝒈 + | | | | 𝝏𝒕 𝜵𝝓 𝜵𝝓 where 𝜆 > 0 , 𝐿 (𝜙 ) = 𝐻(𝜙) , 𝐿 (𝜙 ) = 1 − 𝐻(𝜙) , 𝐻 (𝑥 ) is ( )( ) 𝜹 𝝓 𝝀 𝒆 − 𝝀 𝒆 , (12) the Heaviside function, 𝐾 the Gaussian kernel with a scale 𝝐 𝟏 𝟏 𝟐 𝟐 parameter 𝜉 > 0 , and 𝑓 (𝑥 ), 𝑓 (𝑥 ) approximate intensities in a where 𝑑𝑖𝑣(∙) is the divergence operator, 𝑒 is defined as: neighbourhood of 𝑥 inside and outside the contour 𝒞 . The 𝒆 = 𝑲 (𝒚 − ) 𝒙 |𝑰 (𝒙 ) − 𝒇 (𝒚 )| 𝒅𝒚 . (13) 𝒊 𝝃 𝒊 𝑓 (𝑥 ) is computed using the Euler-Lagrange equations: ( ) ( ) 𝑲 () 𝒙 ∗𝑳 𝝓 𝒙 𝑰 𝒙 𝝃 𝒊 We set the initial level set function with a rectangle form ( ) , 𝒊 = 𝟏, 𝟐 . (5) 𝒇 𝒙 = 𝑲 () 𝒙 ∗𝑳 𝝓 (𝒙 ) 𝝃 𝒊 as: −𝒄 , 𝒙 ∈ 𝒊𝒏𝒅𝒆𝒔𝒊 𝝓 (𝒙 ) = , (14) 𝒄 , 𝒓𝒘𝒊𝒔𝒕𝒉𝒆𝒆𝒐 2.4 Regularization energy term where 𝑐 = 2 in our approach. The overall algorithm is During the evolving procedure, the conventional LSM has summarized in Algorithm 1. been trapped by reinitialization for a long time. Li et al. We applied the proposed method on synthesized images introduced a regularization term to maintain the level set and MR images of the capitate. The MR images were acquired function as the signed distance function [8]. We apply the in vivo using the high-resolution 3D-WATSc (water selective regularization term against the reinitialization problem as: cartilage scans) sequence on a clinical 3T MRI scanner Jianzhang Li et. al., A novel combined level set model for automatic MR image segmentation — 3 (Achieva, Philips Healthcare, Best, The Netherlands). The blurred the actual object boundaries. The Dice coefficient is synthesized image was a manually generated heart shape listed in Table 1. binary image with added Gaussian noise and Salt-and-pepper noise in MATLAB (Version R2019a; The MathWorks, Inc.). Table 1: Dice similarity coefficient on image tests All images have not been preprocessed except cropped to a pixel size of 500 × 500. To comprehensively evaluate the Test group Low High DRLSE Proposed performance of the combined level set model in segmentation threshold threshold method method tasks, we compared the results of multi-threshold method and Synthesized edge-based method DRLSE in [8]. Manual segmentation 0.9699 0.9959 image performed by a radiologist with eight years of experience in MR image A 0.9313 0.9517 0.7605 0.9932 musculoskeletal radiology provided the ground truth. The MR image B 0.9135 0.6477 0.6751 0.9987 curve evolution and results evaluation were conducted in MATLAB. MR image C 0.9117 0.7107 0.6438 0.9995 Algorithm 1 3.2 MR image tests Input: 𝜙 , 𝛼 , 𝜇 , 𝜌 , 𝜆 , 𝜆 , iteration number 𝓃 Output: 𝜙 To illustrate the difficulties in segmenting MR images, the 1. Initialization: 𝜙 ↢ 𝜙 ; 2. for 𝑖 = 0 ↣ 𝓃 do combined level set model was compared with the conventional 3. update 𝑔 , 𝑓 , 𝑓 using Eqs. (2) and (7); multi-threshold method and the DRLSE method. The multi- 4. 𝜙 ↢ 𝜙 + ∆𝑡 ∆𝜙 ; threshold method was conducted in the free, open-source 5. end for platform ITK-SNAP [9]. The results were then transferred into MATLAB for evaluation. The choice of the thresholds is challenging since it is 3 Results either too low leading to an under-segmentation or too high causing an over-segmentation. There were two attempts on setting the thresholds in this section. For MR image A, B and The parameters of the combined level set model were C, the lower value at 16.3%, 7.78% and 18.19% of each assigned as 𝛼 = 0.2 , 𝜇 = 0.001 × 256 , 𝜌 = −3 , 𝜆 = 𝜆 = maximum image grey value was to make sure the least leakage 1. The iteration number of all LSM based approaches was outside the capitate bone. The higher value at 29.2%, 18.19% empirically set at a position where there was no further and 28.57% of each maximum image grey value was intended evolution of the method we put forward. The segmentation to cover all the areas inside the capitate bone. As for two other results are shown in figure 1. For evaluating the accuracy, we LSM approaches, we compared the segmentation results of our utilized the Dice similarity coefficient between obtained proposed model with the DRLSE method. The evolving of contours and ground truth. A higher value of the coefficient each model was terminated after 270, 110 and 115 iterations, means better performance. For a given set of regions 𝒢 respectively. As illustrated in figure 1, the multi-threshold representing the ground truth and 𝒮 representing the method failed on both choices. Our combined level set model segmentation, the similarity between 𝒢 and 𝒮 is expressed as: converged after 270, 110 and 115 iterations, respectively, and | | 𝟐× 𝓖∩𝓢 all desired boundaries were fully covered, while the DRLSE 𝑫𝒊𝒄𝒆 (𝓖, 𝓢 ) = . (15) |𝓖 | |𝓢 | method stopped near the initial 𝜙 due to its sensitivity to the image noise and intensity inhomogeneity. 3.1 Synthesized image tests The dark heart shape area was the object to be segmented. The proposed approach was compared with the DRLSE method. The segmentation with the proposed model stopped after 390 iterations. With the help of local intensity energy term, the evolving curve could merge and analyse the intensities within a small area around interest front points, which provided the correct contour. Nevertheless, as showed in figure 1, the segmentation of the DRLSE method failed after the same iterations due to the massive artificial noises, which Jianzhang Li et. al., A novel combined level set model for automatic MR image segmentation — 4 method, we integrated the edge-based energy term and local 4 Conclusion region-based energy term, which provided additional boundary information and suppressed massive noises and This paper proposed a combined level set model that intensity inhomogeneity, respectively. Experimental results reinforces the segmentation in MR images. In this proposed showed that the proposed combined level set model possesses a much better accuracy with all Dice similarity coefficients > 0.99. Author Statement Research funding: The research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – ES 442/1-1 and RA 2187/4-1. Conflict of interest: Authors state no conflict of interest. Informed consent: Informed consent has been obtained from all individuals included in this study. Ethical approval: The research related to human use complies with all the relevant national regulations, institutional policies and was performed in accordance with the tenets of the Helsinki Declaration, and has been approved by the authors' institutional review board (EK 171/10). References [1] Li C, Huang R, Ding Z, Gatenby JC, Metaxas DN, Gore JC et al. A Level Set Method for Image Segmentation in the Presence of Intensity Inhomogeneities With Application to MRI. IEEE transactions on image processing a publication of the IEEE Signal Processing Society 2011;20(7):2007–16. [2] Vasilevskiy A, Siddiqi K. Flux maximizing geometric flows. IEEE Trans. Pattern Anal. Machine Intell. 2002;24(12):1565–78. [3] Caselles V, Kimmel R, Sapiro G. Geodesic Active Contours. Int J Comput Vision 1997;22(1):61–79. [4] Chan TF, Vese LA. Active contours without edges. IEEE Trans. on Image Process. 2001;10(2):266–77. [5] Lankton S, Tannenbaum A, Lankton S, Tannenbaum A. Localizing region-based active contours. IEEE transactions on image processing a publication of the IEEE Signal Processing Society 2008;17(11):2029–39. [6] Li C, Kao C-Y, Gore JC, Ding Z, Gore JC. Minimization of Region-Scalable Fitting Energy for Image Segmentation. IEEE transactions on image processing a publication of the IEEE Signal Processing Society 2008;17(10):1940–9. [7] Li C, Kao C-Y, Gore JC, Ding Z. Implicit Active Contours Driven by Local Binary Fitting Energy 2007:1–7. [8] Li C, Xu C, Gui C, Fox MD. Distance Regularized Level Set Evolution and Its Application to Image Segmentation. IEEE Figure 1: Image test results. Synthesized Image: from left to right: transactions on image processing a publication of the IEEE Original sketch image defined as ground truth; Segmentation of Signal Processing Society 2010;19(12):3243–54. contours by use of the DRLSE method; Segmentation of contours [9] Yushkevich PA, Piven J, Hazlett HC, Smith RG, Ho S, Gee JC by use of proposed level set model. MR Images: from left to right: et al. User-guided 3D active contour segmentation of anatomical MR image A, B and C. From up to down: Original MR image; structures: significantly improved efficiency and reliability. Segmentation of capitate using the multi-threshold method with NeuroImage 2006;31(3):1116–28. lower threshold; Segmentation of multi- threshold method with higher threshold; Segmentation of DRLSE method; Segmentation of the proposed level set model. Red line: Segmentation contour. Yellow line: Initial zero level set. Cyan line: Ground truth. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Current Directions in Biomedical Engineering de Gruyter

A novel combined level set model for automatic MR image segmentation

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Abstract

DE GRUYTER Current Directions in Biomedical Engineering 2020;6(3): 20203006 Jianzhang Li*, Sven Nebelung, Björn Rath, Markus Tingart and Jörg Eschweiler A novel combined level set model for automatic MR image segmentation Abstract: Medical image processing comes along with object main classes represent the typical LSM, which are edge-based segmentation, which is one of the most important tasks in that [2, 3] and region-based models [4–6]. field. Nevertheless, noise and intensity inhomogeneity in Edge-based models use edge information to detect object magnetic resonance images challenge the segmentation boundaries. This type of model has been found very sensitive procedure. The level set method has been widely used in object to image noise [1]. Region-based models detect the object by detection. The flexible integration of energy terms affords the calculating intensity in the foreground and background on the level set method to deal with variable difficulties. In this paper, image domain. In magnetic resonance (MR) images, the we introduce a novel combined level set model that mainly intensity inhomogeneity occurs from a non-uniform magnetic cooperates with an edge detector and a local region intensity field due to a variety of reasons [7]. Particularly, any intensity descriptor. The noise and intensity inhomogeneities are inhomogeneity may lead to erroneous segmentation outcomes eliminated by the local region intensity descriptor. The edge when the object has a similar intensity as the background. Two detector helps the level set model to locate the object boundaries more precisely. The proposed model was validated similar local region descriptors were proposed to overcome on synthesized images and magnetic resonance images of in such problems [5][6]. Pixels inside the selected region are vivo wrist bones. Comparing with the ground truth, the calculated based on local intensity similarity to avoid proposed method reached a Dice similarity coefficient of > inhomogeneity on global image domain. 0.99 on all image tests, while the compared segmentation By adding different energy functionals, the LSM can deal approaches failed the segmentations. The presented combined with variable image processing scenes. In this paper, we level set model can be used for the object segmentation in propose a combined level set model for object segmentation in magnetic resonance images. MR images of in vivo wrist bones. Our model consists of an Keywords: level set method, MRI, segmentation, intensity edge detector and a local intensity descriptor for the inhomogeneity segmentation purpose, and a regularization term to maintain evolving stability. Experimental results on synthesized images https://doi.org/10.1515/cdbme-2020-3006 and MR images are presented to demonstrate the power and opportunities of the proposed combined level set model. Our study aimed to provide an automatic MR image segmentation 1 Introduction approach with a high segmentation accuracy compared to existing methods. In the medical field, decomposition of an image is challenging due to images’ poor quality, e.g. occlusion, low signal, and contrast, or noises. The Level Set Method (LSM) 2 Material and Method has become a popular technique in recent years [1, 2]. Two 2.1 LSM ______ *Corresponding author: Jianzhang Li: Department of Given a moving curve 𝒞 , the core of the LSM is to Orthopaedic Surgery, RWTH Aachen University Clinic, implicitly describe 𝒞 by the zero level of a higher dimensional Pauwelsstraße 30, 52074, Aachen, Germany. E-mail: function 𝜙: 𝛺 → ℜ as: 𝒞(𝑡) = {𝑥 ∈ 𝛺 | 𝜙 (𝑥, 𝑡 ) = 0} . The jli@ukaachen.de evolving equation can be expressed in the following partial Sven Nebelung: Institute of Diagnostic and Interventional Radiology, University Hospital Düsseldorf, Düsseldorf, Germany. differential equation: Björn Rath: Department of Orthopaedic Surgery, Klinikum Wels- Grieskirchen, Wels, Austria. 𝝏𝝓 = 𝓕 |𝜵𝝓 |, (1) 𝝏𝒕 Markus Tingart, Jörg Eschweiler: Department of Orthopaedic Surgery, RWTH Aachen University Clinic, Aachen, Germany. Open Access. © 2020 Jianzhang Li* et. al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License. Jianzhang Li et. al., A novel combined level set model for automatic MR image segmentation — 2 where ∇ is the gradient operator, and ℱ the speed function that 𝓔 (𝝓 ) = 𝜶 (|𝜵𝝓 (𝒙 )| − 𝟏 ) 𝒅𝒙 , (6) 𝑹𝒆𝒈 controls the motion of the contour. By choosing the function where 𝛼 > 0 is the coefficient. ℱ, the LSM can reach different segmentation goals. A typical choice is to define the ℱ as a boundary detector [3] or a global intensity descriptor [4]. 2.5 Combined level set model and numerical implementation 2.2 Edge detection energy term As described above, the proposed combined level set Edge information is widely applied for segmentation model in this work is defined as: methods [2, 3, 8]. Here, we utilize the edge detection energy 𝓕 = 𝓔 + 𝓔 + 𝓔 . (7) 𝑪𝒐𝒎𝒃𝒊𝒏𝒆𝒅 𝑹𝒆𝒈 𝒅𝑬𝒈𝒆 𝑳𝒂𝒄𝒍𝒐 term to guide the evolving curve. The image 𝐼: 𝛺 → ℜ is a After replacing energy terms, we can obtain the speed given grey level image on the domain 𝛺 , we define a function energy functional ℱ as: 𝑔 as an edge detector by: (| ( )| ) 𝓕 = 𝜶 ∫ 𝜵𝝓 𝒙 − 𝟏 𝒅𝒙 + 𝑪𝒐𝒎𝒃𝒊𝒏𝒆𝒅 𝒈 = , (2) | | 𝟏 𝜵𝑮 ∗𝑰 ( )| | ∑ ( )| ( ) 𝝁 ∫ 𝒈𝜹 𝝓 𝜵𝝓 𝒅𝒙 + 𝝀 ∫ ∫ 𝑲 𝒙 − 𝒚 𝑰 𝒚 − 𝒊 𝝃 𝒊𝟏 𝜴 𝜴 𝜴 where 𝐺 ∗ 𝐼 is the convolved 𝐼 by a Gaussian kernel 𝐺 with ( )| ( ) 𝒇 𝒙 𝑳 𝝓 𝒅𝒚 𝒅𝒙 . (8) 𝒊 𝒊 a standard deviation 𝜎 . For a level set function 𝜙: 𝛺 → ℜ , we define the edge detection energy term ℰ (𝜙 ) by: In practical implementation, the Heaviside function 𝐻 and the Dirac delta function 𝛿 are approximated by following | | 𝓔 (𝝓 ) = 𝝁 ∫ 𝒈𝜹 (𝝓 ) 𝜵𝝓 𝒅𝒙 , (3) 𝒅𝑬𝒈𝒆 𝜴 smoothed functions 𝐻 and 𝛿 respectively, defined as: 𝟏 𝒙 𝟏 𝝅𝒙 | | 𝟏 + + 𝒔𝒊𝒏 , 𝒙 ≤ 𝝐 𝟐 𝝐 𝝅 𝝐 where 𝜇 > 0 , 𝛿(𝑥) is the Dirac delta function. 𝑯 = , (9) 𝟏, 𝒙 > 𝝐 𝟎, 𝒙 < −𝝐 2.3 The local intensity energy term and 𝟏 𝝅𝒙 | | 𝟏 + 𝒄𝒐𝒔( ) , 𝒙 ≤ 𝝐 𝝐𝟐 𝝐 𝜹 = , (10) To enhance the segmentation against image noise and 𝝐 𝟎, |𝒙 | > 𝝐 , intensity inhomogeneity in MR images, a Gaussian distribution kernel works as the local region descriptor. where 𝜖 is the coefficient and usually set to the value of 1.5. Comparing with [5], the Gaussian distribution kernel has such The minimization of ℱ concerning to 𝜙 can be property that weights the intensities around a small interested attained using standard gradient descent method: 𝝏𝝓 𝝏𝓕 area. For a given point 𝑥 ∈ 𝛺 , we define the local intensity = − , (11) 𝝏𝒕 𝝏𝝓 energy term as: 𝓔 ,𝝓 𝒇 (𝒙 ), 𝒇 (𝒙 ) = 𝝀 ∫ ∫ 𝑲 (𝒙 − 𝑳𝒂𝒄𝒍𝒐 𝟏 𝟐 𝒊𝟏 𝒊 𝝃 where 𝜕ℱ ⁄ 𝜕𝜙 is the Gâteaux derivative of the ℱ. Hence, the 𝜴 𝜴 𝒚) |𝑰 (𝒚 ) − 𝒇 (𝒙 )| 𝑳 𝝓 (𝒚 ) 𝒅𝒚 𝒅𝒙 , (4) corresponding gradient flow equation is expressed as: 𝒊 𝒊 𝝏𝝓 𝜵𝝓 𝜵𝝓 = −𝜶 𝜵 𝝓 − 𝒅𝒊𝒗 − 𝝁𝜹 (𝝓 )𝒅𝒊𝒗 𝒈 + | | | | 𝝏𝒕 𝜵𝝓 𝜵𝝓 where 𝜆 > 0 , 𝐿 (𝜙 ) = 𝐻(𝜙) , 𝐿 (𝜙 ) = 1 − 𝐻(𝜙) , 𝐻 (𝑥 ) is ( )( ) 𝜹 𝝓 𝝀 𝒆 − 𝝀 𝒆 , (12) the Heaviside function, 𝐾 the Gaussian kernel with a scale 𝝐 𝟏 𝟏 𝟐 𝟐 parameter 𝜉 > 0 , and 𝑓 (𝑥 ), 𝑓 (𝑥 ) approximate intensities in a where 𝑑𝑖𝑣(∙) is the divergence operator, 𝑒 is defined as: neighbourhood of 𝑥 inside and outside the contour 𝒞 . The 𝒆 = 𝑲 (𝒚 − ) 𝒙 |𝑰 (𝒙 ) − 𝒇 (𝒚 )| 𝒅𝒚 . (13) 𝒊 𝝃 𝒊 𝑓 (𝑥 ) is computed using the Euler-Lagrange equations: ( ) ( ) 𝑲 () 𝒙 ∗𝑳 𝝓 𝒙 𝑰 𝒙 𝝃 𝒊 We set the initial level set function with a rectangle form ( ) , 𝒊 = 𝟏, 𝟐 . (5) 𝒇 𝒙 = 𝑲 () 𝒙 ∗𝑳 𝝓 (𝒙 ) 𝝃 𝒊 as: −𝒄 , 𝒙 ∈ 𝒊𝒏𝒅𝒆𝒔𝒊 𝝓 (𝒙 ) = , (14) 𝒄 , 𝒓𝒘𝒊𝒔𝒕𝒉𝒆𝒆𝒐 2.4 Regularization energy term where 𝑐 = 2 in our approach. The overall algorithm is During the evolving procedure, the conventional LSM has summarized in Algorithm 1. been trapped by reinitialization for a long time. Li et al. We applied the proposed method on synthesized images introduced a regularization term to maintain the level set and MR images of the capitate. The MR images were acquired function as the signed distance function [8]. We apply the in vivo using the high-resolution 3D-WATSc (water selective regularization term against the reinitialization problem as: cartilage scans) sequence on a clinical 3T MRI scanner Jianzhang Li et. al., A novel combined level set model for automatic MR image segmentation — 3 (Achieva, Philips Healthcare, Best, The Netherlands). The blurred the actual object boundaries. The Dice coefficient is synthesized image was a manually generated heart shape listed in Table 1. binary image with added Gaussian noise and Salt-and-pepper noise in MATLAB (Version R2019a; The MathWorks, Inc.). Table 1: Dice similarity coefficient on image tests All images have not been preprocessed except cropped to a pixel size of 500 × 500. To comprehensively evaluate the Test group Low High DRLSE Proposed performance of the combined level set model in segmentation threshold threshold method method tasks, we compared the results of multi-threshold method and Synthesized edge-based method DRLSE in [8]. Manual segmentation 0.9699 0.9959 image performed by a radiologist with eight years of experience in MR image A 0.9313 0.9517 0.7605 0.9932 musculoskeletal radiology provided the ground truth. The MR image B 0.9135 0.6477 0.6751 0.9987 curve evolution and results evaluation were conducted in MATLAB. MR image C 0.9117 0.7107 0.6438 0.9995 Algorithm 1 3.2 MR image tests Input: 𝜙 , 𝛼 , 𝜇 , 𝜌 , 𝜆 , 𝜆 , iteration number 𝓃 Output: 𝜙 To illustrate the difficulties in segmenting MR images, the 1. Initialization: 𝜙 ↢ 𝜙 ; 2. for 𝑖 = 0 ↣ 𝓃 do combined level set model was compared with the conventional 3. update 𝑔 , 𝑓 , 𝑓 using Eqs. (2) and (7); multi-threshold method and the DRLSE method. The multi- 4. 𝜙 ↢ 𝜙 + ∆𝑡 ∆𝜙 ; threshold method was conducted in the free, open-source 5. end for platform ITK-SNAP [9]. The results were then transferred into MATLAB for evaluation. The choice of the thresholds is challenging since it is 3 Results either too low leading to an under-segmentation or too high causing an over-segmentation. There were two attempts on setting the thresholds in this section. For MR image A, B and The parameters of the combined level set model were C, the lower value at 16.3%, 7.78% and 18.19% of each assigned as 𝛼 = 0.2 , 𝜇 = 0.001 × 256 , 𝜌 = −3 , 𝜆 = 𝜆 = maximum image grey value was to make sure the least leakage 1. The iteration number of all LSM based approaches was outside the capitate bone. The higher value at 29.2%, 18.19% empirically set at a position where there was no further and 28.57% of each maximum image grey value was intended evolution of the method we put forward. The segmentation to cover all the areas inside the capitate bone. As for two other results are shown in figure 1. For evaluating the accuracy, we LSM approaches, we compared the segmentation results of our utilized the Dice similarity coefficient between obtained proposed model with the DRLSE method. The evolving of contours and ground truth. A higher value of the coefficient each model was terminated after 270, 110 and 115 iterations, means better performance. For a given set of regions 𝒢 respectively. As illustrated in figure 1, the multi-threshold representing the ground truth and 𝒮 representing the method failed on both choices. Our combined level set model segmentation, the similarity between 𝒢 and 𝒮 is expressed as: converged after 270, 110 and 115 iterations, respectively, and | | 𝟐× 𝓖∩𝓢 all desired boundaries were fully covered, while the DRLSE 𝑫𝒊𝒄𝒆 (𝓖, 𝓢 ) = . (15) |𝓖 | |𝓢 | method stopped near the initial 𝜙 due to its sensitivity to the image noise and intensity inhomogeneity. 3.1 Synthesized image tests The dark heart shape area was the object to be segmented. The proposed approach was compared with the DRLSE method. The segmentation with the proposed model stopped after 390 iterations. With the help of local intensity energy term, the evolving curve could merge and analyse the intensities within a small area around interest front points, which provided the correct contour. Nevertheless, as showed in figure 1, the segmentation of the DRLSE method failed after the same iterations due to the massive artificial noises, which Jianzhang Li et. al., A novel combined level set model for automatic MR image segmentation — 4 method, we integrated the edge-based energy term and local 4 Conclusion region-based energy term, which provided additional boundary information and suppressed massive noises and This paper proposed a combined level set model that intensity inhomogeneity, respectively. Experimental results reinforces the segmentation in MR images. In this proposed showed that the proposed combined level set model possesses a much better accuracy with all Dice similarity coefficients > 0.99. Author Statement Research funding: The research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – ES 442/1-1 and RA 2187/4-1. Conflict of interest: Authors state no conflict of interest. Informed consent: Informed consent has been obtained from all individuals included in this study. Ethical approval: The research related to human use complies with all the relevant national regulations, institutional policies and was performed in accordance with the tenets of the Helsinki Declaration, and has been approved by the authors' institutional review board (EK 171/10). References [1] Li C, Huang R, Ding Z, Gatenby JC, Metaxas DN, Gore JC et al. A Level Set Method for Image Segmentation in the Presence of Intensity Inhomogeneities With Application to MRI. IEEE transactions on image processing a publication of the IEEE Signal Processing Society 2011;20(7):2007–16. [2] Vasilevskiy A, Siddiqi K. Flux maximizing geometric flows. IEEE Trans. Pattern Anal. Machine Intell. 2002;24(12):1565–78. [3] Caselles V, Kimmel R, Sapiro G. Geodesic Active Contours. Int J Comput Vision 1997;22(1):61–79. [4] Chan TF, Vese LA. Active contours without edges. IEEE Trans. on Image Process. 2001;10(2):266–77. [5] Lankton S, Tannenbaum A, Lankton S, Tannenbaum A. Localizing region-based active contours. IEEE transactions on image processing a publication of the IEEE Signal Processing Society 2008;17(11):2029–39. [6] Li C, Kao C-Y, Gore JC, Ding Z, Gore JC. Minimization of Region-Scalable Fitting Energy for Image Segmentation. IEEE transactions on image processing a publication of the IEEE Signal Processing Society 2008;17(10):1940–9. [7] Li C, Kao C-Y, Gore JC, Ding Z. Implicit Active Contours Driven by Local Binary Fitting Energy 2007:1–7. [8] Li C, Xu C, Gui C, Fox MD. Distance Regularized Level Set Evolution and Its Application to Image Segmentation. IEEE Figure 1: Image test results. Synthesized Image: from left to right: transactions on image processing a publication of the IEEE Original sketch image defined as ground truth; Segmentation of Signal Processing Society 2010;19(12):3243–54. contours by use of the DRLSE method; Segmentation of contours [9] Yushkevich PA, Piven J, Hazlett HC, Smith RG, Ho S, Gee JC by use of proposed level set model. MR Images: from left to right: et al. User-guided 3D active contour segmentation of anatomical MR image A, B and C. From up to down: Original MR image; structures: significantly improved efficiency and reliability. Segmentation of capitate using the multi-threshold method with NeuroImage 2006;31(3):1116–28. lower threshold; Segmentation of multi- threshold method with higher threshold; Segmentation of DRLSE method; Segmentation of the proposed level set model. Red line: Segmentation contour. Yellow line: Initial zero level set. Cyan line: Ground truth.

Journal

Current Directions in Biomedical Engineering – de Gruyter

Published: Sep 1, 2020

Keywords: level set method; MRI; segmentation; intensity inhomogeneity

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