Fast marching over the 2D Gabor magnitude domain for tongue body segmentation
© Cui et al.; licensee Springer. 2013
Received: 4 June 2013
Accepted: 12 December 2013
Published: 26 December 2013
Tongue body segmentation is a prerequisite to tongue image analysis and has recently received considerable attention. The existing tongue body segmentation methods usually involve two key steps: edge detection and active contour model (ACM)-based segmentation. However, conventional edge detectors cannot faithfully detect the contour of the tongue body, and the initialization of ACM suffers from the edge discontinuity problem. To address these issues, we proposed a novel tongue body segmentation method, GaborFM, which initializes ACM by performing fast marching over the two-dimensional (2D) Gabor magnitude domain of the tongue images. For the enhancement of the contour of the tongue body, we used the 2D Gabor magnitude-based detector. To cope with the edge discontinuity problem, the fast marching method was utilized to connect the discontinuous contour segments, resulting in a closed and continuous tongue body contour for subsequent ACM-based segmentation. Qualitative and quantitative results showed that GaborFM is superior to the other methods for tongue body segmentation.
Medical image segmentation, analysis, and diagnosis have received much attention in image analysis and computer vision. Several methods of segmentation have been proposed for medical images such as magnetic resonance imaging (MRI), computed tomography (CT), and ultrasound medical imagery. Yezzi et al. proposed the geometric active contour model for segmentation of medical imagery . Tsai et al. proposed a shape-based method using level sets to segment the medical images . For prostate segmentation in transrectal ultrasound (TRUS), Yan et al. proposed a discrete deformable model . Graph cuts , proposed by Boykov et al., have also been used for the segmentation of medical images. As one part of medical image analysis and diagnosis, because of its convenience and non-invasion nature, tongue image diagnosis has received considerable research interest.
Tongue diagnosis has played an important role in traditional Chinese medicine (TCM) for thousands of years [5, 6]. The TCM practitioner can analyze the physiological and pathological conditions of the patient by inspecting the color, shape, and texture of the tongue, making tongue diagnosis very promising for convenient and non-invasive diagnosis. However, traditional tongue diagnosis is a subjective skill which requires years of experience and practice. Moreover, for different practitioners, the diagnosis results may be inconsistent.
To address these problems, computational tongue diagnosis was studied for the digital acquisition and quantitative analysis of tongue images [7–9]. Generally, the computational tongue diagnosis system involves three major modules: tongue body segmentation, feature extraction, syndromes and disease analysis [10–12]. Several imaging systems have been developed for the acquisition of tongue images [13–15]. A number of feature extraction methods have been proposed for extracting features of tongue color [16, 17], coating, and texture [18–21], fissures [22–25], shape [26, 27], tooth marks , petechia spots , and the sublingual vein [30, 31]. Pattern recognition approaches such as the Bayesian network and support vector machine have been used for syndromes and disease analysis [9, 32–34].
Among the modules of computational tongue diagnosis systems, tongue body segmentation is a prerequisite for subsequent feature extraction and diagnostic analysis. Tongue body segmentation usually involves three steps: edge enhancement, contour initialization, and segmentation. For edge enhancement, conventional edge detectors, such as the Gaussian gradient , color gradient , and Canny detector , have been used. Considering the shape and gray-level characteristics of tongue contour, Zuo et al. [10, 11] proposed a polar edge detector to suppress adverse interference from the lip boundary and tongue fissures. For contour initialization, several heuristic , deformable model-based , and watershed-based  methods have been suggested. For segmentation, active contour models (ACM) [11, 13, 35, 37] and the gradient vector flow (GVF) snake  have been used for the final segmentation of the tongue body [14, 39, 40].
Despite the progresses in tongue body segmentation, the existing methods usually suffer from some limitations and cannot meet the performance requirements of a practical computational tongue diagnosis system. For example, in the edge enhancement step, conventional edge detectors usually neglect the characteristics of gray-level variation along the tongue body contour. In the contour initialization step, parts of the tongue body contour might exhibit very weak edges, and the detected contour would be discontinuous, making it hard to obtain a continuous contour for initialization. Moreover, the interference from lips and tongue fissures makes tongue body segmentation more challenging.
Edge enhancement is fundamental for any edge-based active contour and level set based segmentation method. Thus, the enhancement of the tongue body contour by the proposed 2D Gabor magnitude-based edge detection method was first studied. There are several region-based models, such as color-texture segmentations [38, 41] and the C-V model and its extensions , but, tongue texture is very complex and varies with images, so this study used the edge-based model and first investigated the edge enhancement problem.
In the contour initialization stage, the stable segment selection and the fast marching method for contour initialization were used. This initialization strategy allowed the incorporation the various tongue priors to obtain a satisfactory initial contour. For example, segments with less length were abandoned. Considering the symmetrical characteristics of the tongue body contour, we selected pairs of segments with better symmetry. Moreover, to avoid the interference of lips and to remove the shadow parts, two paths were found for any two points. If the difference in length L F of the two paths was low, the inner curve was chosen, otherwise the one with the lower length was chosen.
The contour initialization strategy allowed the employment of many types of tongue priors, such as color, shape, and symmetry, to obtain a satisfactory initial contour, and the GVF snake was able to obtain a local optimal and smooth segmentation results. Geodesic active contours [43, 44] can be used to address the limited capture range problem of the conventional snake model. Most globally optimal active contours [45–48] can obtain the global optimal solution based on the energy function, which generally is only a non-ideal separation of the tongue body from the other parts. Several globally optimal active contours [49, 50], are similar to the fast marching method used in GaborFM, and can obtain the global optimal path to connect the two points. Thus, we did not use the methods for global minimization of the active contour model.
By taking the characteristics of gray-level variation of the real tongue body contour into account, the 2D Gabor magnitude-based detector is more effective than the conventional edge detector for the enhancement of the tongue body contour and the suppression of interference from lip and tongue fissures. For this reason, the 2D Gabor magnitude-based edge detection method was used to address the edge enhancement problem.
To circumvent the edge discontinuity problem, we used fast marching over the 2D Gabor magnitude domain method to connect the discontinuous contour segments into a closed tongue body contour. First, several stable segments were selected based on the edge enhancement result. Then, the contour initialization problem was modeled as the minimal geodesic paths over the 2D Gabor magnitude domain. Finally, the fast marching algorithm was used to obtain a closed tongue body contour for initialization.
The remainder of the paper is organized as follows. Section 2 presents the 2D Gabor magnitude-based edge detector for the enhancement of the tongue body contour. Section 3 describes the scheme for contour initialization and tongue body segmentation. Section 4 provides the qualitative and quantitative results to evaluate the proposed GaborFM method. Finally, Section 5 offers the conclusion.
2. 2D Gabor magnitude-based edge detection
In this section, a 2D Gabor magnitude-based edge detector for the enhancement of the tongue body contour is proposed. It then utilizes the color characteristics of the tongue body to suppress interference from the tongue texture and fissures, and finally uses Otsu's method for edge thresholding.
2.1 2D Gabor magnitude-based detector
where x′ = (x − x 0)cos θ + (y − y 0)sin θ, y′ = (x − x 0)sin θ + (y − y 0)cos θ, (x 0 , y 0) is the center of the function, ω is the radial frequency in radians per unit length, and θ is the orientation of the Gabor functions in radians. The κ is defined by , where δ is the half-amplitude bandwidth of the frequency response, which is between 1 and 1.5 octaves according to neurophysiological findings . When ω and δ are fixed, σ can be derived from σ = κ/ω.
where ‘¯’ denotes the complex conjugate operator. Generally, any profiles in Figure 2b would result in local maxima in M max(x, y). Moreover, the promising performance of the Gabor filter on noise robustness and time-frequency tradeoff makes the proposed 2D Gabor magnitude-based detector suitable for the enhancement of the tongue boundary.
2.2 Edge thresholding
We used a two-step strategy for edge thresholding. First, based on the color characteristics of the tongue body and face, parts of the non-boundary pixels were identified. Then, the binarization of the edge image was obtained by defining the proper threshold.
Unlike the background, due to physiological and pathological factors, the color of the tongue body varies with tongue images. Fortunately, it is relatively easy to distinguish the color of tongue body from that of other facial parts: the pixel values of the tongue body component usually have higher Q value. Let T 1 denote the threshold obtained by Otsu's method. We set the threshold T Q = 2 T 1 and set the pixels with a Q value higher than T Q as tongue body pixels. Morphological operators, mentioned above, were then used to define the tongue body region. A subset of the background and tongue body components was then obtained. As shown in Figure 4b, by assigning these pixels as non-boundary pixels, a modified edge image can be derived for edge binarization.
After edge thresholding, the morphological operators were employed to obtain single-pixel edge curves, resulting in the final binarized edge image shown in Figure 4c.
3. Tongue body segmentation using fast marching and the GVF snake
In this section, first several stable segments from the binarized edge image are selected, and then the fast marching algorithm is used to obtain a closed contour for initialization. Finally, the GVF snake model is adopted for tongue body segmentation.
3.1 Selection of stable segments
Although the 2D Gabor magnitude-based detector is effective, false detection of the tongue body boundary cannot be completely avoided. Moreover, it is also difficult to detect the entire closed tongue body contour by only using the 2D Gabor magnitude-based detector. Fortunately, the location and shape of the tongue body are useful in distinguishing a true boundary from a false boundary. Thus, to suppress the adverse influence of false detection, we used the strategy of selecting stable segments of the tongue body contour.
By referring to the location and shape of the tongue body, the following approach for selecting several stable segments from the binarized edge image is suggested. First, the mean of the coordinate of all the edge pixels and the mean of the coordinate of all the tongue body pixels obtained in Section 2.2 were computed. Then, the center of the tongue body was estimated as the average of and .
3.2 Contour initialization using fast marching
where C is the set of curves with γ(0) = (x A , y A ) and γ(1) = (x C , y C ).
with T A (x A , y A ) = 0, where T A (x, y) denotes the shortest distance L F of (x A, y A) and (x, y). The fast marching methoda was used to solve the Eikonal equation to obtain T A (x, y). The computational complexity of fast marching is O(N logN), where N = mn is the size of the tongue image. Moreover, fast marching stops when the point (x C, y C) is reached, so T A (x, y) can be more efficiently obtained.
with the initial value X t = 0 = (x C , y C ). This ODE problem can be efficiently solved by using second-order Heun's method .
In practice, to avoid the interference of the lips, we found two paths for any two points. If the difference in length L F of the two paths was low, the inner curve was chosen, otherwise the one with the lower length was chosen. Figures 1e and 5c show two examples of the shortest paths constructed by the fast marching method. Satisfactory initialization of the tongue body contours can be obtained by using the fast marching method.
3.3 Gradient vector flow snake
where α and β are weighting parameters that control the snake's tension and rigidity, respectively, x ' (s) and x ' ' (s) denote the first and second derivatives of x(s) with respect to s, and E ext is the external energy.
where | ⋅ | denotes the l 2 norm with . For fast GVF computation, the augmented Lagrangian method (ALM)-based algorithm developed by our group was used .
For tongue body segmentation, μ = 0.1, α = 0.05, and β = 0.01 were chosen.
4. Experimental results
4.1 Database and evaluation criteria
We constructed a tongue image data set of 300 imagesc to evaluate the proposed method. Manual segmentation results were used as the ground truth. All the images were acquired by our tongue image acquisition device. The image size was 768 × 576. Since the images were captured in a semi-enclosed environment under stable lighting conditions, it was not necessary to use any pre-processing method for illumination normalization. All the experiments were executed on a PC with T2250 @1.73 GHz CPU and 2G memory.
4.2 Components of GaborFM
We compared the edge enhancement results obtained using the Sobel, DoG, and 2D Gabor magnitude-based edge detector, as shown in Figure 3. It can be seen that the proposed 2D Gabor magnitude-based detector can faithfully enhance the tongue body contour and is more robust against inference from tongue texture. Thus, compared with Sobel and DoG, the 2D Gabor magnitude-based method is more effective for the enhancement of the tongue body contour.
Gabor + FM: we only used the first two steps (Gabor magnitude-based edge detection and fast marching) for tongue body segmentation and used the contour after fast marching as the contour of the tongue body.
Canny + FM + GVF: we replaced the Gabor magnitude-based edge detection with the Canny detector.
Gabor + FM + GVF: we used all the three steps, GaborFM, for tongue body segmentation.
The HD, MD, and SD of Canny + FM + GVF, Gabor + FM, and GaborFM
Canny + FM + GVF
33.85 ± 39.44
6.83 ± 8.88
9.20 ± 13.69
Gabor + FM
19.15 ± 12.31
4.28 ± 2.18
4.20 ± 3.04
Gabor + FM + GVF
17.79 ± 11.87
4.26 ± 2.17
3.98 ± 2.99
4.3 Comparison with the other segmentation methods
Average CPU time of methods
Average CPU time (s)
Based on the evaluation results above, it can be seen that GaborFM is better than BEDC, Watershed, and PolarSnake for tongue body segmentation. From Table 3, the proposed GaborFM method can achieve the HD, MD distance, and SD of 17.79, 4.26, and 3.98 pixels, respectively. From Table 4, the proposed method can achieve the FPVF and FNVF of 0.96% and 5.44%, respectively. Moreover, the qualitative and quantitative evaluation results also show that GaborFM satisfies the practical performance requirements of tongue body segmentation and can be embedded into a real computational tongue diagnosis system.
In this paper, we proposed a novel method, GaborFM, for automated tongue body segmentation. First, a Gabor magnitude-based detector was developed for edge enhancement. Second, both the color characteristics and the edge enhancement result for the thresholding of the edge image were taken into account. Third, stable segments were selected from the binarized edge image and use the fast marching algorithm over 2D Gabor magnitude domain to obtain a continuous closed curve for contour initialization. Finally, the GVF snake was used for tongue body segmentation. Generally, the proposed method can well address the edge enhancement and the contour discontinuity problems. Experimental results also showed that the proposed method is more effective than other tongue body segmentation methods, i.e., BEDC , Watershed , and PolarSnake .
aTo solve the Eikonal equation, we used the Matlab Toolbox Fast Marching provided by Dr. Gabriel Peyre, which is available to the public at http://www.mathworks.com/matlabcentral/fileexchange/6110-toolbox-fast-marching.
bThe code of GVF snake was provided by the authors of , which is available to the public at http://www.iacl.ece.jhu.edu/static/gvf/. Our Matlab code on ALM-based GVF computation has been released at https://sites.google.com/site/cswmzuo/IALM-GVF_GGVF.rar.
cWe will soon make the tongue image dataset together with the ground truth segmentation results available to the public.
dWe will soon make the tongue image dataset together with the ground truth segmentation results available to the public.
The work is partially supported by the GRF fund from the HKSAR Government, the central fund from the Hong Kong Polytechnic University, the NSFC funds of China (grant nos. 61001037, 61071179, 61271093, and 61102037), and the Fundamental Research Funds for the Central Universities (grant no. HIT.NSRIF.2010051). The authors would like to thank the associate editor and the anonymous reviewers for their constructive suggestions. Thanks to Dr. Edward C. Mignot, Shandong University, for linguistic advice.
- Yezzi A, Kichenassamy S, Kumar A, Olver P, Tannenbaum A: A geometric snake model for segmentation of medical imagery. IEEE Trans. Med. Imag. 1997, 16(2):199-209. 10.1109/42.563665Google Scholar
- Tsai A, Yezzi A Jr, Wells W, Tempany C, Tucker D, Fan A, Grimson WE, Willsky A: A shape-based approach to the segmentation of medical imagery using level sets. IEEE Trans. Med. Imag. 2003, 22(2):137-154. 10.1109/TMI.2002.808355Google Scholar
- Yan P, Xu S, Turkbey B, Kruecher J: Discrete deformable model guided by partial active shape model for TRUS image segmentation. IEEE Trans. Biomedical engineering 2010, 57(5):1158-1166.Google Scholar
- Boykov Y, Funka-Lea G: Graph cuts and efficient N-D image segmentation. Int. J. Comput. Vis. 2006, 70(2):109-131. 10.1007/s11263-006-7934-5Google Scholar
- Xu D: Mutual understanding between traditional Chinese medicine and systems biology: gaps, challenges and opportunities. Int. J. Funct. Inform. Personal. Med. 2009, 2(3):248-260. 10.1504/IJFIPM.2009.030826Google Scholar
- Lukman S, He YL, Hui SC: Computational methods for traditional Chinese medicine: a survey. Computer Methods and Program in Biomedicine 2007, 88(3):283-294. 10.1016/j.cmpb.2007.09.008Google Scholar
- Xu WT, Kanawong R, Xu D, Li S, Ma T, Zhang GX, Duan Y: An automatic tongue detection and segmentation framework for computer-aided tongue image analysis. Columbia, MO, 13–15 June 2011. Proceedings of the IEEE 13th Int. Conf. E-health Networking, Applications and Services 189-192.Google Scholar
- Zhang D: Automated biometrics: technologies and system. Boston: Kluwer; 2000.Google Scholar
- Pang B, Zhang D, Li NM, Wang KQ: Computerized tongue diagnosis based on Bayesian networks. IEEE Trans. Biomedical Engineering 2004, 51(10):1803-1810. 10.1109/TBME.2004.831534Google Scholar
- Zuo W, Wang K, Zhang D, Zhang H: Combination of polar edge detection and active contour model for automated tongue segmentation. Hong Kong, 18–20 December 2004. Proceedings of the 3rd Int. Conf. Image and Graphics 270-273.Google Scholar
- Zhang H, Zuo W, Wang K, Zhang D: A snake-based approach to automated segmentation of tongue image using polar edge detector. Int. J. Imaging Syst. Technol. 2006, 16(4):103-112. 10.1002/ima.20075Google Scholar
- Jung CJ, Jeon YJ, Kim JY, Kim KH: Review on the current trends in tongue diagnosis systems. Integrative Medicine Research 2012, 1(1):13-20. 10.1016/j.imr.2012.09.001Google Scholar
- Cai Y: A novel imaging system for tongue inspection. Anchorage, 21–23 May 2002. Proceedings of the IEEE conf. Instrumentation and Measurement Technology 159-163.Google Scholar
- Li QL, Liu J, Xiao G, Xue YQ: Hyperspectral tongue imaging system used in tongue diagnosis. Shanghai, 16–18 May 2006. Proceedings of the 2nd Int. Conf. Bioinformatics and Biomedical Engineering 2579-2581.Google Scholar
- Dong H, Guo Z, Zeng C, Zhong H, He Y, Wang RK, Liu S: Quantitative analysis on tongue inspection in traditional Chinese medicine using optical coherence tomography. J. Biomed. Opt. 2008, 13(1):011004. 10.1117/1.2870175Google Scholar
- Li C, Yuen P: Tongue image matching using color content. Pattern Recogn. 2002, 35(2):407-419. 10.1016/S0031-3203(01)00021-8Google Scholar
- Wang Y, Yang J, Zhou Y, Wang Y: Region partition and feature matching based color recognition of tongue image. Pattern Recogn. Lett. 2007, 28(1):11-19. 10.1016/j.patrec.2006.06.004Google Scholar
- Bai Y, Shi Y, Wu J, Zhang Y, Wong W, Wu Y, Bai J: Automatic extraction of tongue coating from digital images: a traditional Chinese medicine diagnostic tool. TsingHua Sci Technol 2009, 14(2):170-175. 10.1016/S1007-0214(09)70026-4Google Scholar
- Li W, Hu S, Yao J, Song H: The separation framework of tongue coating and proper in traditional Chinese medicine. Macau, 8–10 December 2009. Proceedings of the 7th Int. Conf. Information, Communications and Signal Processing 1-4.Google Scholar
- Huang W, Yan Z, Xu J, Zhang L: Analysis of the tongue Fur and tongue features by naive bayesian classifier. Taiyuan, 22–24 October 2010. Proceedings of the Int. Conf. Computer Application and System Modeling 304-308. vol. 4,Google Scholar
- Huang B, Zhang D, Li Y, Zhang H, Li N: Tongue coating image retrieval. Harbin, 18–20 January 2011. Proceedings of the 3rd Int. Conf. Advanced Computer Control 292-296.Google Scholar
- Liu L, Zhang D, You J: Detecting wide lines using isotropic nonlinear filter. IEEE Trans. Imaging Processing 2007, 16(6):1584-1595.MathSciNetGoogle Scholar
- Liu L, Zhang D, Ajay K, Wang KQ: Tongue line extraction. Tampa, FL, 8–11 December 2008. Proceedings of the 19th Int. Conf. Pattern Recognition 1-4.Google Scholar
- Liu L, Zhang D: Extracting tongue cracks using the wide line detector. Hong Kong, 4–5 January 2008. Proceedings 1st Int. Conf. Medical Biometrics 49-56.Google Scholar
- Yang Z, Zhang D, Li N: Kernel false-colour transformation and line extraction for fissured tongue image. Journal of Computer-Aided Design and Computer Graphics 2010, 22(5):771-776. 10.3724/SP.J.1089.2010.10754Google Scholar
- Huang B, Wu JS, Zhang D, Li NM: Tongue shape classification by geometric features. Inform. Sci. 2010, 180(2):312-324. 10.1016/j.ins.2009.09.016Google Scholar
- Zhang D, Liu Z, Yan J: Dynamic tongueprint: a novel biometric identifier. Pattern Recogn. 2010, 43(3):1071-1082. 10.1016/j.patcog.2009.09.002MathSciNetGoogle Scholar
- Li WS, Yao JF, Song H: The recognition of the teeth marks of tongue based on the improved level set in TCM. Yantai, 16–18 October 2010. Proceedings of the 3rd Int. Congress on Image and Signal Processing 2700-2704.Google Scholar
- Li JF, Zhang HZ, Wang KQ, Zuo WM: An automated feature extraction method for recognition of petechia spot in tongue diagnosis. Sanya, 13–14 December 2009. Proceedings of the Int. Conf. Future Biomedical Information Engineering 69-72.Google Scholar
- Hoover AD, Kouznetsova V, Goldbaum M: Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response. IEEE Trans. Medical Imaging 2000, 19(3):203-210. 10.1109/42.845178Google Scholar
- Yan Z, Wang K, Li N: Computerized feature quantification of sublingual veins from color sublingual images. Comput. Methods Programs Biomed. 2009, 93(2):192-205. 10.1016/j.cmpb.2008.09.006Google Scholar
- Zhang HZ, Wang KQ, Zhang D, Pang B, Huang B: Computer aided tongue diagnosis system. Shanghai, 1–4 September 2005. Proceedings of the IEEE 27th Annual Conf. Engineering in Medicine and Biology 6754-6757.Google Scholar
- Gao Z, Cui M, Lu GM: A novel computerized system for tongue diagnosis. Leicestershire, 20 November 2008. Proceedings of the Int. Seminar on Future Information Technology and Management Engineering 364-367.Google Scholar
- Lo LC, Hou MC, Chen YL, Chiang JY, Hsu CC: Automatic tongue diagnosis system. Tianjin, 17–19 October 2009. Proceedings of the 2nd Int. Conf. Biomedical Engineering and Informatics 1-5.Google Scholar
- Pang B, Zhang D, Wang KQ: The bi-elliptical deformable contour and its application to automated tongue segmentation in Chinese medicine. IEEE Trans. Medical Imaging 2005, 24(8):946-956.Google Scholar
- Ning J, Zhang D, Wu C, Yue F: Automatic tongue image segmentation based on gradient vector flow and region merging. Neural Comput. & Applic. 2012, 21(8):1819-1826. 10.1007/s00521-010-0484-3Google Scholar
- Yu SY, Yang J, Wang YG, Zhang Y: Color active contour models based tongue segmentation in traditional Chinese medicine. Wuhan, 6–8 July 2007. Proceedings of the 1st Int. C. Bioinformatics and Biomedical Engineering 1065-1068.Google Scholar
- Liapis S, Sifakis E, Tziritas G: Colour and texture segmentation using wavelet frame analysis, deterministic relaxation, and fast marching algorithms. J. Vis. Commun. Image R. 2004, 15: 1-26. 10.1016/S1047-3203(03)00025-7Google Scholar
- Li Q, Wang Y, Liu H, Sun Z: AOTF based hyperspectral tongue imaging system and its applications in computer-aided tongue disease diagnosis. Yantai, 16–18 October 2010. Proceedings of the 3rd Int. Conf. Biomedical Engineering and Informatics 1424-1427.Google Scholar
- Liu Z, Yan J, Zhang D, Li Q: Automated tongue segmentation in hyperspectral images for medicine. Appl. Optics 2007, 46(1):8328-8334.Google Scholar
- Chen J, Pappas TN, Mojsilović A, Rogowitz BE: Adaptive perceptual color-texture image segmentation. IEEE Trans. Image Processing 2005, 14(10):1-13.Google Scholar
- Chan TF, Vese LA: Active contours without edges. IEEE Trans. Image Processing 2001, 10(2):266-277. 10.1109/83.902291Google Scholar
- Caselles V, Kimmel R, Sariro G: Geodesic active contours. Int. J. Comput. Vis. 1997, 2(1):61-79.Google Scholar
- Xu C, Yezzi A Jr, Prince JL: On the relationship between parametric and geometric active contours. vol. 1, Pacific Grove, CA, October 2000. Proceedings of the 34th Asilomar Conference on Signals, Systems and Computers 483-489.Google Scholar
- Chan TF, Esedoglu S, Nikolova M: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 2006, 66(5):1632-1648. 10.1137/040615286MathSciNetGoogle Scholar
- Boykov Y, Kolmogorov V, Cremers D, Delong A: An integral solution to surface evolution PDEs via geo-cuts. Berlin Heidelberg: Springer; 2006.Google Scholar
- Bresson X, Esedoḡlu S, Vandergheynst P, Thiran J-P, Osher S: Fast global minimization of the active contour/snake model. J. Math. Imaging Vis. 2007, 28(2):151-167. 10.1007/s10851-007-0002-0Google Scholar
- Márquez-Neila P, Baumela L, Alvarez L: A morphological approach to curvature-based evolution of curves and surfaces. IEEE Tran. Pattern Anal. Machine Intell. 2014, 36(1):2-17.Google Scholar
- Appleton B, Talbot H: Globally optimal geodesic active contours. J. Math. Imaging Vis. 2005, 23(1):67-86. 10.1007/s10851-005-4968-1MathSciNetGoogle Scholar
- Cohen LD, Kimmel R: Global minimum for active contour models: a minimal path approach. Int. J. Comput. Vis. 1997, 24(1):57-78. 10.1023/A:1007922224810Google Scholar
- Daugman JG: Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters. J. Optical Soc. Amer. 1985, 2(7):1160-1169. 10.1364/JOSAA.2.001160Google Scholar
- Lee TS: Image representation using 2D Gabor wavelet. IEEE Trans. Pattern Analysis and Machine Intelligence 1996, 18(10):959-971. 10.1109/34.541406Google Scholar
- Serrano Á, Martín de Diego I, Conde C, Cabello E: Recent advances in face biometrics with Gabor wavelets: a review. Pattern Recogn. Lett. 2010, 31(5):372-381. 10.1016/j.patrec.2009.11.002Google Scholar
- Riaz F, Hassan A, Rehman S, Qamar U: Texture classification using rotation- and scale-invariant Gabor texture features. IEEE Signal. Process. Lett. 2013, 20(6):607-610.Google Scholar
- Sonka M, Hlavac V, Boyle R: Image processing, analysis, and machine vision. 3rd edition. Stamford: Cengage Learning; 2007.Google Scholar
- Otsu N: A threshold selection method from gray-level histograms. IEEE Trans. System, Man, and Cybernetics 1979, 9(1):62-66.MathSciNetGoogle Scholar
- Sethian JA: Level Set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. Cambridge: Cambridge University Press; 1999.Google Scholar
- Sethian JA: Fast marching methods. SIAM Rev. 1999, 41(2):199-235. 10.1137/S0036144598347059MathSciNetGoogle Scholar
- Pechaud M, Keriven R, Peyre G: Extraction of tubular structures over an orientation domain. Miami, 20–25 June 2009. Proceedings of the IEEE conference on computer vision and pattern recognition, 2009, CVPR 2009 336-342.Google Scholar
- Kass M, Witkin A, Terzopoulos D: Snakes: active contour models. Int. J. Comput. Vis. 1987, 1(4):321-331.Google Scholar
- Xu C, Prince JL: Snakes, shapes, and gradient vector flow. IEEE Trans. Image Processing 1998, 7(3):359-369. 10.1109/83.661186MathSciNetGoogle Scholar
- Ren D, Zuo W, Zhao X, Lin Z, Zhang D: Fast gradient vector flow computation based on augmented Lagrangian method. Pattern Recogn. Lett. 2013, 34(12):219-225.Google Scholar
- Chalana V, Kim Y: A methodology for evaluation of boundary detection algorithms on medical images. IEEE Trans. Med. Imag. 1997, 16(5):642-652. 10.1109/42.640755Google Scholar
- Udupa JK, LeBlanc VR, Schmidt H, Imielinska C, Saha PL, Grevera GJ, Zhuge Y, Currie LM, Moholt P, Jin Y: A methodology for evaluating image segmentation algorithms. Proc. SPIE 2002, 4684: 266-277. 10.1117/12.467166Google Scholar
- Xu C, Prince JL: Generalized gradient vector flow external forces for active contours. Signal Process 1998, 71(2):131-139. 10.1016/S0165-1684(98)00140-6Google Scholar
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