Abstract

Linear feature detection is an important technique in different applications of image processing. To detect linear features in different types of images, a simple but effective algorithm based on a multiple-structuring-element center-surround top-hat transform is proposed. The center-surround top-hat transform is discussed and analyzed. Based on the properties of this transform for image feature detection, multiple structuring elements are constructed corresponding to the possible linear features at different directions. The whole algorithm is divided into four parts. First, the algorithm uses the center-surround top-hat transform to detect all the possible linear features at different directions through constructing multiple structuring elements. Second, the detected linear feature regions at each direction are processed by a closing operation to remove the possible holes or unconnected regions. Third, the processed results of the detected linear feature regions at all directions are combined to form all the possible detected linear feature regions. Fourth, the combined result is refined by using some simple operations to form the final result. Experimental results on different types of images from different applications verified the effective performance of the proposed algorithm. Moreover, the experimental results indicate that the proposed algorithm could be used in different applications.

© 2012 Optical Society of America

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    [CrossRef]
  30. X. Bai and F. Zhou, “Edge detection based on mathematical morphology and iterative thresholding,” in International Conference on Computational Intelligence and Security (IEEE, 2006), pp. 1849–1852.
  31. F. Gonzalez, O. Tubıo, F. Tobajas, V. De Armas, R. Esper-Chaın, and R. Sarmiento, “Morphological processor for real-time image applications,” Microelectronics J. 33, 1115–1122 (2002).
    [CrossRef]
  32. H. Park and R. T. Chin, “Decomposition of arbitrarily shaped morphological structuring elements,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 2–15 (1995).
    [CrossRef]
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    [CrossRef]

2011

H. Peregrina-Barreto, A. M. Herrera-Navarro, L. A. Morales-Hernández, and I. R. Terol-Villalobos, “Morphological rational operator for contrast enhancement,” J. Opt. S. Am. A 28, 455–464 (2011).
[CrossRef]

X. Bai, F. Zhou, and B. Xue, “Multiple linear feature detection through top-hat transform by using multi linear structuring elements,” Optik (2011).
[CrossRef]

X. Bai, F. Zhou, and B. Xue, “Fusion of infrared and visual images through region extraction by using multi scale center-surround top-hat transform,” Opt. Express 19, 8444–8457 (2011).
[CrossRef]

2010

X. Bai and F. Zhou, “Analysis of new top-hat transformation and the application for infrared dim small target detection,” Pattern Recogn. 43, 2145–2156 (2010).
[CrossRef]

J. A. Ferrari and J. L. Flores, “Nondirectional edge enhancement by contrast-reverted low-pass Fourier filtering,” Appl. Opt. 49, 3291–3296 (2010).
[CrossRef]

H. Peregrina-Barreto and I. R. Terol-Villalobos, “Morphological rational multi-scale algorithm for color contrast enhancement,” Proc. SPIE 7532, 75320Q (2010).
[CrossRef]

X. Bai, F. Zhou, and T. Jin, “Enhancement of dim small target through modified top-hat transformation under the condition of heavy clutter,” Signal Process 90, 1643–1654 (2010).
[CrossRef]

2009

C. Sun and P. Vallotton, “Fast linear feature detection using multiple directional non-maximum suppression,” J. Microsc. 234, 147–157 (2009).
[CrossRef]

2008

E. R. Urbach and M. H. F. Wilkinson, “Efficient 2-D grayscale morphological transformations with arbitrary flat structuring elements,” IEEE Trans. Image Process. 17, 1–8 (2008).
[CrossRef]

2006

J. Cha, R. H. Cofer, and S. P. Kozaitis, “Extended Hough transform for linear feature detection,” Pattern Recogn. 39, 1034–1043 (2006).
[CrossRef]

M. Zeng, J. Li, and Z. Peng, “The design of top-hat morphological filter and application to infrared target detection,” Infrared Phys. Technol. 48, 67–76 (2006).
[CrossRef]

I. De, B. Chanda, and B. Chattopadhyay, “Enhancing effective depth-of-field by image fusion using mathematical morphology,” Image Vis. Comput. 24, 1278–1287 (2006).
[CrossRef]

N. Bouaynaya, M. Charif-Chefchaouni, and D. Schonfeld, “Spatially variant morphological restoration and skeleton representation,” IEEE Trans. Image Process. 15, 3579–3591 (2006).
[CrossRef]

2005

J. I. Pastore, E. G. Moler, and V. L. Ballarin, “Segmentation of brain magnetic resonance images through morphological operators and geodesic distance,” Digit. Signal Process 15, 153–160 (2005).
[CrossRef]

F. Zhang, C. Li, and L. Shi, “Detecting and tracking dim moving point target in IR image sequences,” Infrared Phys. Technol. 46, 323–328 (2005).
[CrossRef]

S. Iyer and S. K. Sinha, “A robust approach for automatic detection and segmentation of cracks in underground pipeline images,” Image Vis. Comput. 23, 921–933 (2005).
[CrossRef]

2004

A. C. Jalba, M. H. F. Wilkinson, and J. B. T. M. Roerdink, “Morphological hat-transform scale spaces and their use in pattern classification,” Pattern Recogn. 37, 901–915 (2004).
[CrossRef]

M. Couprie and G. Bertrand, “Topology preserving alternating sequential filter for smoothing two-dimensional and three-dimensional objects,” J. Electron. Imaging 13, 720 (2004).
[CrossRef]

A. C. Jalba, M. H. F. Wilkinson, and J. B. T. M. Roerdink, “Morphological hat-transform scale spaces and their use in pattern classification,” Pattern Recogn. 37, 901–915 (2004).
[CrossRef]

V.-P. Onana, E. Trouvé, G. Mauris, J.-P. Rudant, and E. Tonyé, “Detection of linear features in synthetic-aperture radar images by use of the localized Radon transform and prior information,” Appl. Opt. 43, 264–273 (2004).
[CrossRef]

2003

C. Sun and S. Pallottino, “Circular shortest path in images,” Pattern Recogn. 36, 709–719 (2003).
[CrossRef]

2002

T. Chen, Q. H. Wu, R. Rahmani-Torkaman, and J. Hughes, “A pseudo top-hat mathematical morphological approach to edge detection in dark regions,” Pattern Recogn. 35, 199–210 (2002).
[CrossRef]

F. Gonzalez, O. Tubıo, F. Tobajas, V. De Armas, R. Esper-Chaın, and R. Sarmiento, “Morphological processor for real-time image applications,” Microelectronics J. 33, 1115–1122 (2002).
[CrossRef]

2001

J.-H. Jang and K.-S. Hong, “Linear band detection based on the Euclidean distance transform and a new line segment extraction method,” Pattern Recogn. 34, 1751–1764 (2001).
[CrossRef]

E. Magli, G. Olmo, and L. Lo Presti, “On-board selection of relevant images: an application to linear feature recognition,” IEEE Trans. Image Process. 10, 543–553 (2001).
[CrossRef]

2000

P. T. Jackway, “Improved morphological top-hat,” Electron. Lett. 36, 1194–1195 (2000).
[CrossRef]

1999

1996

M. Van Droogenbroeck and H. Talbot, “Fast computation of morphological operations with arbitrary structuring elements,” Pattern Recogn. Lett. 17, 1451–1460 (1996).
[CrossRef]

1995

H. Park and R. T. Chin, “Decomposition of arbitrarily shaped morphological structuring elements,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 2–15 (1995).
[CrossRef]

1986

J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 679–698 (1986).
[CrossRef]

1979

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man Cybern. 9, 62–66 (1979).
[CrossRef]

Bai, X.

X. Bai, F. Zhou, and B. Xue, “Multiple linear feature detection through top-hat transform by using multi linear structuring elements,” Optik (2011).
[CrossRef]

X. Bai, F. Zhou, and B. Xue, “Fusion of infrared and visual images through region extraction by using multi scale center-surround top-hat transform,” Opt. Express 19, 8444–8457 (2011).
[CrossRef]

X. Bai, F. Zhou, and T. Jin, “Enhancement of dim small target through modified top-hat transformation under the condition of heavy clutter,” Signal Process 90, 1643–1654 (2010).
[CrossRef]

X. Bai and F. Zhou, “Analysis of new top-hat transformation and the application for infrared dim small target detection,” Pattern Recogn. 43, 2145–2156 (2010).
[CrossRef]

X. Bai and F. Zhou, “Edge detection based on mathematical morphology and iterative thresholding,” in International Conference on Computational Intelligence and Security (IEEE, 2006), pp. 1849–1852.

Ballarin, V. L.

J. I. Pastore, E. G. Moler, and V. L. Ballarin, “Segmentation of brain magnetic resonance images through morphological operators and geodesic distance,” Digit. Signal Process 15, 153–160 (2005).
[CrossRef]

Bertrand, G.

M. Couprie and G. Bertrand, “Topology preserving alternating sequential filter for smoothing two-dimensional and three-dimensional objects,” J. Electron. Imaging 13, 720 (2004).
[CrossRef]

Bouaynaya, N.

N. Bouaynaya, M. Charif-Chefchaouni, and D. Schonfeld, “Spatially variant morphological restoration and skeleton representation,” IEEE Trans. Image Process. 15, 3579–3591 (2006).
[CrossRef]

Canny, J.

J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 679–698 (1986).
[CrossRef]

Cha, J.

J. Cha, R. H. Cofer, and S. P. Kozaitis, “Extended Hough transform for linear feature detection,” Pattern Recogn. 39, 1034–1043 (2006).
[CrossRef]

Chanda, B.

I. De, B. Chanda, and B. Chattopadhyay, “Enhancing effective depth-of-field by image fusion using mathematical morphology,” Image Vis. Comput. 24, 1278–1287 (2006).
[CrossRef]

Charif-Chefchaouni, M.

N. Bouaynaya, M. Charif-Chefchaouni, and D. Schonfeld, “Spatially variant morphological restoration and skeleton representation,” IEEE Trans. Image Process. 15, 3579–3591 (2006).
[CrossRef]

Chattopadhyay, B.

I. De, B. Chanda, and B. Chattopadhyay, “Enhancing effective depth-of-field by image fusion using mathematical morphology,” Image Vis. Comput. 24, 1278–1287 (2006).
[CrossRef]

Chen, T.

T. Chen, Q. H. Wu, R. Rahmani-Torkaman, and J. Hughes, “A pseudo top-hat mathematical morphological approach to edge detection in dark regions,” Pattern Recogn. 35, 199–210 (2002).
[CrossRef]

Chin, R. T.

H. Park and R. T. Chin, “Decomposition of arbitrarily shaped morphological structuring elements,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 2–15 (1995).
[CrossRef]

Cofer, R. H.

J. Cha, R. H. Cofer, and S. P. Kozaitis, “Extended Hough transform for linear feature detection,” Pattern Recogn. 39, 1034–1043 (2006).
[CrossRef]

Couprie, M.

M. Couprie and G. Bertrand, “Topology preserving alternating sequential filter for smoothing two-dimensional and three-dimensional objects,” J. Electron. Imaging 13, 720 (2004).
[CrossRef]

De, I.

I. De, B. Chanda, and B. Chattopadhyay, “Enhancing effective depth-of-field by image fusion using mathematical morphology,” Image Vis. Comput. 24, 1278–1287 (2006).
[CrossRef]

De Armas, V.

F. Gonzalez, O. Tubıo, F. Tobajas, V. De Armas, R. Esper-Chaın, and R. Sarmiento, “Morphological processor for real-time image applications,” Microelectronics J. 33, 1115–1122 (2002).
[CrossRef]

Esper-Chain, R.

F. Gonzalez, O. Tubıo, F. Tobajas, V. De Armas, R. Esper-Chaın, and R. Sarmiento, “Morphological processor for real-time image applications,” Microelectronics J. 33, 1115–1122 (2002).
[CrossRef]

Ferrari, J. A.

Flores, J. L.

Gonzalez, F.

F. Gonzalez, O. Tubıo, F. Tobajas, V. De Armas, R. Esper-Chaın, and R. Sarmiento, “Morphological processor for real-time image applications,” Microelectronics J. 33, 1115–1122 (2002).
[CrossRef]

Herrera-Navarro, A. M.

H. Peregrina-Barreto, A. M. Herrera-Navarro, L. A. Morales-Hernández, and I. R. Terol-Villalobos, “Morphological rational operator for contrast enhancement,” J. Opt. S. Am. A 28, 455–464 (2011).
[CrossRef]

Hong, K.-S.

J.-H. Jang and K.-S. Hong, “Linear band detection based on the Euclidean distance transform and a new line segment extraction method,” Pattern Recogn. 34, 1751–1764 (2001).
[CrossRef]

Hughes, J.

T. Chen, Q. H. Wu, R. Rahmani-Torkaman, and J. Hughes, “A pseudo top-hat mathematical morphological approach to edge detection in dark regions,” Pattern Recogn. 35, 199–210 (2002).
[CrossRef]

Iyer, S.

S. Iyer and S. K. Sinha, “A robust approach for automatic detection and segmentation of cracks in underground pipeline images,” Image Vis. Comput. 23, 921–933 (2005).
[CrossRef]

Jackway, P. T.

P. T. Jackway, “Improved morphological top-hat,” Electron. Lett. 36, 1194–1195 (2000).
[CrossRef]

Jalba, A. C.

A. C. Jalba, M. H. F. Wilkinson, and J. B. T. M. Roerdink, “Morphological hat-transform scale spaces and their use in pattern classification,” Pattern Recogn. 37, 901–915 (2004).
[CrossRef]

A. C. Jalba, M. H. F. Wilkinson, and J. B. T. M. Roerdink, “Morphological hat-transform scale spaces and their use in pattern classification,” Pattern Recogn. 37, 901–915 (2004).
[CrossRef]

Jang, J.-H.

J.-H. Jang and K.-S. Hong, “Linear band detection based on the Euclidean distance transform and a new line segment extraction method,” Pattern Recogn. 34, 1751–1764 (2001).
[CrossRef]

Jin, T.

X. Bai, F. Zhou, and T. Jin, “Enhancement of dim small target through modified top-hat transformation under the condition of heavy clutter,” Signal Process 90, 1643–1654 (2010).
[CrossRef]

Jing, H.

Kozaitis, S. P.

J. Cha, R. H. Cofer, and S. P. Kozaitis, “Extended Hough transform for linear feature detection,” Pattern Recogn. 39, 1034–1043 (2006).
[CrossRef]

Li, C.

F. Zhang, C. Li, and L. Shi, “Detecting and tracking dim moving point target in IR image sequences,” Infrared Phys. Technol. 46, 323–328 (2005).
[CrossRef]

Li, J.

M. Zeng, J. Li, and Z. Peng, “The design of top-hat morphological filter and application to infrared target detection,” Infrared Phys. Technol. 48, 67–76 (2006).
[CrossRef]

Liu, L.

Lo Presti, L.

E. Magli, G. Olmo, and L. Lo Presti, “On-board selection of relevant images: an application to linear feature recognition,” IEEE Trans. Image Process. 10, 543–553 (2001).
[CrossRef]

Magli, E.

E. Magli, G. Olmo, and L. Lo Presti, “On-board selection of relevant images: an application to linear feature recognition,” IEEE Trans. Image Process. 10, 543–553 (2001).
[CrossRef]

Mauris, G.

Moler, E. G.

J. I. Pastore, E. G. Moler, and V. L. Ballarin, “Segmentation of brain magnetic resonance images through morphological operators and geodesic distance,” Digit. Signal Process 15, 153–160 (2005).
[CrossRef]

Morales-Hernández, L. A.

H. Peregrina-Barreto, A. M. Herrera-Navarro, L. A. Morales-Hernández, and I. R. Terol-Villalobos, “Morphological rational operator for contrast enhancement,” J. Opt. S. Am. A 28, 455–464 (2011).
[CrossRef]

Olmo, G.

E. Magli, G. Olmo, and L. Lo Presti, “On-board selection of relevant images: an application to linear feature recognition,” IEEE Trans. Image Process. 10, 543–553 (2001).
[CrossRef]

Onana, V.-P.

Otsu, N.

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man Cybern. 9, 62–66 (1979).
[CrossRef]

Pallottino, S.

C. Sun and S. Pallottino, “Circular shortest path in images,” Pattern Recogn. 36, 709–719 (2003).
[CrossRef]

Park, H.

H. Park and R. T. Chin, “Decomposition of arbitrarily shaped morphological structuring elements,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 2–15 (1995).
[CrossRef]

Pastore, J. I.

J. I. Pastore, E. G. Moler, and V. L. Ballarin, “Segmentation of brain magnetic resonance images through morphological operators and geodesic distance,” Digit. Signal Process 15, 153–160 (2005).
[CrossRef]

Peng, Z.

M. Zeng, J. Li, and Z. Peng, “The design of top-hat morphological filter and application to infrared target detection,” Infrared Phys. Technol. 48, 67–76 (2006).
[CrossRef]

Peregrina-Barreto, H.

H. Peregrina-Barreto, A. M. Herrera-Navarro, L. A. Morales-Hernández, and I. R. Terol-Villalobos, “Morphological rational operator for contrast enhancement,” J. Opt. S. Am. A 28, 455–464 (2011).
[CrossRef]

H. Peregrina-Barreto and I. R. Terol-Villalobos, “Morphological rational multi-scale algorithm for color contrast enhancement,” Proc. SPIE 7532, 75320Q (2010).
[CrossRef]

Rahmani-Torkaman, R.

T. Chen, Q. H. Wu, R. Rahmani-Torkaman, and J. Hughes, “A pseudo top-hat mathematical morphological approach to edge detection in dark regions,” Pattern Recogn. 35, 199–210 (2002).
[CrossRef]

Roerdink, J. B. T. M.

A. C. Jalba, M. H. F. Wilkinson, and J. B. T. M. Roerdink, “Morphological hat-transform scale spaces and their use in pattern classification,” Pattern Recogn. 37, 901–915 (2004).
[CrossRef]

A. C. Jalba, M. H. F. Wilkinson, and J. B. T. M. Roerdink, “Morphological hat-transform scale spaces and their use in pattern classification,” Pattern Recogn. 37, 901–915 (2004).
[CrossRef]

Rudant, J.-P.

Sarmiento, R.

F. Gonzalez, O. Tubıo, F. Tobajas, V. De Armas, R. Esper-Chaın, and R. Sarmiento, “Morphological processor for real-time image applications,” Microelectronics J. 33, 1115–1122 (2002).
[CrossRef]

Schonfeld, D.

N. Bouaynaya, M. Charif-Chefchaouni, and D. Schonfeld, “Spatially variant morphological restoration and skeleton representation,” IEEE Trans. Image Process. 15, 3579–3591 (2006).
[CrossRef]

Shi, L.

F. Zhang, C. Li, and L. Shi, “Detecting and tracking dim moving point target in IR image sequences,” Infrared Phys. Technol. 46, 323–328 (2005).
[CrossRef]

Sinha, S. K.

S. Iyer and S. K. Sinha, “A robust approach for automatic detection and segmentation of cracks in underground pipeline images,” Image Vis. Comput. 23, 921–933 (2005).
[CrossRef]

Soille, P.

P. Soille, Morphological Image Analysis: Principle and Applications (Springer, 2003).

Sun, C.

C. Sun and P. Vallotton, “Fast linear feature detection using multiple directional non-maximum suppression,” J. Microsc. 234, 147–157 (2009).
[CrossRef]

C. Sun and S. Pallottino, “Circular shortest path in images,” Pattern Recogn. 36, 709–719 (2003).
[CrossRef]

Talbot, H.

M. Van Droogenbroeck and H. Talbot, “Fast computation of morphological operations with arbitrary structuring elements,” Pattern Recogn. Lett. 17, 1451–1460 (1996).
[CrossRef]

Terol-Villalobos, I. R.

H. Peregrina-Barreto, A. M. Herrera-Navarro, L. A. Morales-Hernández, and I. R. Terol-Villalobos, “Morphological rational operator for contrast enhancement,” J. Opt. S. Am. A 28, 455–464 (2011).
[CrossRef]

H. Peregrina-Barreto and I. R. Terol-Villalobos, “Morphological rational multi-scale algorithm for color contrast enhancement,” Proc. SPIE 7532, 75320Q (2010).
[CrossRef]

Tobajas, F.

F. Gonzalez, O. Tubıo, F. Tobajas, V. De Armas, R. Esper-Chaın, and R. Sarmiento, “Morphological processor for real-time image applications,” Microelectronics J. 33, 1115–1122 (2002).
[CrossRef]

Tonyé, E.

Trouvé, E.

Tubio, O.

F. Gonzalez, O. Tubıo, F. Tobajas, V. De Armas, R. Esper-Chaın, and R. Sarmiento, “Morphological processor for real-time image applications,” Microelectronics J. 33, 1115–1122 (2002).
[CrossRef]

Urbach, E. R.

E. R. Urbach and M. H. F. Wilkinson, “Efficient 2-D grayscale morphological transformations with arbitrary flat structuring elements,” IEEE Trans. Image Process. 17, 1–8 (2008).
[CrossRef]

Vallotton, P.

C. Sun and P. Vallotton, “Fast linear feature detection using multiple directional non-maximum suppression,” J. Microsc. 234, 147–157 (2009).
[CrossRef]

Van Droogenbroeck, M.

M. Van Droogenbroeck and H. Talbot, “Fast computation of morphological operations with arbitrary structuring elements,” Pattern Recogn. Lett. 17, 1451–1460 (1996).
[CrossRef]

Wang, C.

Wilkinson, M. H. F.

E. R. Urbach and M. H. F. Wilkinson, “Efficient 2-D grayscale morphological transformations with arbitrary flat structuring elements,” IEEE Trans. Image Process. 17, 1–8 (2008).
[CrossRef]

A. C. Jalba, M. H. F. Wilkinson, and J. B. T. M. Roerdink, “Morphological hat-transform scale spaces and their use in pattern classification,” Pattern Recogn. 37, 901–915 (2004).
[CrossRef]

A. C. Jalba, M. H. F. Wilkinson, and J. B. T. M. Roerdink, “Morphological hat-transform scale spaces and their use in pattern classification,” Pattern Recogn. 37, 901–915 (2004).
[CrossRef]

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Figures (14)

Fig. 1.
Fig. 1.

Simulated image.

Fig. 2.
Fig. 2.

Constructed structuring element. (a) Bb; (b) ΔB.

Fig. 3.
Fig. 3.

Result of center-surround top-hat transform.

Fig. 4.
Fig. 4.

Result of WTH.

Fig. 5.
Fig. 5.

(a) Inner structuring element Bb0 and (b) marginal structuring element ΔB0.

Fig. 6.
Fig. 6.

Inner and marginal structuring elements at directions 45°, 90°, and 135°.

Fig. 7.
Fig. 7.

Procedure of the algorithm.

Fig. 8.
Fig. 8.

Linear structuring elements used in closing.

Fig. 9.
Fig. 9.

Examples of brick image with one crack line, neurite outgrowth image, and biomedical image with noise: (a) original images; (b) detection results.

Fig. 10.
Fig. 10.

Examples of fingerprint image and skin-and-hair image: (a) original images; (b) detection results.

Fig. 11.
Fig. 11.

Simple example of hair detection: (a) original image; (b) result of Canny edge detector; (c) result of Otsu’s thresholding method; (d) result of classical top-hat transform-based algorithm; (e) result of MDNMS; (f) result of proposed algorithm.

Fig. 12.
Fig. 12.

Example of a document image: (a) original image; (b) result of Canny edge detector; (c) result of Otsu’s thresholding method; (d) result of classical top-hat transform-based algorithm; (e) result of MDNMS; (f) result of proposed algorithm.

Fig. 13.
Fig. 13.

Example of crow skin image: (a) original image; (b) result of Canny edge detector; (c) result of Otsu’s thresholding method; (d) result of classical top-hat transform-based algorithm; (e) result of MDNMS; (f) result of proposed algorithm.

Fig. 14.
Fig. 14.

Example of digital subtraction angiography image: (a) original image; (b) result of Canny edge detector; (c) result of Otsu’s thresholding method; (d) result of classical top-hat transform-based algorithm; (e) result of MDNMS; (f) result of proposed algorithm.

Tables (1)

Tables Icon

Table 1. Quantitative Comparisons Using Pc and Pp

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

fB=maxi,j(f(xi,yj)+B(i,j)),
fB=mini,j(f(x+i,y+j)B(i,j)).
B(i,j)=0,i,j,
fB=maxi,j(f(xi,yj)),
fB=mini,j(f(x+i,y+j)).
fB=(fB)B,
fB=(fB)B.
WTH(x,y)=f(x,y)fB(x,y),
BTH(x,y)=fB(x,y)f(x,y).
fBoi(x,y)=(fΔB)Bb,
fBoi(x,y)=(fΔB)Bb.
NWTH(x,y)=f(x,y)fBoi(x,y),
NBTH(x,y)=fBoi(x,y)f(x,y).
NWTH(x,y)=f(x,y)min(fBoi(x,y),f(x,y)),
NBTH(x,y)=max(fBoi(x,y),f(x,y))f(x,y).
NWTH0(x,y)=f(x,y)min(fBoi(x,y),f(x,y))=f(x,y)min((fΔB0)Bb0,f(x,y)).
NBTH0(x,y)=max(fBoi(x,y),f(x,y))f(x,y)=max((fΔB0)Bb0,f(x,y))f(x,y).
NWTHt(x,y)=f(x,y)min(fBoi(x,y),f(x,y))=f(x,y)min((fΔBt)Bbt,f(x,y)),
NBTHt(x,y)=max(fBoi(x,y),f(x,y))f(x,y)=max((fΔBt)Bbt,f(x,y))f(x,y).
L0=NWTH0B0,L45=NWTH45B45,L90=NWTH90B90,L135=NWTH135B135,
L=L0+L45+L90+L135.
CompletenessPc:PcNcNa,
CorrectnessPp:PpNcNb.

Metrics