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A method based on rank-ordered filter to detect edges in cellular image
Pattern Recognition Letters 30 (2009) 634–640
Contents lists available at ScienceDirect
Pattern Recognition Letters journal homepage: www.elsevier.com/locate/patrec
A method based on rank-ordered filter to detect edges in cellular image Xiaoyin Xu a,*, Zhong Yang b, Yaming Wang b a b
Department of Radiology, Brigham and Women’s Hospital, Boston, MA 02115, USA Department of Anesthesia, Brigham and Women’s Hospital, Boston, MA 02115, USA
a r t i c l e
i n f o
Article history: Received 9 May 2008 Received in revised form 19 November 2008 Available online 9 January 2009 Communicated by J.A. Robinson Keywords: Median filter Order filter Total variation Edge detection Cellular image processing
a b s t r a c t To extract morphological features about nuclei from microscopy cellular image, it is usually required to find the edges of nuclei at first. Standard edge detection methods may not produce satisfactory results due to the varying brightness and background in cellular image. It is important to extract close, smooth, and correct edges in order to compute features like compactness, convexity, roundness, and etc. We present a new method to detect edges of nuclei in microscopy images. The method is based on using median filtering to compute the total variation with respect to the central pixel in a filter window. This step exploits one important feature of median filter, i.e., within the filter window, the total variation (TV) with respect to the median is always less than or equal to the TV with respect to the original center pixel. In other words, median filtering looks for the output that minimizes the total variation within the filtering window. The resulting image has enhanced contrast along the boundary of nuclei. As the final step, we use popular edge detection methods such as Canny detector and Laplacian of Gaussian to find edges of nuclei in the image. Examples from processing real cellular image obtained by light microscope show that the method obtains better edges in terms of connectivity, smoothness, and closely following the boundary of nuclei. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction In cellular imaging, it is important to analyze nucleus morphology by tracking its boundary, shape (Chicurel, 2002; Zimmer et al., 2002). In this work, we focus on finding the boundary of nuclei by edge detection. Once the boundaries of nuclei are found, more features can be extracted about the nuclei, such as their form factor, convexity, compactness, and roundness. These features can be utilized in biomedical quantitative analysis. Popular edge detection methods include Laplacian of Gaussian, Sobel detector, Prewitt detector, and Canny detector (Canny, 1986; Gonzalez and Woods, 2002). With different degrees of complexity, above methods look for an abrupt change in intensity, which indicates the boundary between two regions in an image. Like any detection methods, there are two types of errors in edge detection, missing a true edge and incorrectly detecting a non-existent edge (phantom edge). Above errors cause disconnected edges and spurious lines in the result. Postprocessing is usually required to connect broken edges and remove phantom edges. The advantage of above mentioned methods is their simplicity and fast computation. Among them, Canny edge detector is widely considered the most successful method. In this paper, we describe a method to preprocess cellular image to enhance the boundaries of nuclei. The method * Corresponding author. Tel.: +1 617 525 9596; fax: +1 617 278 6961. E-mail address: [email protected] (X. Xu). 0167-8655/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.patrec.2008.12.009
processes an image by median filtering at first and then computes the total variation with respect to the center pixel in the filter window. The resulting image is then processed by standard edge detection methods such as Canny or Laplacian of Gaussian to find the edges. By design, median filter removes an unrepresentative pixel with the most representative pixel in the filter window. This feature essentially reduces variations in the filter window. We exploit this fact to enhance contrast around the boundaries of nuclei, as will be shown by examples in Section 3. We then apply edge detection on the preprocessed images. Results show that we can achieve better performance in terms of connectivity of the edges and fewer spurious edges detected in the background. To illustrate the scheme of our method, we chose to use Canny detector because it is considered one of the most popular edge detection methods and easy to implement. Canny detector was developed based on the optimal criteria of edge detection: good detection, good localization, and one response to one edge (Canny, 1986). It is straightforward to generate our approach to other edge detection methods. Other more sophisticated edge detection methods include statistical approach (Rakesh et al., 2004; Restrepo et al., 1994), hypothesis testing (Qie and Bhandarkar, 1992), genetic algorithms (Bhandarkar et al., 1994), neural networks (Srinivasan et al., 1995), Bayesian methods (Hebertand Malagre, 1994) and active contour (Xu et al., 1998, 1987). However, with varying computational complexity, these methods tend to take longer time to complete.
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In image processing, median filter is usually used to remove impulse noise (Pitas and Venetsanopoulos, 1990). Compared with convolutional filters, the filter is more robust in that a single very unrepresentative pixel in the filter window will not affect the median value significantly. Also, since the median must actually be one of the pixels in the filter window, the median filter does not create new pixel values when the filter crosses an edge. For this reason the median filter is better at preserving sharp discontinuities than spatial averaging filters (Jain, 1989). However, median filter is prone to alter pixels undisturbed by the noise (Jain, 1989; Arce and Foster, 1989) thereby causing a number of artifacts including edge jitter (Bovik et al., 1985; Nodes and Gallagher, 1984) and streaking (Bovik, 1985). Modified forms of the median filter have been proposed to overcome these shortcomings (Florencio et al., 1994; Chen and Wu, 2001; Yang et al., 1995; Song et al., 1994; Ko and Lee, 1991; Sun and Neuvo, 1994; Abreu et al., 1996; Wang and Zhang, 1999; Hore et al., 2002; Xu et al., 2004; Lin and Yu, 2004). Our method is based on the fact that median filtering reduces the total variation with respect to the center pixel in the filter window. So for each pixel, we calculated the total variation with respect to the center pixel after median filtering and form a new image. As will be shown by examples, this new image has improved contrast along the boundaries of nuclei. Total variation has been widely used to denoise (Vogel and Oman, 1996; Chan et al., 1999; Rudin et al., 1992) and deblur images (Candela et al., 2003; Vogel et al., 1998). In the above applications, total variation is employed as a regularization term to a cost function (Acar and Vogel, 1994; Li and Santosa, 1996). The cost function is then minimized to find solutions to the image denoising and deblurring problems. In this work, instead of using total variation as a regularization term, we look into the resulting image of total variation after median filtering directly and use the image for the purpose of edge detection. The paper is organized as follows. Section 2 discusses the median filter and its property of obtaining minimum total variation with respect to the center pixel in the filter window. Section 3 demonstrates the performance of our algorithm using cellular images. Finally, discussions and conclusions are given in Section 4.
X¼SþB
ð1Þ
where X, S, and B represent the observed image, the noise-free image, and background, respectively. In the ideal case, the background is identically zero. However, due to imperfection in illumination and noise in charge-coupled device (CCD) camera, B has nonzero values. Here we assume B is an independent and identical distributed stochastic variable. The true image S is inhomogeneous because nuclei have different sizes and are in different phases of the cell cycle. Another reason of inhomogeneity of S is the structure of nuclei, like chromatin, is of different brightness. Therefore the challenge is to find the edges of a random signal S in the presence of B. To process cellular images, we make two assumptions that nuclei (1) have closed boundaries, (2) occupy areas larger than a few pixels, presumably larger than the size of filter windows used in preprocessing. The first assumption indicates that the 2D microscopic image captures whole nuclei, instead of parts of nuclei, the only exception are the nuclei located on the boundaries the image. In other words, assuming the boundary can be parameterized as vðtÞ; 0 6 t 6 1, then vð0Þ ¼ vð1Þ if the boundary is closed. The second assumption is based on the fact that the resolution of the microscope is high enough such that a single nucleus occupies tens of pixels. Fig. 1 displays part of a raw image obtained by bright-field microscope. From the figure, we can see that the above assumptions about nuclei are valid. In this work, we exploit an important feature of median filtering to get better result in edge detection, as we will show next. We apply Wiener filter to remove the background B under the assumption that B is white with a variance of r2B , its power spectra is PB ¼ r2B . In noise removal, Wiener filter calculates the local mean and standard deviation of the image pixels in a moving window. Fig. 2a shows the result of Wiener filtering of the original image of Fig. 1. 2.3. Edge enhancement
2. Method
b of size M N, the median filter is applied over Given an image X b a window surrounding the current pixel Xðm; nÞ as
2.1. Nuclei staining In many cases, biological specimens do not have high enough contrast to be satisfactorily imaged by optical microscopy using bright-field illumination. To increase the contrast, nuclei are often stained with reactive organic dyes. The images used in our work were acquired by an Olympus IX-70 microscope with an exposure time of ranging from 10 ms to 20 ms and a 10 objective lens. The nuclei were stained with 40 -6-Diamidino-2-phenylindole (DAPI, 10 lg/ml) which binds to DNA and yields blue fluorescence when excited.
b Yðm; nÞ ¼ medianf Xðm i; n jÞ; for ði; jÞ 2 Wðm; nÞg; for m ¼ 1; . . . ; M;
n ¼ 1; . . . ; N
ð2Þ ð3Þ
2.2. Image model and background removal Today’s microscope can acquire images of nuclei at unprecedented resolution and depth (Stephens and Allan, 2003; Periasamy, 2001). In cellular imaging, usually a large number of nuclei exist in a tissue sample. To extract features about the nuclei, it is important to find edges of nuclei. Many edge detection schemes have been developed over years. However challenges remain on correctly and accurately detecting edges of nuclei. One reason is that a cellular image is often of uneven brightness and it is difficult to separate nuclei from the background. We model the observed cellular image as multiple bright objects appearing on a dark background
Fig. 1. Live nuclei image obtained by an bright-field microscope. The background is not homogeneous. Variation in the background is hard to see due to the large difference between the brightest and the darkest pixels in the image. Scale bar is 100 lm.
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Fig. 2. (a) Wiener filter removes noise and (b) after Wiener filtering, the image is enhanced by the new method for edge detection.
where Wðm; nÞ is a predetermined window centered at position ðm; nÞ. Usually, Wðm; nÞ is chosen to be of odd size (Jain, 1989). Median filter has a feature that can be used to enhance edges. At first we define total variation (TV) with respect to the center pixel as
b TVð Xðm; nÞÞ
X ði;jÞ2Wðm;nÞ
b Xði; jÞ Xðm; nÞ
ð4Þ
where Xðm; nÞ is the central pixel in the filter window. Then we have
X ði;jÞ2Wðm;nÞ
b Xði; jÞ Yðm; nÞ 6
X ði;jÞ2Wðm;nÞ
b Xði; jÞ Xðm; nÞ:
ð5Þ
Eq. (5) shows that median filtering minimizes the total variation with respect to the center pixel. In other words, the median value of a set of numbers minimizes the average of the absolute variation. In image processing we can assume that the pixels of an images have a discrete probably distribution FðxÞ where x ¼ x1 ; . . . ; xmedian ; . . . ; xN and for the pixels within a window we have
Pðx 6 xmedian Þ ¼ Pðx P xmedian Þ ¼
xX median
FðxÞ
ð6Þ
x¼x1
We can demonstrate this behavior using an 1D example. Consider a set of numbers {1, 3, 4, 6, 8, 9, 18}, their median is 6 and mean is 7. The absolute variation is {5, 3, 2, 0, 2, 3, 12} with a total of 27 using the median and {6, 4, 3, 1, 1, 2, 11} with a total of 28 using the mean, implying that the median is more representative of the whole data set than the mean is. b for edge detection, we proceed in two To enhance an image X steps
~ Yðm; nÞ ¼ medianfXðm i; n jÞ; for ði; jÞ 2 Uðm; nÞg X b jÞ Yðm; nÞ Zðm; nÞ ¼ Xði;
ð7Þ ð8Þ
ði;jÞ2Uðm;nÞ
where m ¼ 1; . . . ; M; n ¼ 1; . . . ; N. In Eq. (7) we apply median filter e and stores the result in matrix Y. For each pixel position, Eq. on X (8), calculates the TV with respect to the Y. Note that the two filter windows W and U in steps 1 and 2 may be of different sizes. In our case, we set the sizes of W and U to 3 3. An example of the enhanced image is shown in Fig. 2b. From the figure we noticed that the nuclei tend to have sharper contrast to the background, which should improve edge detection by other methods. 2.4. Edge detection In the next step, we use standard methods, such as Canny detector, to find edges in Z. Among many edge detectors, Canny detector
(Canny, 1986) is regarded as the most successful method. After applying the Canny detector, we used binary morphological operators of ‘spur’ followed by ‘clean’ to remove endpoints of lines and isolated 1’s in the resulting image. 3. Examples We used cellular images to test and demonstrate the performance of our algorithm. Resolution of the sample images used in this section is 0.114 lm in both x and y direction. The grayscale raw image has pixel value between [0, 65535]. Fig. 3 compares the processing results by Otsu’s method, Canny’s edge detector, and our method. Fig. 3a displays the segmentation results given by the Otsu’s method. Because of the uneven background of the original image, the optimal threshold determined by the Otsu’s method was too low to correctly segment the nuclei located to the bottom of the image. Such a result is expected as Otsu’s method searches for a global threshold and the uneven background may introduce bias in calculating the threshold. Fig. 3b displays the edges given by directly applying Canny edge detector on the original image without any preprocessing but with the binary morphological operators described above. We then use Canny detector to find edges after applying Wiener filter and followed by the binary morphological operations, Fig. 3c. The result of our method, followed by the same sequence of binary morphological operations, is shown in Fig. 3d. Comparing the two results, we note that the edges in Fig. 3c have broken sections while the edges in Fig. 3d are connected. Another advantage of the new method is that it generates fewer spurious edges outside the nuclei, comparing Fig. 3d with c. In both figures, the spurious edges can be removed in postprocessing. It is interesting to compare the results of Fig. 3b–d to note that, without preprocessing, Canny edge detector generated a few false edges, as manifested by the small circles in Fig. 3b, and missed a few valid nuclei, while the result of Fig. 3c also has a few false edges and misses. The result of Fig. 3d, by visual inspection, does not have false edges but also has a few missed nuclei. All three approaches have difficulty to accurately segment the clustered nuclei located to the lower left corner of the images. To quantitatively evaluate the detection results of Fig. 3b–d we manually counted the nuclei and compared the result with those shown in the figure. We found that by applying Canny detector without any preprocessing the algorithm detected 79 nuclei, of which 21 were false positive, and missed 33 true nuclei. Applying Canny detector on the Wiener filtered images gave better results as it detected 54 true nuclei, no false detection, and missed 29 true nuclei, Fig. 3c. After preprocessing the image by the new method, Canny detector found 63 true nuclei, no false
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Fig. 3. (a) Segmentation results by Otsu’s method (Otsu, 1979), (b) applying Canny edge detector directly on the original image, (c) applying by Canny detector after Wiener filtering, (d) edges obtained by the new method. Images of (b–d) have been post-processed by binary morphological operations of ‘spur’ and ‘clean’.
Fig. 4. (a) Original nuclei image, (b) preprocessed image, (c) result of Canny detector following Wiener filtering, overlaid on the original image, (d) result of Canny detector using the preprocessed image, U is 3 3. The edges are overlaid on the original image. Scale bar is 100 lm.
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Fig. 5. (a) Original nuclei image, (b) result of applying Canny detector directly on the original image, (c) result of Canny detector following Wiener filtering and (d) result of Canny detector using the preprocessed image, U is 3 3. The edges are overlaid on the original image.
Fig. 6. (a) Original nuclei image, (b) preprocessed image, (c) result of Canny detector following Wiener filtering and (d) result of Canny detector using the preprocessed image, U is 3 3. The edges are overlaid on the original image.
X. Xu et al. / Pattern Recognition Letters 30 (2009) 634–640
detection, and missed 20. Overall the new method performs better in terms of achieving a higher detection rate with a lower rate of false alarm. In all the examples, the thresholds in Canny detector are determined automatically by using the function ‘‘edge” in MatLabÒ. A second example is shown in Fig. 4. The original and the preprocessed images are shown in Fig. 4a and b, respectively. We applied the Canny edge detector on the original image after removing noise by a Wiener filter, Fig. 4c. For comparison, the result of edge detection by our method is shown in Fig. 4d. We then counted the detected nuclei in Fig. 4c and d. The new method correctly detected 72 nuclei while missing 16. The conventional Canny edge detector of Fig. 4c found 60 true nuclei and missed 28 of them. Comparing the results of standard edge detector with those of our method, we noticed that the new method has a higher probability of detecting nuclei by finding enclosed edges. We use another example to examine this effect, Fig. 5. The figure shows nuclei image acquired in a different experiment. The original image shown in Fig. 5a. In this example we did not perform binary morphological operations in order to observe how the detected edges differ. The edges given by Canny detector without any preprocessing, with Wiener filtering, and with preprocessing by the new method are displayed in Fig. 5b–d, respectively. Comparing the three resulting images, we note that some nuclei boundaries are broken in Fig. 5b and c, while the new method tends to generate enclosed boundaries. In our comparison we found the new method improves nuclei boundary detection. However the new method has a drawback that it may create double boundaries for some, especially big, nuclei, as can be seen in Fig. 5d. This artifact can be easily removed by binary morphological operation. Another observation about the new method is that the boundaries given by it appear larger than the boundaries created by standard method. This does not represent a problem in our work since our objective is to detect and count the nuclei in the images. On the other hand, due to the point spread function of the microscope and degree of staining, nuclei do not display sharp boundaries in this type of images. Therefore different detection methods may not generate the exactly same edges, for example, we notice that the boundaries given by standard Canny detector may appear smaller than some nuclei in Fig. 5b and c. Another example is shown in Fig. 6. The original image and intermediate result given by our method are shown in Fig. 6a and b, respectively. Comparing the two images we observe that the intermediate result has a sharper contrast around the boundaries of the nuclei. If we use Wiener filter to remove noise and then apply Canny operator we obtain the result of Fig. 6c which has broken boundaries for the two nuclei located to the top and left of the image. Applying Canny operator on Fig. 6b, however, generates enclosed boundaries, Fig. 6d.
4. Discussions and conclusion In this paper we have introduced a preprocessing method based on median filtering and total variation with respect to the center pixel in the filter window. By exploiting the fact that median filter minimizes the TV with respect to the center pixel, the preprocessing method aims to enhance boundaries of nuclei in microscopic image. The result can then be processed by widely used edge detector such as Laplacian of Gaussian and Canny method. Using real cellular images, we have shown that the new method improves edge detection by standard methods in terms of increasing connectivity of edges and avoiding false edges in the background. The approach developed in this work aims to improve existing methods like Canny detector, especially on some difficult cases in edge detection such as low contrast of some nuclei. Our method is easy to implement and has a fast computational speed. In the
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current form the primary objective of the method is to detect nuclei by counting the number of enclosed boundaries. We note that in some cases the boundaries given by the method appear slightly outward displaced, as seen in Figs. 5 and 6. Due to the finite size of the point spread function of the microscope and various degrees of staining, the true nuclei boundaries cannot be objectively and precisely determined in this type image since the signal intensities do not have abrupt changes at the boundaries. Therefore edge detection methods may generate slightly different boundaries in the results, especially for the methods based on calculating the gradients of the images. It is noted that the method is developed under the two assumptions about the cellular images. Improved performance can be achieved by judiciously combining this method with other approaches in identifying edges.
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Contents lists available at ScienceDirect
Pattern Recognition Letters journal homepage: www.elsevier.com/locate/patrec
A method based on rank-ordered filter to detect edges in cellular image Xiaoyin Xu a,*, Zhong Yang b, Yaming Wang b a b
Department of Radiology, Brigham and Women’s Hospital, Boston, MA 02115, USA Department of Anesthesia, Brigham and Women’s Hospital, Boston, MA 02115, USA
a r t i c l e
i n f o
Article history: Received 9 May 2008 Received in revised form 19 November 2008 Available online 9 January 2009 Communicated by J.A. Robinson Keywords: Median filter Order filter Total variation Edge detection Cellular image processing
a b s t r a c t To extract morphological features about nuclei from microscopy cellular image, it is usually required to find the edges of nuclei at first. Standard edge detection methods may not produce satisfactory results due to the varying brightness and background in cellular image. It is important to extract close, smooth, and correct edges in order to compute features like compactness, convexity, roundness, and etc. We present a new method to detect edges of nuclei in microscopy images. The method is based on using median filtering to compute the total variation with respect to the central pixel in a filter window. This step exploits one important feature of median filter, i.e., within the filter window, the total variation (TV) with respect to the median is always less than or equal to the TV with respect to the original center pixel. In other words, median filtering looks for the output that minimizes the total variation within the filtering window. The resulting image has enhanced contrast along the boundary of nuclei. As the final step, we use popular edge detection methods such as Canny detector and Laplacian of Gaussian to find edges of nuclei in the image. Examples from processing real cellular image obtained by light microscope show that the method obtains better edges in terms of connectivity, smoothness, and closely following the boundary of nuclei. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction In cellular imaging, it is important to analyze nucleus morphology by tracking its boundary, shape (Chicurel, 2002; Zimmer et al., 2002). In this work, we focus on finding the boundary of nuclei by edge detection. Once the boundaries of nuclei are found, more features can be extracted about the nuclei, such as their form factor, convexity, compactness, and roundness. These features can be utilized in biomedical quantitative analysis. Popular edge detection methods include Laplacian of Gaussian, Sobel detector, Prewitt detector, and Canny detector (Canny, 1986; Gonzalez and Woods, 2002). With different degrees of complexity, above methods look for an abrupt change in intensity, which indicates the boundary between two regions in an image. Like any detection methods, there are two types of errors in edge detection, missing a true edge and incorrectly detecting a non-existent edge (phantom edge). Above errors cause disconnected edges and spurious lines in the result. Postprocessing is usually required to connect broken edges and remove phantom edges. The advantage of above mentioned methods is their simplicity and fast computation. Among them, Canny edge detector is widely considered the most successful method. In this paper, we describe a method to preprocess cellular image to enhance the boundaries of nuclei. The method * Corresponding author. Tel.: +1 617 525 9596; fax: +1 617 278 6961. E-mail address: [email protected] (X. Xu). 0167-8655/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.patrec.2008.12.009
processes an image by median filtering at first and then computes the total variation with respect to the center pixel in the filter window. The resulting image is then processed by standard edge detection methods such as Canny or Laplacian of Gaussian to find the edges. By design, median filter removes an unrepresentative pixel with the most representative pixel in the filter window. This feature essentially reduces variations in the filter window. We exploit this fact to enhance contrast around the boundaries of nuclei, as will be shown by examples in Section 3. We then apply edge detection on the preprocessed images. Results show that we can achieve better performance in terms of connectivity of the edges and fewer spurious edges detected in the background. To illustrate the scheme of our method, we chose to use Canny detector because it is considered one of the most popular edge detection methods and easy to implement. Canny detector was developed based on the optimal criteria of edge detection: good detection, good localization, and one response to one edge (Canny, 1986). It is straightforward to generate our approach to other edge detection methods. Other more sophisticated edge detection methods include statistical approach (Rakesh et al., 2004; Restrepo et al., 1994), hypothesis testing (Qie and Bhandarkar, 1992), genetic algorithms (Bhandarkar et al., 1994), neural networks (Srinivasan et al., 1995), Bayesian methods (Hebertand Malagre, 1994) and active contour (Xu et al., 1998, 1987). However, with varying computational complexity, these methods tend to take longer time to complete.
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In image processing, median filter is usually used to remove impulse noise (Pitas and Venetsanopoulos, 1990). Compared with convolutional filters, the filter is more robust in that a single very unrepresentative pixel in the filter window will not affect the median value significantly. Also, since the median must actually be one of the pixels in the filter window, the median filter does not create new pixel values when the filter crosses an edge. For this reason the median filter is better at preserving sharp discontinuities than spatial averaging filters (Jain, 1989). However, median filter is prone to alter pixels undisturbed by the noise (Jain, 1989; Arce and Foster, 1989) thereby causing a number of artifacts including edge jitter (Bovik et al., 1985; Nodes and Gallagher, 1984) and streaking (Bovik, 1985). Modified forms of the median filter have been proposed to overcome these shortcomings (Florencio et al., 1994; Chen and Wu, 2001; Yang et al., 1995; Song et al., 1994; Ko and Lee, 1991; Sun and Neuvo, 1994; Abreu et al., 1996; Wang and Zhang, 1999; Hore et al., 2002; Xu et al., 2004; Lin and Yu, 2004). Our method is based on the fact that median filtering reduces the total variation with respect to the center pixel in the filter window. So for each pixel, we calculated the total variation with respect to the center pixel after median filtering and form a new image. As will be shown by examples, this new image has improved contrast along the boundaries of nuclei. Total variation has been widely used to denoise (Vogel and Oman, 1996; Chan et al., 1999; Rudin et al., 1992) and deblur images (Candela et al., 2003; Vogel et al., 1998). In the above applications, total variation is employed as a regularization term to a cost function (Acar and Vogel, 1994; Li and Santosa, 1996). The cost function is then minimized to find solutions to the image denoising and deblurring problems. In this work, instead of using total variation as a regularization term, we look into the resulting image of total variation after median filtering directly and use the image for the purpose of edge detection. The paper is organized as follows. Section 2 discusses the median filter and its property of obtaining minimum total variation with respect to the center pixel in the filter window. Section 3 demonstrates the performance of our algorithm using cellular images. Finally, discussions and conclusions are given in Section 4.
X¼SþB
ð1Þ
where X, S, and B represent the observed image, the noise-free image, and background, respectively. In the ideal case, the background is identically zero. However, due to imperfection in illumination and noise in charge-coupled device (CCD) camera, B has nonzero values. Here we assume B is an independent and identical distributed stochastic variable. The true image S is inhomogeneous because nuclei have different sizes and are in different phases of the cell cycle. Another reason of inhomogeneity of S is the structure of nuclei, like chromatin, is of different brightness. Therefore the challenge is to find the edges of a random signal S in the presence of B. To process cellular images, we make two assumptions that nuclei (1) have closed boundaries, (2) occupy areas larger than a few pixels, presumably larger than the size of filter windows used in preprocessing. The first assumption indicates that the 2D microscopic image captures whole nuclei, instead of parts of nuclei, the only exception are the nuclei located on the boundaries the image. In other words, assuming the boundary can be parameterized as vðtÞ; 0 6 t 6 1, then vð0Þ ¼ vð1Þ if the boundary is closed. The second assumption is based on the fact that the resolution of the microscope is high enough such that a single nucleus occupies tens of pixels. Fig. 1 displays part of a raw image obtained by bright-field microscope. From the figure, we can see that the above assumptions about nuclei are valid. In this work, we exploit an important feature of median filtering to get better result in edge detection, as we will show next. We apply Wiener filter to remove the background B under the assumption that B is white with a variance of r2B , its power spectra is PB ¼ r2B . In noise removal, Wiener filter calculates the local mean and standard deviation of the image pixels in a moving window. Fig. 2a shows the result of Wiener filtering of the original image of Fig. 1. 2.3. Edge enhancement
2. Method
b of size M N, the median filter is applied over Given an image X b a window surrounding the current pixel Xðm; nÞ as
2.1. Nuclei staining In many cases, biological specimens do not have high enough contrast to be satisfactorily imaged by optical microscopy using bright-field illumination. To increase the contrast, nuclei are often stained with reactive organic dyes. The images used in our work were acquired by an Olympus IX-70 microscope with an exposure time of ranging from 10 ms to 20 ms and a 10 objective lens. The nuclei were stained with 40 -6-Diamidino-2-phenylindole (DAPI, 10 lg/ml) which binds to DNA and yields blue fluorescence when excited.
b Yðm; nÞ ¼ medianf Xðm i; n jÞ; for ði; jÞ 2 Wðm; nÞg; for m ¼ 1; . . . ; M;
n ¼ 1; . . . ; N
ð2Þ ð3Þ
2.2. Image model and background removal Today’s microscope can acquire images of nuclei at unprecedented resolution and depth (Stephens and Allan, 2003; Periasamy, 2001). In cellular imaging, usually a large number of nuclei exist in a tissue sample. To extract features about the nuclei, it is important to find edges of nuclei. Many edge detection schemes have been developed over years. However challenges remain on correctly and accurately detecting edges of nuclei. One reason is that a cellular image is often of uneven brightness and it is difficult to separate nuclei from the background. We model the observed cellular image as multiple bright objects appearing on a dark background
Fig. 1. Live nuclei image obtained by an bright-field microscope. The background is not homogeneous. Variation in the background is hard to see due to the large difference between the brightest and the darkest pixels in the image. Scale bar is 100 lm.
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Fig. 2. (a) Wiener filter removes noise and (b) after Wiener filtering, the image is enhanced by the new method for edge detection.
where Wðm; nÞ is a predetermined window centered at position ðm; nÞ. Usually, Wðm; nÞ is chosen to be of odd size (Jain, 1989). Median filter has a feature that can be used to enhance edges. At first we define total variation (TV) with respect to the center pixel as
b TVð Xðm; nÞÞ
X ði;jÞ2Wðm;nÞ
b Xði; jÞ Xðm; nÞ
ð4Þ
where Xðm; nÞ is the central pixel in the filter window. Then we have
X ði;jÞ2Wðm;nÞ
b Xði; jÞ Yðm; nÞ 6
X ði;jÞ2Wðm;nÞ
b Xði; jÞ Xðm; nÞ:
ð5Þ
Eq. (5) shows that median filtering minimizes the total variation with respect to the center pixel. In other words, the median value of a set of numbers minimizes the average of the absolute variation. In image processing we can assume that the pixels of an images have a discrete probably distribution FðxÞ where x ¼ x1 ; . . . ; xmedian ; . . . ; xN and for the pixels within a window we have
Pðx 6 xmedian Þ ¼ Pðx P xmedian Þ ¼
xX median
FðxÞ
ð6Þ
x¼x1
We can demonstrate this behavior using an 1D example. Consider a set of numbers {1, 3, 4, 6, 8, 9, 18}, their median is 6 and mean is 7. The absolute variation is {5, 3, 2, 0, 2, 3, 12} with a total of 27 using the median and {6, 4, 3, 1, 1, 2, 11} with a total of 28 using the mean, implying that the median is more representative of the whole data set than the mean is. b for edge detection, we proceed in two To enhance an image X steps
~ Yðm; nÞ ¼ medianfXðm i; n jÞ; for ði; jÞ 2 Uðm; nÞg X b jÞ Yðm; nÞ Zðm; nÞ ¼ Xði;
ð7Þ ð8Þ
ði;jÞ2Uðm;nÞ
where m ¼ 1; . . . ; M; n ¼ 1; . . . ; N. In Eq. (7) we apply median filter e and stores the result in matrix Y. For each pixel position, Eq. on X (8), calculates the TV with respect to the Y. Note that the two filter windows W and U in steps 1 and 2 may be of different sizes. In our case, we set the sizes of W and U to 3 3. An example of the enhanced image is shown in Fig. 2b. From the figure we noticed that the nuclei tend to have sharper contrast to the background, which should improve edge detection by other methods. 2.4. Edge detection In the next step, we use standard methods, such as Canny detector, to find edges in Z. Among many edge detectors, Canny detector
(Canny, 1986) is regarded as the most successful method. After applying the Canny detector, we used binary morphological operators of ‘spur’ followed by ‘clean’ to remove endpoints of lines and isolated 1’s in the resulting image. 3. Examples We used cellular images to test and demonstrate the performance of our algorithm. Resolution of the sample images used in this section is 0.114 lm in both x and y direction. The grayscale raw image has pixel value between [0, 65535]. Fig. 3 compares the processing results by Otsu’s method, Canny’s edge detector, and our method. Fig. 3a displays the segmentation results given by the Otsu’s method. Because of the uneven background of the original image, the optimal threshold determined by the Otsu’s method was too low to correctly segment the nuclei located to the bottom of the image. Such a result is expected as Otsu’s method searches for a global threshold and the uneven background may introduce bias in calculating the threshold. Fig. 3b displays the edges given by directly applying Canny edge detector on the original image without any preprocessing but with the binary morphological operators described above. We then use Canny detector to find edges after applying Wiener filter and followed by the binary morphological operations, Fig. 3c. The result of our method, followed by the same sequence of binary morphological operations, is shown in Fig. 3d. Comparing the two results, we note that the edges in Fig. 3c have broken sections while the edges in Fig. 3d are connected. Another advantage of the new method is that it generates fewer spurious edges outside the nuclei, comparing Fig. 3d with c. In both figures, the spurious edges can be removed in postprocessing. It is interesting to compare the results of Fig. 3b–d to note that, without preprocessing, Canny edge detector generated a few false edges, as manifested by the small circles in Fig. 3b, and missed a few valid nuclei, while the result of Fig. 3c also has a few false edges and misses. The result of Fig. 3d, by visual inspection, does not have false edges but also has a few missed nuclei. All three approaches have difficulty to accurately segment the clustered nuclei located to the lower left corner of the images. To quantitatively evaluate the detection results of Fig. 3b–d we manually counted the nuclei and compared the result with those shown in the figure. We found that by applying Canny detector without any preprocessing the algorithm detected 79 nuclei, of which 21 were false positive, and missed 33 true nuclei. Applying Canny detector on the Wiener filtered images gave better results as it detected 54 true nuclei, no false detection, and missed 29 true nuclei, Fig. 3c. After preprocessing the image by the new method, Canny detector found 63 true nuclei, no false
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Fig. 3. (a) Segmentation results by Otsu’s method (Otsu, 1979), (b) applying Canny edge detector directly on the original image, (c) applying by Canny detector after Wiener filtering, (d) edges obtained by the new method. Images of (b–d) have been post-processed by binary morphological operations of ‘spur’ and ‘clean’.
Fig. 4. (a) Original nuclei image, (b) preprocessed image, (c) result of Canny detector following Wiener filtering, overlaid on the original image, (d) result of Canny detector using the preprocessed image, U is 3 3. The edges are overlaid on the original image. Scale bar is 100 lm.
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Fig. 5. (a) Original nuclei image, (b) result of applying Canny detector directly on the original image, (c) result of Canny detector following Wiener filtering and (d) result of Canny detector using the preprocessed image, U is 3 3. The edges are overlaid on the original image.
Fig. 6. (a) Original nuclei image, (b) preprocessed image, (c) result of Canny detector following Wiener filtering and (d) result of Canny detector using the preprocessed image, U is 3 3. The edges are overlaid on the original image.
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detection, and missed 20. Overall the new method performs better in terms of achieving a higher detection rate with a lower rate of false alarm. In all the examples, the thresholds in Canny detector are determined automatically by using the function ‘‘edge” in MatLabÒ. A second example is shown in Fig. 4. The original and the preprocessed images are shown in Fig. 4a and b, respectively. We applied the Canny edge detector on the original image after removing noise by a Wiener filter, Fig. 4c. For comparison, the result of edge detection by our method is shown in Fig. 4d. We then counted the detected nuclei in Fig. 4c and d. The new method correctly detected 72 nuclei while missing 16. The conventional Canny edge detector of Fig. 4c found 60 true nuclei and missed 28 of them. Comparing the results of standard edge detector with those of our method, we noticed that the new method has a higher probability of detecting nuclei by finding enclosed edges. We use another example to examine this effect, Fig. 5. The figure shows nuclei image acquired in a different experiment. The original image shown in Fig. 5a. In this example we did not perform binary morphological operations in order to observe how the detected edges differ. The edges given by Canny detector without any preprocessing, with Wiener filtering, and with preprocessing by the new method are displayed in Fig. 5b–d, respectively. Comparing the three resulting images, we note that some nuclei boundaries are broken in Fig. 5b and c, while the new method tends to generate enclosed boundaries. In our comparison we found the new method improves nuclei boundary detection. However the new method has a drawback that it may create double boundaries for some, especially big, nuclei, as can be seen in Fig. 5d. This artifact can be easily removed by binary morphological operation. Another observation about the new method is that the boundaries given by it appear larger than the boundaries created by standard method. This does not represent a problem in our work since our objective is to detect and count the nuclei in the images. On the other hand, due to the point spread function of the microscope and degree of staining, nuclei do not display sharp boundaries in this type of images. Therefore different detection methods may not generate the exactly same edges, for example, we notice that the boundaries given by standard Canny detector may appear smaller than some nuclei in Fig. 5b and c. Another example is shown in Fig. 6. The original image and intermediate result given by our method are shown in Fig. 6a and b, respectively. Comparing the two images we observe that the intermediate result has a sharper contrast around the boundaries of the nuclei. If we use Wiener filter to remove noise and then apply Canny operator we obtain the result of Fig. 6c which has broken boundaries for the two nuclei located to the top and left of the image. Applying Canny operator on Fig. 6b, however, generates enclosed boundaries, Fig. 6d.
4. Discussions and conclusion In this paper we have introduced a preprocessing method based on median filtering and total variation with respect to the center pixel in the filter window. By exploiting the fact that median filter minimizes the TV with respect to the center pixel, the preprocessing method aims to enhance boundaries of nuclei in microscopic image. The result can then be processed by widely used edge detector such as Laplacian of Gaussian and Canny method. Using real cellular images, we have shown that the new method improves edge detection by standard methods in terms of increasing connectivity of edges and avoiding false edges in the background. The approach developed in this work aims to improve existing methods like Canny detector, especially on some difficult cases in edge detection such as low contrast of some nuclei. Our method is easy to implement and has a fast computational speed. In the
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current form the primary objective of the method is to detect nuclei by counting the number of enclosed boundaries. We note that in some cases the boundaries given by the method appear slightly outward displaced, as seen in Figs. 5 and 6. Due to the finite size of the point spread function of the microscope and various degrees of staining, the true nuclei boundaries cannot be objectively and precisely determined in this type image since the signal intensities do not have abrupt changes at the boundaries. Therefore edge detection methods may generate slightly different boundaries in the results, especially for the methods based on calculating the gradients of the images. It is noted that the method is developed under the two assumptions about the cellular images. Improved performance can be achieved by judiciously combining this method with other approaches in identifying edges.
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