Multi-scale toggle operator for constructing image sharpness measure

Optics & Laser Technology 44 (2012) 2004–2014

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Multi-scale toggle operator for constructing image sharpness measure Xiangzhi Bai n, Fugen Zhou, Bindang Xue Image Processing Center, Beijing University of Aeronautics and Astronautics, 100191 Beijing, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 January 2012 Received in revised form 5 March 2012 Accepted 27 March 2012 Available online 14 April 2012

To construct effective image sharpness measure with good discrimination ability, a multi-scale toggle operator based algorithm is proposed in this paper. Firstly, toggle operator is used to extract image details. And, the multi-scale theory is used in toggle operator to extract multi-scale image details. Then, the final image details are obtained through applying the pixel-wise maximum operation on the extracted multi-scale image details. Finally, the mean value of the obtained final image details is used as the constructed image sharpness measure. Experimental results on different images show that, the proposed image sharpness measure could correctly quantify the clarity of image and is suitable for discriminating the image clarity change. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Image sharpness measure Toggle operator Mathematical morphology

1. Introduction Image clarity is key to an image based application, such as microscopy imaging, verifying the performance of image processing and so on [1–4]. Image sharpness is an important measure of image clarity. Measuring image sharpness is very useful for identifying and producing the clear image in optical image related applications. The effective image sharpness measure should well quantify the image clarity and has good ability for identifying the small clarity change of an image. This would be very important for identifying and producing clear image in different applications. To effectively measure the clarity of an image, different types of measures have been used. Gray values in clear image are different in different areas, which indicate different standard deviation values. So, standard deviation value could be used as sharpness measure. However, the standard deviation value could not well discriminate the small clarity change. Entropy based measures [5,6] are effective for measuring image clarity; however, they are not sensitive to the gray value change. Using the spatial information [7] is a direct way to measure the clarity of an image. But, this measure may give incorrect results for some images with similar clarity. Gradient information is the obvious and direct features of an image. A clear image indicates clear image gradient with large gray values in the gradient map of the image. So, using the gradient information could construct effective image sharpness measure [8]. But, the discrimination ability of this measure for small image clarity change is not good. These mean, although some image sharpness measures are proposed,

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they may not have good ability to discriminate the image clarity change. So, it would be meaningful to propose a good image sharpness measure with effective ability for discriminating the image clarity change. Images with good clarity are usually very clear. Clear images contain more image details. Thus, well identifying and utilizing these image details would produce good image sharpness measure. Morphological operators [9–15] could extract image details for different applications. Especially, morphological toggle operator [9,13,14] using dilation and erosion could be well used to identify the real image details through the selective output corresponding to different rules. Moreover, multi-scale theory [15,16] of mathematical morphology could also be used by toggle operator to extract image details at different scales of image. In our previous work [13], we used toggle operator to achieve the edge preserved image fusion, which indicates that the toggle operator could be well used to identify the edge features. These edge features represent the image details which are important image features for identifying image sharpness. Therefore, based on toggle operator, an effective image sharpness measure may be constructed. Many of the work in this paper are different from our previous work in [13]. (1) Ref. [13] used toggle operator for image fusion, this paper uses toggle operator to construct image sharpness measure; (2) Ref. [13] extracts important features from different original images to form one final fusion image, this paper extracts the bright and dark image details of the original image to quantify the clarity of the original image; (3) Because the ideas and purposes are very different, the implementations of Ref. [13] and this paper are different. In this paper, a new image sharpness measure based on the morphological toggle operator is proposed. Firstly, dilation and

X. Bai et al. / Optics & Laser Technology 44 (2012) 2004–2014

erosion based toggle operator for feature extraction and the multiscale extension are discussed. Secondly, the identified multi-scale image details through the multi-scale toggle operator are used to calculate the final image details for constructing the image sharpness measure. Finally, the mean value of the calculated final image details is used as the proposed image sharpness measure. The main contributions of this paper are: (1) giving the way of using morphological toggle operator to extract image details for sharpness measuring; (2) proposing a new image sharpness measure based on the toggle operator; (3) improving the ability of the measure to discriminate small image clarity change. Standard images, medical images, mineral images and other images from different applications are used to verify the effectiveness of the proposed image sharpness measure, and the results show that the proposed image sharpness measure performs well for measuring image sharpness and has good ability to discriminate the image clarity. Therefore, this measure would be very useful for different applications related to image clarity measurement, including image fusion, image enhancement, auto-focusing and so on.

2. Morphological toggle operator Mathematical morphology has been an important theory and been widely used in image analysis and pattern recognition [9–16]. The base of mathematical morphology is geometry and set theory [9]. The basic operations are defined based on two sets: the image f(x, y) to be processed and the used structuring element B(u, v) to process f(x, y). (x, y) and (u, v) represent the pixel coordinates of f and B, respectively. B represents the symmetrical of B. The two basic morphological operations, which are dilation and erosion and denoted by f B and fY B, are defined as follows:  f  Bðx,yÞ ¼ maxðf ðxu,yvÞ þ Bðu,vÞÞ u,v

f YBðx,yÞ ¼ minðf ðx þu,y þvÞBðu,vÞÞ u,v

Based on dilation and erosion, one type of toggle operator is defined as follows [9,13,14]: 8 > < f  Bðx,yÞ, if f  Bðx,yÞf ðx,yÞ o f ðx,yÞf YBðx,yÞ TOðx,yÞ ¼ f YBðx,yÞ, if f  Bðx,yÞf ðx,yÞ 4 f ðx,yÞf YBðx,yÞ > : f ðx,yÞ, else Moreover, to identify the important image features, another definition of TO can be defined as follows: 8 > < f  Bðx,yÞ, if f  Bðx,yÞf ðx,yÞ of ðx,yÞf YBðx,yÞnT TOðx,yÞ ¼ f YBðx,yÞ, if f  Bðx,yÞf ðx,yÞnT 4 f ðx,yÞf YBðx,yÞ : > : f ðx,yÞ, else This definition of TO indicates that, the value of each pixel (x, y) in this definition of TO is an output of the same pixel selected from the result of dilation or erosion or the original image. This outputted value is very close to the gray value of the same pixel in the original image, which is specified by the positive value nT. Therefore, this TO could preserve the original image information better, and it would be useful for identifying the important image features. nT is a positive value, which could be selected following some prior knowledge.

3. Image sharpness measure based on toggle operator 3.1. Multi-scale toggle operator Structuring element is one important parameter in morphological operator [13–16], including toggle operator [13,14]. The result of dilation or erosion corresponds to the structuring

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element B being used. Thus, the final result of the toggle operator, which is the selective output from the result of dilation or erosion or the original image, also corresponds to the structuring element B being used. Therefore, TO identifies the image features at the scale corresponding to the size of the structuring element B. However, only one structuring element is used in toggle operator. And, image features may exist at different scales of image. So, to identify all the image features, multi-scale structuring elements with different sizes should be used in toggle operator [13]. Let B1, B2, y, Bn represent n scales of structuring elements with the same shape and increasing sizes. Bi ¼ B1  B1 . . .  B1 , 1r i rn: |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} dilationi times

Using the multi-scale structuring element Bi(u, v), the multiscale dilation and erosion are given below. f  Bi ðx,yÞ ¼ maxðf ðxu,yvÞ þ B i ðu,vÞÞ, u,v

f YBi ðx,yÞ ¼ minðf ðx þ u,yþ vÞBi ðu,vÞÞ: u,v

Based on the multi-scale dilation and erosion, the multi-scale toggle contrast operator using the multi-scale structuring element Bi(u, v) is calculated as follows: 8 > < f  Bi ðx,yÞ, if f  Bi ðx,yÞf ðx,yÞ of ðx,yÞf YBi ðx,yÞnT TOi ðx,yÞ ¼ f YBi ðx,yÞ, if f  Bi ðx,yÞf ðx,yÞnT 4f ðx,yÞf YBi ðx,yÞ : > : f ðx,yÞ, else Through calculating the multi-scale toggle operator, the multiscale image features could be identified by the toggle operator. 3.2. Multi-scale feature extraction based on toggle operator The results of dilation and erosion satisfy the following relationships [9,13]: f  Bðx,yÞ Zf ðx,yÞ, f YBðx,yÞ r f ðx,yÞ: The gray values of the pixels in the result of TO are from the result of the dilation or erosion or the original image. So, some gray values of the result of TO may be larger than the gray values of the same pixels in the original image, whereas some gray values of the result of TO may be smaller than the gray values of the same pixels in the original image. These gray values could be extracted as image details for different applications [13]. The gray values of the result of TO which are larger than the gray values of the same pixels in the original image come from the result of the dilation operation. These gray values represent the bright image details identified by TO. And, these bright image details can be calculated through comparing the result of TO and the original image as follows: DTOðf Þðx,yÞ ¼ max ðTOðf Þðx,yÞf ðx,yÞ,0Þ: DTO extracts bright image details identified by TO. These bright image details are the output gray values produced by the dilation operation. Similarly, the gray values of the result of TO which are smaller than the gray values of the same pixels in the original image come from the result of the erosion operation. These gray values represent the dark image details identified by TO. And, these dark image details can be calculated through comparing the result of TO and the original image as follows: ETOðf Þðx,yÞ ¼ max ðTOðf Þðx,yÞf ðx,yÞ,0Þ: ETO extracts dark image details identified by TO. These dark image details are the output gray values produced by the erosion operation.

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Image details in an image include bright and dark image details. DTO and ETO represent the bright and dark image details identified by TO, respectively. These identified bright and dark image details are the important image features to quantify the clarity of an image. The identified bright and dark image details by DTO and ETO correspond to the structuring element being used. To extract all the useful image details, the multi-scale toggle operator should be used to identify the multi-scale image details. The identified multi-scale bright and dark image details through toggle operator at each scale i can be calculated following the expression of DTO and ETO as follows, respectively.

ETOðf Þðx,yÞ ¼ max ðf ðx,yÞTOi ðf Þðx,yÞ,0Þ:

DTOðf Þðx,yÞ ¼ max ðTOi ðf Þðx,yÞf ðx,yÞ,0Þ,

EFðx,yÞ ¼ maxðETOi ðx,yÞÞ:

3.3. Constructing sharpness measure The extracted bright image details at each scale have larger gray values than other scales. So, the final bright image details extracted from all the scales can be calculated as follows. DFðx,yÞ ¼ maxðDTOi ðx,yÞÞ: i

Similarly, the extracted dark image details at each scale have larger gray values than other scales. So, the final dark image details extracted from all the scales can be calculated as follows: i

Fig. 1. The original Lena image and the processed images by different filters.

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Image details include bright and dark image details. So, the final identified image details can be obtained through combing the final bright and dark image details as follows: FIDðx,yÞ ¼ DFðx,yÞ þ EFðx,yÞ: A clear image should have more image details, that is, large mean value of FID. Therefore, the mean value of FID could be used to measure the clarity of an image, which is named as morphological toggle operator based image sharpness measure and denoted by MTOICM as follows:

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3 to 5 scales would be enough and could achieve an effective result. In this paper, we use the scale number n ¼4. Flat structuring element is the simple and widely used structuring element. And, the structuring element with square shape is the widely and well used type of structuring element. Therefore, in this paper, we use the flat structuring element with square shape.

4. Experimental results

MTOICM ¼ meanðFIDðx,yÞÞ: x,y

3.4. Parameter specification The main parameters used in this paper are the structuring element and scale number. The scale number n decides the quantity of the identified image details. Large scale number indicates more identified image details from more scales. However, there is no need to use a very large scale number, and a large scale number will dramatically increase the calculation time. We have tried the measure on different images. The results show that,

To verify the effectiveness of MTOICM, the standard images, mineral images, medical images and other images from different applications are used in this paper. Also, to do the comparison, some widely used measures including the standard deviation value (STD), entropy value (E) [5,6], spatial frequency (SF) [8] and linear index of fuzziness (LIF) [7] are calculated in this paper. STD is one classical and widely used measure. A clear image contains more image details, which indicates a bigger STD value. A clear image should have more useful information. Therefore, E which is an entropy based measure should be a large value for a clear image. SF is a gradient based measure. A clear image should have

Table 1 Values of different measures calculated using the original Lena image and the processed images.

Original image Median filter smoothed image Average filter smoothed image Gaussian filter smoothed image Unsharp filter smoothed image High-boost filter smoothed image

STD

E

SF

LIF

MTOICM

47.9403 47.6691 47.0611 47.5474 54.0754 57.2267

7.44851 7.4204 7.41644 7.43182 7.66401 6.75729

14.1438 11.1994 9.6067 11.7284 42.4476 26.3517

0.5395 0.5467 0.5518 0.5465 0.5132 0.4771

2.2949 1.5553 1.4341 1.8817 10.1558 6.3390

0.35

0.4

0.3

0.35 0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0

STD

E

SF

LIF

MTOICM

0.2

STD

E

SF

LIF

STD

E

SF

LIF

MTOICM

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.15 0.1 0.05 0 STD

E

SF

LIF

MTOICM

MTOICM

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 STD

E

SF

LIF

MTOICM

Fig. 2. D values of different measures on the original Lena image and the corresponding processed image by each filter for discrimination ability comparison.

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a big SF value. Spatial information based image sharpness measure is also one type of important measures. And, LIF is an effectively and widely used spatial information based measure. 0.5 0.4 0.3 0.2 0.1 0 STD

E

SF

LIF

MTOICM

Fig. 3. The MD value of each measure for the overall discrimination ability comparison using the Lena image.

The spatial features of a clear image should be clear, which means the uncertainty of these features should be small. Then, a small value of LIF indicates a clear image. These measures are widely used in various applications. To verify the performance of the proposed measure, different images and the corresponding processed images by median filter, average filter, Gaussian filter, unsharp filter and high-boost filter are used to calculate the values of different measures. Median filter is one effective non-linear filter in image processing. After the smoothing with a median filter, an image should not be clearer than the original image. Moreover, the difference between the smoothed image and the original image is small. Then, if this small difference could be well discriminated, the used image sharpness measure would be very effective. Thus, using the smoothed image by median filter and the original image, to

Fig. 4. The original Cameraman image and the processed images by different filters.

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calculate the values of different measures and do the comparison, could not only verify the effective performance of the proposed measure, but also show how good the measure in discriminating the small change of image clarity. Therefore, median filter could be used to do the comparison. Moreover, to do the comparison on more processed images by different filters, the result images processed by other widely used image filters, including the average filter, Gaussian filter, unsharp filter and high-boost filter are also used in the comparison results. The average filter and Gaussian filter are two important and widely used low-pass filters. These filters may smooth image details, and the smoothed images become unclear. The unsharp filter and high-boost filter are two important and well used high-pass filters. These filters will sharpen the original image and enhance image details. The processed images by these two filters become clear. So, using the processed images by these filters would completely demonstrate the effective performance of MTOICM. The size of the used median filter and average filter is 3  3. The Gaussian filter has size 3  3 and sigma 0.5. The parameter alpha of unsharp filter is 0.2. The boost parameter in high-boost filter is 2. All the parameters in these filters are well and widely used. Fig. 1 is an example of the standard Lena image (a) is the original image, (b) is the processed image by median filter, (c) is the processed image by average filter, (d) is the processed image

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by Gaussian filter, (e) is the processed image by unsharp filter, (f) is the processed image by high-boost filter. The difference between the original and processed images is small. The original image is clearer than the processed images by median filter, average filter and Gaussian filter. The processed images by unsharp filter and high-boost filter are clearer than the original image. The values of different measures on the original and processed images by different filters are shown in Table 1. Table 1 shows that, because the original image is clearer than the processed images by median filter, average filter and Gaussian filter, the values of STD, E, SF and MTOICM on the original image are larger than the values of STD, E, SF and MTOICM on the processed images by median filter, average filter and Gaussian filter, respectively. And, because the processed images by unsharp filter and high-boost filter are clearer than the original image, the values of STD, E, SF and MTOICM on the original image are smaller than the values of STD, E, SF and MTOICM on the processed images by unsharp filter and high-boost filter, respectively. Also, the value of LIF on the original image is smaller than the value of LIF on the processed images by median filter, average filter and Gaussian filter, while the value of LIF on the original image is larger than the value of LIF on the processed images by unsharp filter and high-boost filter. So, all the measures including the MTOICM can correctly quantify the clarity of image.

Table 2 Values of different measures calculated using the original Cameraman image and the processed images.

Original image Median filter smoothed image Average filter smoothed image Gaussian filter smoothed image Unsharp filter smoothed image High-boost filter smoothed image

STD

E

SF

LIF

MTOICM

61.9962 61.6281 61.0505 61.6150 65.2509 64.2603

7.04664 6.99555 7.02707 7.04207 7.15365 6.16699

14.3603 12.7048 11.514 13.0665 27.0701 21.7039

0.5755 0.5785 0.5786 0.5767 0.5650 0.5584

2.8505 2.0760 2.2529 2.5764 6.8733 4.7753

0.3

0.25

0.25

0.2

0.2

0.15

0.15 0.1

0.1

0.05

0.05 0

0 STD

E

SF

LIF

MTOICM

0.1

0.6

0.08

0.5

STD

E

SF

STD

E

SF

LIF

MTOICM

0.4

0.06

0.3 0.04

0.2

0.02

0.1

0

0 STD

E

SF

LIF

MTOICM

STD

E

LIF

MTOICM

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 SF

LIF

MTOICM

Fig. 5. D values of different measures on the original Cameraman image and the corresponding processed image by each filter for discrimination ability comparison.

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Moreover, the difference between the values of MTOICM on the original image and the processed images is large. This means, MTOICM is good at discriminating image clarity change. To 0.35

quantitatively compare the ability of different measures for discriminating the image clarity change, a quantitative value calculated using the following formula on the original and the processed image by each filter is used. D ¼ 9MOri 2MPr 9=maxðMOri , MPr Þ:

0.3 0.25 0.2 0.15 0.1 0.05 0 STD

E

SF

LIF

MTOICM

Fig. 6. The MD value of each measure for the overall discrimination ability comparison using the Cameraman image.

MOri is the value of one measure calculated using the original image. MPr is the value of the same measure calculated using the processed image by one filter. D is the absolute difference between the values of MOri and MPr dividing by the maximum value of the two values. If D is large, 9MOri MPr9 would be large corresponding to the value max (MOri, MPr). Then, the ability of the corresponding measure to discriminate the image clarity change would be good. Therefore, a large D value indicates good ability of the corresponding measure to discriminate the image

Fig. 7. The original living room image and the processed images by different filters.

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clarity change. The D values of different measures on the original and processed Lena images by each filter are calculated and shown in Fig. 2. Fig. 2(a) shows the D values of different measures on the original and processed Lena image by median filter, (b) shows the D values of different measures on the original and processed Lena image by average filter, (c) shows the D values of different measures on the original and processed Lena image by Gaussian filter, (d) shows the D values of different measures on the original and processed Lena image by unsharp filter, (e) shows the D values of different measures on the original and processed Lena image by high-boost filter. Fig. 2 shows that, the D value of MTOICM is the largest among these measures on all the processed images by different filters. So, MTOICM is good for image clarity measuring and its discrimination ability is the best among these measures. Moreover, to compare the overall performance of each measure on all the processed Lena images by different filters, the mean value of all the D values corresponding to each measure on the images in Fig. 1 is calculated and used. And, let MD represent this mean value. The MD value of each measure using the Lena image is shown in Fig. 3. A bigger MD value indicates a better performance of the corresponding measure to discriminate the image clarity change. Fig. 3 shows that, the MD value of MTOICM is larger than other measures. So, the overall performance of

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MTOICM for sharpness measurement and the discrimination ability of MTOICM for image clarity change are better than other measures on all the processed images by different filters. Therefore, MTOICM is an effective image sharpness measure. Fig. 4 is an example of Cameraman image. (a) is the original image, (b) is the processed image by median filter, (c) is the processed image by average filter, (d) is the processed image by Gaussian filter, (e) is the processed image by unsharp filter, (f) is the processed image by high-boost filter. The values of different measures on the original Cameraman image and the processed image by each filter are shown in Table 2. The original image is not very clear. Also, the original image is clearer than the processed images by the median filter, average filter and Gaussian filter, while the processed images by the unsharp filter and highboost filter are clearer than the original image. Moreover, because the filters have small size, the difference between the original image and processed images by different filters is small. Table 2 also verifies this. It shows that, the values of each measure on the original and the processed image by each filter are different. And, almost all the measures including MTOICM correctly quantify the clarity of image. To quantitatively comparing the discrimination ability of different measures, the D values of different measures on the original and processed Cameraman images by each filter are calculated and shown in Fig. 5. Fig. 5(a) shows the D values of

Table 3 Values of different measures calculated using the original living room image and the processed images.

Original image Median filter smoothed image Average filter smoothed image Gaussian filter smoothed image Unsharp filter smoothed image High-boost filter smoothed image

STD

E

SF

LIF

MTOICM

44.3747 43.3072 42.4288 43.5014 55.5811 49.6922

7.29511 7.24915 7.37427 7.40764 7.54692 6.57121

19.9242 14.0008 11.4039 15.7459 60.9496 34.0162

0.5359 0.5568 0.5594 0.5491 0.5038 0.5107

3.8482 1.9695 1.7075 2.7409 18.5075 9.5659

0.5

0.6

0.4

0.5 0.4

0.3

0.3 0.2

0.2

0.1

0.1

0

0 STD

E

SF

LIF

MTOICM

0.3

STD

E

SF

STD

E

SF

LIF

MTOICM

0.8 0.7

0.25

0.6 0.2

0.5 0.4

0.15

0.3

0.1

0.2 0.05

0.1

0

0 STD

E

SF

LIF

MTOICM

LIF

MTOICM

0.6 0.5 0.4 0.3 0.2 0.1 0 STD

E

SF

LIF

MTOICM

Fig. 8. D values of different measures on the original living room image and the corresponding processed image by each filter for discrimination ability comparison.

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different measures on the original and processed Cameraman image by median filter, (b) shows the D values of different measures on the original and processed Cameraman image by average filter, (c) shows the D values of different measures on the original and processed Cameraman image by Gaussian filter, (d) shows the D values of different measures on the original and processed Cameraman image by unsharp filter, (e) shows the D values of different measures on the original and processed Cameraman image by high-boost filter. Fig. 5 shows that, the D value of MTOICM is larger than other measures for all the filters. So, the ability of MTOICM to discriminate the small image clarity change is the best among these measures. And, to compare the overall performance of each measure on all the processed Cameraman images by different filters, the MD value of each measure using the Cameraman image is shown in Fig. 6. It shows that, the MD value of MTOICM is larger than other

0.6 0.5 0.4 0.3 0.2 0.1 0 STD

E

SF

LIF

MTOICM

Fig. 9. The MD value of each measure for the overall discrimination ability comparison using the living room image.

measures, which means the overall performance of MTOICM for both the sharpness measurement and the discrimination ability for image clarity change is the best among these measures on all the processed images by different filters. Therefore, MTOICM is effective for image sharpness measuring. Fig. 7 is an example of living room image obtained from a video. (a) is the original image, (b) is the processed image by median filter, (c) is the processed image by average filter, (d) is the processed image by Gaussian filter, (e) is the processed image by unsharp filter, (f) is the processed image by high-boost filter. The values of different measures on the original and processed images by each filter are shown in Table 3. The original image is also not very clear. The processed images by the median filter, average filter and Gaussian filter are blurred, and the processed images by the unsharp filter and high-boost filter are clear. Because the size of the filters is small, the difference between the original and the processed images is small. This is also illustrated in Table 3. However, the discrimination ability of MTOICM is better than other measures. To clearly show the good discrimination ability of MTOICM, the D values of different measures on the original and processed living room images by each filter are calculated and shown in Fig. 8. Fig. 8(a) shows the D values of different measures on the original and processed living room image by median filter, (b) shows the D values of different measures on the original and processed living room image by average filter, (c) shows the D values of different measures on the original and processed living room image by Gaussian filter, (d) shows the D values of different measures on the original and processed living room image by unsharp filter, (e) shows the D

Fig. 10. The original mineral image and the processed images by different filters.

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values of different measures on the original and processed living room image by high-boost filter. Fig. 8 demonstrates that, corresponding to each filter, the D value of MTOICM is the largest among these measures. So, MTOICM can correctly measure the clarity of images and the discrimination ability of MTOICM is better than others for all the filters. And, the MD value of each measure using the original and processed living room images is shown in Fig. 9 to compare the overall performance of the discrimination ability of each measure for image clarity change on all the filters. Fig. 9 shows that, the MD value of MTOICM is larger than other measures. Therefore, MTOICM performs better than other measures and is an effective image sharpness menasure. Figs. 10 and 12 give some other examples of mineral images. In these examples, (a) lists the original mineral images, (b) lists the

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processed images by median filter, (c) lists the processed images by average filter, (d) lists the processed images by Gaussian filter, (e) lists the processed images by unsharp filter, (f) lists the processed images by high-boost filter. Fig. 11 shows the MD values of different measures on the original and processed images by different filters in Fig. 10 to compare the overall performance of each measure. Although most of the measures can discriminate the clarity change of the images, the MD value of MTOICM is the largest. So, the performance of MTOICM is better than other measures. Fig. 12 is another example of mineral image. The MD values of different measures on the original and processed images by different filters are shown in Fig. 13 to compare the overall performance of each measure. Again, the MD value of MTOICM is the largest, which indicates that the proposed MTOICM achieves the best performance among these measures for all the filters. 0.6

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Fig. 11. The MD value of each measure for the overall discrimination ability comparison using the mineral image.

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Fig. 13. The MD value of each measure for the overall discrimination ability comparison using another mineral image.

Fig. 12. Another example of the original mineral image and the processed images by different filters.

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The images used in the experiment are from different applications. These results verified that, the proposed measure MTOICM could correctly quantify the clarity of image and could be used to discriminate the small image clarity change. Therefore, the proposed MTOICM is an effective image sharpness measure.

are grateful to Dr. Yan Li at Peking University, Beijing, China, for her helpful discussions and comments.

5. Conclusions

[1] Bai X, Zhou F, Xue B. Fusion of infrared and visual images through region extraction by using multi scale center-surround top-hat transform. Optics Express 2011;19:8444–57. [2] Yang C. Improving the sharpness of an image with non-uniform illumination. Optics & Laser Technology 2005;37:235–8. [3] Ferraro P, Paturzo M, Memmolo P, Finizio A. Controlling depth of focus in 3D image reconstructions by flexible and adaptive deformation of digital holograms. Optics Letters 2009;34:2787–9. [4] Mann C, Yu L, Lo C, Kim M. High-resolution quantitative phase-contrast microscopy by digital holography. Optics Express 2005;13:8693–8. [5] Bai X, Gu S, Zhou F. Entropy powered image fusion based on multi scale tophat transform. In: Proceedings of the 3rd International Congress on Image and Signal Processing. Yantai, China; 2010. pp. 1083–7. [6] Barbieri Andre L, de Arruda GF, Rodrigues Francisco A, Bruno Odemir M, da Fontoura Costa Luciano. An entropy-based approach to automatic image segmentation of satellite images. Physica A 2011;390:512–8. [7] Lai R, Yang Y, Wang B, Zhou H. A quantitative measure based infrared image enhancement algorithm using plateau histogram. Optics Communications 2010;283:4283–8. [8] Aslantas V, Kurban R. A comparison of criterion functions for fusion of multifocus noisy images. Optics Communications 2009;282:3231–42. [9] Soille P. Morphological image analysis-principle and applications. Germany: Springer; 2003. [10] Mura M, Benediktsson J, Bovolo F, Bruzzone L. An unsupervised technique based on morphological filters for change detection in very high resolution images. IEEE Geoscience and Remote Sensing Letters 2008;5:433–7. [11] Bai X, Zhou F. Analysis of new top-hat transformation and the application for infrared dim small target detection. Pattern Recognition 2010;43:2145–56. [12] Oliveira M, Leite NA. Multi-scale directional operator and morphological tools for reconnecting broken ridges in fingerprint images. Pattern Recognition 2008;41:367–77. [13] Bai X, Zhou F, Xue B. Edge preserved image fusion based on multiscale toggle contrast operator. Image and Vision Computing 2011;29:829–39. [14] Dorini L, Leite N. A scale–space toggle operator for morphological segmentation. In: Proceedings of the 8th International Symposium on Mathematical Morphology. Rio de Janeiro, Brazil; October 10–13, 2007. pp. 101–12. [15] Bai X, Zhou F, Xue B. Image enhancement using multi scale image features extracted by top-hat transform. Optics & Laser Technology 2012;44:328–36. [16] Jackway P, Deriche M. Scale–space properties of the Multi-scale morphological dilation–erosion. IEEE Transactions on Pattern Analysis and Machine Intelligence 1996;18:38–51.

Image sharpness measure is one type of useful measure in different image processing based applications. To construct effective image sharpness measure with good clarity discrimination ability, a morphological toggle operator based image sharpness measure MTOICM is proposed in this paper. Toggle operator can identify useful image details which are important information to represent the clarity of an image. Multi-scale image details are identified through multi-scale toggle operators, which are used to construct the final image sharpness measure. With the toggle operator, useful image features which can identify the image clarity are extracted. With the multi-scale theory, the useful image features at different scales are extracted. Therefore, the constructed image sharpness measure is effective and sensitive to the image clarity change. Experimental results on different images show that, the proposed morphological toggle operator based image sharpness measure can be used to measure image clarity, and the discrimination ability of MTOICM is better than others. So, MTOICM can be used in different applications related to image clarity measurement, including image fusion, image enhancement, auto-focusing and so on.

Acknowledgments The authors would like to thank the anonymous reviewers and editor for their very constructive comments and suggestions. This work is partly supported by the National Natural Science Foundation of China (60902056) Fundamental Research Funds for the Central Universities (YWF-11-03-Q-065), and State Key Laboratory of Virtual Reality Technology and System. The authors

References