SCIENCE CHINA Information Sciences

. RESEARCH PAPER .

January 2013, Vol. 56 012303:1–012303:12 doi: 10.1007/s11432-011-4475-5

Vector morphological operators in HSV color space LEI Tao1,2 ∗ , WANG Yi1 , FAN YangYu1 & ZHAO Jiong1,3 1School

of Electronics and Information, Northwestern Polytechnical University, Xi’an 710129, China; of Information and Electronics Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China; 3Telecommunication Engineering Institute, Air Force Engineering University, Xi’an 710077, China

2School

Received November 28, 2010; accepted April 18, 2011; published online January 1, 2012

Abstract In HSV color space, the current vector morphological operators have low capability to reduce color noise caused by hue and saturation in color image processing. Because they sort the color pixels according to the hierarchical ordering of V, S, H, which is against the equal principle of the three channels in color image processing. A novel vector ordering based on the combination of H, S and V is proposed in this paper, and the associated vector morphological erosion, dilation and composite filtering operators are defined. Compared with the popular vector morphological operators, experimental results show that the new operators can reduce the color noise effectively without any new color pixels while preserving the image details. And the filtered images have higher peak signal-to-noise ratio (PSNR) and lower mean absolute error (MSE). Keywords

HSV color space, vector morphological operators, vector ordering, image details

Citation Lei T, Wang Y, Fan Y Y, et al. Vector morphological operators in HSV color space. Sci China Inf Sci, 2013, 56: 012303(12), doi: 10.1007/s11432-011-4475-5

1

Introduction

Mathematical morphology is an application of the grids theory in spatial structure. Before determining the morphological operators, the total ordering of the morphological operators in the whole grid has to be introduced. In the theory of grayscale morphology, the morphological operators have a fixed ordering based on the grey value [1]. Unfortunately, the theory of grayscale morphology cannot be directly applied to color image processing, as a color image contains three channels: red, blue and green. If we define a complete grid (M, cp ), where cp denotes vector ordering less or equal, for two color pixels pi and pj , then pi cp pj [2, 3]. Furthermore, vector morphological erosion and dilation can be defined according to cp , and applied to the color image processing. Much work has been done on the vector morphological operators, see for example, Serra [4], Goutsias [5], Angulo [6,7], Luengo-Oraz [8,9], Vardavolia [10] and Louverdis [11]. Goutsias [5] believes that, when examining the vector ordering, we can define the color morphological erosion, dilation, opening and closing in the same way as we define gray morphological operators, and many gray morphological algorithms can be directly extended to treat color image processing. But Goutsias did not present a specified vector ordering. Some other researchers proposed different vector orderings in various color spaces [9–17]. The hue-saturation-value (HSV) color space is intimately related to the way human beings perceive color. ∗ Corresponding

author (email: [email protected])

c Science China Press and Springer-Verlag Berlin Heidelberg 2012 

info.scichina.com

www.springerlink.com

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Vardavolia and Louverdis [10–13] proposed two vector orderings in HSV color space respectively. A kind of morphological operators with hue and saturation taken into account simultaneously has been introduced in HLS color space, and the use of morphological algorithms in L*a*b* color space was also discussed by Hanbury. Furthermore, a total ordering on the color vectors in L*a*b* color space was defined by using a weighting function and lexicographical order [14]. Anglulo [6] proposed an approach of vector ordering based on LSH color space. In addition, Peters Π proposed a method of vector morphology for anglevalued images and applied this method to process color image in L*u*v* color space [15]. Considering RGB color space is widely used in the devices of capturing and displaying the images and computational system, Witte [17] proposed vector ordering in RGB color space. For the aforementioned vector orderings, we can clearly find out that the vector morphological operators proposed by Louverdis outperform other operators. In view of hue’s rotundity, the vector ordering proposed by Vardavolia [10] only considers the brightness and saturation. Although the vector ordering can be adopted to define vector morphological operators, it does not match the rule of practical vector ordering. With this in mind, Louverdis presented a new vector ordering for color image processing. However, experiments show that the color pixels’ ordering presented in [10] is similar to that in [11], because there is a hierarchical order within the three color components, i.e. brightness, saturation and hue. And the third layer (hue) could not affect the color pixels’ ordering. Considering the hue’s rotundity, Hanbury et al. [18] studied the morphological operators of unit circle. And they introduced many vector morphological operators based on various color spaces and employed them in color image processing. As the hue takes an important role in the vector ordering, the morphological operators are suitable for the hue dominated color images [16–20]. On the basis of Hanbury’s research, Aptoula [21–23] further explored the sequence of hue. He divided the hue into three color sectors with the red, the blue and the green as the reference hue, and achieved good processing effect by computing the distance between the current hue and the reference hue. As the current vector ordering does not take into account the relationship among the three components, a new vector ordering in HSV color space is reported. The ordering of color pixel is determined by vector model and angle. In the three components of HSV color space, S and V determine the module value of color pixels, and H determines the angle value. The module and angle, as two independent components, can be used to construct a new function for vector ordering. Finally, the vector morphological erosion, dilation, opening, closing and the composite filters are defined by the new vector ordering. Experimental results show that the novel vector morphological operators can get better processing effect on color images.

2

Related work

At present, RGB model has been widely used in image acquisition and display; it is a color model with generality and practicality. Based on a Cartesian coordinate system, RGB model can be defined as a cube, in which RGB values are at three corners: black at the origin; and white at the corner farthest from the origin. In this model, the gray scale extends from black to white along the line joining these two points, and the different colors in this model are points on or inside the cube, as shown in Figure 1. Witte [17] proposed a method of vector ordering in RGB space (DRGB). For one color pixel p(x, y), d1 represents the Euclidean distance between p(x, y) and o(0, 0, 0), and d2 refers to the Euclidean distance between p(x, y) and w(1, 1, 1).  d1 (o, p) = (pr − 0)2 + (pg − 0)2 + (pb − 0)2 , (1)  (2) d2 (w, p) = (1 − pr )2 + (1 − pg )2 + (1 − pb )2 . The ordering relationship of color pixels is defined as ⎧ p p ⎪ ⎪ d11 (o, p1 ) < d12 (o, p2 ), ⎨ p1 < p2 ⇔ DRGB

or ⎪ ⎪ ⎩ dp1 (o, p ) = dp2 (o, p ) and dp1 (w, p ) > dp2 (w, p ), 1 2 1 2 1 1 2 2

(3)

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p1 = p2 ⇔ dp11 (o, p1 ) = dp12 (o, p2 ) and dp21 (w, p1 ) = dp22 (w, p2 ), DRGB

(4)

where s2 , ⎪ ⎪ ⎪ ⎪ or ⎪ ⎪ ⎪ ⎩ v = v and s = s and h < h , 1 2 1 2 1 2

p1 (h1 , s1 , v1 ) = p2 (h2 , s2 , v2 ) ⇔ v1 = v2 and s1 = s2 and h1 = h2 . VSH

(7)

(8)

The new vector ordering is denoted by VSH. Compared with VS, VSH includes three layers. Brightness is a subjective descriptor that cannot be measured in practice. It embodies the achromatic notion of intensity and is one of the key factors for describing color sensation, and therefore the brightness is placed in the first layer; saturation has a strict ordering relationship, so it comes in the second layer; since hue lacks a fixed ordering, it is put in the last layer. VSH can get better effect on color pixel ordering than VS, but it still has some drawbacks for the ordering of hue dominated color band. For example, if v1 < v2 , s1 < s2 , |v2 −v1 | ≈ 0 and |v2 −v1 | < |s2 −s1 |, then we can get p1 (h1 , s1 , v1 ) VSH p2 (h2 , s2 , v2 ) according to the practical intuition of human for color, as shown in Figure 6. And Figure 7 (a) shows the ordering results of the color bands (Figure 3) by using VSH. In Figure 7(a), we can see that VSH vector ordering has good effect on the brightness dominated color band, but has poor effect on the other color bands. Thus, two new methods of vector ordering, SVH (based on the hierarchical idea of VSH, the ordering of the three components is S, V, H) and HVS (the

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Figure 1

Figure 3

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RGB color model.

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Figure 2

DRGB vector ordering.

Original color bands. (a) Hue dominated color band; (b) brightness dominated color band; (c) saturation

dominated color band.

Figure 4

Ordering results of Figure 3 using DRGB. (a) Ordering result of Figure 3(a) using DRGB; (b) ordering result

of Figure 3(b) using DRGB; (c) ordering result of Figure 3(c) using DRGB.

Figure 5

HSV color model.

Figure 6

VSH vector ordering.

ordering of the three components is H, V, S) are introduced here. SVH and HVS are adopted to sort the color bands in Figure 3. The ordering results are shown in Figure 7(b)–(c).

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Figure 7

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Ordering results of color bands based on hierarchical vector ordering. (a) Ordering results of Figure 3 using

VSH; (b) ordering results of Figure 3 using SVH; (c) ordering results of Figure 3 using HVS.

As shown in Figure 7, VSH, SVH and HVS can only be used to process brightness dominated color band, saturation dominated color band, and hue dominated color band, respectively. In other words, the three vector orderings have low robustness. Therefore, they are not suitable for the processing of natural color band. In order to improve VSH, SVH and HVS, a new vector ordering is reported below.

3

A novel vector ordering

As the three color components H, S, V play different roles in the current vector ordering, these vector orderings have low robustness for color image processing except some special color images one component with stronger contrast, and a new vector ordering which can describe color pixels’ ordering objectively should be established. Among the current vector orderings, VSH has the best performance. Here, a vector ordering with four-layer components based on VSH is presented as follows: (the new vector ordering is denoted by HHV (vector ordering based on the value of brightness plus hue), vector less and equal are denoted by s2 , ⎪ ⎪ ⎪ ⎪ ⎪ or ⎪ ⎪ ⎪ ⎩ v1 + h1 = v2 + h2 and v1 = v2 and s1 = s2 and h1 > h2 . Through analyzing the vector ordering HHV, we can find that HHV still chooses a hierarchical strategy. But unlike the other vector orderings, the sum of brightness and hue is more unique. It means that most of the color pixels can be ordered accurately at the first layer while a small number of color pixels can be ordered at the second layer. As a result, the computations of the last two layers are redundant. Figure 8(c) is the ordering result of a 5×5 color block (Figure 8(a)) based on HHV. Figure 8(b) presents the sorting result of the first layer in HHV. Figure 8(c) shows that the color pixels’ ordering has been achieved uniquely by the first layer in HHV. In order to improve the real-time of HHV, the simplified HHV vector ordering is presented, and denoted by SHHV, as follows: ⎧ ⎪ ⎪ ⎨ v1 + h1 < v2 + h2 , p1 < p2 ⇔ SHHV

or ⎪ ⎪ ⎩ v + h = v + h and s > s . 1 1 2 2 1 2

(10)

SHHV includes two layers, the first layer is the combination of brightness and hue, the second layer selects the saturation. Using SHHV to sort the color bands in Figure 3, we get Figure 9.

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Figure 8

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Ordering result of color pixels using the component of the first hierarchy of HHV. (a) A color block size of 5 × 5;

(b) ordering result of the first hierarchy of HHV; (c) ordering result of Figure 8(a).

Figure 9

Ordering results of Figure 3 using SHHV. (a) Ordering result of Figure 3(a) using SHHV; (b) ordering result of

Figure 3(b) using SHHV; (c) ordering result of Figure 3(c) using SHHV.

The hue and brightness play a dominant role in the color bands (see Figure 3(a) and (b)) which can be ordered by SHHV accurately. The results are shown in Figure 9(a) and (b). Nevertheless, SHHV has no effect on the saturation dominated color band, as shown in Figure 9(c). To solve the problem that the three components are not placed in the same layer, a function pi = f (hi , si , vi ) is introduced, which can reveal the practical ordering relationships among the vectors. Through analyzing the HSV color model in Figures 5 and 6, we can see that any vector in HSV color space can be represented by module and angle of vectors. The module can be denoted by vi2 + s2i , and the angular distance can be denoted by D(hi , h0 ). We can define h1  h2 ⇔ D(h1 , h0 )  D(h2 , h0 ) in terms of the circle morphological theory, where ∀h1 , h2 ∈ [0, 1], for D(h, h0 ), as follows:  |h − h0 | < 0.5, |h − h0 | D(h, h0 ) = (11) h, h0 ∈ [0, 1]. 1 − |h − h0 | |h − h0 |  0.5,  Then, the three components are changed into two. Accordingly, we can get p = vi2 + s2i + D(hi , h0 ). i   2 2 2 2 As vi + si and D(hi , h0 ) are two independent components, pi = vi + si + D(hi , h0 ) is more optimal than HHV. The new vector ordering is denoted by HVSD. ⎧   2 + s2 + D(h , h ) < ⎪ v v22 + s22 + D(h2 , h0 ), ⎪ 1 0 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ or ⎨   (12) p1 < p2 ⇔ v12 + s21 + D(h1 , h0 ) = v22 + s22 + D(h2 , h0 ) and v1 < v2 , ⎪ HVSD ⎪ ⎪ ⎪ or ⎪ ⎪  ⎪ ⎩ v 2 + s2 + D(h , h ) = v 2 + s2 + D(h , h ) and v = v and s > s . 1 0 2 0 1 2 1 2 1 1 2 2

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Figure 10

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Ordering results of Figure 3 using SHVSD. (a) Ordering result of Figure 3(a) using SHVSD; (b) ordering result

of Figure 3(b) using SHVSD; (c) ordering result of Figure 3(c) using SHVSD.

Figure 11

Ordering results of the first hierarchical of SHVSD for Figure 10.

It has been proven that the first layer in HHV is enough for the color pixels’ ordering. Similarly, we can simplify HVSD to reduce the computations, and then improve the real time of algorithms. The simplified HHVSD can be represented by SHVSD, as follows:   p1 < p2 ⇔ v12 + s21 + D(h1 , h0 ) < v22 + s22 + D(h2 , h0 ). (13) SHVSD

SHVSD is used to sort the color bands in Figure 3, and the processed color bands are shown in Figures 10 and 11. It is clear that SHVSD vector ordering has good effect on vector ordering of various color bands. No matter it is the special color images or natural color images, the ideal vector ordering results can be acquired by using SHVSD.

4

Vector morphological operators

Let f denote a color image corrupted by color impulse noise and let f (x, y) be its color pixel value at (x, y), b(x, y) is the value of the structure element at (x, y), Df and Db are the domains of f and b. Vector morphological erosion and dilation of f by b, denoted by f Θcp b and f ⊕cp b, are defined as (see [1, 2]) (14) f Θ b(s, t) = min{f (s + x, t + y) − b(x, y)|(s + x), (t + y) ∈ Df , (x, y) ∈ Db }, cp

f ⊕ b(s, t) = max{f (s − x, t − y) + b(x, y)|(s − x), (t − y) ∈ Df , (x, y) ∈ Db }. cp

(15)

Based on the definition of vector erosion and dilation, vector opening and vector closing of f by b, denoted by f ◦cp b and f •cp b, are defined as f ◦ b = (f Θ b) ⊕ b,

(16)

f • b = (f ⊕ b) Θ b.

(17)

cp cp

cp cp

cp cp

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Figure 12

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“Hat” and noisy images corrupted by 25% salt and pepper noise. (a) “Hat”; (b) noisy image.

As for the grayscale morphology, because an opening operation can remove the bright image structures while a closing have the same filtering effect but on the dark image structures [24]. According to the duality of morphological operators, an opening and closing is composed of clammed sequential morphological filters. Such as vector open-close and close-open filters (γV O represents vector opening operation, ϕV O represents vector closing operation, V O represents vector ordering).

◦ ϕV O γV O (f, b) = f b • b, (18) cp cp

(19) γV O ϕV O (f, b) = f • b ◦ b. cp

cp

Based on the definition of the alternative sequential filters to the gray image, the vector alternative sequential filters (the sequential open-close and sequential close-open) are proposed, defined as follows: ψVi O (f, b1 , b2 , . . . , bi ) = ϕV O (γV O (· · · ϕV O (γV O (ϕV O (γV O (f, b1 ), b1 ), b2 ), b2 ) . . . , bi ), bi ),

(20)

φiV O (f, b1 , b2 , . . . , bi )

(21)

= γV O (ϕV O (· · · γV O (ϕV O (γV O (ϕV O (f, b1 ), b1 ), b2 ), b2 ) . . . , bi ), bi ).

The structure element bi satisfies equation b2 = b1 ⊕ b1 , b2 = b1 ⊕ b1 , . . . , bi = b( i − 1) ⊕ b1 , and i can be determined by the size of image and noisy level.

5

Experimental results

In experiments, in order to evaluate the proposed vector morphological operators, we choose the tested image “Hat” (512-by-512, 24-bit). The performance parameters of computer are CPU: Intel(R) Core(TM)2 Duo E7500, 2.93 GHz, RAM memory: 2 GB. The proposed vector morphological operators and the other operators are used to process the color image corrupted by salt and pepper noise. Different filtered images are shown in Figure 13. Figure 12(a) is original image, Figure 12(b) is noisy image corrupted by 25% salt 2 (in [15]), ϕHLS γHLS and and pepper noise. Figure 13(a)–(n) are filtered images by ϕLab γLab and ψLab 2 2 2 ψHLS (in [14]), ϕRGB γRGB and ψRGB (in [17]), ϕV S γV S and ψV S (in [10]), ϕV SH γV SH and ψV2 SH (in 2 , ϕSHV SD γSHV SD and [11]), and the proposed vector morphological operators ϕSHHV γSHHV , ψSHHV 2 ψSHV SD (The structure element b1 is a square of size 3 × 3, and b2 is a square of size 5 × 5, since the average hue or the most frequent hue is chosen usually as the conference hue, we choose h0 = 0). Figure 13(a)–(d) show the filtered images via the methods proposed in [14] and [15], which have poor effect for the noise removing since that the saturation and brightness are ignored in vector ordering. Figure 13(e)–(f) show the filtered images by the method in [17], which are clearer than Figure 13(a)–(d). Vector morphological operators based on VS and VSH cannot reduce noise efficiently while preserving image details, as shown in Figure 13(g)–(j). Compared with the current vector operators, the proposed morphological operators have better effect on color noise reduction, as shown in Figure 13(k)–(n), where

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Figure 13

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Restoration performance comparison on the “Hat” image degraded 25% salt and pepper noise. (a) Using

2 2 2 ; (c) using ϕHLS γHLS ; (d) using ψHLS ; (e) using ϕRGB γRGB ; (f) using ψRGB ; (g) using ϕLab γLab ; (b) using ψLab 2 ; (i) using ϕ 2 2 ϕV S γV S ; (h) using ψV V SH γV SH ; (j) using ψV SH ; (k) using ϕSHHV γSHHV ; (l) using ψSHHV ; (m) using S

2 ϕSHV SD γSHV SD ; (n) using ψSHV SD .

SHVSD is the best method of vector ordering, and Figure 13(m) and (n) illustrate the advantage of SHVSD. The restoration performance is quantitatively measured by the mean absolute error (MAE) and peak signal-to-noise ratio (PSNR) [25, 26]. MSE =

M  N  1 f (i, j) − g(i, j)2 , 3M × N i=1 j=1

3 × M × N × 2552 , PSNR = 10 log10 M N 2 i=1 j=1 f (i, j) − g(i, j)

(22)

(23)

where, f is the original image and g is the restored image of size M × N . The PSNR and MSE values for the proposed vector morphological operators and the other compared are shown in Tables 1 and 2, Figures 14 and 15. From the above experimental results, we can find that the filtered color images have good effect by 2 using ϕSHV SD γSHV SD and ψSHV SD , and have lower MSE and higher PSNR than the popular vector morphological operators.

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Table 1

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Comparative results of MSE of different vector morphological operators for various percentages of impulse noise

(salt and pepper) for the image “Hat” Noise level

Methods 5%

10%

15%

20%

25%

30%

35%

Ref. [15]

ϕLab γLab

615.013

1093.21

1773.49

2471.88

2985.80

3642.34

4239.31

2 ψLab

979.357

1528.35

2165.32

2688.61

3156.77

3521.46

4299.43

Ref. [14]

ϕHLS γHLS

579.965

2736.47

5176.78

7008.26

8549.92

9831.06

10777.4

2 ψHLS

458.946

3157.61

6205.21

8032.37

10154.7

11114.2

11513.8

ϕRGB γRGB

245.320

968.562

2285.11

3346.48

3964.92

4564.03

5096.06

2 ψRGB

289.932

1093.90

2921.00

3970.52

4287.20

4400.34

4861.61

ϕV S γV S

383.257

727.608

1085.59

1514.12

1705.26

2101.78

2523.07

2 ψV S

449.076

778.113

1141.26

1454.22

1758.30

2129.66

2402.19

ϕV SH γV SH

383.197

729.284

1068.41

1508.81

1698.68

2064.29

2427.57

2 ψV SH

454.934

776.342

1106.83

1455.54

1760.43

2125.13

2350.20

ϕSHHV γSHHV

155.480

280.525

487.187

682.663

909.616

1184.70

1635.86

2 ψSHHV

248.091

348.122

544.930

752.898

930.798

1099.25

1316.73

ϕSHV SD γSHV SD

125.714

189.923

244.523

330.486

535.441

988.716

1552.47

2 ψSHV SD

220.332

271.558

319.009

403.142

386.037

472.793

991.035

Ref. [17]

Ref. [10]

Ref. [11]

SHHV

SHVSD

Table 2

Comparative results of PSNR of different vector morphological operators for various percentages of impulse noise

(salt and pepper) for the image “Hat” Noise level

Methods Ref. [15]

Ref. [14]

Ref. [17]

Ref. [10]

Ref. [11]

SHHV

SHVSD

6

5%

10%

15%

20%

25%

30%

35%

ϕLab γLab

20.2419

17.7437

15.6425

14.2005

13.3801

12.5169

11.8578

2 ψLab

18.2213

16.2885

14.7755

13.8355

13.1383

12.6635

11.7966

ϕHLS γHLS

20.4967

13.7588

10.9901

9.67469

8.81117

8.20479

7.80564

2 ψHLS

21.5131

13.1372

10.2032

9.08236

8.06412

7.67200

7.51858

ϕRGB γRGB

24.2334

18.2695

14.5417

12.8849

12.1484

11.5373

11.0584

2 ψRGB

23.5078

17.7410

13.4754

12.1423

11.8090

11.6959

11.2629

ϕV S γV S

22.2958

19.5118

17.7741

16.3291

15.8128

14.9049

14.1114

2 ψV S

21.6076

19.2203

17.5569

16.5044

15.6798

14.8476

14.3247

ϕV SH γV SH

22.2965

19.5018

17.8433

16.3444

15.8296

14.9830

14.2790

2 ψV SH

21.5513

19.2302

17.6899

16.5005

15.6746

14.8569

14.4197

ϕSHHV γSHHV

26.2140

23.6510

21.2538

19.7887

18.5422

17.3947

15.9933

2 ψSHHV

24.1846

22.7134

20.7673

19.3634

18.4422

17.7198

16.9358

ϕSHV SD γSHV SD

27.1369

25.3450

24.2476

22.9392

20.8436

18.1801

16.2206

2 ψSHV SD

24.7000

23.7921

23.0927

22.0762

22.2645

21.3840

18.1699

Conclusions

In this study, we have analyzed the advantages and disadvantages of vector ordering in different color spaces, and proposed a novel vector ordering rule which breaks the idea of hierarchical order from the classical vector ordering. Based on the new vector ordering, the associated vector morphological operators and the combined operators have also been defined. They were used to reduce the noise in the color images and good effect was achieved. The experiment results show that the new vector morphology

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Figure 14

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Comparative results of MSE of different vector morphological operators for various percentages of impulse noise

(salt and pepper) for the image “Hat”. (a) Opening-closing operator; (b) alternative sequential opening-closing operator (i = 2).

Figure 15

Comparative results of PSNR of different vector morphological operators for various percentages of impulse

noise (salt and pepper) for the image “hat”. (a) Opening-closing operator; (b) alternative sequential opening-closing operator (i = 2).

filtering operators outperformed the current color morphological operators in robustness. And the final filtered image has a higher peak signal-to-noise ratio (PSNR) and a lower mean absolute error (MSE).

Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos. 60872159, 60903127) and Aoxiang Star Project at Northwestern Polytechnical University, Xi’an, Shanxi, China. We thank Kodak for providing free test images.

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Sci China Inf Sci

January 2013 Vol. 56 012303:12

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