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Optimal parameter estimation of ε-filter employing distribution distance
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Journal of the Franklin Institute 349 (2012) 2570–2584 www.elsevier.com/locate/jfranklin
Optimal parameter estimation of e-filter employing distribution distance Noriaki Suetakea,n, Go Tanakab, Hayato Hashiic, Eiji Uchinoa,d a
Graduate School of Science and Engineering, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8512, Japan Graduate School of Natural Sciences, Nagoya City University, Yamanohata 1, Mizuho-cho, Mizuho-ku, Nagoya 467-8601, Japan c Eizo Nanao Corporation, 153 Shimokashiwano, Hakusan, Ishikawa 924-8566, Japan d Fuzzy Logic Systems Institute, 680-41 Kawazu, Iizuka 820-0067, Japan
b
Received 28 March 2011; received in revised form 19 March 2012; accepted 19 July 2012 Available online 31 July 2012
Abstract As the filter which can effectively remove the small amplitude noises on digital images, the e-filter has been proposed. In order to effectively use this filter, a smoothing parameter e-filter should be appropriately estimated before applying it. To address this problem, the authors proposed the parameter estimation method based on Hellinger distance (HD). In the method, HD between a residual signal and assumed noise distribution was evaluated, and a parameter e of the e-filter was estimated by finding the value giving minimum distance. However, the enough discussion on use of HD has not been made. In this paper, it is attempted to utilize not only the HD, but also various distribution distances in the parameter estimation, and their performances and characteristics are compared and analyzed experimentally. Furthermore, the parameter estimation method is extended to be applicable for the vector e-filter for the color images. Consequently, through the experiments, it is shown that L1-norm or maximum norm is appropriate as the distribution distance used in the parameter estimation methods from the view points of the simplicity of the calculation and MSE performance in the filtering. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction The e-filter proposed by Harashima et al. [1] is an edge-preserving smoothing filter. This filter can effectively remove small amplitude noise, and has been employed in noise n
Corresponding author. E-mail address: [email protected] (N. Suetake).
0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.07.006
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removal in many practical scenes so far [2–9]. In order to effectively use the e-filter, its smoothing parameter e has to be appropriately estimated before processing. However, the estimation has been done empirically up to now. Therefore, it is needed to develop the estimation method of e, which can determine its value effectively and efficiently. In order to realize efficient e estimation, the method based on the non-correlation property between output and residual signals was proposed by Matsumoto and coworker [10,11]. In their method, for image signals, parameter estimation is achieved using the non-correlation property between restored and residual image signals—a difference between input and restored image signals. However, there are some cases where the non-correlation property is not suitable for the estimation of e. That is, the effectiveness of their method heavily depends on the sort of images. To address this issue, we have proposed a parameter estimation method based on the Hellinger distance (HD) [12] between an assumed noise model and the residual signal obtained as a difference of input and restored signals [13]. In the method, the parameter estimation was done finding e value giving the minimum distance. However, the enough discussion on use of HD was not made. Hence, the analysis and the discussion about the metrics for the parameter estimation should be achieved deeply. In this paper, we attempt to utilize not only the HD, but also various distribution distances, and compare their performances and characteristics in the parameter estimation of e-filter. Furthermore, we extend the parameter estimation method to be applicable for the vector e-filter [4], which is effective for the removal of noise superimposed on color digital images. Through experiments, the effectiveness and validity of the present methods are proved. The rest of this paper is organized as follows: Section 2 gives a brief overview of the e-filter and the vector e-filter. Details of the present method is explained in Section 3. Then, experimental results for some digital images and conclusions are described in Sections 4 and 5, respectively. 2. e-Filter and vector e-filter The e-filter [1] used in the image filtering is explained. An e-filter output I e at a pixel (x,y) is obtained as follows: r r X X
I e ðx,yÞ ¼ Iðx,yÞ
½aði,jÞF ðIðx,yÞIðx þ i,y þ jÞÞ,
ð1Þ
i ¼ r j ¼ r
where I is an input image corrupted by noise. r is a window size and a positive integer. The following function and filter coefficient are used as F and a, respectively, in this paper z, erzre, F ðzÞ ¼ ð2Þ 0 otherwise and aði,jÞ ¼
1 , ð2r þ 1Þ2
ð3Þ
where e is a smoothing parameter and a nonnegative real number. The filter coefficient shown in Eq. (3) gives the effect of the mean filter to the e-filter. The e-filter becomes a mean filter when e is set to an enough large value (e.g., 255).
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In the vector e-filter [4] for the color image processing, Eqs. (1) and (2) become as follows: r r X X ½aði,jÞ F 0 ðIðx,yÞIðx þ i,y þ jÞÞ ð4Þ I e ðx,yÞ ¼ Iðx,yÞ i ¼ r j ¼ r
and
( 0
F ðzÞ ¼
z, 0
JzJre, otherwise,
ð5Þ
where I stands for a vector composed of RGB components, i.e., I ¼ ðR,G,BÞ. In this paper, for instance, R component of I is represented as IR . 3. Parameter estimation method employing distribution distance In the present method, the optimal smoothing parameter e of the e-filter is estimated based on a distribution distance between a noise model assumed beforehand and the residual signal obtained as a difference of input and restored signals. The flow of the method is shown in Fig. 1. When the distribution distance is small, it is judged as fine noise removal is achieved with the e value. 3.1. Parameter estimation of e-filter The optimal value en of the e-filter is obtained as follows: en ¼ arg min DistðH res ðeÞ,H ass ðs^ est ÞÞ,
ð6Þ
e
where ‘‘Dist’’ is a function to calculate distribution distance. H res stands for the probability density function of a residual signal I res , and is obtained as a normalized histogram of I res . Here, I res —a difference between an input image I and a restored image I e obtained by the e-filter—is obtained as follows: I res ðx,yÞ ¼ Iðx,yÞI e ðx,yÞ:
ð7Þ
ε -filter
MAD estimation
Distribution distance calculation
Assumed noise distribution
Residual signal distribution
Fig. 1. Flow of the present parameter estimation method for e-filter.
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In Eq. (6), H ass indicates the probability density function of an assumed additive noise. In the present method, an assumption on the distribution of an additive noise is imposed and its parameter should be previously estimated from a noisy input image. In this study, the Gaussian noise is assumed as the additive noise and its standard deviation is estimated as sest using the standardized median of absolute deviation from the median (MAD) [14]. MAD is calculated in each block cropped as a square with k k pixels on an input image. MAD in the i-th block s^ i is calculated as follows: s^ i ¼ 1:483 medfjIi,j medfIi,j : 1rjrk2 gj : 1rjrk2 g,
ð8Þ
where ‘‘med’’ is the function to obtain a median and Ii,j is the j-th pixel value in the i-th block. Consequently, sest is obtained as follows: sest ¼ min s^ i :
ð9Þ
i
As the Dist in Eq. (6), some metrics are considered to use and are tested in this paper. Concretely, HD, Jensen–Shannon divergence (JSD) [15], L1-norm (L1) [16], maximum norm (L1 ) [16], and symmetric Kullback–Leibler divergence (SKLD) [17,18] are considered to use as the Dist. These are defined as follows: ( )1=2 255 X 1=2 res ass HD ¼ 1 ½H ðlÞH ðlÞ , ð10Þ l ¼ 255
255 X 1 2H res ðlÞ 2H ass ðlÞ res ass H ðlÞ log res JSD ¼ þ H ðlÞ log res , 2 H ðlÞ þ H ass ðlÞ H ðlÞ þ H ass ðlÞ l ¼ 255 L1 ¼
255 X
jH res ðlÞH ass ðlÞj,
ð11Þ
ð12Þ
l ¼ 255
L1 ¼ maxjH res ðlÞH ass ðlÞj
ð13Þ
l
and 255 X
SKLD ¼
½H res ðlÞH ass ðlÞ log
l ¼ 255
H res ðlÞ , H ass ðlÞ
ð14Þ
where ‘‘log’’ stands for the natural logarithm, and l in Eq. (13) is the integer in ½255,255. Each distribution distance becomes small when two distributions are similar to each other. 3.2. Parameter estimation of vector e-filter Fig. 2 shows the flow of the parameter estimation of vector e-filter. Concerning the parameter estimation of the vector e-filter, the optimal value en is obtained as follows: X res ass est DistðHK ðeÞ,HK ðsK ÞÞ, ð15Þ en ¼ arg min e
K2fR,G,Bg
res where HK stands for the probability density function of a residual signal IKres and is obtained as a normalized histogram of IKres in the similar manner to the case of the e-filter.
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Vector ε -filter
Dist. Calc.
MAD estimation
Dist. Calc.
Dist. Calc.
Assumed noise Residual signal distribution distribution
Fig. 2. Flow of the present parameter estimation method for vector e-filter.
IKres is obtained as follows: IKres ðx,yÞ ¼ IK ðx,yÞIKe ðx,yÞ:
ð16Þ
ass In Eq. (15), HK also means the probability density function of an assumed additive noise. The estimated standard deviation of the additive noise sest K is also obtained in the similar manner to the case of the e-filter.
4. Experimental results The effectiveness and validity of the present method are verified by experiments employing various digital images. The mean square error (MSE) is used as an index for quantitative evaluation. When a monochrome image is processed, MSE is calculated from an original noisefree image I ori and a restored image I e . For a color image, MSE is calculated from I ori and I e . MSE becomes small when a restored image is similar to an original one. 4.1. Relationship between e of e-filter and metrics Relationships between e and MSE are shown in Fig. 3 when Lenna and Barbara images included in the standard image database (SIDBA) [19,20] are employed. Each image constitutes of 256 256 pixels. The e giving the minimum MSE in Fig. 3 means the optimal value eopt . Concerning Lenna and Barbara images, from Fig. 3, eopt is around 40, 70, and 100 in the cases where the standard deviation of the Gaussian noise s is 10, 20, and 30, respectively. In the setup of the e-filter, Eqs. (2) and (3) are used as F and a, respectively. The filter window size r is empirically set as 2. Relationships e vs. a correlation coefficient (CC) calculated between I e and I res are shown in Fig. 4. In Matsumoto’s method [10], it is assumed that I ori ð I e Þ is uncorrelated
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Fig. 3. e vs. MSE: (a) Lenna and (b) Barbara.
Fig. 4. e vs. R: (a) Lenna and (b) Barbara.
with additive noise, and the e giving the minimum CC is regarded as the optimal value. However, CC does not always reflect MSE well as seen in the results for Lenna when s is 20 and 30, and for Barbara with all ss. It shows that Matsumoto’s method does not work well in some cases. Relationships e vs. HD, e vs. JSD, e vs. L1, e vs. L1 , and e vs. SKLD are shown in Figs. 5–9, respectively. The block size k in s estimation is empirically set as 30. As shown in these figures, en s obtained based on distribution distances are sufficiently similar to eopt s although en s are little bit different from eopt s. The distribution distance-based parameter estimation is highly promising in comparison with Matsumoto’s method. 4.2. Performance of parameter estimation for e-filter For example, Fig. 10 shows partial images of Barbara. Fig. 10(a) shows the input image corrupted by the Gaussian noise with s ¼ 10. Fig. 10(b) shows the optimal noise removal result which can be obtained by the e-filter. Fig. 10(c) and (d) is filtering results in the cases
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Fig. 5. e vs. HD: (a) Lenna and (b) Barbara.
Fig. 6. e vs. JSD: (a) Lenna and (b) Barbara.
Fig. 7. e vs. L1: (a) Lenna and (b) Barbara.
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Fig. 8. e vs. L1 : (a) Lenna and (b) Barbara.
Fig. 9. e vs. SKLD: (a) Lenna and (b) Barbara.
where the e value estimated by Matsumoto’s method and the present method using L1 , respectively. From the figures, the superiority of the present method in comparison with Matsumoto’s method is confirmed in image quality. Experimental results for various images and ss are shown in Tables 1–3. The Lenna and Barbara images, and seven images included in the SIDBA ‘‘airplane,’’ ‘‘boat,’’ ‘‘bridge,’’ ‘‘cameraman,’’ ‘‘girl,’’ ‘‘lax,’’ and ‘‘woman’’ are employed here. Each image constitutes of 256 256 pixels. In these tables, ‘‘OPT’’ and ‘‘CC’’ denote the optimal result and Matsumoto’s method, respectively. Further, ‘‘HD,’’ ‘‘JSD,’’ ‘‘L1,’’ ‘‘L1 ,’’ and ‘‘SKLD’’ stand for the present method using the distribution distances HD, JSD, L1, L1 , and SKLD, respectively. From Tables 1–3, the present parameter estimation is good for many images although en is not perfectly same to eopt . Especially, from Table 1, it can be said that L1 or L1 is the suitable distribution distance measure among tested measures in the e value estimation. In the cases where Barbara and bridge are used as input images of concern, MSEs of the results obtained by the present method are especially large comparing with the optimal
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Fig. 10. Experimental results for Barbara (partial image). (a) Noisy input image with s ¼ 10, (b) optimal filtering result (eopt ¼ 33). Noise removal results obtained by the e-filter with actual parameter estimation: (c) Matsumoto’s method (e ¼ 199), and (d) the present method using L1 as the Dist function (en ¼ 55). Table 1 Experimental results with s ¼ 10. Image
Airplane Barbara Boat Bridge Cameraman Girl Lax Lenna Woman
e(MSE) OPT
CC
HD
JSD
L1
L1
SKLD
36(38) 33(61) 36(39) 30(77) 34(36) 37(35) 32(64) 38(37) 37(39)
36(38) 199(468) 38(39) 217(495) 47(43) 53(42) 232(479) 38(37) 220(162)
43(40) 63(122) 42(40) 71(194) 42(39) 46(38) 50(87) 48(41) 42(41)
42(40) 63(122) 41(40) 70(189) 42(39) 45(37) 48(83) 48(41) 41(40)
40(39) 58(105) 37(39) 64(163) 36(36) 40(35) 46(79) 44(39) 38(40)
39(39) 55(96) 37(39) 64(163) 35(36) 41(35) 44(76) 43(38) 37(39)
42(40) 63(122) 41(40) 70(189) 42(39) 46(38) 48(83) 48(41) 42(41)
ones. The reason is supposed that the estimation accuracy of s for Barbara and bridge is lower than for other images. This issue also occurs in the cases where s is 20 and 30 as shown in Tables 2 and 3. However, in most cases shown in Tables 1–3, the present e estimation method is superior to Matsumoto’s method. Therefore, the effectiveness and validity of the present method are confirmed.
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Table 2 Experimental results with s ¼ 20. Image
Airplane Barbara Boat Bridge Cameraman Girl Lax Lenna Woman
e(MSE) OPT
CC
HD
JSD
L1
L1
SKLD
74(107) 66(181) 74(99) 63(214) 73(99) 82(72) 65(177) 78(92) 76(97)
250(282) 237(480) 234(171) 251(510) 255(343) 134(89) 253(492) 255(180) 250(176)
82(111) 96(249) 81(101) 102(313) 80(100) 96(76) 85(206) 82(93) 88(103)
82(111) 92(235) 79(100) 102(313) 77(99) 92(75) 85(206) 80(93) 87(102)
71(107) 85(213) 71(100) 92(277) 70(99) 81(72) 76(187) 70(95) 73(97)
72(107) 81(202) 67(103) 90(270) 82(102) 72(74) 77(189) 67(98) 69(100)
82(111) 95(246) 81(101) 102(313) 77(99) 92(75) 85(206) 82(93) 87(102)
Table 3 Experimental results with s ¼ 30. Image
Airplane Barbara Boat Bridge Cameraman Girl Lax Lenna Woman
e(MSE) OPT
CC
HD
JSD
L1
L1
SKLD
109(184) 101(307) 120(159) 98(347) 110(181) 125(110) 99(293) 117(148) 117(157)
255(303) 255(501) 254(193) 255(534) 255(372) 255(126) 255(513) 255(202) 255(199)
129(198) 156(426) 130(162) 140(426) 71(294) 80(190) 130(342) 145(162) 146(171)
124(193) 150(412) 130(162) 136(416) 70(301) 79(195) 127(335) 139(157) 140(168)
106(184) 148(407) 114(159) 118(372) 63(360) 73(233) 113(305) 122(149) 127(160)
104(185) 126(350) 110(161) 115(366) 70(301) 76(213) 102(293) 104(156) 115(158)
125(194) 150(412) 130(162) 136(416) 71(294) 80(190) 129(340) 139(157) 140(168)
Fig. 11. Effectiveness of the vector processing in the e-filter: (a) original test image, (b) input noisy image with s ¼ 10, (c) noise removal result obtained by the CWEF, and (d) noise removal result obtained by the VEF.
4.3. Effectiveness of vector e-filter comparing with component-wise e-filter The effectiveness of the vector processing in the e-filter is shown using a test image in Fig. 11(a). Fig. 11(b) is an input noisy image with the Gaussian noise of which standard deviation s is 10. Fig. 11(c) and (d) is ‘‘optimal’’ noise removal results obtained by the component-wise e-filter (CWEF) and the vector e-filter (VEF), respectively. In CWEF,
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the process shown in Eqs. (1)–(3) is applied to each RGB component, and then CWEF has three parameters eR , eG , and eB . Here, optimal parameters are determined by MSE. In the test image, edge intensity ðDR,DG,DBÞ is ð40,25,50Þ, and JðDR,DG,DBÞJ is 68.7. ðenR ,enG ,enB Þ in the CWEF is ð41,46,43Þ, and en in the VEF is 68. That is, CWEF destroys edges in R and G components as shown in Fig. 11(c). On the other hand, VEF preserves the edge as shown in Fig. 11(d). The VEF is superior to the CWEF in edge preservation. Hence, only the VEF is handled in the processing of color images, hereafter. 4.4. Relationship between e of vector e-filter and metrics Relationships between e and metrics are shown using the balloon image as an example. The image constitutes of 256 256 pixels. The Gaussian noise of which s is 5, 10, or 15 is superimposed on the original image, and the VEF is applied to it. Fig. 12(a) shows the relationship between e and MSE. The e giving the minimum MSE in the figure is the
Fig. 12. e vs. metrics for the balloon image: (a) MSE, (b) CC, (c) HD, (d) JSD, (e) L1, (f) L1 , and (g) SKLD.
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optimal value eopt . Concerning the balloon, eopt is around 30, 50, and 80 in the cases where s is 5, 10, and 15, respectively. Matsumoto’s method is used as a method for comparison, in the similar manner to Sections 4.1 and 4.2. In the setup of VEF used both in Matsumoto’s and the present method, Eqs. (3) and (5) are used as a and F 0 , respectively. Further, r and k, which are the filter window size and the block size in s estimation, are empirically set as 2 and 30, respectively. Relationships e vs. a correlation coefficient (CC) calculated between I e and I res are shown in Fig. 12(b). In Matsumoto’s method, it is assumed that I ori ( I e ) is uncorrelated with the additive noise, and the e giving the minimum CC is regarded as the optimal value. However, CC does not always reflect MSE well as seen in the results for the balloon when s is 10 and 15. It shows that Matsumoto’s method does not work well in some cases. Relationships e vs. HD, e vs. JSD, e vs. L1, e vs. L1 , and e vs. SKLD are shown in Fig. 12(c)–(g), respectively. As shown in these figures, en s obtained based on distribution
Fig. 13. Experimental results for the balloon image. (a) Noisy input image with s ¼ 10, (b) optimal VEF result (eopt ¼ 50). Noise removal results obtained by VEF with actual parameter estimation: (c) Matsumoto’s method (e ¼ 228), (d) the present method using HD as the Dist function (en ¼ 49).
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distances are almost the same as eopt s. The distribution distance-based parameter estimation is superior to Matsumoto’s method. Fig. 13 shows noise removal results of the balloon image obtained by the VEF with some parameters when s is 10. The e value determined by Matsumoto’s method is too large, and the resulting image is blurred as shown in Fig. 13(c). On the other hand, the resulting image obtained by the VEF with the present parameter estimation is good as shown in Fig. 13(d). It is almost the same as the optimal one. 4.5. Performance of parameter estimation for vector e-filter The balloon image and another 11 images included in the SIDBA ‘‘aerial,’’ ‘‘airplane,’’ ‘‘couple,’’ ‘‘earth,’’ ‘‘girl,’’ ‘‘Lenna,’’ ‘‘mandrill,’’ ‘‘milkdrop,’’ ‘‘parrots’’, ‘‘pepper,’’ and Table 4 Experimental results with s ¼ 5. Image
Aerial Airplane Balloon Couple Earth Girl Lenna Mandrill Milkdrop Parrots Pepper Sailboat
e(MSE) OPT
CC
HD
JSD
L1
L1
SKLD
26(20) 26(10) 26(7) 23(11) 26(12) 24(13) 26(12) 26(20) 25(8) 27(9) 26(12) 27(12)
10(25) 25(10) 26(7) 10(21) 10(23) 27(14) 25(12) 10(25) 28(9) 24(9) 25(12) 23(12)
70(87) 33(11) 27(7) 14(16) 33(13) 28(14) 49(18) 59(51) 29(9) 33(10) 44(16) 36(13)
66(78) 33(11) 27(7) 15(15) 33(13) 24(13) 44(16) 58(50) 29(9) 35(10) 43(15) 36(13)
69(85) 36(12) 28(7) 15(15) 35(14) 28(14) 51(18) 59(51) 29(9) 37(10) 47(17) 39(14)
67(80) 41(13) 29(7) 18(13) 36(14) 29(14) 52(19) 59(51) 46(11) 42(11) 56(20) 41(15)
66(78) 33(11) 27(7) 15(15) 33(13) 22(13) 44(16) 58(50) 29(9) 35(10) 42(15) 36(13)
Table 5 Experimental results with s ¼ 10. Image
Aerial Airplane Balloon Couple Earth Girl Lenna Mandrill Milkdrop Parrots Pepper Sailboat
e(MSE) OPT
CC
HD
JSD
L1
L1
SKLD
47(59) 51(27) 50(18) 46(29) 50(32) 49(30) 52(29) 48(61) 55(19) 53(22) 53(29) 52(32)
10(100) 10(100) 228(49) 10(85) 10(100) 10(93) 10(99) 10(99) 53(19) 10(97) 10(97) 10(100)
85(119) 53(28) 49(18) 18(74) 54(32) 35(39) 67(32) 74(89) 54(19) 42(26) 57(29) 60(34)
85(119) 52(27) 49(18) 18(74) 53(32) 34(41) 67(32) 74(89) 54(19) 41(27) 45(30) 60(34)
81(111) 55(28) 46(19) 19(72) 53(32) 35(39) 67(32) 73(87) 53(19) 41(27) 58(29) 62(35)
83(115) 58(28) 47(18) 20(69) 54(32) 34(41) 66(32) 72(85) 75(21) 44(25) 66(31) 65(36)
85(119) 52(27) 49(18) 18(74) 53(32) 33(43) 67(32) 74(89) 54(19) 41(27) 44(31) 60(34)
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Table 6 Experimental results with s ¼ 15. Image
Aerial Airplane Balloon Couple Earth Girl Lenna Mandrill Milkdrop Parrots Pepper Sailboat
e(MSE) OPT
CC
HD
JSD
L1
L1
SKLD
69(103) 76(49) 75(31) 68(49) 73(54) 73(47) 77(48) 70(109) 80(31) 79(38) 76(48) 75(58)
10(225) 10(224) 10(226) 10(183) 10(223) 10(206) 10(222) 10(222) 10(218) 10(219) 10(216) 10(222)
108(163) 78(49) 72(32) 22(168) 75(54) 40(116) 90(51) 93(137) 74(32) 43(106) 59(61) 87(60)
106(159) 77(49) 71(32) 23(165) 74(54) 40(116) 88(50) 93(137) 72(32) 44(101) 53(74) 87(60)
102(151) 78(49) 65(35) 24(162) 71(54) 40(116) 84(49) 91(133) 67(34) 42(111) 57(64) 84(59)
100(147) 78(49) 63(37) 24(162) 70(54) 40(116) 83(49) 88(127) 73(32) 43(106) 59(61) 86(60)
106(159) 77(49) 71(32) 22(168) 74(54) 41(111) 88(50) 93(137) 72(32) 45(96) 53(74) 87(60)
‘‘sailboat’’ are employed here. Each image constitutes of 256 256 pixels. s is set as 5, 10, or 15 here. Experimental results for the 12 images are shown in Tables 4–6. In most cases shown in Tables 4–6, estimation results obtained by the present method are superior to those obtained by Matsumoto’s method. However, estimation results and MSE results obtained by the present method are sometimes not good. These are, for instance, aerial and couple est cases shown in Table 5, and due to failure of sest K estimation using the MAD. When sK estimation is satisfactory, the parameter estimation based on distribution distances is fine. 5. Conclusions In this paper, in order to realize efficient parameter setting of the e-filter and the vector e-filter, we showed the parameter estimation method using the distribution distance between signal distributions. In the present method, the parameter estimation was done by finding the e value giving the minimum distribution distance, which was calculated between the assumed noise model and the residual signal obtained as a difference of the input and restored signals. As the distribution distances, concretely, HD, JSD, L1, L1 , and SKLD were considered and tested. Through comparison experiments with Matsumoto’s method employing non-correlation information, the superiority and effectiveness of the present method were confirmed. From the view point of the simplicity of the calculation, it can be said that L1 or L1 is suitable for the distribution distance in the present method. Future works are improvement of the estimation accuracy of the standard deviation of the Gaussian noise and development of an efficient search algorithm to find the optimal value en .
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[2] K. Arakawa, H. Harashima, H. Miyakawa, Statistical analysis of e-separating nonlinear digital filters, Electronics and Communications in Japan, Part 1 66 (January (1)) (1983) 10–18. [3] K. Arakawa, H. Harashima, H. Miyakawa, Recursive e-nonlinear digital filter, Electronics and Communications in Japan, Part 1 66 (November (11)) (1983) 17–24. [4] H. Kondo, K. Hanari, K. Shimizu, K. Arakawa, Colored face image processing by vector e-filter—Removal of wrinkles, in: IEICE General Conference No. D-11-143, March 1998 (in Japanese). [5] H. Watabe, K. Arakawa, Y. Arakawa, Nonlinear inverse filter using e-filter and its application to image restoration, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E83-A (February (2)) (2000) 283–290. [6] K. Arakawa, K. Matsuura, H. Watabe, Y. Arakawa, A method of noise reduction for speech signals using component separating e-filters, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences J85-A (October (10)) (2002) 1059–1069 (Japanese Edition). [7] K. Arakawa, T. Okada, e-Separating nonlinear filter bank and its application to face image beautification, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences J88-A (November (11)) (2005) 1216–1225 (Japanese Edition). [8] M. Matsumoto, Band-pass e-filter for edge enhancement and noise removal, IEICE Transactions on Information and Systems E93-D (February (2)) (2010) 367–375. [9] M. Muneyasu, A. Taguchi, Nonlinear Digital Signal Processing, Asakura Publishing Co., Ltd., Tokyo, 1999 (in Japanese). [10] M. Matsumoto, S. Hashimoto, Estimation of optimal parameter in e-filter based on signal-noise decorrelation, IEICE Transactions on Information and Systems E92-D (June (6)) (2009) 1312–1315. [11] M. Matsumoto, Adaptive e-filter based on signal-noise decorrelation, in: 2009 International Symposium on Intelligent Signal Processing and Communication Systems, Kanazawa, Japan, December 2009, pp. 611–614. [12] D. Pollard, A User’s Guide to Measure Theoretic Probability, Cambridge University Press, Cambridge, UK, 2003. [13] N. Suetake, G. Tanaka, H. Hashii, E. Uchino, Hellinger distance-based parameter tuning for e-filter, IEICE Transactions on Information and Systems E92-D (September (9)) (2010) 2647–2650. [14] J. Astola, P. Kuosmanen, Fundamentals of Nonlinear Digital Filtering, CRC Press, Boca Raton, FL, 1997. [15] Y. Qiao, N. Minematsu, The General Form of Divergence Invariant to Transformations, IEICE Technical Report, SP2008-51, July 2008. [16] G.H. Golub, C.F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, MD, 1996. [17] S. Kullback, R.A. Leibler, On information and sufficiency, Annals of Mathematical Statistics 22 (March (1)) (1951) 79–86. [18] S. Kullback, Information Theory and Statistics, Dover Publications, New York, NY, 1968. [19] M. Sakauchi, Y. Ohsawa, M. Sone, M. Onoe, Management of the Standard Image Database for Image Processing Researches (SIDBA), ITEJ Technical Report, vol. 8, no. 38, December 1984, pp. 7–12 (in Japanese). [20] /http://www.ess.ic.kanagawa-it.ac.jp/app_images_j.htmlS.
Journal of the Franklin Institute 349 (2012) 2570–2584 www.elsevier.com/locate/jfranklin
Optimal parameter estimation of e-filter employing distribution distance Noriaki Suetakea,n, Go Tanakab, Hayato Hashiic, Eiji Uchinoa,d a
Graduate School of Science and Engineering, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8512, Japan Graduate School of Natural Sciences, Nagoya City University, Yamanohata 1, Mizuho-cho, Mizuho-ku, Nagoya 467-8601, Japan c Eizo Nanao Corporation, 153 Shimokashiwano, Hakusan, Ishikawa 924-8566, Japan d Fuzzy Logic Systems Institute, 680-41 Kawazu, Iizuka 820-0067, Japan
b
Received 28 March 2011; received in revised form 19 March 2012; accepted 19 July 2012 Available online 31 July 2012
Abstract As the filter which can effectively remove the small amplitude noises on digital images, the e-filter has been proposed. In order to effectively use this filter, a smoothing parameter e-filter should be appropriately estimated before applying it. To address this problem, the authors proposed the parameter estimation method based on Hellinger distance (HD). In the method, HD between a residual signal and assumed noise distribution was evaluated, and a parameter e of the e-filter was estimated by finding the value giving minimum distance. However, the enough discussion on use of HD has not been made. In this paper, it is attempted to utilize not only the HD, but also various distribution distances in the parameter estimation, and their performances and characteristics are compared and analyzed experimentally. Furthermore, the parameter estimation method is extended to be applicable for the vector e-filter for the color images. Consequently, through the experiments, it is shown that L1-norm or maximum norm is appropriate as the distribution distance used in the parameter estimation methods from the view points of the simplicity of the calculation and MSE performance in the filtering. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction The e-filter proposed by Harashima et al. [1] is an edge-preserving smoothing filter. This filter can effectively remove small amplitude noise, and has been employed in noise n
Corresponding author. E-mail address: [email protected] (N. Suetake).
0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.07.006
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removal in many practical scenes so far [2–9]. In order to effectively use the e-filter, its smoothing parameter e has to be appropriately estimated before processing. However, the estimation has been done empirically up to now. Therefore, it is needed to develop the estimation method of e, which can determine its value effectively and efficiently. In order to realize efficient e estimation, the method based on the non-correlation property between output and residual signals was proposed by Matsumoto and coworker [10,11]. In their method, for image signals, parameter estimation is achieved using the non-correlation property between restored and residual image signals—a difference between input and restored image signals. However, there are some cases where the non-correlation property is not suitable for the estimation of e. That is, the effectiveness of their method heavily depends on the sort of images. To address this issue, we have proposed a parameter estimation method based on the Hellinger distance (HD) [12] between an assumed noise model and the residual signal obtained as a difference of input and restored signals [13]. In the method, the parameter estimation was done finding e value giving the minimum distance. However, the enough discussion on use of HD was not made. Hence, the analysis and the discussion about the metrics for the parameter estimation should be achieved deeply. In this paper, we attempt to utilize not only the HD, but also various distribution distances, and compare their performances and characteristics in the parameter estimation of e-filter. Furthermore, we extend the parameter estimation method to be applicable for the vector e-filter [4], which is effective for the removal of noise superimposed on color digital images. Through experiments, the effectiveness and validity of the present methods are proved. The rest of this paper is organized as follows: Section 2 gives a brief overview of the e-filter and the vector e-filter. Details of the present method is explained in Section 3. Then, experimental results for some digital images and conclusions are described in Sections 4 and 5, respectively. 2. e-Filter and vector e-filter The e-filter [1] used in the image filtering is explained. An e-filter output I e at a pixel (x,y) is obtained as follows: r r X X
I e ðx,yÞ ¼ Iðx,yÞ
½aði,jÞF ðIðx,yÞIðx þ i,y þ jÞÞ,
ð1Þ
i ¼ r j ¼ r
where I is an input image corrupted by noise. r is a window size and a positive integer. The following function and filter coefficient are used as F and a, respectively, in this paper z, erzre, F ðzÞ ¼ ð2Þ 0 otherwise and aði,jÞ ¼
1 , ð2r þ 1Þ2
ð3Þ
where e is a smoothing parameter and a nonnegative real number. The filter coefficient shown in Eq. (3) gives the effect of the mean filter to the e-filter. The e-filter becomes a mean filter when e is set to an enough large value (e.g., 255).
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In the vector e-filter [4] for the color image processing, Eqs. (1) and (2) become as follows: r r X X ½aði,jÞ F 0 ðIðx,yÞIðx þ i,y þ jÞÞ ð4Þ I e ðx,yÞ ¼ Iðx,yÞ i ¼ r j ¼ r
and
( 0
F ðzÞ ¼
z, 0
JzJre, otherwise,
ð5Þ
where I stands for a vector composed of RGB components, i.e., I ¼ ðR,G,BÞ. In this paper, for instance, R component of I is represented as IR . 3. Parameter estimation method employing distribution distance In the present method, the optimal smoothing parameter e of the e-filter is estimated based on a distribution distance between a noise model assumed beforehand and the residual signal obtained as a difference of input and restored signals. The flow of the method is shown in Fig. 1. When the distribution distance is small, it is judged as fine noise removal is achieved with the e value. 3.1. Parameter estimation of e-filter The optimal value en of the e-filter is obtained as follows: en ¼ arg min DistðH res ðeÞ,H ass ðs^ est ÞÞ,
ð6Þ
e
where ‘‘Dist’’ is a function to calculate distribution distance. H res stands for the probability density function of a residual signal I res , and is obtained as a normalized histogram of I res . Here, I res —a difference between an input image I and a restored image I e obtained by the e-filter—is obtained as follows: I res ðx,yÞ ¼ Iðx,yÞI e ðx,yÞ:
ð7Þ
ε -filter
MAD estimation
Distribution distance calculation
Assumed noise distribution
Residual signal distribution
Fig. 1. Flow of the present parameter estimation method for e-filter.
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In Eq. (6), H ass indicates the probability density function of an assumed additive noise. In the present method, an assumption on the distribution of an additive noise is imposed and its parameter should be previously estimated from a noisy input image. In this study, the Gaussian noise is assumed as the additive noise and its standard deviation is estimated as sest using the standardized median of absolute deviation from the median (MAD) [14]. MAD is calculated in each block cropped as a square with k k pixels on an input image. MAD in the i-th block s^ i is calculated as follows: s^ i ¼ 1:483 medfjIi,j medfIi,j : 1rjrk2 gj : 1rjrk2 g,
ð8Þ
where ‘‘med’’ is the function to obtain a median and Ii,j is the j-th pixel value in the i-th block. Consequently, sest is obtained as follows: sest ¼ min s^ i :
ð9Þ
i
As the Dist in Eq. (6), some metrics are considered to use and are tested in this paper. Concretely, HD, Jensen–Shannon divergence (JSD) [15], L1-norm (L1) [16], maximum norm (L1 ) [16], and symmetric Kullback–Leibler divergence (SKLD) [17,18] are considered to use as the Dist. These are defined as follows: ( )1=2 255 X 1=2 res ass HD ¼ 1 ½H ðlÞH ðlÞ , ð10Þ l ¼ 255
255 X 1 2H res ðlÞ 2H ass ðlÞ res ass H ðlÞ log res JSD ¼ þ H ðlÞ log res , 2 H ðlÞ þ H ass ðlÞ H ðlÞ þ H ass ðlÞ l ¼ 255 L1 ¼
255 X
jH res ðlÞH ass ðlÞj,
ð11Þ
ð12Þ
l ¼ 255
L1 ¼ maxjH res ðlÞH ass ðlÞj
ð13Þ
l
and 255 X
SKLD ¼
½H res ðlÞH ass ðlÞ log
l ¼ 255
H res ðlÞ , H ass ðlÞ
ð14Þ
where ‘‘log’’ stands for the natural logarithm, and l in Eq. (13) is the integer in ½255,255. Each distribution distance becomes small when two distributions are similar to each other. 3.2. Parameter estimation of vector e-filter Fig. 2 shows the flow of the parameter estimation of vector e-filter. Concerning the parameter estimation of the vector e-filter, the optimal value en is obtained as follows: X res ass est DistðHK ðeÞ,HK ðsK ÞÞ, ð15Þ en ¼ arg min e
K2fR,G,Bg
res where HK stands for the probability density function of a residual signal IKres and is obtained as a normalized histogram of IKres in the similar manner to the case of the e-filter.
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Vector ε -filter
Dist. Calc.
MAD estimation
Dist. Calc.
Dist. Calc.
Assumed noise Residual signal distribution distribution
Fig. 2. Flow of the present parameter estimation method for vector e-filter.
IKres is obtained as follows: IKres ðx,yÞ ¼ IK ðx,yÞIKe ðx,yÞ:
ð16Þ
ass In Eq. (15), HK also means the probability density function of an assumed additive noise. The estimated standard deviation of the additive noise sest K is also obtained in the similar manner to the case of the e-filter.
4. Experimental results The effectiveness and validity of the present method are verified by experiments employing various digital images. The mean square error (MSE) is used as an index for quantitative evaluation. When a monochrome image is processed, MSE is calculated from an original noisefree image I ori and a restored image I e . For a color image, MSE is calculated from I ori and I e . MSE becomes small when a restored image is similar to an original one. 4.1. Relationship between e of e-filter and metrics Relationships between e and MSE are shown in Fig. 3 when Lenna and Barbara images included in the standard image database (SIDBA) [19,20] are employed. Each image constitutes of 256 256 pixels. The e giving the minimum MSE in Fig. 3 means the optimal value eopt . Concerning Lenna and Barbara images, from Fig. 3, eopt is around 40, 70, and 100 in the cases where the standard deviation of the Gaussian noise s is 10, 20, and 30, respectively. In the setup of the e-filter, Eqs. (2) and (3) are used as F and a, respectively. The filter window size r is empirically set as 2. Relationships e vs. a correlation coefficient (CC) calculated between I e and I res are shown in Fig. 4. In Matsumoto’s method [10], it is assumed that I ori ð I e Þ is uncorrelated
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Fig. 3. e vs. MSE: (a) Lenna and (b) Barbara.
Fig. 4. e vs. R: (a) Lenna and (b) Barbara.
with additive noise, and the e giving the minimum CC is regarded as the optimal value. However, CC does not always reflect MSE well as seen in the results for Lenna when s is 20 and 30, and for Barbara with all ss. It shows that Matsumoto’s method does not work well in some cases. Relationships e vs. HD, e vs. JSD, e vs. L1, e vs. L1 , and e vs. SKLD are shown in Figs. 5–9, respectively. The block size k in s estimation is empirically set as 30. As shown in these figures, en s obtained based on distribution distances are sufficiently similar to eopt s although en s are little bit different from eopt s. The distribution distance-based parameter estimation is highly promising in comparison with Matsumoto’s method. 4.2. Performance of parameter estimation for e-filter For example, Fig. 10 shows partial images of Barbara. Fig. 10(a) shows the input image corrupted by the Gaussian noise with s ¼ 10. Fig. 10(b) shows the optimal noise removal result which can be obtained by the e-filter. Fig. 10(c) and (d) is filtering results in the cases
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Fig. 5. e vs. HD: (a) Lenna and (b) Barbara.
Fig. 6. e vs. JSD: (a) Lenna and (b) Barbara.
Fig. 7. e vs. L1: (a) Lenna and (b) Barbara.
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Fig. 8. e vs. L1 : (a) Lenna and (b) Barbara.
Fig. 9. e vs. SKLD: (a) Lenna and (b) Barbara.
where the e value estimated by Matsumoto’s method and the present method using L1 , respectively. From the figures, the superiority of the present method in comparison with Matsumoto’s method is confirmed in image quality. Experimental results for various images and ss are shown in Tables 1–3. The Lenna and Barbara images, and seven images included in the SIDBA ‘‘airplane,’’ ‘‘boat,’’ ‘‘bridge,’’ ‘‘cameraman,’’ ‘‘girl,’’ ‘‘lax,’’ and ‘‘woman’’ are employed here. Each image constitutes of 256 256 pixels. In these tables, ‘‘OPT’’ and ‘‘CC’’ denote the optimal result and Matsumoto’s method, respectively. Further, ‘‘HD,’’ ‘‘JSD,’’ ‘‘L1,’’ ‘‘L1 ,’’ and ‘‘SKLD’’ stand for the present method using the distribution distances HD, JSD, L1, L1 , and SKLD, respectively. From Tables 1–3, the present parameter estimation is good for many images although en is not perfectly same to eopt . Especially, from Table 1, it can be said that L1 or L1 is the suitable distribution distance measure among tested measures in the e value estimation. In the cases where Barbara and bridge are used as input images of concern, MSEs of the results obtained by the present method are especially large comparing with the optimal
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Fig. 10. Experimental results for Barbara (partial image). (a) Noisy input image with s ¼ 10, (b) optimal filtering result (eopt ¼ 33). Noise removal results obtained by the e-filter with actual parameter estimation: (c) Matsumoto’s method (e ¼ 199), and (d) the present method using L1 as the Dist function (en ¼ 55). Table 1 Experimental results with s ¼ 10. Image
Airplane Barbara Boat Bridge Cameraman Girl Lax Lenna Woman
e(MSE) OPT
CC
HD
JSD
L1
L1
SKLD
36(38) 33(61) 36(39) 30(77) 34(36) 37(35) 32(64) 38(37) 37(39)
36(38) 199(468) 38(39) 217(495) 47(43) 53(42) 232(479) 38(37) 220(162)
43(40) 63(122) 42(40) 71(194) 42(39) 46(38) 50(87) 48(41) 42(41)
42(40) 63(122) 41(40) 70(189) 42(39) 45(37) 48(83) 48(41) 41(40)
40(39) 58(105) 37(39) 64(163) 36(36) 40(35) 46(79) 44(39) 38(40)
39(39) 55(96) 37(39) 64(163) 35(36) 41(35) 44(76) 43(38) 37(39)
42(40) 63(122) 41(40) 70(189) 42(39) 46(38) 48(83) 48(41) 42(41)
ones. The reason is supposed that the estimation accuracy of s for Barbara and bridge is lower than for other images. This issue also occurs in the cases where s is 20 and 30 as shown in Tables 2 and 3. However, in most cases shown in Tables 1–3, the present e estimation method is superior to Matsumoto’s method. Therefore, the effectiveness and validity of the present method are confirmed.
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Table 2 Experimental results with s ¼ 20. Image
Airplane Barbara Boat Bridge Cameraman Girl Lax Lenna Woman
e(MSE) OPT
CC
HD
JSD
L1
L1
SKLD
74(107) 66(181) 74(99) 63(214) 73(99) 82(72) 65(177) 78(92) 76(97)
250(282) 237(480) 234(171) 251(510) 255(343) 134(89) 253(492) 255(180) 250(176)
82(111) 96(249) 81(101) 102(313) 80(100) 96(76) 85(206) 82(93) 88(103)
82(111) 92(235) 79(100) 102(313) 77(99) 92(75) 85(206) 80(93) 87(102)
71(107) 85(213) 71(100) 92(277) 70(99) 81(72) 76(187) 70(95) 73(97)
72(107) 81(202) 67(103) 90(270) 82(102) 72(74) 77(189) 67(98) 69(100)
82(111) 95(246) 81(101) 102(313) 77(99) 92(75) 85(206) 82(93) 87(102)
Table 3 Experimental results with s ¼ 30. Image
Airplane Barbara Boat Bridge Cameraman Girl Lax Lenna Woman
e(MSE) OPT
CC
HD
JSD
L1
L1
SKLD
109(184) 101(307) 120(159) 98(347) 110(181) 125(110) 99(293) 117(148) 117(157)
255(303) 255(501) 254(193) 255(534) 255(372) 255(126) 255(513) 255(202) 255(199)
129(198) 156(426) 130(162) 140(426) 71(294) 80(190) 130(342) 145(162) 146(171)
124(193) 150(412) 130(162) 136(416) 70(301) 79(195) 127(335) 139(157) 140(168)
106(184) 148(407) 114(159) 118(372) 63(360) 73(233) 113(305) 122(149) 127(160)
104(185) 126(350) 110(161) 115(366) 70(301) 76(213) 102(293) 104(156) 115(158)
125(194) 150(412) 130(162) 136(416) 71(294) 80(190) 129(340) 139(157) 140(168)
Fig. 11. Effectiveness of the vector processing in the e-filter: (a) original test image, (b) input noisy image with s ¼ 10, (c) noise removal result obtained by the CWEF, and (d) noise removal result obtained by the VEF.
4.3. Effectiveness of vector e-filter comparing with component-wise e-filter The effectiveness of the vector processing in the e-filter is shown using a test image in Fig. 11(a). Fig. 11(b) is an input noisy image with the Gaussian noise of which standard deviation s is 10. Fig. 11(c) and (d) is ‘‘optimal’’ noise removal results obtained by the component-wise e-filter (CWEF) and the vector e-filter (VEF), respectively. In CWEF,
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the process shown in Eqs. (1)–(3) is applied to each RGB component, and then CWEF has three parameters eR , eG , and eB . Here, optimal parameters are determined by MSE. In the test image, edge intensity ðDR,DG,DBÞ is ð40,25,50Þ, and JðDR,DG,DBÞJ is 68.7. ðenR ,enG ,enB Þ in the CWEF is ð41,46,43Þ, and en in the VEF is 68. That is, CWEF destroys edges in R and G components as shown in Fig. 11(c). On the other hand, VEF preserves the edge as shown in Fig. 11(d). The VEF is superior to the CWEF in edge preservation. Hence, only the VEF is handled in the processing of color images, hereafter. 4.4. Relationship between e of vector e-filter and metrics Relationships between e and metrics are shown using the balloon image as an example. The image constitutes of 256 256 pixels. The Gaussian noise of which s is 5, 10, or 15 is superimposed on the original image, and the VEF is applied to it. Fig. 12(a) shows the relationship between e and MSE. The e giving the minimum MSE in the figure is the
Fig. 12. e vs. metrics for the balloon image: (a) MSE, (b) CC, (c) HD, (d) JSD, (e) L1, (f) L1 , and (g) SKLD.
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optimal value eopt . Concerning the balloon, eopt is around 30, 50, and 80 in the cases where s is 5, 10, and 15, respectively. Matsumoto’s method is used as a method for comparison, in the similar manner to Sections 4.1 and 4.2. In the setup of VEF used both in Matsumoto’s and the present method, Eqs. (3) and (5) are used as a and F 0 , respectively. Further, r and k, which are the filter window size and the block size in s estimation, are empirically set as 2 and 30, respectively. Relationships e vs. a correlation coefficient (CC) calculated between I e and I res are shown in Fig. 12(b). In Matsumoto’s method, it is assumed that I ori ( I e ) is uncorrelated with the additive noise, and the e giving the minimum CC is regarded as the optimal value. However, CC does not always reflect MSE well as seen in the results for the balloon when s is 10 and 15. It shows that Matsumoto’s method does not work well in some cases. Relationships e vs. HD, e vs. JSD, e vs. L1, e vs. L1 , and e vs. SKLD are shown in Fig. 12(c)–(g), respectively. As shown in these figures, en s obtained based on distribution
Fig. 13. Experimental results for the balloon image. (a) Noisy input image with s ¼ 10, (b) optimal VEF result (eopt ¼ 50). Noise removal results obtained by VEF with actual parameter estimation: (c) Matsumoto’s method (e ¼ 228), (d) the present method using HD as the Dist function (en ¼ 49).
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distances are almost the same as eopt s. The distribution distance-based parameter estimation is superior to Matsumoto’s method. Fig. 13 shows noise removal results of the balloon image obtained by the VEF with some parameters when s is 10. The e value determined by Matsumoto’s method is too large, and the resulting image is blurred as shown in Fig. 13(c). On the other hand, the resulting image obtained by the VEF with the present parameter estimation is good as shown in Fig. 13(d). It is almost the same as the optimal one. 4.5. Performance of parameter estimation for vector e-filter The balloon image and another 11 images included in the SIDBA ‘‘aerial,’’ ‘‘airplane,’’ ‘‘couple,’’ ‘‘earth,’’ ‘‘girl,’’ ‘‘Lenna,’’ ‘‘mandrill,’’ ‘‘milkdrop,’’ ‘‘parrots’’, ‘‘pepper,’’ and Table 4 Experimental results with s ¼ 5. Image
Aerial Airplane Balloon Couple Earth Girl Lenna Mandrill Milkdrop Parrots Pepper Sailboat
e(MSE) OPT
CC
HD
JSD
L1
L1
SKLD
26(20) 26(10) 26(7) 23(11) 26(12) 24(13) 26(12) 26(20) 25(8) 27(9) 26(12) 27(12)
10(25) 25(10) 26(7) 10(21) 10(23) 27(14) 25(12) 10(25) 28(9) 24(9) 25(12) 23(12)
70(87) 33(11) 27(7) 14(16) 33(13) 28(14) 49(18) 59(51) 29(9) 33(10) 44(16) 36(13)
66(78) 33(11) 27(7) 15(15) 33(13) 24(13) 44(16) 58(50) 29(9) 35(10) 43(15) 36(13)
69(85) 36(12) 28(7) 15(15) 35(14) 28(14) 51(18) 59(51) 29(9) 37(10) 47(17) 39(14)
67(80) 41(13) 29(7) 18(13) 36(14) 29(14) 52(19) 59(51) 46(11) 42(11) 56(20) 41(15)
66(78) 33(11) 27(7) 15(15) 33(13) 22(13) 44(16) 58(50) 29(9) 35(10) 42(15) 36(13)
Table 5 Experimental results with s ¼ 10. Image
Aerial Airplane Balloon Couple Earth Girl Lenna Mandrill Milkdrop Parrots Pepper Sailboat
e(MSE) OPT
CC
HD
JSD
L1
L1
SKLD
47(59) 51(27) 50(18) 46(29) 50(32) 49(30) 52(29) 48(61) 55(19) 53(22) 53(29) 52(32)
10(100) 10(100) 228(49) 10(85) 10(100) 10(93) 10(99) 10(99) 53(19) 10(97) 10(97) 10(100)
85(119) 53(28) 49(18) 18(74) 54(32) 35(39) 67(32) 74(89) 54(19) 42(26) 57(29) 60(34)
85(119) 52(27) 49(18) 18(74) 53(32) 34(41) 67(32) 74(89) 54(19) 41(27) 45(30) 60(34)
81(111) 55(28) 46(19) 19(72) 53(32) 35(39) 67(32) 73(87) 53(19) 41(27) 58(29) 62(35)
83(115) 58(28) 47(18) 20(69) 54(32) 34(41) 66(32) 72(85) 75(21) 44(25) 66(31) 65(36)
85(119) 52(27) 49(18) 18(74) 53(32) 33(43) 67(32) 74(89) 54(19) 41(27) 44(31) 60(34)
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Table 6 Experimental results with s ¼ 15. Image
Aerial Airplane Balloon Couple Earth Girl Lenna Mandrill Milkdrop Parrots Pepper Sailboat
e(MSE) OPT
CC
HD
JSD
L1
L1
SKLD
69(103) 76(49) 75(31) 68(49) 73(54) 73(47) 77(48) 70(109) 80(31) 79(38) 76(48) 75(58)
10(225) 10(224) 10(226) 10(183) 10(223) 10(206) 10(222) 10(222) 10(218) 10(219) 10(216) 10(222)
108(163) 78(49) 72(32) 22(168) 75(54) 40(116) 90(51) 93(137) 74(32) 43(106) 59(61) 87(60)
106(159) 77(49) 71(32) 23(165) 74(54) 40(116) 88(50) 93(137) 72(32) 44(101) 53(74) 87(60)
102(151) 78(49) 65(35) 24(162) 71(54) 40(116) 84(49) 91(133) 67(34) 42(111) 57(64) 84(59)
100(147) 78(49) 63(37) 24(162) 70(54) 40(116) 83(49) 88(127) 73(32) 43(106) 59(61) 86(60)
106(159) 77(49) 71(32) 22(168) 74(54) 41(111) 88(50) 93(137) 72(32) 45(96) 53(74) 87(60)
‘‘sailboat’’ are employed here. Each image constitutes of 256 256 pixels. s is set as 5, 10, or 15 here. Experimental results for the 12 images are shown in Tables 4–6. In most cases shown in Tables 4–6, estimation results obtained by the present method are superior to those obtained by Matsumoto’s method. However, estimation results and MSE results obtained by the present method are sometimes not good. These are, for instance, aerial and couple est cases shown in Table 5, and due to failure of sest K estimation using the MAD. When sK estimation is satisfactory, the parameter estimation based on distribution distances is fine. 5. Conclusions In this paper, in order to realize efficient parameter setting of the e-filter and the vector e-filter, we showed the parameter estimation method using the distribution distance between signal distributions. In the present method, the parameter estimation was done by finding the e value giving the minimum distribution distance, which was calculated between the assumed noise model and the residual signal obtained as a difference of the input and restored signals. As the distribution distances, concretely, HD, JSD, L1, L1 , and SKLD were considered and tested. Through comparison experiments with Matsumoto’s method employing non-correlation information, the superiority and effectiveness of the present method were confirmed. From the view point of the simplicity of the calculation, it can be said that L1 or L1 is suitable for the distribution distance in the present method. Future works are improvement of the estimation accuracy of the standard deviation of the Gaussian noise and development of an efficient search algorithm to find the optimal value en .
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