Tuning of Fuzzy Sharpening Filters for Biomedical Image Enhancement

MENDEL — Soft Computing Journal, Volume 24, No.1, June 2018, Brno, Czech RepublicX TUNING OF FUZZY SHARPENING FILTERS FOR BIOMEDICAL IMAGE ENHANCEMENT Jaromir Kukal1∗, Abduljalil Sireis2, Zuzana Krbcova2 1Czech Technical University in Prague Faculty of Nuclear Sciences and Physical Engineering Department of Software Engineering Trojanova 13, 120 00 Prague 2 Czech Republic jaromir.kuk[email protected] 2University of Chemistry and Technology in Prague Faculty of Chemical Engineering Department of Computing and Control Engineering Technicka 5, 166 26 Prague 6 Czech Republic Abstract: Various approaches are used for image smoothing and sharpening. The class of fuzzy filters is widely used in the case of spiky noise due to their non–linear behavior. A lot of popular fuzzy filters are realizable in Lukasiewicz algebra with square root. Frequently applied low-pass fuzzy filters were selected from literature and used for the image sharpening with dyadic weights. The first aim of the paper is to find the optimum sharpening with the best Signal–to–Noise Ratio criterion for various noise types and offer general suggestions for fuzzy filter selection. Our results are directly applicable to tomographic images from MRI, PET and SPECT scanners. Keywords: image smoothing, image sharpening, Lukasiewicz algebra, fuzzy image processing, filter bank, MRI, PET, SPECT 1 Introduction There are many approaches for designing linear and nonlinear 2D image enhancement. As typical for biomedical SPECT, PET, and MRI scans, the spiky noise has to be eliminated by statistically robust filters. The paper is based on fuzzy image processing in Lukasiewicz algebra with square root [1, 2] which enables to realize robust low-pass and sharpening filters. The enhancement quality is measured by Signal to Noise Ratio (SNR) and various combinations of integer masks and sharpening approaches are applied to biomedical image with Gaussian and spiky noise. There is only a finite number of combinations and the most suitable ones are discussed. Lukasiewicz algebra with square root is remembered in the second section. The next section summarizes several operations which are useful for data processing. The frame of fuzzy local image processing and computer experiment design are established in the fourth section. The fifth section summarizes the main results of optimal sharpening with the highest possible SNR [3] which is followed by concluding remarks. 2 Lukasiewicz Algebra with Square Root The main idea behind the scope of the paper is to compare real properties of various fuzzy filters. First, it is necessary to specify the collection of permitted operations. Supposing intensity normalization into [0, 1] interval, we decide to apply Many–Valued Algebra (MVA) [4]. Let L= [0,1] be supporting set. The Lukasiewicz algebra with square root [2] is defined as LAsqrt =hL,∧,∨,⊗,→,0,1,sqrti(1) where •a∧b= min(a, b), •a∨b= max(a, b), •0 = min L, 121 ISSN: 1803-3814 MENDEL — Soft Computing Journal, Volume 24, No.1, June 2018, Brno, Czech RepublicX •1 = max L, •a⊗b= max(a+b−1,0), •a→b= min(1 −a+b, 1), •sqrt(a) = √a= (1 + a)/2. Derived operators of LAsqrt are • ¬a= 1 −a, •a⊕b=¬(¬a⊗¬b) = min(a+b, 1), •ab=a⊗¬b= max(a−b, 0). Given algebra is frequently used in fuzzy image processing [5]. 3 Permitted Operations in LAsqrt The main question is how to design selected operations for signal and image processing in given algebra. Fortunately, LAsqrt is able to realize both traditional operators [5] and constrainded linear transforms with dyadic weights. Let a, b, c ∈[0,1] be fuzzy variables, kbe order in sorted list of nvalues, H∈N0be dyadic exponent, and α=m/2Nbe dyadic weight. There are several cases: •When a+b≤1, standard addition is realizable as a+b=a⊕b. •When a≥b, standard subtraction is realizable as a−b=ab. •Using functions ϕ(x) = sqrt(x)(1/2) = x/2 and trim(x) = min(1,max(0, x)) we can realize operations based on traditional multiplication by dyadic weight. The last case can be proven as follows. First, x/2H=ϕ[H](x) where ϕ[0](x) = x, ϕ[H](x) = ϕ(ϕ[H−1](x)).Therefore, for mx ≤2Nwe directly have m·x 2H= m M k=1 ϕ[H](x) (2) which will be useful for realization of several linear filters. Traditional image sharpening is unconstrained and based on formula a+α·(b−c) which has to be trimmed into [0,1] interval in the case of fuzzy processing. Therefore, we directly calculate trim(a+α·(b−c)) = a⊕α·(bc)α·(cb).(3) Finally, when the data list of nvalues is sorted as x(1) ≤x(2) ≤. . . ≤x(n)(4) the kth value x(k)can be expressed in Disjoint Normal Form (DNF) and directly calculated using operators ∨,∧. 4 Fuzzy Local Image Processing Using simple tricks from previous section, we can perform selected procedures of local image processing. Basic terms and operations are recalled first. Let M, N ∈Nbe height and width of normalized image X∈[0,1]M×N. The local image processing operates on list of values [6] from pixel neighborhood. The values are represented by vector x∈[0,1]mof length m∈N. It can be connected with vector y∈[0,1]nof length n∈Nusing operator xty= (x1, x2, . . . , xm, y1, y2, . . . , yn)T.(5) Aggregation operator [3] is therefore defined as N G k=1 xk=x1tx2t... txn.(6) 122 Tuning of Fuzzy Sharpening Filters for Biomedical Image Enhancement MENDEL — Soft Computing Journal, Volume 24, No.1, June 2018, Brno, Czech RepublicX Multiple evidence of values can be obtained by multiplication operator nx= N G k=1 x(7) which will be also useful for mask applications. The integer mask of radius r∈Nis represented by matrix W∈N0(2r+1)×(2r+1) with central symmetry. The local processing with mask Wof image part B∈[0,1](2r+1)×(2r+1) generates the weighted list of values [3] as vector x= 2r+1 G i=1 2r+1 G j=1 (wi,j bi,j) (8) of length nand any permitted operation from LAsqrt can be used for local processing. The local fuzzy image processing is frequently based on the evaluation of sample statistics: •Erosion [3] as E(x) = Vn k=1 xk, •Dilation [3] as D(x) = Wn k=1 xk, •Median [7, 8] for odd nas M(x) = x(k)where k= (n+ 1)/2, •Median for even nas M(x) = ϕ(x(k))⊕ϕx(k+1)) where k=n/2, •First quartile [9] as Q1(x) = M(x(1), . . . , x(k)) where k=bn/2c, •Third quartile [9] as Q3(x) = M(x(k), . . . , x(n)) where k=dn/2e+ 1, •Average [10] for dyadic n= 2Has A(x) = Ln k=1 ϕ[H](x(k)). More sophisticated statistical calculations are based on Walsh list [11] which was originally published as W(x) = G i≤j xi+xj 2(9) but can be expressed in fuzzy form W(x) = n G i=1 n G j=i (ϕ(xi)⊕ϕ(xj)).(10) The other useful local operators are [3]: •Hodges–Lehmann Median [11] as H(x) = M(W(x)), •Opening O(x) as erosion followed by dilation, •Closing C(x) as dilation followed be erosion, •OC Mean OCM(x) = ϕ(O(x)) ⊕ϕ(C(x)), •OC Filter OC(x) as opening followed by closing, •CO Filter CO(x) as closing followed by opening, •OCCO Mean OCCOM(x) = ϕ(OC(x)) ⊕ϕ(CO(x)), •Constrained Filter CF(x) = (A(x)∨Q1(x)) ∧Q3(x) and the central weight of mask Wneeds to be replaced by zero. 123 J. Kukal et al. MENDEL — Soft Computing Journal, Volume 24, No.1, June 2018, Brno, Czech RepublicX Previous approaches can be used for Fuzzy Low–Pass (FLP) [12] filtering which should improve the image quality. Another possibility is to apply FLP filter in image sharpening process using xcas intensity behind the mask center. Resulted sharpening operator S(x) = FLP(x) + α·(xc−FLP(x)) (11) is also realizable in LAsqrt for dyadic sharpening gain α≥0. 5 Sharpening Filters Portfolio Various low-pass fuzzy filters with various but symmetric masks will be used for image sharpening with dyadic gain. Resulting bank of fuzzy filters is determined by filtering approaches and mask weights. 5.1 Fuzzy Filtering Approaches Various fuzzy sharpening approaches were used for 2D image enhancement. List of fuzzy low–pass filters is included in Tab. 1. Image sharpening gain was studied for α∈ {k/32 : k= 0,...,160}for various images and noise types. Table 1: Low–Pass Filter Portfolio Index Filter Description 1 M Median 2 A Average 3 H Hodges–Lehmann 4 O Opening 5 C Closing 6 OCM Opening-Closing Mean 7 OC Opening followed by Closing 8 CO Closing followed by Opening 9 OCCOM OC and CO Mean 10 CF Constrained Filter 5.2 Compact 3×3Masks In this study we used five small compact masks as M1=  010 111 010  ,M2=  010 141 010  , M3=  111 111 111  ,M4=  111 181 111  ,M5=  121 242 121   but only M2,M4,M5are applicable in average filter. 5.3 Compact 5×5Masks Seven large compact masks were also used as M6=       00100 01110 11111 01110 00100       ,M7=       00100 01110 11411 01110 00100       , M8=       01110 11111 11111 11111 01110       ,M9=       0 1 1 1 0 1 1 1 1 1 111211 1 1 1 1 1 0 1 1 1 0       , 124 Tuning of Fuzzy Sharpening Filters for Biomedical Image Enhancement MENDEL — Soft Computing Journal, Volume 24, No.1, June 2018, Brno, Czech RepublicX M10 =       11111 11111 11111 11111 11111       ,M11 =       11111 11111 11811 11111 11111       , M12 =       1 4 6 4 1 4 16 24 16 4 6 24 36 24 6 4 16 24 16 4 1 4 6 4 1       but only M7,M9,M11,M12 are applicable in average filter. 5.4 Ring Masks The dyadic property was also obtained by removing central points. Resulting ring masks are M13 =  010 101 010  ,M14 =  111 101 111  , M15 =       00100 01010 10001 01010 00100       ,M16 =       01110 11011 10001 11011 01110       . 6 Optimal Sharpening in LAsqrt Numerical experiments on two representative images were performed to obtain the best possible increasing of Signal to Noise Ratio (SNR) as generally recommended criterion. Pixel intensity of ideal image was normalized into interval [0, 1], first. The random noise was added and resulting image was constrained to original range [0, 1] as model of real image in the second step. Various types of fuzzy sharpening filters with various masks and sharpening gains were applied to real image to obtain a set of enhanced images. The SNR criterion was designed using the logarithmic scale as SNR = 10 log10 Psignal Pnoise (12) where Psignal, Pnoise are average powers of the signal and additive noise. The quality of sharpening was studied as improvement ∆SNR =SNR −SNR0in dB where SNR0, SNR reflect the image quality before and after the sharpening. A slice of human brain 2D MRI ROI of size 40×40 was used in the first case together with additive Gaussian noise σ= 0.05 as example of low-level noise. Results of sharpening are included in Tabs. 2,3 as optimal increasing of SNR value and adequate sharpening gain. The symbol * means that the filter cannot be realized in the fuzzy system. The best results were obtained for averaging filter F2with four-point ring mask M13. The same mask also produced acceptable increasing of SNR for Hodges–Lehmann median F3, median F1, and constraind filter F10 with α= 13/32 in majority of cases. The open–closing filter F6is also acceptable with five-point mask M1and α= 1/16. The noised MRI image and the result of the best filter F2are depicted in Fig. 1. The same image of size 40×40 was used in the second case with spiky noise generated as follows. Additive Gaussian noise with σ= 0.1 was corrupted by pepper-salt noise with probability 0.05. Results of sharpening are included in Tabs. 4,5 as optimal SNR increasing values and adequate sharpening gains. The best results were obtained for Hodges–Lehmann median filter F3with binomial mask M12 and α= 0. The median F1, OCCOM F9, and constrained F10 filters over–performed the average filter F2in SNR improvement. The same MRI image corrupted by spiky noise and the result of the best filter F3are depicted in Fig. 1. 7 Conclusions The fuzzy low–pass and sharpening filters have been investigated in the case of MRI images with Gaussian and spiky noise. The best results in ∆SNR were obtained for Hodges–Lehmann low–pass filter with binomial 5 ×5 125 J. Kukal et al. MENDEL — Soft Computing Journal, Volume 24, No.1, June 2018, Brno, Czech RepublicX Figure 1: Performance of fuzzy filters: Gaussian–noised image (left top), smoothing by dyadic average (right top), spiky–noised image (left bottom), smoothing by Hodges–Lehmann median (bottom right) mask in the case of pepper–salt noise. The same filtering approach was also suitable in the case of Gaussian noise however the best choice is dyadic average low–pass filter with four–point ring 3 ×3 mask in the Gaussian case. This study can be useful for future biomedical applications with unknown noise type and range when the Hodges–Lehmann low–pass filter with binomial 5 ×5 mask frequently overperforms traditional linear, median, and morphological filters and the sharpening is omitted. Acknowledgement: This paper is supported by SGS17/196/OHK4/3T/14 grant of Czech Technical University in Prague, grant GA17-05840S of the Czech Science Foundation, and MSMT No 21-SVV/2018. References [1] Hajek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998) [2] Novak, V., Perfilieva, I., Mockor, J.: Mathematical Principles of Fuzzy Logic. Kluwer Academic Publishers, Boston (1999) [3] Gonzalez, R.C., Woods, R.E.: Digital Image Processing (3rd edition). Prentice Hall, New York (2008) 126 Tuning of Fuzzy Sharpening Filters for Biomedical Image Enhancement MENDEL — Soft Computing Journal, Volume 24, No.1, June 2018, Brno, Czech RepublicX Table 2: Sharpening quality as ∆SNR [dB] for Gaussian noise MASK F1F2F3F4F5F6F7F8F9F10 1 1.37 * 1.71 0.66 0.42 1.18 0.76 0.67 0.86 * 2 0.43 1.78 1.24 0.66 0.42 1.18 0.76 0.67 0.86 1.77 3 1.51 * 1.66 0.46 0.21 0.75 0.40 0.25 0.43 * 4 0.21 1.53 1.26 0.46 0.21 0.75 0.40 0.25 0.43 1.50 5 1.35 1.69 1.73 0.46 0.21 0.75 0.40 0.25 0.43 1.68 6 1.13 * 1.07 0.29 0.13 0.47 0.23 0.14 0.24 * 7 0.89 0.94 1.12 0.29 0.13 0.47 0.23 0.14 0.24 0.94 8 0.86 * 0.67 0.18 0.09 0.33 0.14 0.08 0.15 * 9 0.39 0.54 0.77 0.18 0.09 0.33 0.14 0.08 0.15 0.55 10 0.74 * 0.54 0.13 0.03 0.19 0.09 0.03 0.08 * 11 0.55 0.41 0.62 0.13 0.03 0.19 0.09 0.03 0.08 0.41 12 1.24 1.04 1.25 0.13 0.03 0.19 0.09 0.03 0.08 1.04 13 1.67 1.85 1.79 0.54 0.31 0.90 0.57 0.48 0.61 1.85 14 1.65 1.55 1.66 0.43 0.20 0.69 0.41 0.29 0.44 1.55 15 0.81 0.58 0.63 0.21 0.09 0.09 0.18 0.12 0.19 0.58 16 0.63 0.36 0.42 0.17 0.09 0.09 0.14 0.09 0.15 0.36 Table 3: Sharpening gain α(in 1/32) for Gaussian noise MASK F1F2F3F4F5F6F7F8F9F10 1 9 * 8 16 19 2 17 18 14 * 2 0 0 0 16 19 2 17 18 14 0 3 11 * 11 22 25 13 23 25 22 * 4 0 0 0 22 25 13 23 25 22 1 5 5 7 6 22 25 13 23 25 22 8 6 15 * 16 26 28 20 27 28 26 * 7 9 14 11 26 28 20 27 28 26 14 8 19 * 21 28 29 23 28 29 28 * 9 7 17 10 28 29 23 28 29 28 17 10 20 * 22 29 30 26 29 31 29 * 11 15 22 18 29 30 26 29 31 29 22 12 12 16 13 29 30 26 29 31 29 16 13 14 13 13 17 21 6 19 20 18 13 14 14 14 14 22 25 14 23 24 21 14 15 21 22 22 27 28 28 27 28 27 22 16 22 25 24 28 29 29 28 29 28 25 [4] Azevedo, F.: Constraint Solving over Multi-Valued Logics: Application to Digital Circuits. 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Kukal et al. MENDEL — Soft Computing Journal, Volume 24, No.1, June 2018, Brno, Czech RepublicX Table 4: Sharpening quality as ∆SNR [dB] for spiky noise MASK F1F2F3F4F5F6F7F8F9F10 1 6.67 * 7.39 3.95 0.93 5.15 6.28 6.00 8.09 * 2 2.42 4.40 4.13 3.95 0.93 5.15 6.28 6.00 8.09 6.33 3 8.38 * 8.51 3.13 0.63 4.64 4.73 3.60 6.84 * 4 1.85 4.71 4.57 3.13 0.63 4.64 4.73 3.60 6.84 7.34 5 7.08 6.66 8.08 3.13 0.63 4.64 4.73 3.60 6.84 7.37 6 8.23 * 8.26 2.88 0.61 4.67 4.14 3.31 6.36 * 7 7.48 6.18 7.76 2.88 0.61 4.67 4.14 3.31 6.36 7.32 8 7.99 * 7.93 2.09 0.43 3.55 2.96 2.23 5.34 * 9 5.13 5.60 5.42 2.09 0.43 3.55 2.96 2.23 5.34 7.07 10 7.70 * 7.60 1.71 0.37 3.15 2.26 1.79 4.56 * 11 7.36 6.19 7.03 1.71 0.37 3.15 2.26 1.79 4.56 6.95 12 8.55 7.17 8.61 1.71 0.37 3.15 2.26 1.79 4.56 7.41 13 7.21 5.80 7.25 3.74 0.73 4.54 6.07 5.75 7.37 5.80 14 8.56 6.94 8.34 3.48 0.80 5.12 5.36 4.92 7.32 7.08 15 7.24 5.95 7.16 3.23 1.02 5.01 4.83 4.16 6.25 6.04 16 7.10 6.10 7.07 2.69 0.91 5.07 3.96 3.44 5.98 6.20 Table 5: Sharpening gain α(in 1/32) for spiky noise MASK F1F2F3F4F5F6F7F8F9F10 1 0 * 0 6 11 0 4 2 0 * 2 0 0 0 6 11 0 4 2 0 0 3 1 * 1 11 20 0 8 9 2 * 4 0 0 0 11 20 0 8 9 2 0 5 0 0 0 11 20 0 8 9 2 0 6 2 * 1 13 21 0 10 11 3 * 7 0 0 0 13 21 0 10 11 3 0 8 2 * 2 17 25 4 15 16 6 * 9 0 0 0 17 25 4 15 16 6 0 10 3 * 3 20 26 6 18 18 8 * 11 0 0 0 20 26 6 18 18 8 0 12 0 0 0 20 26 6 18 18 8 0 13 4 7 4 6 14 0 3 2 0 7 14 3 5 3 9 16 0 6 5 1 5 15 4 6 4 10 15 0 8 8 3 6 16 4 5 4 14 19 0 11 11 5 5 128 Tuning of Fuzzy Sharpening Filters for Biomedical Image Enhancement