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Image Processing & Communications, vol. 21, no. 4, pp.5-12 DOI: 10.1515/ipc-2016-0019 5 RECONSTRUCTED QUANTIZED COEFFICIENTS MODELED WITH GENERALIZED GAUSSIAN DISTRIBUTION WITH EXPONENT 1/3 ROBERT K RUPI ŃSKI West-Pomeranian University of Technology in Szczecin, Chair of Signal Processing and Multimedia Engineering, ul. 26-Kwietnia 10, 71-126 Szczecin, Poland, e-mail:[email protected] Abstract. DCT, WHT (Walsh-Hadamard Transform) and DST (Dis- Generalized Gaussian distribution (GGD) in- crete Sine Transform) could be modeled with GGD and it cludes specials cases when the shape parame- was discussed by Clarke [5]. Zero-mean GGD was ap- ter equals p = 1 and p = 2. It corresponds plied to the tangential wavelet coefficients for compress- to Laplacian and Gaussian distributions respec- ing three-dimensional triangular mesh data by Lavu et tively. For p → ∞, f (x) becomes a uniform al. [18]. Sharifi et al. [27] applied GGD to 16 frequency distribution, and for p → 0, f (x) approaches subbands of the original and the difference frames of a an impulse function. Chapeau-Blondeau et video sequence. Achim et al. [1] modeled the ultrasound al. [4] considered another special case p = 0.5. image wavelet coefficients by the generalized Laplacian The article discusses more peaky case in which density. The image segmentation algorithm based on the GGD p = 1/3. wavelet transform with the application of GGD was presented in [31]. GGD and asymmetric GGD (AGGD) Key words. Generalized Gaussian distribution, were fitted to certain regular statistical properties of natu- maximum likelihood estimation, quantization, ral images to get the natural scene statistics (NSS) model reconstruction. in [33]. Wang et al. [32] applied GGD to approximate an atmosphere point spread function (APSF) kernel to pro- 1 Introduction pose the efficient method to remove haze from a single image. Song et al. [28] constructed a GGD based model Generalized Gaussian distribution has been widely used to to introduce more facial details into the initial image synmodel distributions ranging from a highly peaked to a uni- thesis. The statistical properties of the stereoscopic image form one. It has been also applied in many different areas, of the reorganized discrete cosine transform (RDCT) subex. watermarking [9]. It is very often applied to model the band coefficients were modeled with GGD to propose the transform coefficients such as discrete cosine transform stereoscopic image quality assessment in [20]. (DCT) or wavelet ones. The coefficients of the transforms Many methods have been designed to estimate the paUnauthenticated Download Date | 6/16/17 5:10 PM 6 R. Krupiński Tab. 1: The shape parameters of GGD for "Cameraman" Ci,j i=0 1 2 3 4 5 6 7 j=0 – 0.32 0.41 0.42 0.39 0.40 0.49 0.51 1 0.29 0.27 0.32 0.38 0.41 0.46 0.52 0.49 2 0.26 0.29 0.34 0.40 0.41 0.46 0.52 0.54 3 0.27 0.31 0.31 0.37 0.40 0.46 0.53 0.53 4 0.31 0.31 0.33 0.42 0.38 0.44 0.53 0.59 5 0.36 0.32 0.39 0.40 0.45 0.47 0.47 0.52 6 0.35 0.37 0.37 0.42 0.46 0.49 0.50 0.54 7 0.39 0.37 0.40 0.44 0.49 0.52 0.54 0.55 Tab. 2: The shape parameters of GGD for “Cameraman”, where at least 95% of DCT coefficients were none-zero after dequantization Ci,j i=0 1 2 3 4 5 6 j=0 – 0.32 0.41 0.42 0.39 0.40 0.49 1 0.29 0.27 0.32 0.38 0.41 0.46 – 2 0.26 0.29 0.34 0.40 0.41 – – 3 0.27 0.31 0.31 0.37 0.40 – – 4 0.31 0.31 0.33 0.42 – – – 5 0.36 0.32 0.39 – – – – 6 0.35 0.37 – – – – – 7 0.39 0.37 – – – – – Fig. 1: The "Cameraman" image pixels. DCT was performed for a block 8×8. The estimation of the shape parameters of GGD was applied to DCT rameters of GGD. A review of the different approaches coefficients. Ci,j denotes a coefficient in i row and j colto the shape parameter estimation problems can be found umn in the block 8 × 8 of DCT coefficients. The indexes in [30]. The disadvantage of this distribution is that these of DCT coefficients Ci,j vary i, j ∈< 0, 7 >, where C0,0 methods are complex. Therefore, authors proposed the corresponds to DC coefficient. Table 2 was derived from approximated approach [15, 17]. Table 1 in order to check which distributions are avail- Chapeau-Blondeau et al. [4] showed that by restricting able to the decoder. If, after dequantization, at least 95% the power of distribution to a value of 0.5 the calculations of DCT coefficients were none-zero, the distribution was can be simplified and the equations can be presented in a taken into account. Otherwise, the shape parameters were closed form. This allowed to improve the image recon- skipped, which resulted in the reduced table. struction based on the quantized DCT coefficients [14]. The article extends this approach to more peaky distribution, where the power coefficient of GGD goes toward 0. By assuming a source signal with GGD with a power parameter p = 1/3, equations for the centroid reconstruction in a closed form can be obtained, whereas for a GGD model it cannot be done. The maximum likelihood (ML) It can be noticed that some distributions are appropriate to model with GGD with a power parameter p = 1/3. It depends on the source image where for the "Lenna" and "Barbara" images the distributions were more close to GGD 0.5 [14]. The article is organized in the following manner. In method of discrete GGD p = 1/3 is derived, which re- Section 2 the continuous GGD p = 1/3 is presented and quires the estimation of only one parameter. in Section 3 the discrete GGD p = 1/3 is discussed. In Table 1 contains the results collected for the estimation Section 4 the biased reconstruction of quantized coeffiof the shape parameters of GGD for the “Cameraman” cients assuming GGD p = 1/3 is introduced. The experiimage. The image was monochromatic, 256 × 256 size in mental results are presented in Section 5. Unauthenticated Download Date | 6/16/17 5:10 PM 7 Image Processing & Communications, vol. 21, no. 4, pp. 5-12 2 Continuous generalized Gaussian density function with exponent 1/3 takes the form N 1 X |xi |1/3 3N i=1 λ= !−3 (5) Probability density function of the continuous random where N denotes the number of observations. variable of GGD is [3, 5, 7, 8] p λ·p e−[λ·|x|] f (x) = 2 · Γ p1 0 λ=0.5 λ=1 λ=2 0.1 (1) 0.08 tz−1 e−t dt, z > 0 [21], p is the shape f(x) where Γ(z) = R∞ 0.12 parameter and λ is connected to the variance of the distri- 0.06 0.04 bution. 0.02 The special case of the density function of GGD with 0 −10 exponent p = 1/3 of the continuous random variable is f (x) = λ −[λ·|x|]1/3 e 12 −5 0 x 5 10 (2) Fig. 2: Density function of GGD with exponent p = 1/3 of the continuous random variable for three selected paThe cumulative distribution is obtained by integrating rameters λ Equation (3) Fig. 2 depicts density function of GGD with exponent Zx F (x) = f (z)dz (3) p = 1/3 of the continuous random variable for three selected parameters λ. −∞ which results in the cumulative GGD p = 1/3 1 −g · (2 + 2g + g 2 ), for x ≤ 0 4e F (x) = 1 −h 1 − 4 e · (2 + 2h + h2 ), for x > 0 Probability density function of the continuous zeromean random variable is usually assumed for the coef(4) ficients before quantization available to the encoder. 1/3 g = (−λ · x) , 1/3 h = (λ · x) . The advantage of fixing the power parameter is that the where cumulative GGD p = 1/3 can be presented in a closed 3 Discrete generalized Gaussian density function with exponent 1/3 form. Many methods to estimate parameters has been de- In JPEG and MPEG reconstruction [10–12, 23, 25, 26] signed. The most common approach is to use the max- the coefficients available to the decoder are reconstructed imum likelihood estimators. The maximum likelihood to the bin center. The reconstructed values are yi = i · Q, function and estimators are discussed in [6, 19, 29, 34]. where i is both the bin index and the quantized value. The The maximum likelihood estimator for continuous GGD parameter Q is the quantization factor (the length of the p = 1/3 can be obtained by finding the maximum like- interval). The value yi represents a reconstructed value, lihood function of Equation (2) and maximizing it with which is also the bin center. respect to λ. After certain transformations the estimator Integrating function f (x) (Equation (2)) over the interUnauthenticated Download Date | 6/16/17 5:10 PM 8 R. Krupiński val (Q · i − 0.5 · Q, Q · i + 0.5 · Q) Pi = λ 12 (i+0.5)·Q Z e−[λ·|x|] 1/3 form dx N0 · e−A · Q + P0 (6) N (i−0.5)·Q + gives the probability density function of the discrete random variable of GGD 1/3 Pi = 41 e−Ci · (2 + 2Ci + Ci2 )− − 14 e−Bi · (2 + 2Bi + Bi2 ) i 6= 0 P0 = 1 − 12 e−A · (2 + 2A + A2 ) i = 0 1 e−Bi · Bi3 − e−Ci · Ci3 1X = 0, λ i=1 Pi (8) where N0 denotes the number of observations equal zero and N1 denotes the number of observations not equal (7) zero. The equation N = N0 + N1 must be held. The estimated λ parameter is received only from the discrete ob- servations (the quantized values available to the decoder) 1/3 A = (0.5 · λ · Q) , without prior knowledge of the coefficients before quantiwhere Bi = (λ · (|yi | + 0.5 · Q))1/3 , 1/3 zation available to the encoder. Therefore, it is expected to Ci = (λ · (|yi | − 0.5 · Q)) . Fig. 3 depicts the density function of GGD with expo- restore GGD p = 1/3 of the continuous random variable nent p = 1/3 and λ = 1 for discrete random variable available to the encoder before the quantization process. Pi (Equation (7)) and continuous random variable f (x) (Equation (2)). It should be noted that the distribution of coefficients available to the decoder is discrete and consists of scaled deltas in the bin centers. 4 The reconstruction of coefficients The reconstructed coefficients can be biased based on the assumed model. Different models have been applied 0.25 P i in [2, 16, 24]. Based on the observation that the DCT co- f(x) 0.2 efficients have a peak at zero and decrease exponentially, Ahumada et al. [2] made adjustments to the reconstructed 0.15 coefficients. The reconstruction for Laplace distribution with centroid was presented in [16, 24]. 0.1 According to the equation for the centroid reconstruc- 0.05 tion of the distribution in [22], the equation for the cen0 −50 0 x 50 Fig. 3: Density function of GGD with exponent p = 1/3 and λ = 1 for discrete random variable Pi (Equation (7)) and continuous random variable f (x) (Equation (2)) The estimator of discrete GGD 1/3 of the maximum likelihood method can be found similarly as for the Laplacian discrete source [24] and GGD 0.5 [14]. The maxi- troid reconstruction of GGD 1/3 takes the form yˆi = sgn(yi ) · 1 Li · λ Mi (9) where Li = e−Ci · (Ci5 + 5 · Ci4 + 20 · Ci3 + +60 · Ci2 + 120 · Ci + 120)− −e−Bi · (Bi5 + 5 · Bi4 + 20 · Bi3 + +60 · Bi2 + 120 · Bi + 120), Mi = e−Ci · (2 + 2Ci + Ci2 ) − e−Bi · (2 + 2Bi + Bi2 ). The value reconstructed with centroid minimizes the mum likelihood function of Equation (7) is set and then Mean Square Error (MSE) in the interval (Q · i − 0.5 · it is maximized with respect to λ. After certain transfor- Q, Q · i + 0.5 · Q). mations the ML estimator of discrete GGD 1/3 takes the Another advantage of fixing the power parameter p = Unauthenticated Download Date | 6/16/17 5:10 PM 9 Image Processing & Communications, vol. 21, no. 4, pp. 5-12 4 1/3 is that the reconstruction equation can be presented in 10 λ=0.1 λ=2.3 λ=4.5 a closed form. 2 10 Experiments ∆RMSE 5 In the first simulation, the performance of maximum like- 0 10 −2 10 lihood estimator is evaluated (Equation (8)). The input sequence is generated with the GGD generator [13] with −4 10 0 200 400 600 800 1000 N p = 1/3. Then the sequence is quantized and dequan- tized, which corresponds to lossy compression. The aim Fig. 5: Difference between RMSE of the estimators Equaof the estimator is to reproduce the initial distribution be- tion (5) and Equation (8) for the quantization factor Q = fore the quantization only on the dequantized coefficients. 20 The sequence range N ∈< 31, 1000 > and the quantization steps Q ∈< 2, 20 > are considered. The simulation performs better than the estimator based on the contin- was repeated 1000 times. Relative Mean Square Error uous distribution. Figures show the positive values that confirms the expectations. The higher value of λ, the es- (RMSE) was calculated from the equation RM SE = timator (8) is getting better in terms of RMSE. It can be M 1 X (λ̂ − λ)2 M i=1 λ2 (10) also noticed that the higher value of quantization factor Q, the estimator (8) is getting better in terms of RMSE. where λ̂ is a value estimated by the model and λ is a real In the next simulation, the set of DCT coefficients is value of a lambda parameter. M denotes the number of generated with the GGD generator and the shape paramrepetitions. eter p = 1/3. The input sequence is created xi with calculating the inverse of DCT. These DCT coefficients are 2 10 quantized and dequantized, and the restored sequence yi is calculated with the inverse of DCT. MSE is calculated 0 10 ∆RMSE for the lossy transformation. −2 10 M SE = −4 10 10 0 5 10 Q 15 (11) where yˆi is reconstructed sequence and xi is the input sig- λ=0.1 λ=2.3 λ=4.5 −6 N 1 X (yˆi − xi )2 N i=1 20 nal. The reconstructed DCT coefficients are modified with Fig. 4: Difference between RMSE of the estimators Equa- Equation (9) for the λ parameter estimated from Equation (5) and Equation (8) for the sequence length N = tions (5) and (8). Then the restored sequence yˆ is calcui 1000 lated with the inverse of DCT and MSE. Figures 4 and 5 depict the difference between RMSE Figures 6 and 7 depict the difference between MSE of of the estimators Equation (5) and Equation (8). It is ex- normally reconstructed signal yi and modified reconstrucpected that the estimator based on the discrete distribution tion yˆi ((5) or (8)). It can be noticed that the modified Unauthenticated Download Date | 6/16/17 5:10 PM 10 R. Krupiński reconstruction yˆi gives smaller MSE than MSE of nor- transformation. The reconstructed DWT coefficients are mally reconstructed signal yi (the positive values in the modified with Equation (9) for the λ parameter estimated figures). It should be noted that the estimator (8) gives from Equations (5) and (8). Then the restored sequence yˆi is calculated with the inverse of DWT and MSE. smaller MSE than the estimator (5). 1.5 1.5 λd=0.1 λd=0.1 λc=0.1 λc=0.1 λ =2.3 d λ =2.3 λc=2.3 d λc=2.3 λd=4.5 1 λd=4.5 1 λc=4.5 ∆MSE ∆MSE λc=4.5 0.5 0.5 0 0 5 10 Q 15 20 0 0 5 10 Q 15 20 Fig. 6: Difference between MSE of normally recon- Fig. 8: Difference between MSE of normally reconstructed signal yi and modified reconstruction yˆi (λc (5) structed signal y and modified reconstruction yˆ (λ (5) i i c or λd (8)) for the sequence length N = 1000 (IDCT) or λd (8)) for the sequence length N = 1000 (IDWT) 1.6 1.4 1.2 1.5 λc=0.1 λd=0.1 λd=2.3 λc=0.1 λc=2.3 λd=2.3 λd=4.5 λc=2.3 λc=4.5 1 λd=4.5 λc=4.5 0.8 ∆MSE ∆MSE 1 λd=0.1 0.6 0.4 0.5 0.2 0 0 200 400 600 800 1000 N 0 0 200 400 600 800 1000 N Fig. 7: Difference between MSE of normally reconstructed signal yi and modified reconstruction yˆi (λc (5) Fig. 9: Difference between MSE of normally reconstructed signal yi and modified reconstruction yˆi (λc (5) or λd (8)) for the quantization factor Q = 20 (IDCT) or λd (8)) for the quantization factor Q = 20 (IDWT) In the last simulation, the set of detailed Discrete Wavelet Transform (DWT) coefficients is generated with Figures 8 and 9 depict the difference between MSE of the GGD generator and the shape parameter p = 1/3. The normally reconstructed signal yi and modified reconstrucinput sequence is created xi with calculating the inverse tion yˆi ((5) or (8)). It can be noticed that the modified of DWT, whereas the approximation DWT coefficients are reconstruction yˆi gives smaller MSE than MSE of norset to zero. These DWT coefficients are quantized and de- mally reconstructed signal yi (the positive values in the quantized, and then the restored sequence yi is calculated figures). It should be noted that the estimator (8) gives with the inverse of DWT. MSE is calculated for the lossy smaller MSE than the estimator (5). Unauthenticated Download Date | 6/16/17 5:10 PM 11 Image Processing & Communications, vol. 21, no. 4, pp. 5-12 6 Summary [6] Deutch, R. (1965). Estimation theory. PrenticeHall, Englewood Cliffs, N.J. Chapeau-Blondeau et al. [4] considered a special case of GGD where the power parameter is p = 0.5. In this arti- [7] Du, Y. (1991). Ein sphärisch invariantes Verbund- cle, the special case of GGD is considered where p = 1/3. dichtemodell für Bildsignale. AEU. Archiv für Elek- This distribution is more peaky comparing to p = 0.5. Se- tronik und Übertragungstechnik, 45(3), 148-159. lecting such a model allows to simplify calculations and denote the closed form equations. The density function of GGD with exponent p = 1/3 of the continuous random variable and the probability density function of the [8] Farvardin, N., Modestino, J. (1984). Optimum quantizer performance for a class of non-Gaussian memoryless sources. IEEE Transactions on Information Theory, 30(3), 485-497. discrete random variable of GGD p = 1/3 are given. Assuming the GGD p = 1/3 model of quantized coef- [9] Hernandez, J. R., Amado, M., Perez-Gonzalez, F. ficients, the modified reconstruction equation is defined. (2000). DCT-domain watermarking techniques for The simulations showed that MSE can be improved by still images: Detector performance analysis and a the application of this modified reconstruction for the sig- new structure. IEEE transactions on image process- nals that are transformed with either DWT or DCT, where ing, 9(1), 55-68. the information is lost by quantization. [10] ISO/IEC. (1993). Coding of moving pictures and associated audio for digital storage media up to References about 1,5 Mbits/s. International Standard 11172 [1] Achim, A., Bezerianos, A., Tsakalides, P. (2001). [11] ISO/IEC. (1994). Generic coding of moving pic- Novel Bayesian multiscale method for speckle re- tures and associated audio information. Interna- moval in medical ultrasound images. IEEE transac- tional Standard 13818 tions on medical imaging, 20(8), 772-783. [12] ISO/IEC. (1998). Information technology – cod[2] Ahumada, A.J., Horng, R. (1994, January). Smoothing DCT compression artifacts. 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