RECONSTRUCTED QUANTIZED COEFFICIENTS MODELED WITH GENERALIZED GAUSSIAN DISTRIBUTION WITH EXPONENT 1/3

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1 Image Processing & Communications, vol. 2, no. 4, pp.5-2 DOI:.55/ipc RECONSTRUCTED QUANTIZED COEFFICIENTS MODELED WITH GENERALIZED GAUSSIAN DISTRIBUTION WITH EXPONENT /3 ROBERT KRUPIŃSKI West-Pomeranian University of Technology in Szczecin, Chair of Signal Processing and Multimedia Engineering, ul. 26-Kwietnia, 7-26 Szczecin, Poland, rkrupinski@wp.pl Abstract. Generalized Gaussian distribution (GGD) includes specials cases when the shape parameter equals p = and p = 2. It corresponds to Laplacian and Gaussian distributions respectively. For p, f(x) becomes a uniform distribution, and for p, f(x) approaches an impulse function. Chapeau-Blondeau et al. [4] considered another special case p =.5. The article discusses more peaky case in which GGD p = /3. Key words. Generalized Gaussian distribution, maximum likelihood estimation, quantization, reconstruction. Introduction Generalized Gaussian distribution has been widely used to model distributions ranging from a highly peaked to a uniform one. It has been also applied in many different areas, ex. watermarking [9]. It is very often applied to model the transform coefficients such as discrete cosine transform (DCT) or wavelet ones. The coefficients of the transforms DCT, WHT (Walsh-Hadamard Transform) and DST (Discrete Sine Transform) could be modeled with GGD and it was discussed by Clarke [5]. Zero-mean GGD was applied to the tangential wavelet coefficients for compressing three-dimensional triangular mesh data by Lavu et al. [8]. Sharifi et al. [27] applied GGD to 6 frequency subbands of the original and the difference frames of a video sequence. Achim et al. [] modeled the ultrasound image wavelet coefficients by the generalized Laplacian density. The image segmentation algorithm based on the wavelet transform with the application of GGD was presented in [3]. GGD and asymmetric GGD (AGGD) were fitted to certain regular statistical properties of natural images to get the natural scene statistics (NSS) model in [33]. Wang et al. [32] applied GGD to approximate an atmosphere point spread function (APSF) kernel to propose the efficient method to remove haze from a single image. Song et al. [28] constructed a GGD based model to introduce more facial details into the initial image synthesis. The statistical properties of the stereoscopic image of the reorganized discrete cosine transform (RDCT) subband coefficients were modeled with GGD to propose the stereoscopic image quality assessment in [2]. Many methods have been designed to estimate the pa-

2 6 R. Krupiński Tab. : The shape parameters of GGD for "Cameraman" C i,j j= i= Tab. 2: The shape parameters of GGD for Cameraman, where at least 95% of DCT coefficients were none-zero after dequantization C i,j j= i= rameters of GGD. A review of the different approaches to the shape parameter estimation problems can be found in [3]. The disadvantage of this distribution is that these methods are complex. Therefore, authors proposed the approximated approach [5, 7]. Chapeau-Blondeau et al. [4] showed that by restricting the power of distribution to a value of.5 the calculations can be simplified and the equations can be presented in a closed form. This allowed to improve the image reconstruction based on the quantized DCT coefficients [4]. The article extends this approach to more peaky distribution, where the power coefficient of GGD goes toward. By assuming a source signal with GGD with a power parameter p = /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) method of discrete GGD p = /3 is derived, which requires the estimation of only one parameter. Table contains the results collected for the estimation of the shape parameters of GGD for the Cameraman image. The image was monochromatic, size in Fig. : The "Cameraman" image pixels. DCT was performed for a block 8 8. The estimation of the shape parameters of GGD was applied to DCT coefficients. C i,j denotes a coefficient in i row and j column in the block 8 8 of DCT coefficients. The indexes of DCT coefficients C i,j vary i, j <, 7 >, where C, corresponds to DC coefficient. Table 2 was derived from Table in order to check which distributions are available to the decoder. If, after dequantization, at least 95% of DCT coefficients were none-zero, the distribution was taken into account. Otherwise, the shape parameters were skipped, which resulted in the reduced table. It can be noticed that some distributions are appropriate to model with GGD with a power parameter p = /3. It depends on the source image where for the "Lenna" and "Barbara" images the distributions were more close to GGD.5 [4]. The article is organized in the following manner. In Section 2 the continuous GGD p = /3 is presented and in Section 3 the discrete GGD p = /3 is discussed. In Section 4 the biased reconstruction of quantized coefficients assuming GGD p = /3 is introduced. The experimental results are presented in Section 5.

3 Image Processing & Communications, vol. 2, no. 4, pp Continuous generalized Gaussian density function with exponent /3 Probability density function of the continuous random variable of GGD is [3, 5, 7, 8] f(x) = λ ( p )e [λ x ]p () 2 Γ p takes the form ( ) 3 N λ = x i /3 3N i= where N denotes the number of observations λ=.5 λ= λ=2 (5) where Γ(z) = t z e t dt, z > [2], p is the shape parameter and λ is connected to the variance of the distribution. The special case of the density function of GGD with exponent p = /3 of the continuous random variable is f(x) x f(x) = λ 2 e [λ x ]/3 (2) The cumulative distribution is obtained by integrating Equation (3) F (x) = x f(z)dz (3) which results in the cumulative GGD p = /3 { F (x) = 4 e g (2 + 2g + g 2 ), for x 4 e h (2 + 2h + h 2 ), for x > (4) where g = ( λ x)/3, h = (λ x) /3. The advantage of fixing the power parameter is that the cumulative GGD p = /3 can be presented in a closed form. Many methods to estimate parameters has been designed. The most common approach is to use the maximum likelihood estimators. The maximum likelihood function and estimators are discussed in [6, 9, 29, 34]. The maximum likelihood estimator for continuous GGD p = /3 can be obtained by finding the maximum likelihood function of Equation (2) and maximizing it with respect to λ. After certain transformations the estimator Fig. 2: Density function of GGD with exponent p = /3 of the continuous random variable for three selected parameters λ Fig. 2 depicts density function of GGD with exponent p = /3 of the continuous random variable for three selected parameters λ. Probability density function of the continuous zeromean random variable is usually assumed for the coefficients before quantization available to the encoder. 3 Discrete generalized Gaussian density function with exponent /3 In JPEG and MPEG reconstruction [ 2, 23, 25, 26] the coefficients available to the decoder are reconstructed to the bin center. The reconstructed values are y i = i Q, where i is both the bin index and the quantized value. The parameter Q is the quantization factor (the length of the interval). The value y i represents a reconstructed value, which is also the bin center. Integrating function f(x) (Equation (2)) over the inter-

4 8 R. Krupiński val (Q i.5 Q, Q i +.5 Q) P i = λ 2 (i+.5) Q (i.5) Q e [λ x ]/3 dx (6) gives the probability density function of the discrete random variable of GGD /3 P i = 4 e Ci (2 + 2C i + Ci 2) 4 e Bi (2 + 2B i + Bi 2) i P = 2 e A (2 + 2A + A 2 ) i = (7) A = (.5 λ Q) /3, where B i = (λ ( y i +.5 Q)) /3, C i = (λ ( y i.5 Q)) /3. Fig. 3 depicts the density function of GGD with exponent p = /3 and λ = for discrete random variable P i (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 P i f(x) 5 5 x Fig. 3: Density function of GGD with exponent p = /3 and λ = for discrete random variable P i (Equation (7)) and continuous random variable f(x) (Equation (2)) The estimator of discrete GGD /3 of the maximum likelihood method can be found similarly as for the Laplacian discrete source [24] and GGD.5 [4]. The maximum likelihood function of Equation (7) is set and then it is maximized with respect to λ. After certain transformations the ML estimator of discrete GGD /3 takes the form + λ N i= N e A Q P + e Bi B 3 i e Ci C 3 i P i =, (8) where N denotes the number of observations equal zero and N denotes the number of observations not equal zero. The equation N = N + N must be held. The estimated λ parameter is received only from the discrete observations (the quantized values available to the decoder) without prior knowledge of the coefficients before quantization available to the encoder. Therefore, it is expected to restore GGD p = /3 of the continuous random variable available to the encoder before the quantization process. 4 The reconstruction of coefficients The reconstructed coefficients can be biased based on the assumed model. Different models have been applied in [2, 6, 24]. Based on the observation that the DCT coefficients have a peak at zero and decrease exponentially, Ahumada et al. [2] made adjustments to the reconstructed coefficients. The reconstruction for Laplace distribution with centroid was presented in [6, 24]. According to the equation for the centroid reconstruction of the distribution in [22], the equation for the centroid reconstruction of GGD /3 takes the form ŷ i = sgn(y i ) λ Li M i (9) where L i = e Ci (Ci C4 i + 2 C3 i + +6 Ci C i + 2) e Bi (Bi B4 i + 2 B3 i + +6 Bi B i + 2), M i = e Ci (2 + 2C i + Ci 2) e Bi (2 + 2B i + Bi 2). The value reconstructed with centroid minimizes the Mean Square Error (MSE) in the interval (Q i.5 Q, Q i +.5 Q). Another advantage of fixing the power parameter p =

5 Image Processing & Communications, vol. 2, no. 4, pp /3 is that the reconstruction equation can be presented in a closed form. 4 2 λ=. λ=2.3 λ=4.5 5 Experiments RMSE In the first simulation, the performance of maximum likelihood estimator is evaluated (Equation (8)). The input sequence is generated with the GGD generator [3] with p = /3. Then the sequence is quantized and dequantized, which corresponds to lossy compression. The aim of the estimator is to reproduce the initial distribution before the quantization only on the dequantized coefficients. The sequence range N < 3, > and the quantization steps Q < 2, 2 > are considered. The simulation was repeated times. (RMSE) was calculated from the equation RMSE = M M (ˆλ λ) 2 i= Relative Mean Square Error λ 2 () where ˆλ is a value estimated by the model and λ is a real value of a lambda parameter. M denotes the number of repetitions. RMSE λ=. λ=2.3 λ= Q Fig. 4: Difference between RMSE of the estimators Equation (5) and Equation (8) for the sequence length N = Figures 4 and 5 depict the difference between RMSE of the estimators Equation (5) and Equation (8). It is expected that the estimator based on the discrete distribution N Fig. 5: Difference between RMSE of the estimators Equation (5) and Equation (8) for the quantization factor Q = 2 performs better than the estimator based on the continuous distribution. Figures show the positive values that confirms the expectations. The higher value of λ, the estimator (8) is getting better in terms of RMSE. It can be also noticed that the higher value of quantization factor Q, the estimator (8) is getting better in terms of RMSE. In the next simulation, the set of DCT coefficients is generated with the GGD generator and the shape parameter p = /3. The input sequence is created x i with calculating the inverse of DCT. These DCT coefficients are quantized and dequantized, and the restored sequence y i is calculated with the inverse of DCT. MSE is calculated for the lossy transformation. MSE = N N (ŷ i x i ) 2 () i= where ŷ i is reconstructed sequence and x i is the input signal. The reconstructed DCT coefficients are modified with Equation (9) for the λ parameter estimated from Equations (5) and (8). Then the restored sequence ŷ i is calculated with the inverse of DCT and MSE. Figures 6 and 7 depict the difference between MSE of normally reconstructed signal y i and modified reconstruction ŷ i ((5) or (8)). It can be noticed that the modified

6 R. Krupiński reconstruction ŷ i gives smaller MSE than MSE of normally reconstructed signal y i (the positive values in the figures). It should be noted that the estimator (8) gives smaller MSE than the estimator (5). transformation. The reconstructed DWT coefficients are modified with Equation (9) for the λ parameter estimated from Equations (5) and (8). Then the restored sequence ŷ i is calculated with the inverse of DWT and MSE. MSE.5 λ =. d =. λ d =2.3 λ =2.3 c λ =4.5 d λ =4.5 c MSE.5 λ =. d =. λ d =2.3 λ =2.3 c λ =4.5 d = Q Fig. 6: Difference between MSE of normally reconstructed signal y i and modified reconstruction ŷ i ( (5) or λ d (8)) for the sequence length N = (IDCT) Q Fig. 8: Difference between MSE of normally reconstructed signal y i and modified reconstruction ŷ i ( (5) or λ d (8)) for the sequence length N = (IDWT).6 MSE λ =. d =. λ =2.3 d λ =2.3 c λ =4.5 d λ =4.5 c MSE.5 λ =. d =. λ =2.3 d λ =2.3 c λ =4.5 d λ =4.5 c N Fig. 7: Difference between MSE of normally reconstructed signal y i and modified reconstruction ŷ i ( (5) or λ d (8)) for the quantization factor Q = 2 (IDCT) In the last simulation, the set of detailed Discrete Wavelet Transform (DWT) coefficients is generated with the GGD generator and the shape parameter p = /3. The input sequence is created x i with calculating the inverse of DWT, whereas the approximation DWT coefficients are set to zero. These DWT coefficients are quantized and dequantized, and then the restored sequence y i is calculated with the inverse of DWT. MSE is calculated for the lossy N Fig. 9: Difference between MSE of normally reconstructed signal y i and modified reconstruction ŷ i ( (5) or λ d (8)) for the quantization factor Q = 2 (IDWT) Figures 8 and 9 depict the difference between MSE of normally reconstructed signal y i and modified reconstruction ŷ i ((5) or (8)). It can be noticed that the modified reconstruction ŷ i gives smaller MSE than MSE of normally reconstructed signal y i (the positive values in the figures). It should be noted that the estimator (8) gives smaller MSE than the estimator (5).

7 Image Processing & Communications, vol. 2, no. 4, pp Summary Chapeau-Blondeau et al. [4] considered a special case of GGD where the power parameter is p =.5. In this article, the special case of GGD is considered where p = /3. This distribution is more peaky comparing to p =.5. Selecting such a model allows to simplify calculations and denote the closed form equations. The density function of GGD with exponent p = /3 of the continuous random variable and the probability density function of the discrete random variable of GGD p = /3 are given. Assuming the GGD p = /3 model of quantized coefficients, the modified reconstruction equation is defined. The simulations showed that MSE can be improved by the application of this modified reconstruction for the signals that are transformed with either DWT or DCT, where the information is lost by quantization. References [] Achim, A., Bezerianos, A., Tsakalides, P. (2). Novel Bayesian multiscale method for speckle removal in medical ultrasound images. IEEE transactions on medical imaging, 2(8), [2] Ahumada, A.J., Horng, R. (994, January). Smoothing DCT compression artifacts. In SID Symposium Digest of Technical Papers (Vol. 25, pp ). Society for Information Display [3] Box, G. E., Tiao, G. C. (2). Bayesian inference in statistical analysis (Vol. 4). John Wiley & Sons [4] Chapeau-Blondeau, F., Monir, A. (22). Numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent /2. IEEE transactions on signal processing, 5(9), [5] Clarke, R.J. (985). Transform coding of images. Astrophysics [6] Deutch, R. (965). Estimation theory. Prentice- Hall, Englewood Cliffs, N.J. [7] Du, Y. (99). Ein sphärisch invariantes Verbunddichtemodell für Bildsignale. AEU. Archiv für Elektronik und Übertragungstechnik, 45(3), [8] Farvardin, N., Modestino, J. (984). Optimum quantizer performance for a class of non-gaussian memoryless sources. IEEE Transactions on Information Theory, 3(3), [9] Hernandez, J. R., Amado, M., Perez-Gonzalez, F. (2). DCT-domain watermarking techniques for still images: Detector performance analysis and a new structure. IEEE transactions on image processing, 9(), [] ISO/IEC. (993). Coding of moving pictures and associated audio for digital storage media up to about,5 Mbits/s. International Standard 72 [] ISO/IEC. (994). Generic coding of moving pictures and associated audio information. International Standard 388 [2] ISO/IEC. (998). Information technology coding of audio-visual objects. International Standard 4496 [3] Kokkinakis, K., Nandi, A. K. (25). Exponent parameter estimation for generalized Gaussian probability density functions with application to speech modeling. Signal Processing, 85(9), [4] Krupiński, R., Purczyński, J. (27). Modeling the distribution of DCT coefficients for JPEG reconstruction. Signal Processing: Image Communication, 22(5), [5] Krupiński, R. (25). Approximated fast estimator for the shape parameter of generalized Gaussian distribution for a small sample size. Bulletin of

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