Statistical Distributions of the Lapped Transform Coefficients for Images
Discrete Cosine Transform (DCT) based transform coding is very popular in image, video and speech compression due to its good energy compaction and decorrelating properties. However, at low bit rates, the reconstructed images generally suffer from visually annoying blocking artifacts as a result of coarse quantization. Lapped transform was proposed as an alternative to the DCT with reduced blocking artifacts and increased coding gain. Lapped transforms are popular for their good performance, robustness against oversmoothing and availability of fast implementation algorithms. However, there is no proper study reported in the literature regarding the statistical distributions of block Lapped Orthogonal Transform (LOT) and Lapped Biorthogonal Transform (LBT) coefficients. This study performs two goodness-of-fit tests, the Kolmogorov-Smirnov (KS) test and the 2- test, to determine the distribution that best fits the LOT and LBT coefficients. The experimental results show that the distribution of a majority of the significant AC coefficients can be modeled by the Generalized Gaussian distribution. The knowledge of the statistical distribution of transform coefficients greatly helps in the design of optimal quantizers that may lead to minimum distortion and hence achieve optimal coding efficiency.
[1] F. Bellifemine, A. Capellino, A. Chimienti, R. Picco, and R. Ponti,
"Statistical analysis of the 2d-dct coefficients of the differential signal for images," Signal Processing, Image Communication, vol. 4, pp. 477-488, 1992.
[2] M. Bhaskaranand and J. D. Gibson, "Distributions of 3d dct coeffcients
for video," in Proc. of IEEE International Conference on Acoustics, Speech and Signal Processing, 2009, pp. 793 - 796.
[3] J. H. Chang, J. W. Shin, N. S. Kim, and S. K. Mitra, "Image probability
distribution based on generalized gamma function," IEEE Signal Processing Letters, vol. 12, no. 4, pp. 325-328, 2004.
[4] J. D. Eggerton and M. D. Srinath, "Statistical distributions of image dct
coefficients," Computer Electrical Engineering, vol. 12, pp. 137-145,1986.
(a) C01 (b) C10
Fig. 3. Logarithmic histograms of the block LOT (M=8) coefficients for
Lena image and the best Gaussian, Laplacian, Gamma, Generalized Gaussian pdfs fitted to this histogram in log domain.
(a) C01 (b) C10
Fig. 4. Logarithmic histograms of the block LBT (M=8) coefficients for
Lena image and the best Gaussian, Laplacian, Gamma, Generalized Gaussian pdfs fitted to this histogram in log domain.
[5] N. S. Jayant and P. Noll, Digital Coding of waveforms. Prentice Hall,1984.
[6] R. L. Joshi and T. R. Fischer, "Comparison of generalized gaussian
and laplacian modeling in dct image coding," IEEE Signal Processing Letters, vol. 2, no. 5, pp. 81-82, 1995.
[7] S. M. Kay, Fundamentals of statistical signal processing: estimation
theory. Prentice-Hall Ltd, Englewood Cliffs, USA, 1993.
[8] S. Mallat, "A theory for multiresolution signal decomposition: The
wavelet representation," IEEE Transactions on Pattern Recognition Machine Intelligence, vol. 11, pp. 674-693, 1989.
[9] H. S. Malvar, "The lot : Transform coding without blocking effects,"
IEEE Transactions on Accoustics, Speech and Signal Processing, vol. 37,
no. 4, pp. 553-559, 1989.
[10] ÔÇöÔÇö, Signal Processing with Lapped Transforms. Norwood, MA :
Artech House, 1992.
[11] ÔÇöÔÇö, "Lapped biorthogonal transform for transform coding with reduced blocking and ringing artifacts," in IEEE International Conference
on Acoustics, Speech and Signal Processing, vol. 3, 1997, pp. 2421-2424.
[12] , "Biorthogonal and nonuniform lapped transforms for transform
coding with reduced blocking and ringing artifacts," IEEE Transactions
on Signal Processing, vol. 46, no. 4, pp. 1043-1053, April 1998.
[13] "Fast progressive image coding without wavelets," in Data Compression
Conference, 2000, pp. 243-252.
[14] F. Muller, "Distribution shape of two dimensional dct coefficients of
natural images," Electronics Letters, vol. 29, no. 22, pp. 1935-1936,
1993.
[15] R. C. Reininger and J. D. Gibson, "Distributions of the two dimensional
dct coefficients for images," IEEE Transactions on Communications,
vol. 31, no. 6, pp. 835-839, 1983.
[16] V. K. Rohatgi and A. K. E. Saleh, An Introduction to Probability and
Statistics. John Wiley and Sons, 2001.
[17] K. Sharifi and A. Leon-Garcia, "Estimation of shape parameter for
generalized gaussian distributions in subband decompositions of video,"
IEEE Transactions on Circuits and Systems for Video Technology, vol. 5,
no. 1, p. 5256, 1995.
[18] S. R. Smoot and L. A. Rowe, “Study of dct coefficient distributions,”
SPIE, vol. 2657, pp. 403–411, 1996.
[1] F. Bellifemine, A. Capellino, A. Chimienti, R. Picco, and R. Ponti,
"Statistical analysis of the 2d-dct coefficients of the differential signal for images," Signal Processing, Image Communication, vol. 4, pp. 477-488, 1992.
[2] M. Bhaskaranand and J. D. Gibson, "Distributions of 3d dct coeffcients
for video," in Proc. of IEEE International Conference on Acoustics, Speech and Signal Processing, 2009, pp. 793 - 796.
[3] J. H. Chang, J. W. Shin, N. S. Kim, and S. K. Mitra, "Image probability
distribution based on generalized gamma function," IEEE Signal Processing Letters, vol. 12, no. 4, pp. 325-328, 2004.
[4] J. D. Eggerton and M. D. Srinath, "Statistical distributions of image dct
coefficients," Computer Electrical Engineering, vol. 12, pp. 137-145,1986.
(a) C01 (b) C10
Fig. 3. Logarithmic histograms of the block LOT (M=8) coefficients for
Lena image and the best Gaussian, Laplacian, Gamma, Generalized Gaussian pdfs fitted to this histogram in log domain.
(a) C01 (b) C10
Fig. 4. Logarithmic histograms of the block LBT (M=8) coefficients for
Lena image and the best Gaussian, Laplacian, Gamma, Generalized Gaussian pdfs fitted to this histogram in log domain.
[5] N. S. Jayant and P. Noll, Digital Coding of waveforms. Prentice Hall,1984.
[6] R. L. Joshi and T. R. Fischer, "Comparison of generalized gaussian
and laplacian modeling in dct image coding," IEEE Signal Processing Letters, vol. 2, no. 5, pp. 81-82, 1995.
[7] S. M. Kay, Fundamentals of statistical signal processing: estimation
theory. Prentice-Hall Ltd, Englewood Cliffs, USA, 1993.
[8] S. Mallat, "A theory for multiresolution signal decomposition: The
wavelet representation," IEEE Transactions on Pattern Recognition Machine Intelligence, vol. 11, pp. 674-693, 1989.
[9] H. S. Malvar, "The lot : Transform coding without blocking effects,"
IEEE Transactions on Accoustics, Speech and Signal Processing, vol. 37,
no. 4, pp. 553-559, 1989.
[10] ÔÇöÔÇö, Signal Processing with Lapped Transforms. Norwood, MA :
Artech House, 1992.
[11] ÔÇöÔÇö, "Lapped biorthogonal transform for transform coding with reduced blocking and ringing artifacts," in IEEE International Conference
on Acoustics, Speech and Signal Processing, vol. 3, 1997, pp. 2421-2424.
[12] , "Biorthogonal and nonuniform lapped transforms for transform
coding with reduced blocking and ringing artifacts," IEEE Transactions
on Signal Processing, vol. 46, no. 4, pp. 1043-1053, April 1998.
[13] "Fast progressive image coding without wavelets," in Data Compression
Conference, 2000, pp. 243-252.
[14] F. Muller, "Distribution shape of two dimensional dct coefficients of
natural images," Electronics Letters, vol. 29, no. 22, pp. 1935-1936,
1993.
[15] R. C. Reininger and J. D. Gibson, "Distributions of the two dimensional
dct coefficients for images," IEEE Transactions on Communications,
vol. 31, no. 6, pp. 835-839, 1983.
[16] V. K. Rohatgi and A. K. E. Saleh, An Introduction to Probability and
Statistics. John Wiley and Sons, 2001.
[17] K. Sharifi and A. Leon-Garcia, "Estimation of shape parameter for
generalized gaussian distributions in subband decompositions of video,"
IEEE Transactions on Circuits and Systems for Video Technology, vol. 5,
no. 1, p. 5256, 1995.
[18] S. R. Smoot and L. A. Rowe, “Study of dct coefficient distributions,”
SPIE, vol. 2657, pp. 403–411, 1996.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:63666", author = "Vijay Kumar Nath and Deepika Hazarika and Anil Mahanta and ", title = "Statistical Distributions of the Lapped Transform Coefficients for Images", abstract = "Discrete Cosine Transform (DCT) based transform coding is very popular in image, video and speech compression due to its good energy compaction and decorrelating properties. However, at low bit rates, the reconstructed images generally suffer from visually annoying blocking artifacts as a result of coarse quantization. Lapped transform was proposed as an alternative to the DCT with reduced blocking artifacts and increased coding gain. Lapped transforms are popular for their good performance, robustness against oversmoothing and availability of fast implementation algorithms. However, there is no proper study reported in the literature regarding the statistical distributions of block Lapped Orthogonal Transform (LOT) and Lapped Biorthogonal Transform (LBT) coefficients. This study performs two goodness-of-fit tests, the Kolmogorov-Smirnov (KS) test and the 2- test, to determine the distribution that best fits the LOT and LBT coefficients. The experimental results show that the distribution of a majority of the significant AC coefficients can be modeled by the Generalized Gaussian distribution. The knowledge of the statistical distribution of transform coefficients greatly helps in the design of optimal quantizers that may lead to minimum distortion and hence achieve optimal coding efficiency.
", keywords = "Lapped orthogonal transform, Lapped biorthogonal transform, Image compression, KS test, ", volume = "5", number = "12", pages = "2121-8", }
