# A mathematical analysis of the DCT coefficient distributions for images

@article{Lam2000AMA, title={A mathematical analysis of the DCT coefficient distributions for images}, author={Edmund Y. Lam and Joseph W. Goodman}, journal={IEEE transactions on image processing : a publication of the IEEE Signal Processing Society}, year={2000}, volume={9 10}, pages={ 1661-6 } }

Over the past two decades, there have been various studies on the distributions of the DCT coefficients for images. However, they have concentrated only on fitting the empirical data from some standard pictures with a variety of well-known statistical distributions, and then comparing their goodness of fit. The Laplacian distribution is the dominant choice balancing simplicity of the model and fidelity to the empirical data. Yet, to the best of our knowledge, there has been no mathematical…

## 728 Citations

### Gaussian DCT Coefficient Models

- Mathematics
- 2009

It has been known that the distribution of the discrete cosine transform (DCT) coefficients of most natural images follow a Laplace distribution. However, recent work has shown that the Laplace…

### Gg DCT Coefficient Models

- MathematicsInt. J. Wavelets Multiresolution Inf. Process.
- 2010

This work derives a comprehensive collection of formulas for the distribution of the actual DCT coefficients from a model that contains both Laplace and Gaussian distributions as particular cases is the generalized Gaussian (GG) distribution.

### Modeling of the DCT coefficients of images

- Engineering2008 IEEE International Symposium on Circuits and Systems
- 2008

It is shown that the SNIG PDF is a more effective prior as compared to the generalized Gaussian, alpha-stable, and Laplacian PDFs for modelling the full-frame DCT coefficients of natural images.

### Analyzing the Effect of JPEG Compression on Local Variance of Image Intensity

- Computer ScienceIEEE Transactions on Image Processing
- 2016

The theoretical analysis presented in this paper provides some new insights into the behavior of local variance under JPEG compression, which has the potential to be used in some areas of image processing and analysis, such as image enhancement, image quality assessment, and image filtering.

### Transparent Composite Model for DCT Coefficients: Design and Analysis

- Computer ScienceIEEE Transactions on Image Processing
- 2014

The modeling performance and the data reduction feature of the GMTCM make it a desirable choice for modeling discrete or integer DCT coefficients in the real-world image or video applications, as summarized in a few of the authors' further studies on quantization design, entropy coding design, and image understanding and management.

### Modelling of the video DCT coefficients

- Engineering2008 International Conference on Electrical and Computer Engineering
- 2008

In this paper, the normal inverse Gaussian probability density function (PDF) is presented as a highly suitable prior for modelling the DCT coefficients of videos. The parameters of the proposed…

### Analysis of the DCT coefficient distributions for document coding

- Computer ScienceIEEE Signal Processing Letters
- 2004

This letter presents an analysis of their DCT coefficient distributions, and shows that a Gaussian distribution can be a realistic model, and can use a generalized Gaussian model to incorporate the Laplacian distribution found for natural images.

### First absolute moment and variance estimators used in JPEG reconstruction

- EngineeringIEEE Signal Processing Letters
- 2004

Two estimators (alternatives to the maximum likelihood one) of the discrete Laplacian distribution modeling DCT are presented and reconstructed values of images are adjusted based on a distribution with estimated parameters.

### Title Analysis of the DCT coefficient distributions for documentcoding

- Mathematics
- 2004

It is known that the distribution of the discrete cosine transform (DCT) coefficients of most natural images follow a Laplacian distribution, and this knowledge has been employed to improve decoder…

### Statistical Model of Quantized DCT Coefficients: Application in the Steganalysis of Jsteg Algorithm

- Computer Science, EngineeringIEEE Transactions on Image Processing
- 2014

By formulating the hidden data detection as a hypothesis testing, this paper studies the most powerful likelihood ratio test for the steganalysis of Jsteg algorithm and establishes theoretically its statistical performance.

## References

SHOWING 1-10 OF 16 REFERENCES

### Distributions of the Two-Dimensional DCT Coefficients for Images

- EngineeringIEEE Trans. Commun.
- 1983

From a simulation of the DCT coding System it is shown that the assumption that the coefficients are Laplacian yields a higher actual output signal-to-noise ratio and a much better agreement between theory and simulation than the Gaussian assumption.

### Statistical analysis of the DCT coefficients and their quantization error

- EngineeringConference Record of The Thirtieth Asilomar Conference on Signals, Systems and Computers
- 1996

A detailed probabilistic analysis of the DCT coefficients and the error incurred during the quantization and reconstruction of natural images is presented. The distribution of both the DCT…

### On the distribution of the DCT coefficients

- EngineeringProceedings of ICASSP '94. IEEE International Conference on Acoustics, Speech and Signal Processing
- 1994

A statistical study to improve the image compression rate in discrete cosine transform (DCT) based methods and especially for compression methods which are defined in JPEG norm using a battery of parametric and non-parametric fit-tests and laws such as those of Cauchy, Laplace, Gauss and a mixture distribution of Laplace and Gaussian.

### Distribution shape of two-dimensional DCT coefficients of natural images

- Engineering
- 1993

The univariate distribution of DCT coefficients of natural images is investigated. The probability density function (PDF) of the coefficients is modelled with the generalised Gaussian function (GGF)…

### DCT coefficient distributions

- EngineeringElectronic Imaging
- 1996

An analysis of a broad suite of images confirms previous findings that a Laplacian distribution can be used to model the luminance ac coefficients and the distribution model is applied to improve dynamic generation of quantization matrices.

### A Theory for Multiresolution Signal Decomposition: The Wavelet Representation

- Computer ScienceIEEE Trans. Pattern Anal. Mach. Intell.
- 1989

It is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2/Sup j/ can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions.

### A posteriori restoration of block transform-compressed data

- Computer ScienceProceedings DCC '95 Data Compression Conference
- 1995

Two known a posteriori enhancement techniques, transform coefficient adjustment and low-pass filtering, are adapted to ICT restoration based on a quantitative model of distortion statistics, and combined to achieve significant objective and subjective improvement in restored image fidelity compared with original data.

### Adaptive image restoration using a generalized Gaussian model for unknown noise

- Computer ScienceIEEE Trans. Image Process.
- 1995

A model adaptive method is proposed for restoring blurred and noise corrupted images. The generalized p-Gaussian family of probability density functions is used as the approximating parametric noise…

### JPEG: Still Image Data Compression Standard

- Computer Science
- 1992

This chapter discusses JPEG Syntax and Data Organization, the history of JPEG, and some of the aspects of the Human Visual Systems that make up JPEG.