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Table Of Integrals Series And Products

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Fast and robust fuzzy c-means clustering algorithms incorporating local information for image segmentation

In this paper, we provide a unified framework for identifying the source digital camera from its images and for revealing digitally altered images using photo-response nonuniformity noise (PRNU), which is a unique stochastic fingerprint of imaging sensors. The PRNU is obtained using a maximum-likelihood estimator derived from a simplified model of the sensor output. Both digital forensics tasks are then achieved by detecting the presence of sensor PRNU in specific regions of the image under investigation. The detection is formulated as a hypothesis testing problem. The statistical distribution of the optimal test statistics is obtained using a predictor of the test statistics on small image blocks. The predictor enables more accurate and meaningful estimation of probabilities of false rejection of a correct camera and missed detection of a tampered region. We also include a benchmark implementation of this framework and detailed experimental validation. The robustness of the proposed forensic methods is tested on common image processing, such as JPEG compression, gamma correction, resizing, and denoising.

/pdf/determining-image-origin-and-integrity-using-sensor-noise-375kg5mshh.pdf

Determining Image Origin and Integrity Using Sensor Noise

Consider a discrete source producing a sequence of message letters from a finite alphabet. A single-letter distortion measure is given by a non-negative matrix (d ij ). The entryd ij measures the ?cost? or ?distortion? if letteriis reproduced at the receiver as letterj. The average distortion of a communications system (source-coder-noisy channel-decoder) is taken to bed= ? i.j P ij d ij whereP ij is the probability ofibeing reproduced asj. It is shown that there is a functionR(d) that measures the ?equivalent rate? of the source for a given level of distortion. For coding purposes where a leveldof distortion can be tolerated, the source acts like one with information rateR(d). Methods are given for calculatingR(d), and various properties discussed. Finally, generalizations to ergodic sources, to continuous sources, and to distortion measures involving blocks of letters are developed. In this paper a study is made of the problem of coding a discrete source of information, given afidelity criterionor ameasure of the distortionof the final recovered message at the receiving point relative to the actual transmitted message. In a particular case there might be a certain tolerable level of distortion as determined by this measure. It is desired to so encode the information that the maximum possible signaling rate is obtained without exceeding the tolerable distortion level. This work is an expansion and detailed elaboration of ideas presented earlier [1], with particular reference to the discrete case. We shall show that for a wide class of distortion measures and discrete sources of information there exists a functionR(d) (depending on the particular distortion measure and source) which measures, in a sense, the equivalent rateRof the source (in bits per letter produced) whendis the allowed distortion level. Methods will be given for evaluatingR(d) explicitly in certain simple cases and for evaluatingR(d) by a limiting process in more complex cases. The basic results are roughly that it is impossible to signal at a rate faster thanC/R(d) (source letters per second) over a memoryless channel of capacityC(bits per second) with a distortion measure less than or equal tod. On the other hand, by sufficiently long block codes it is possible to approach as closely as desired the rateC/R(d) with distortion leveld. Finally, some particular examples, using error probability per letter of message and other simple distortion measures, are worked out in detail.

Coding Theorems for a Discrete Source With a Fidelity CriterionInstitute of Radio Engineers, International Convention Record, vol. 7, 1959.

Synthetic aperture radar (SAR) imagery has found important applications due to its clear advantages over optical satellite imagery one of them being able to operate in various weather conditions. However, due to the physics of the radar imaging process, SAR images contain unwanted artifacts in the form of a granular look which is called speckle. The assumptions of the classical SAR image generation model lead to a Rayleigh distribution model for the histogram of the SAR image. However, some experimental data such as images of urban areas show impulsive characteristics that correspond to underlying heavy-tailed distributions, which are clearly non-Rayleigh. Some alternative distributions have been suggested such as the Weibull, log-normal, and the k-distribution which had success in varying degrees depending on the application. Recently, an alternative model namely the /spl alpha/-stable distribution has been suggested for modeling radar clutter. In this paper, we show that the amplitude distribution of the complex wave, the real and the imaginery components of which are assumed to be distributed by the /spl alpha/-stable distribution, is a generalization of the Rayleigh distribution. We demonstrate that the amplitude distribution is a mixture of Rayleighs as is the k-distribution in accordance with earlier work on modeling SAR images which showed that almost all successful SAR image models could be expressed as mixtures of Rayleighs. We also present parameter estimation techniques based on negative order moments for the new model. Finally, we test the performance of the model on urban images and compare with other models such as Rayleigh, Weibull, and the k-distribution.

Modeling SAR images with a generalization of the Rayleigh distribution

Synthetic aperture radar (SAR) images are inherently affected by a signal dependent noise known as speckle, which is due to the radar wave coherence. In this paper, we propose a novel adaptive despeckling filter and derive a maximum a posteriori (MAP) estimator for the radar cross section (RCS). We first employ a logarithmic transformation to change the multiplicative speckle into additive noise. We model the RCS using the recently introduced heavy-tailed Rayleigh density function, which was derived based on the assumption that the real and imaginary parts of the received complex signal are best described using the alpha-stable family of distribution. We estimate model parameters from noisy observations by means of second-kind statistics theory, which relies on the Mellin transform. Finally, we compare the proposed algorithm with several classical speckle filters applied on actual SAR images. Experimental results show that the homomorphic MAP filter based on the heavy-tailed Rayleigh prior for the RCS is among the best for speckle removal

/pdf/sar-image-filtering-based-on-the-heavy-tailed-rayleigh-model-of4kvyl689.pdf

SAR image filtering based on the heavy-tailed Rayleigh model

Recently, the dual-tree complex wavelet transform has been proposed as an analysis tool featuring near shift-invariance and improved directional selectivity compared to the standard wavelet transform. Within this framework, we describe a novel technique for removing noise from digital images. We design a bivariate maximum a posteriori estimator, which relies on the family of isotropic α-stable distributions. Using this relatively new statistical model we are able to better capture the heavy-tailed nature of the data as well as the interscale dependencies of wavelet coefficients. We test our algorithm for the Cauchy case, in comparison with several recently published methods. The simulation results show that our proposed technique achieves state-of-the-art performance in terms of root mean squared (RMS) error.

Image denoising using bivariate α-stable distributions in the complex wavelet domain

https://hal.inria.fr/docs/00/07/05/14/PDF/RR-5493.pdf

SAR Image Filtering Based on the Heavy-Tailed

Over the last few years, there has been a great interest in /spl alpha/-stable distributions for modeling impulsive data. As a critical step in modeling with /spl alpha/-stable distributions, the problem of estimating the parameters of stable distributions have been addressed by several works in the literature. However, many of these works consider only the special case of symmetric stable random variables. This is an important restriction, however, since most real-life signals are skewed. The existing techniques on estimating skewed distribution parameters are either computationally too expensive, require lookup tables, or have poor convergence properties. We introduce three novel classes of estimators for the parameters of general stable distributions, which are generalizations of the methods previously suggested for parameter estimation of symmetric stable distributions. These estimators exploit expressions we develop for fractional lower order, negative order, and logarithmic moments and tail statistics. We also introduce simple transformations that allow one to use existing symmetric stable parameter estimation techniques. Techniques suggested in this paper provide the only closed-form solutions we are aware of for parameters that may be efficiently computed. Simulation results show that at least one of our new estimators has better performance than the existing techniques over most of the parameter space. Furthermore, our techniques require substantially less computation.

Ercan E. Kuruoglu

Papers

Modeling SAR images with a generalization of the Rayleigh distribution

SAR image filtering based on the heavy-tailed Rayleigh model

Image denoising using bivariate α-stable distributions in the complex wavelet domain

SAR Image Filtering Based on the Heavy-Tailed

Density parameter estimation of skewed /spl alpha/-stable distributions