1. Alexey Roenko1, Vladimir Lukin1, Sergey Abramov1, Igor Djurovic2
Automation of Analysis for Correlated Noise in Images Based on DCT Coefficient Statistics
Alexey Roenko1, Vladimir Lukin1, Sergey Abramov1, Igor Djurovic2
1 Department of Transmitters, Receivers and Signal Processing, National Aerospace University
2 Electrical Engineering Department, University of Montenegro
Introduction
It has been recently proposed to estimate either the parameter p in (1) or some other parameter connected to p as a statistics of DCT-coefficients in image block. In [10] it was shown that p can be derived from analysis of data sample kurtosis. In turn, kurtosis deals with a fourth central moment for the data sample μ4 and/or with a percentile coefficient of kurtosis. If μ4 is considerably larger than 3, this indicates with high probability that a given sample of DCT coefficients has a heavy tailed distribution and belongs to heterogeneous image blocks. An example of the histogram of the μ4 estimates is presented in Fig. 1.
The carried out tests have proven that the distribution mode relates to the estimates obtained in homogeneous image blocks. Histogram maximum value is determined by image complexity (it is smaller for more complex images; tail heaviness is larger, respectively). Distribution mode for any parameter estimated for DCT coefficients in blocks is connected with spatial correlation of the noise. This opens opportunities to evaluation of noise spatial correlation automatically.
Analysis for multiplicative noise
The main idea for processing this kind of noise states that if noise is i.i.d. then it has distribution of DCT coefficients in homogeneous blocks close to Gaussian irrespectively to noise type (Poissonian, pure multiplicative, film-grain, etc.). Similarly, if noise is spatially correlated, then under condition of spatially invariant properties of this noise the considered statistics of DCT coefficients in homogeneous image blocks is the same. Then the estimates of the parameter p are close to each other and they form histogram maximum. The parameter estimates in heterogeneous blocks form heavy tail, right hand or left hand one depending upon a considered parameter.
To verify this hypothesis, we considered real-life SAR images. For single-look SAR images acquired by TerraSAR-X spaceborne sensor, it is known that images are corrupted by fully developed speckle that has pure multiplicative nature, non-Gaussian pdf, and certain degree of spatial correlation [4]. Two examples of single-look image fragments of size 512x512 pixels are represented in Fig. 3 (both from Rosenheim set).
Noise is one of major factors that degrades quality of various types of acquired images and video [1, 2]. Because of this, noise has to be either removed by filtering (denoising) or reconstruction or taken into account by other image processing operations (compression, segmentation, enhancement, etc). For these purposes, one has to know a priori or to pre- determine noise type and its basic statistical and spectral-correlation characteristics [3].
Available a priori information can be incomplete or limited. It can be known that noise is signal- dependent and it might be spatially correlated but nothing else is known about noise statistics and spatial correlation properties. Such practical situations happen especially often for specific type of images [4] available in Internet or some databases but having incomplete description.
Good way to deal with such uncertainties is to apply automatic techniques for determination of noise type and blind estimation of its main characteristics [4]. One of such approaches is based on several rather simple operations [4, 5]. The first one determines is noise i.i.d. or spatially correlated [6]. In the former case, only noise statistics is to be estimated. In the latter case, spatial correlation of the noise is to be estimated as well.
In the work the processing (analysis) approach on the basis of DCT coefficient statistics in 8x8 pixel blocks is used for the aforementioned purpose. Then, the analysis is done for the case of pure additive noise only as well as for the case of pure multiplicative noise, namely, speckle in single- and multi-look images formed by synthetic aperture radar (SAR). Moreover, it is shown that the approach is quite general and can be exploited for other types of signal-dependent noise.
a) b)
Figure 3. Single-look TerraSAR-X image fragments
The histogram in Fig.1 is just for the SAR image in Fig.3a. As it is seen, M (coordinate X of histogram mode) for it differs from 3 (equals to 3.6) showing that speckle has certain spatial correlation (histogram maximum equals to 0.042). For the image in Fig. 3b, M=3.35. Histogram maximum is larger and equals to demonstrating that the percentage of homogeneous blocks is larger in this image than in image in Fig.3a.
Totally, twelve single-look SAR images acquired by TerraSAR-X (six from Rosenheim and six from Uluru datasets) have been analyzed. The values of M for them vary from 3.24 to 3.63 with the tendency to be larger for images with more complex structure. Such “stability” of M shows, on one hand, that speckle is really spatially correlated. On another hand, it shows that M can be used for rough characterization of speckle spatial correlation.
Multi-look SAR images have been analyzed as well. Efficient number of looks in acquired images was reported to be about 2.2. Careful analysis of speckle spatial correlation carried out in homogeneous fragments has clearly demonstrated that correlation degree (main lobe width of 2D autocorrelation function) is larger than for single-look data. Even larger speckle correlation degree was observed for SAR images with larger efficient number of looks.
Figure 1. Histogram of the μ4 estimates for DCT coefficients in 8x8 blocks
DCT-coefficients properties
Analysis for additive noise case
It has been shown that probability density function (pdf) of DCT-coefficients deviates from Gaussian or Laplacian models [6-9]. The distributions possess heavy tails and, therefore, Generalized Gaussian Distribution (GGD) [12] or some other heavy tail pdf families can describe the observed pdfs better.
We concentrate on using the GGD model for which pdf can be uniquely defined by two parameters (under assumption that location parameter is equal to zero). A first parameter (shape parameter) is p(0;∞) and it characterizes distribution tail heaviness where smaller p relates to heavier tail [12]. p=1 corresponds to Laplace distribution and p=2 – to Gaussian pdf as the special cases. A second parameter is the scale parameter σGGD.
(1)
where
In order to analyze the image properties, it is divided on 8x8 overlapped or non-overlapped pixel blocks. DCT is applied for each block and some statistics is calculated for 63 DCT coefficients (excluding DC DCT coefficient). In case of AWGN, DCT-coefficients have Gaussian statistics in homogeneous image blocks and obey non-Gaussian (heavy-tailed) distributions in heterogeneous image blocks. If noise is spatially correlated, distribution of 63 DCT coefficients becomes non-Gaussian.
The method described was tested component-wise for color images of the database TID2008 [15]. This database contains images corrupted by i.i.d. (white) additive noise and by spatially correlated additive noise. Fig. 2 presents dependence of the mode M of the histogram of the estimates of μ4 on image index n in the database (there are 25 test images in TID2008). These dependences have been obtained for the noise variance equal to 65 and for green component of color images. The dependences obtained for red and blue components are similar as well as dependence obtained for noise variance equal to 130. As seen, by setting a proper threshold, we can easily discriminate images corrupted by i.i.d. noise and the considered type of spatially correlated noise.
References
Due to the lack of free space references are not represented in poster but can be found in the conference proceedings paper.
Contacts
Figure 2. Dependences of the mode M for green component of color images in TID2008 on image index n for images corrupted by AWGN (lower curve) and spatially correlated noise (upper curve) with variance 100
Vladimir Lukin –
Igor Djurovic –
MECO’2014, Budva, June 15th – 19th , Montenegro