Pattern Recognition in SAR Images using Fractional Random Fields and its Possible Application to the Problem of the Detection of Oil Spills in Open Sea

TL;DR

This paper explores a novel method for detecting oil spills in open sea using fractional random fields and wavelet filters applied to SAR images, demonstrating preliminary experimental success.

Contribution

It introduces a new approach combining fractional random fields and wavelet filters for oil spill detection in SAR images, with initial experimental validation.

Findings

Preliminary results show promise in oil spill detection.

Wavelet filters enhance detection accuracy.

Method leverages self-similar, long-range dependence properties.

Abstract

In this note we deal with the detection of oil spills in open sea via self similar, long range dependence random fields and wavelet filters. We show some preliminary experimental results of our technique with Sentinel 1 SAR images.

Equations

$$S_X(\omega )=\frac{C}{\Vert \omega {\Vert }^a}\text{mathchar}46$$

$$Y_1(p)=(h_1\ast X)(p)\text{ and }{Y}_2(p)=(h_2\ast X)(p)\text{mathchar}46$$

$$S_{Y_1}(\omega )=|H_1(\omega ){|}^2{S}_X(\omega )$$

$$S_{Y_2}(\omega )=16|H_1(2\omega ){|}^2{S}_X(\omega )\text{mathchar}46$$

$${\sigma }_2^2=2^{a+2}{\sigma }_1^2\text{mathchar}46$$

Full text

Keywords: Fractional Processes, Wavelets .

AMS class: 60G22, 60G35 (Primary), 42C40, 60G20 (Secondary)

Assuming that an intensity SAR image is modeled by a self similar random process [1] we propose a possible rather simple analysis scheme for the detection of oils spills in open sea. Moreover as this is essentially a smoothness or complexity analysis technique of the image it can be potentially applied to the recognition and classification of other patterns. To explode notions related to self similarity and fractals related to SAR image processing (and optical images) is not completely new. In e.g. [4, 5] computing an approximate Box/Hausdorff fractal dimension of the neighborhood of each pixel some possible algorithms for oil spill detection are proposed. Indeed, since ”fractalïs a more or less vague term, several underlying models with varying results and computational complexity can be proposed. As suggested in [4, 5] box counting techniques have a rather good performance in solving this detection problem. However this may be not the case from the the point of view of the required computational time to perform this task. In contrast, Fourier analysis methods or similar as wavelet analysis [1] are potentially much simpler and fast since there exist several optimal FFT algorithms. Here, among all possible models of self similar random fields to model the image we propose the class of the stationary (or locally stationary) $\dfrac{1}{f}$-processes [1]. These fields were introduced by A. Kolmogrov in a first statistical study of turbulent flows. An $X=\{X(p),p\}$ random field of these class, with $p$ in $\mathbb{R}^{2}$, has an spectral density of the form (or asymptotically near the frequency $\omega=0$) [7]

[TABLE]

The exponent $a$ gives a quantitative measure of the the correlation of the random field. For greater $a$, $X$ is more correlated and in particular for appropriate values of $a$, $X$ displays Long Range Dependence. In contrast, for $a\longrightarrow 0$, $X$ is nearly white noise. The parameter $a$ is linearly related to the so called Hurst parameter [1], which measures the self similarity of the random field. Finally, at least in the Gaussian case, $a$ is also related to the Hausdorff dimension of the graph of $X$ and the roughness of it.

In this context, we shall assume that the portion of image containing the oil spill fluid will be approximately behaving as a $\dfrac{1}{f}$-random field with a certain parameter $a$. Waves are supposed to be attenuated in contact in the presence of oil, which under suitable conditions, will traduce in a smoother and highly correlated surface in comparison to non contaminated open sea water. The proposed detection scheme is basically to test if the possible oil spill area follows model (1) for some $a$. To do this, given any finite energy filter $h_{1}(p)$ with a Fourier transform $H_{1}(\omega)$ we consider a scaled version of it $h_{2}(p):=h_{1}\left(\dfrac{p}{2}\right)$ and if $X$ is our image/field we obtain two filtered (convoluted) versions of it, namely:

[TABLE]

The corresponding spectral densities then are given by :

[TABLE]

and

[TABLE]

Assuming stationarity [7] and after some Fourier domain calculations, one gets that if $X$ has indeed an spectral density as (1) then the variances $\sigma_{1}$ and $\sigma_{2}$ of $Y_{1}$ and $Y_{2}$ respectively, are related by:

[TABLE]

Equation (2) is the key to obtain estimates for $a$. In particular, this scale relation makes the estimation procedure easy to implement by wavelet filters [3, 6]. Furthermore, recalling that if $R_{X}$ is the correlation function of $X$ then $R_{X}$ is related by its Fourier Transform with $S_{X}$, i.e. $\widehat{R_{X}}=S_{X}$, we can prove a kind of reciprocal of this result:

The divergence of the integral of the correlation function is one of the usual forms to describe random fields with long range dependence. Thus, if an equation like (2) holds for some fixed $a$ and any finite energy filter then $X$ displays long range dependence, and moreover, with additional conditions equation 1 holds. In practice, $a$ is unknown and one can only test with a finite number of filters. However, based in these calculations, in this work we propose that for a finite suitable parametrized family of wavelet filters $h(s,p),s=s_{1},\mathchar 46\relax\mathchar 46\relax\mathchar 46\relax,s_{n}$ and corresponding estimates of $a(s)$ we can design a rather simple first warning detection scheme which in a first stage heavily relies only on linear, matrix and spectral numerical operations.

Here we present some practical examples with real images (see Figure 1). We note that $a(s,p)$ defines in some way a transformation or map of the original $X$. To illustrate the idea, if for certain fixed values $s_{1},s_{2},s_{3}$ we generate an RGB auxiliary image assigning to each color component a value linearly related to the corresponding estimates of $a(s)$ for each filter. In the zones of the image where these values are, ideally, equal the related RGB intensities must be equal as well generating a nearly dark grey flat area where $a(s)$ equals a constant. Obviously if more filters $(n\geq 3)$ are involved more accurate the following decision will become. The second stage (classification and decision) takes the transformed image $a(s,p)$ as an input into a combination of a neural network and a classical statistical hypothesis [2] testing previously trained with several images samples and its respective transforms $a(s,p)$. In this work we present some promising experimental results with this technique.

Acknowledgements

This work was funded by the Universidad de Buenos Aires, Grant. No. 20020170100266BA, CONICET and CONAE, under Project No. 5 of the Anuncio de Oportunidad para el desarrollo de aplicaciones y puesta apunto de metodologías para el área oceanográfica utlizando imágenes SAR, Buenos Aires, Argentina.

Referencias

• [1] Cohen S., Istas J., Fractional Fields and Applications, Springer, 2013.
• [2] Duda R., Hart P., and Stork D. Pattern Classification, Wiley, 2nd. edition. 2000.
• [3] Mallat S. A Wavelet tour of signal processing, Academic Press, 2008.
• [4] Marghany M., Hashim M., Fractal dimension algorithm for detection Oil Spills using RADARSAT 1 SAR, in Computational Science and its applications ICCSA 2007. Lecture Notes in Computer Science, Vol. 4705. Springer,

• [5] Marghany M., Hashim M. Discrimination between oil spill and look alike using Fractal dimension algorithm form RADARSAT 1 SAR and AIRSAR/POLSAR data, International Journal of Physical Sciences , 6(7), pp. 1711-1719, 2011.
• [6] Meyer Y. Wavelets and operators. Cambridge Studies in Advance Mathematics No. 37, Cambridge University Press, 1992.
• [7] Rosenblatt M., Stationary Sequences and Random Fields, Birkhäuser, 1985.