The analysis of ocean surface is widely performed using synthetic aperture
radar (SAR) imagery as it yields information for wide areas under challenging
weather conditions, during day or night, etc. Speckle noise constitutes however
the main reason for reduced performance in applications such as classification,
ship detection, target tracking and so on. This paper presents an investigation
into the despeckling of SAR images of the ocean that include ship wake
structures, via sparse regularisation using the Cauchy proximal operator. We
propose a closed-form expression for calculating the proximal operator for the
Cauchy prior, which makes it applicable in generic proximal splitting
algorithms. In our experiments, we simulate SAR images of moving vessels and
their wakes. The performance of the proposed method is evaluated in comparison
to the L1 and TV norm regularisation functions. The results show a superior
performance of the proposed method for all the utilised images generated.Comment: 6 pages, 2 Figures. This work has been presented in IGARSS 202

Achim, Alin

Karakuş, Oktay

Rizaev, Igor

English

arXiv

The analysis of ocean surface is widely performed using synthetic aperture radar (SAR) imagery as it yields information for wide areas under challenging weather conditions, during day or night, etc. Speckle noise constitutes however the main reason for reduced performance in applications such as classification, ship detection, target tracking and so on. This paper presents an investigation into the despeckling of SAR images of the ocean that include ship wake structures, via sparse regularisation using the Cauchy proximal operator. We propose a closed form expression for calculating the proximal operator for the Cauchy prior, which makes it applicable in generic proximal splitting algorithms. In our experiments, we simulate SAR images of moving vessels and their wakes. The performance of the proposed method is evaluated in comparison to the L1 and TV norm regularisation functions. The results show a superior performance of the proposed method for all the utilised images generated

Karakus, O.

Rizaev, I.

Achim, A.

Online Research @ Cardiff

A simulation study to evaluate the performance of the cauchy proximal operator in despeckling SAR images of the sea surface

The analysis of ocean surface is widely performed using synthetic aperture radar (SAR) imagery as it yields information for wide areas under challenging weather conditions, during day or night, etc. Speckle noise constitutes however the main reason for reduced performance in applications such as classification, ship detection, target tracking and so on. This paper presents an investigation into the despeckling of SAR images of the ocean that include ship wake structures, via sparse regularisation using the Cauchy proximal operator. We propose a closed-form expression for calculating the proximal operator for the Cauchy prior, which makes it applicable in generic proximal splitting algorithms. In our experiments, we simulate SAR images of moving vessels and their wakes. The performance of the proposed method is evaluated in comparison to the L1 and TV norm regularisation functions. The results show a superior performance of the proposed method for all the utilised images generated

Explore Bristol Research

                          Karakuş, O., Rizaev, I., & Achim, A. (2020). A Simulation Study toEvaluate the Performance of the Cauchy Proximal Operator inDespeckling SAR Images of the Sea Surface. arXiv.Early version, also known as pre-printLink to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/1A Simulation Study to Evaluate the Performance ofthe Cauchy Proximal Operator in Despeckling SARImages of the Sea SurfaceOktay Karakuş, Igor Rizaev, Alin Achim,AbstractThe analysis of ocean surface is widely performed using synthetic aperture radar (SAR) imagery as it yieldsinformation for wide areas under challenging weather conditions, during day or night, etc. Speckle noise constituteshowever the main reason for reduced performance in applications such as classification, ship detection, target trackingand so on. This paper presents an investigation into the despeckling of SAR images of the ocean that include shipwake structures, via sparse regularisation using the Cauchy proximal operator. We propose a closed form expressionfor calculating the proximal operator for the Cauchy prior, which makes it applicable in generic proximal splittingalgorithms. In our experiments, we simulate SAR images of moving vessels and their wakes. The performance ofthe proposed method is evaluated in comparison to the L1 and TV norm regularisation functions. The results showa superior performance of the proposed method for all the utilised images generated.Index TermsCauchy proximal operator, Simulated SAR images, Ship wakes, Despeckling.I. INTRODUCTIONSYNTHETIC aperture radar (SAR) images of the sea surface provide useful information in a number ofapplications, including meteorology, environmental monitoring, wave structure analysis, ship monitoring, orenergy generation. A common and important problem hampering statistical inferences from SAR imagery is thepresence of multiplicative speckle noise. This may lead to loss of crucial details in SAR images and can causeproblems in the analysis of these images, e.g. in feature detection, segmentation or classification [1], [2], [3].Since the availability of real SAR images is limited, more attention is being paid to SAR simulation. SimulatedSAR images are especially useful when evaluating despeckling algorithms, in as much as a speckle-free imagecan be constructed. In order to form a SAR image, two types of modelling requires consideration: (i) sea surfacemodelling, and (ii) modelling of reflected SAR signal from the waves. There are numerous sea spectra available thathave been developed based on experimental measurements [4] for the modelling of the sea surface. Here, we adoptElfouhaily et al. spectrum [5], whilst for modelling the ship wake, the Kelvin wake model is implemented on thebasis of Michell theory [6]. The elevation model of the water surfaces is generally considered as a superposition ofship-generated waves and wind-generated waves. For the simulation of SAR images, a two-scale composite model[7], [8] is used, including the tilt and hydrodynamic modulations, and the velocity bunching [9].The Cauchy distribution employed in this work is a member of the α-stable distribution family and known for itsability to model heavy-tailed data. As a prior, it has a sparsity-enforcing behaviour, similar to its generalised-Gaussian counterpart, the Laplace distribution [10] (i.e. the L1 norm), and it has generally been utilised indespeckling studies by modelling sub-band transform coefficients [11], [12].In this paper, we propose a despeckling study on simulated SAR images of the sea surface including ship wakestructures. The proposed methodology incorporates the Cauchy distribution as a regularisation function. Furthermore,we derive the Cauchy proximal operator, which makes Cauchy regularisation applicable in standard proximalsplitting algorithms such as forward backward (FB). Simulated SAR images incorporate a moving vessel with twodifferent moving directions, and the performance of our despeckling approach is then compared to methods thatThis work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) under grant EP/R009260/1 (AssenSAR).Oktay Karakuş, Igor Rizaev and Alin Achim are with the Visual Information Laboratory, University of Bristol, Bristol BS1 5DD, U.K.(e-mail: o.karakus@bristol.ac.uk; i.g.rizaev@bristol.ac.uk; alin.achim@bristol.ac.uk)arXiv:2012.06657v1  [eess.IV]  11 Dec 20202use regularisation functions such as L1 and total variation (TV ) norms under log-normal speckle noise of differentnumber of looks.The rest of the paper is organised as follows: we first present our methodology for creating simulated SARimages of the sea surface and ship wakes in Section II. In Section III, we introduce the Cauchy proximal operatorand the despeckling methodology. In Section IV, we present the experimental analysis, followed by concludingremarks and future work directions in Section V.II. SIMULATION OF OCEAN SAR IMAGESThe irregular sea surface model Zsea(x, y, z, t) with summation of many independent harmonic waves is formu-lated as ∑i∑jAij cos [ki(x cos θj + y sin θj)− ωit+ rij ] (1)where ki and ωi are the wavenumber and wave circular frequency, respectively, whilst Aij is the amplitudeAij =√2S(ki)D(ki, θj)dkidθj (2)where S(ki) is the omnidirectional wave spectrum, D(ki, θj) refers to the angular spreading function, dki anddθj represent the sampling intervals. The omnidirectional Elfouhaily et al. [5] spectrum is expressed as S(k) =k−3[Bl +Bh], with the gravity or the long-wave part of the spectrumBl =12αpcpcLPMJp exp[− Ω√10(√kkp− 1)], (3)the capillary or the short-wave part of the spectrumBh =12αmcmcLPMJp exp[−14(kkm− 1)2], (4)and the spreading function takes the formD(k, θ) =12π[1 + ∆(k) cos(2θ)] . (5)For modelling the Kelvin wake elevation, the fluid velocity potential ϕship and the ship elevation model rela-tionship are usedZship =Usg∂ϕship∂x(6)where the approximated form of fluid velocity potential (with parameters described in [6]) is presented asϕship(x, y, z) = −16BLπUsFr6Re∫ ∞0C(τ, x, z)eiyτdτ (7)The scattering of the SAR signal from the disturbed surface of waves is a complex process, which involves thescanning platform geometry parameters (e.g. microwave signal parameters) and physical properties of the surface.According to the two-scale composite model (TSM), and taking into account the tilt and hydrodynamic modulations,the mean normalised radar cross-section (NRCS) is described as [6], [8]:σ̄(x, y) = 8πk4e cos4 µiW (kBx, kBy)|T |2 ×[1 + 2Re∫M(k)F (k)eikxdk](8)where ke is the radar wavenumber, µi represents the local incidence angle, W (·) refers to the energy density spectrumof the surface roughness with components kBx, kBy [6]. M(·) represents the complex modulation transfer function(MTF), and F (·) is the Fourier transform of the sea surface model. The final intensity image is obtained by applyingthe correction for nonuniform displacements of model facets in the azimuthal direction via the velocity bunching(VB) mechanism [9].3III. THE PROPOSED DESPECKLING METHODLet us have an observed SAR image G with multiplicative speckle noise VG = FV, (9)where F is the speckle-free SAR image. The multiplicative image formation model given in (9) is often manipulatedas an additive one by taking the logarithm of both sides log(G) = log(FV ), which leads tog = f + v, (10)where g, f and v refer to the logarithms of G, F and V , respectively. The discrete wavelet transform (DWT) is alinear operation, which can consequently be applied to (10) to avoid the undesired effects of filtering operations. Ateach resolution level and for all orientations, we get additive terms corresponding to noisy wavelet coefficients (Γ(i))that can be written as the sum of the transformations of the speckle-free signal (Φ(i)) and the noise components(ν(i)) asΓ(i) = Φ(i) + ν(i). (11)The despeckling model in this paper is depicted in Figure 1, where blocks W and W−1 represent the forwardand inverse discrete wavelet transform operators.Fig. 1. Signal-dependent additive despeckling model in transform domain [13].In order to obtain the estimate of the speckle-free representation of the SAR image in the transform domain,Φ̂(i), the minimisation of the cost function with the Cauchy penalty term is then performed for each orientationi = 1, 2, 3 [14],arg minΦ(i){‖Γ(i) − Φ(i)‖22 −∑j,klog(γγ2 + φ2(i),j,k)}(12)where φ(i),j,k refers to Φ(j, k) for orientation i.In order to solve the minimisation problem in (12) by using proximal splitting methods such as FB, the proximaloperator of the Cauchy regulariser should be defined. Basically, for any ω-Lipchitz gradient function h(·) andω > 0, the proximal operator is defined as [15]proxωh(x) = arg minu{h(u) + ‖u− x‖2/2ω}. (13)Then, we substitute the function h with the negative logarithm of the Cauchy distribution − log(γ/(γ2 + Φ2(i))),leading to the Cauchy proximal operator as [16]arg minu{− log(γγ2 + u2)+‖u− x‖22ω}(14)The solution to this minimisation problem can be obtained by taking the first derivative of (14) w.r.t. u andsetting it to zero. Hence, we haveu3 − xu2 + (γ2 + 2ω)u− xγ2 = 0. (15)Wan et al. [17] proposed a solution to the denoising problem of a Cauchy signal under Gaussian noise, anddefined this solution as “Cauchy shrinkage”. Similarly, the minimisation problem in (14) can be solved with thesame approach as in [17], however with different parameterisation. Hence, the solution to (15) can be obtainedthrough Cardano’s method, which is given in Algorithm 1.Having the Cauchy proximal operator, the minimisation for despeckling given in (12), is subsequently solved viaFB algorithm for any iteration k as [16]u(k) = Φ(k)(i) − ω(Φ(k)(i) − Γ(i)), (16)Φ(k+1)(i) = PROXCAUCHY(u(k), γ, ω). (17)4Algorithm 1 Procedure for Cauchy Proximal Operator1: procedure PROXCAUCHY(x, γ, ω)2: p← γ2 + 2ω − x233: q ← xγ2 + 2x327 −x3(γ2 + 2µ)4: s← 3√q2 +√p327 +q245: t← 3√p2 −√p327 +q246: return z ← x3 + s+ t7: end procedureIV. EXPERIMENTAL RESULTS(a) Original image (b) Speckled image (L=3) (c) L1 (d) TV (e) Cauchy(f) Speckled image (L=7) (g) L1 (h) TV (i) CauchyFig. 2. Visual despeckling results for Image-1.The proposed method was tested for two different simulated SAR images, the main parameters of which areas follows. The size of scene is 512 × 512 m, the facet dimension for sea and ship wave models was set to 2m. For both images, the wind speed was 5 m/s with direction of 45 degrees relative to the azimuth of the SARplatform. The parameters of the ship were selected as length L = 52 m, beam B = 5.7 m, and draft D = 3.5 m,and the Froude number, Fr = 0.5. The ship heading was 0 and 45 degrees relative to the azimuth for Image-1 andImage-2, respectively. The SAR platform parameters were: the platform altitude of 4.5 km, the platform velocity of190 m/s, the SAR signal frequency of 9.65 GHz (X band) with VV polarisation, the incidence angle of 35 degrees,and lastly the azimuth and range image resolutions were both set to 2 m.Both speckle-free simulated SAR images were then multiplied with the log-normal noise [18] with the number oflooks, L chosen to be 3, 5 and 7. Speckle images for all three noise cases were processed by using the despecklingmethod for L1, TV and Cauchy regularisation functions. The performance of the methods were then compared interms of peak signal-to-noise ratio (PSNR) and signal-to-mean squared error (S/MSE) values, which are given inTable I. In Figure 2, despeckling results for Image-1 are depicted for L values of 3 and 7.Table I shows that the proposed Cauchy-based method achieved the best despeckling results for L values of 5 and7, whereas the TV achieved better results for the case of L = 3. However, examining the visual results in Figure2, we can clearly see that despite its lower PSNR and S/MSE values for L = 3, the proposed method reconstructsboth ship wake and sea surface structures and shows similar characteristics to the original speckle-free SAR image5in Figure 2-(a) when compared to TV . Even though TV preserves ship wake structures as can be seen from theresults in Figure 2-(d) and 2-(i), it discards the sea surface details whilst the final despeckled image is very blurry.TABLE IDESPECKLING PERFORMANCE MEASURESL = 3 L = 5 L = 7PSNR S/MSE PSNR S/MSE PSNR S/MSEImage-1 Noisy 22.363 4.785 24.484 6.995 25.920 8.474L1 22.652 5.245 22.977 5.571 23.136 5.730TV 25.461 8.056 25.834 8.429 25.990 8.586Cauchy 25.204 7.799 26.089 8.685 26.569 9.166Image-2 Noisy 25.126 4.792 27.274 6.987 28.712 8.460L1 25.409 5.184 25.726 5.501 25.881 5.656TV 28.106 7.884 28.471 8.249 28.624 8.403Cauchy 27.800 7.577 28.646 8.424 29.106 8.885V. CONCLUSIONIn this paper, we proposed a despeckling method, which we tested on simulated SAR imagery of the sea surface.Specifically, a closed form expression for calculating the proximal operator of Cauchy prior was proposed, whichmakes it applicable in proximal splitting algorithms such as FB. In addition, we presented a simulator for SARimages of the ocean surface that can incorporate ship wake structures. The performance of the Cauchy regularisationwith FB algorithm showed better despeckling results both in terms of PSNR and S/MSE measures as well as visuallyfor the simulated SAR images.REFERENCES[1] E. E. Kuruoglu and J. Zerubia, “Modeling SAR images with a generalization of the Rayleigh distribution,” IEEE Transactions on ImageProcessing, vol. 13, no. 4, pp. 527–533, 2004.[2] A. Achim, E. E. Kuruoglu, and J. Zerubia, “SAR image filtering based on the heavy-tailed Rayleigh model,” IEEE Transactions onImage Processing, vol. 15, no. 9, pp. 2686–2693, 2006.[3] O. Karakuş, E. E. Kuruoğlu, and M. A. Altınkaya, “Generalized Bayesian model selection for speckle on remote sensing images,”IEEE Transactions on Image Processing, vol. 28, no. 4, pp. 1748–1758, 2018.[4] A. Arnold-Bos, A. Khenchaf, and A. Martin, “Bistatic radar imaging of the marine environment—part i: Theoretical background,”IEEE Transactions on Geoscience and Remote Sensing, vol. 45, no. 11, pp. 3372–3383, 2007.[5] T. Elfouhaily, B. Chapron, K. Katsaros, and D. Vandemark, “A unified directional spectrum for long and short wind-driven waves,”Journal of Geophysical Research: Oceans, vol. 102, no. C7, pp. 15 781–15 796, 1997.[6] G. Zilman, A. Zapolski, and M. Marom, “On detectability of a ship’s Kelvin wake in simulated SAR images of rough sea surface,”IEEE Transactions on Geoscience and Remote Sensing, vol. 53, no. 2, pp. 609–619, 2014.[7] W. R. Alpers, D. B. Ross, and C. L. Rufenach, “On the detectability of ocean surface waves by real and synthetic aperture radar,”Journal of Geophysical Research: Oceans, vol. 86, no. C7, pp. 6481–6498, 1981.[8] R. Romeiser and W. Alpers, “An improved composite surface model for the radar backscattering cross section of the ocean surface:2. model response to surface roughness variations and the radar imaging of underwater bottom topography,” Journal of GeophysicalResearch: Oceans, vol. 102, no. C11, pp. 25 251–25 267, 1997.[9] L. Zurk and W. J. Plant, “Comparison of actual and simulated synthetic aperture radar image spectra of ocean waves,” Journal ofGeophysical Research: Oceans, vol. 101, no. C4, pp. 8913–8931, 1996.[10] A. Mohammad-Djafari, “Bayesian approach with prior models which enforce sparsity in signal and image processing,” EURASIPJournal on Advances in Signal Processing, vol. 2012, no. 1, p. 52, 2012.[11] A. Achim, P. Tsakalides, and A. Bezerianos, “SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling,”IEEE Trans. Geosci. and Remote Sensing, vol. 41, pp. 1773–1784, Aug. 2003.[12] G. Chen and X. Liu, “Wavelet-based SAR image despeckling using Cauchy pdf modeling,” in 2008 IEEE Radar Conference. IEEE,2008, pp. 1–5.[13] F. Argenti, A. Lapini, T. Bianchi, and L. Alparone, “A tutorial on speckle reduction in synthetic aperture radar images,” IEEE Geoscienceand remote sensing magazine, vol. 1, no. 3, pp. 6–35, 2013.[14] O. Karakuş and A. Achim, “On solving SAR imaging inverse problems using nonconvex regularization with a Cauchy-based penalty,”IEEE Transactions on Geoscience and Remote Sensing, pp. 1–13, 2020.[15] P. L. Combettes and J.-C. Pesquet, “Proximal splitting methods in signal processing,” in Fixed-point algorithms for inverse problemsin science and engineering. Springer, 2011, pp. 185–212.6[16] O. Karakuş, P. Mayo, and A. Achim, “Convergence guarantees for non-convex optimisation with Cauchy-based penalties,” IEEETransactions on Signal Processing, vol. 68, pp. 6159–6170, 2020.[17] T. Wan, N. Canagarajah, and A. Achim, “Segmentation of noisy colour images using cauchy distribution in the complex waveletdomain,” IET Image Processing, vol. 5, no. 2, pp. 159–170, 2011.[18] L. Gagnon and A. Jouan, “Speckle filtering of SAR images: a comparative study between complex-wavelet-based and standard filters,”in Wavelet Applications in Signal and Image Processing V, vol. 3169. International Society for Optics and Photonics, 1997, pp. 80–91.

A Simulation Study to Evaluate the Performance of the Cauchy Proximal Operator in Despeckling SAR Images of the Sea Surface

https://research-information.bris.ac.uk/files/274656743/2012.06657v1.pdf

A Simulation Study to Evaluate the Performance of the Cauchy Proximal Operator in Despeckling SAR Images of the Sea Surface

Abstract

Similar works

Full text

Available Versions

Online Research @ Cardiff

Explore Bristol Research