# Exploiting the Dual-Tree Complex Wavelet Transform for Ship Wake Detection
in SAR Imagery

Wanli Ma, Alin Achim, Oktay Karakuş This work was supported by the UK
Engineering and Physical Sciences Research Council (EPSRC) under grant
EP/R009260/1 (AssenSAR). Wanli Ma, Alin Achim and Oktay Karakuş are with the
Visual Information Laboratory, University of Bristol, Bristol BS1 5DD, U.K.
(e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we analyse synthetic aperture radar (SAR) images of the sea
surface using an inverse problem formulation whereby Radon domain information
is enhanced in order to accurately detect ship wakes. This is achieved by
promoting linear features in the images. For the inverse problem-solving
stage, we propose a penalty function, which combines the dual-tree complex
wavelet transform (DT-CWT) with the non-convex Cauchy penalty function. The
solution to this inverse problem is based on the forward-backward (FB)
splitting algorithm to obtain enhanced images in the Radon domain. The
proposed method achieves the best results and leads to significant improvement
in terms of various performance metrics, compared to state-of-the-art ship
wake detection methods. The accuracy of detecting ship wakes in SAR images
with different frequency bands and spatial resolution reaches more than 90%,
which clearly demonstrates an accuracy gain of 7% compared to the second-best
approach.

###### Index Terms:

SAR image, ship wake detection, inverse problem, DT-CWT, FB splitting

##  I Introduction

Thanks to the development of remote sensors technology, synthetic aperture
radar (SAR) has become a widespread and effective imaging modality to acquire
vital information for analysing and tracking vessels on the ocean surface. In
SAR images of the ocean surface, a moving vessel generates different wake
structures, which can be identified as: (i) turbulent wake, (ii) Narrow
V-wake, and (iii) Kelvin wake. Due to speckle and the complex nature of the
SAR imagery, the success of ship and wake detection remain limited, and new
approaches have become necessary to address the aforementioned issues. In the
literature, various works have suggested algorithms to leverage the
information in SAR imagery for the purpose of wake detection.

Considering the linear characteristics of ship wakes, their detection within
SAR images is well suited to be performed in the Radon domain, which is the
gold standard in line detection along with the Hough transform. Thus, Radon
transform based methods have dominated the literature for ship wake detection
solutions [1, 2, 3, 4, 5, 6]. Different from the previous approaches, Karakuş
et. al. [7, 8] addressed ship wake detection as an inverse problem by
expressing the observed SAR image in terms of the Radon domain information,
and solution to the aforementioned inverse problem has been obtained by using
sparse penalty functions such as the generalized minimax concave (GMC) penalty
[9]. The approach in [7, 8] has then been implemented with a non-convex Cauchy
based penalty function, with the ensuing algorithm guaranteed to convergence
[10]. This led to higher ship wake detection accuracy in [11, 12] compared to
the GMC, $L_{1}$ and total variation (TV) penalties.

The dual-tree complex wavelet transform (DT-CWT) offers added benefits over
the discrete wavelet transform (DWT). It calculates the complex transform of a
signal using two separate DWT decompositions; one tree generates the real part
of the transformation and another tree generates the imaginary part. DT-CWT is
thus able to reduce the translation sensitivity and improve the directional
selectivity of DWT. In the SAR imaging literature, owing to its advantages
compared to the DWT, it has shown benefits in applications including
despeckling [13, 14], sea ice detection [15], and image enhancement [16].

In this paper, we propose a novel algorithm for detecting ship wakes, which is
based on imaging inverse problems to enhance linear features in SAR imagery.
We achieve this by combining the non-convex Cauchy based penalty function [11]
with the DT-CWT. The proposed method, which benefits from the near shift-
invariance property of the DT-CWT, along with the heavy-tailed nature of the
Cauchy penalty function, is then compared to methods based on several state-
of-the-art penalty functions, such as GMC [7], $TV$, Cauchy [11], and the
method of Graziano et. al. [17]. All the corresponding minimisation problems
are solved via the forward-backward (FB) proximal splitting algorithm.

The rest of the paper is organised as follows: we first present the
theoretical preliminaries in Section II. The proposed method is discussed in
Section III, whilst Section IV presents the experimental analysis. Section V
concludes the paper with a brief summary and future work directions.

##  II THEORETICAL PRELIMINARIES

###  II-A Inverse Problem Formulation

Since ship wakes are regarded as linear structures, the SAR image formation
model can be defined in terms of the inverse Radon transform as [7]

$Y=\mathcal{C}X+N$ (1)

where $Y$ is the observed SAR image and $X(r,\theta)$ represents the
information in the transform domain. Also, $\mathcal{C}$ refers to the inverse
Radon transform, and $N$ is additive noise.

Under the assumption of an independent and identically distributed (iid)
Gaussian noise, we express the data fidelity term (i.e. the likelihood) as

$\displaystyle\mathcal{L}(Y,\mathcal{C}X)=\|Y-\mathcal{C}X\|_{2}^{2}$ (2)

Since recovering the object of interest $X$ from the observation $Y$ is an
ill-posed inverse problem, we must scrutinise the prior information on $X$ to
obtain a stable and unique reconstruction result. Having the prior knowledge
$p(X)$, the problem of estimating $X$ from the observed SAR image $Y$ by using
the signal model in (1) turns into a minimisation problem as

$\displaystyle\hat{X}=\arg\min_{X}\bigg{\\{}\|Y-\mathcal{C}X\|_{2}^{2}-\log
p(X)\bigg{\\}}$ (3)

where $\psi(X)=-\log p(X)$ is the penalty function, and thus the cost function
takes the form $F(X)=\mathcal{L}(Y,\mathcal{C}X)+\psi(X)$.

###  II-B Dual-Tree Complex Wavelet Transform (DT-CWT)

The DT-CWT is an efficient improvement of the classical DWT, which addresses
the poor performance for complex and/or modulated signals, such as radar,
speech, etc. As shown in [18], two real DWTs constitute a DT-CWT, which has
real and imaginary parts given by the first real DWT and the second real DWT,
respectively. Figure 1 depicts the forward DT-CWT operation.

Figure 1: Analysis filter bank for the DT-CWT [18]

As long as we express two real DWTs as real ($\Psi_{h}(t)$) and imaginary
($\Psi_{g}(t)$) pair, the generic expression of the DT-CWT can be written as
[18]

$\Psi(\mathrm{t})=\Psi_{h}(t)+j\Psi_{g}(t)$ (4)

There is no data flow between $\Psi_{h}(t)$ and $\Psi_{g}(t)$, they thus can
be calculated separately. Since in this paper we deal with images, a 2-D Dual-
Tree CWT needs to be implemented as

$\Psi(x,y)=\Psi(x)\Psi(y)$ (5)

Combining (4) and (5), we have

$\Psi(x,y)=\left[\Psi_{h}(x)+j\Psi_{g}(x)\right]\left[\Psi_{h}(y)+j\Psi_{g}(y)\right]$
(6)

##  III THE PROPOSED METHOD

###  III-A Cauchy and DT-CWT Based Penalty Function

The Cauchy distribution is a member of the symmetric $\alpha$-stable family,
and the probability density function (pdf) of the Cauchy distribution is
expressed as

$\displaystyle p(x)\propto\left[\frac{\gamma}{x^{2}+\gamma^{2}}\right]$ (7)

where the parameter $\gamma$ is the scale parameter which determines the
spread of the Cauchy distribution around zero. When replacing the prior
information in (3) with the Cauchy prior, we obtain the following minimisation
problem as [10]

$\hat{X}_{\text{Cauchy}}:=\arg\min_{\boldsymbol{X}}\left\\{\|Y-\operatorname{CX}\|_{2}^{2}-\log\frac{\gamma}{X^{2}+\gamma^{2}}\right\\}$
(8)

where the regularization term $-\log\frac{\gamma}{X^{2}+\gamma^{2}}$ is the
non-convex Cauchy-based penalty function.

In addition to its role in modelling the heavy-tailed noise in various signal
and image processing applications, Cauchy based penalty function has rarely
been used as the signal prior in the literature. Based on the derivation of
its proximal operator in [10], the work in [11] proposed the Cauchy penalty
function for addressing four SAR imaging inverse problems: super-resolution,
image reconstruction, despeckling and ship wake detection. The Cauchy based
penalty function has shown improved performance over various important penalty
functions in [11] such as $L_{1}$, TV, and GMC.

Although some efforts on ship wake detection based on inverse problem solving
have been done by employing penalty functions such as GMC [8], and Cauchy
[11], both penalty functions are in need of further improvements mainly for
better tackling noise artefacts, which sometimes induce false wake detections.
DT-CWT appears a promising tool for efficiently enhancing the corresponding
SAR image for the purpose of wake detection, thanks to its properties of
reducing the translation sensitivity and improving the directional selectivity
[18]. Hence, in order to benefit from its advantages, a second penalty
function base on the DT-CWT is designed and added to the cost function

$F(X)=\|Y-\mathcal{C}X\|_{2}^{2}-\log\frac{\gamma}{X^{2}+\gamma^{2}}+\lambda\|\mathcal{B}X\|_{1}$
(9)

where $\mathcal{B}$ is the forward DT-CWT operator, and $\|\cdot\|_{1}$ is
$L_{1}$ norm which is a convex function. The inclusion of an additional term,
$\|\mathcal{B}X\|_{1}$ involving the DT-CWT transform of $X$, contributes to
the enhancement of the Radon transform $X$ of the SAR image $Y$, and promotes
sparsity through the use of L1 norm.

The solution for minimising the cost function in (9) can be based on proximal
splitting algorithms such as the Forward-Backward (FB) [19] and Alternating
Direction Method of Multipliers (ADMM) [20]. Thanks to its simpler
implementation and success in image processing applications, in this work we
prefer to utilise the FB algorithm. The FB splitting algorithm first
necessitates to split the cost function into two functions as

$\displaystyle\hat{X}_{\text{Cauchy-
DTCWT}}=\arg\min_{X}\bigg{\\{}F(X)=g(X)+h(X)\bigg{\\}}.$ (10)

To this end, having a convex data fidelity term and the differentiable Cauchy
penalty function, we choose functions $g(\cdot)$ and $h(\cdot)$

$\displaystyle g(X)$
$\displaystyle=\|Y-\mathcal{C}X\|_{2}^{2}-\log\frac{\gamma}{X^{2}+\gamma^{2}},$
(11) $\displaystyle h(X)$ $\displaystyle=\lambda\|\mathcal{B}X\|_{1}.$ (12)

The splitting of the cost function mentioned above only requires the proximal
operator of $h(X)$. Thus, for any $\mu>0$, the iterative FB solution is
obtained by

$X^{(\ell+1)}=\operatorname{prox}_{\mu h}\left(X^{(\ell)}-\mu\nabla
g\left(X^{(\ell)}\right)\right)$ (13)

where the gradient of the function $g$ becomes

$\displaystyle\nabla
g(X)=\mathcal{C}^{T}(\mathcal{C}X-Y)+\frac{2X}{\gamma^{2}+X^{2}}.$ (14)

The proximal operator of the $L_{1}$ norm-based penalty functions is the well-
known soft thresholding function

$\displaystyle\operatorname{prox}_{\mu
h}(X)=\max\left(0,1-\frac{\mu\lambda}{BX}\right).$ (15)

Consequently, the iterative solution to the proposed minimisation problem in
(9) is as shown in Algorithm 1 below. We set the maximum number of iterations
(maxIter) to 500. The error term $\varepsilon^{(i)}$ is set to 0.005, which
can be calculated as:
$\epsilon^{(i)}=\frac{\left\|X^{(i)}-X^{(i-1)}\right\|}{\left\|X^{(i-1)}\right\|}$.

Algorithm 1 FB Algorithm for the proposed penalty function

Input: SAR imagery $Y$ and $\mu,\gamma,\lambda$
Output: Radon image X
Set: $\boldsymbol{i}=0$ and $\boldsymbol{X}^{(0)}=\\{\boldsymbol{0}\\}$

1:do:

2:$Z^{(i)}=X^{(i)}-\mu\nabla(g(X^{(i)})).$

3:$\mathbf{W}=\operatorname{soft}\left\\{\mathbf{B}\left(\mathbf{Z}^{(i)}\right),\boldsymbol{\mu}*\lambda\right\\}$

4:$X^{(i+1)}={\mathcal{B}}^{-1}(\mathbf{W})$

5:$i++$

6: while: $\varepsilon^{(i)}>10^{-3}$ or $i<$ maxIter

###  III-B Ship Wake Detection

As soon as we have the reconstructed Radon information from the optimisation
procedure detailed above, in order to detect the wake structures, we use the
same detection method as previously proposed in [7]. The computational steps
of the whole ship wake detection algorithm in this paper are presented in
Figure 2. For further details please see [7].

Figure 2: Ship wake detection Method.

For the validation in Step 4, we use the merit index
$\mathrm{F}_{\mathrm{t}}^{\mathrm{I}}=\frac{\overline{\mathrm{I}}_{\mathrm{t}}}{\overline{\mathrm{I}}}-1$
[17], in which ${\overline{\mathrm{I}}_{\mathrm{t}}}$ is the mean value along
the half wake, and ${\overline{\mathrm{I}}}$ is the average intensity of the
SAR image. For the turbulent wake validation,
$\mathrm{F}_{\mathrm{t}}^{\mathrm{I}}$ is allowed to be a negative value. For
other wake structures $\mathrm{F}_{\mathrm{t}}^{\mathrm{I}}$ must be greater
than zero. We also set a margin of 10% to reduce the possibility of false
detections, thus for the Narrow-V and Kelvin wakes,
$\mathrm{F}_{\mathrm{t}}^{\mathrm{I}}$ needs to be greater than 0.1.

##  IV Experimental results

###  IV-A SAR Image Data

In this paper, 22 real SAR images from five different satellite platforms
including TerraSAR-X, ALOS2, COSMO-SkyMed, Sentinel-1, and ICEYE were used.
Seven of them are X-band SAR images from TerraSAR-X, with a resolution of
3$\times$3 m. We also tested four L-band SAR images from ALOS2, which also
have a resolution of 3$\times$3 m. The COSMO-SkyMed (5 images) and ICEYE (1
image) also provide X-band SAR images. The ICEYE [21] image utilised in this
work is an HH-polarised, Stripmap product, which has 3-m resolution for both
azimuth and range directions. Sentinel-1 (5 images) utilises C-band frequency
and the resolution of the images is 10$\times$10 m and hence includes less
details than the images from the other satellite platforms used. In Table I,
each SAR image is shown with their visible and invisible wakes (1 and 0
values, respectively) corresponding to a visual analysis.

TABLE I: Wake information of the SAR imagery data sets. T: Turbulent, N1 & N2: Narrow V-wake, K1 & K2: Kelvin wake. Image |  Image | Wake Types
---|---|---
|  Source |  T |  N1 |  N2 |  K1 |  K2
1 |  TerraSAR-X |  1 |  1 |  0 |  0 |  1
2 |  TerraSAR-X |  1 |  1 |  0 |  1 |  0
3 |  ALOS2 |  1 |  1 |  0 |  0 |  0
4 |  ALOS2 |  1 |  1 |  0 |  1 |  0
5 |  COSMO-SkyMed |  1 |  1 |  0 |  0 |  0
6 |  COSMO-SkyMed |  1 |  1 |  0 |  1 |  0
7 |  COSMO-SkyMed |  1 |  1 |  0 |  1 |  0
8 |  COSMO-SkyMed |  1 |  1 |  0 |  0 |  0
9 |  Sentinel-1 |  1 |  1 |  0 |  0 |  0
10 |  Sentinel-1 |  1 |  1 |  0 |  0 |  0
11 |  TerraSAR-X |  1 |  1 |  0 |  1 |  0
12 |  TerraSAR-X |  1 |  1 |  0 |  1 |  0
13 |  TerraSAR-X |  1 |  1 |  0 |  1 |  0
14 |  TerraSAR-X |  1 |  1 |  0 |  1 |  0
15 |  COSMO-SkyMed |  1 |  1 |  0 |  0 |  0
16 |  ALOS2 |  1 |  1 |  0 |  0 |  0
17 |  ALOS2 |  1 |  1 |  0 |  0 |  0
18 |  Sentinel-1 |  1 |  1 |  0 |  0 |  0
19 |  Sentinel-1 |  1 |  1 |  0 |  0 |  0
20 |  Sentinel-1 |  1 |  1 |  0 |  0 |  0
21 |  ICEYE |  1 |  1 |  0 |  1 |  0
22 |  TerraSAR-X |  1 |  1 |  0 |  0 |  0

###  IV-B Performance Analysis

We compared the proposed method to four state-of-the-art methods: (1) the
detection algorithm proposed by Graziano et. al. [17], (2) the method based on
GMC penalty function [7], (3) total variation (TV) norm-based method, (4)
method using a non-convex Cauchy penalty function [11]. All 22 images
mentioned above were centred on the ship and masked after appropriate pre-
processing stages. For each image, ship wake structures are classified into
three categories: (1) Turbulent Wake, (2) narrow-V Wake (2 arms) and (3)
Kelvin wake (2 arms) [22], and the detection performance metrics are
calculated depending on correct detection/discard of these wakes. The ship
wake detection procedure was implemented by using the AssenSAR Wake Detector
Matlab software [23] available as open-source, and for the DT-CWT
implementation we used the sample functions published in [24]. The detailed
performance analysis for all images is shown in Table III and Table III with
the best results in bold.

TABLE II: Detection performance for all SAR images in terms of the receiver
operating characteristic (ROC) metrics

Methods |  TP |  TN |  FP |  FN
---|---|---|---|---
Graziano et. al. [17] |  32.73$\%$ |  38.18$\%$ |  25.45$\%$ |  3.64$\%$
GMC [8] |  43.64$\%$ |  34.55$\%$ |  18.18$\%$ |  3.64$\%$
TV |  38.18$\%$ |  33.64$\%$ |  26.36$\%$ |  1.82$\%$
Cauchy [11] |  37.27$\%$ |  47.27$\%$ |  10.00$\%$ |  6.36$\%$
Proposed |  41.82$\%$ |  48.18$\%$ |  5.45$\%$ |  5.45$\%$

TABLE III: Detection performance of different methods for all SAR images

Methods |  % Accuracy |  $F_{1}$ |  LR+ |  Youden’s J
---|---|---|---|---
Graziano et. al. [17] |  70.91$\%$ |  0.69 |  2.25 |  0.50
GMC [8] |  77.27$\%$ |  0.80 |  2.68 |  0.58
TV |  70.91$\%$ |  0.73 |  2.17 |  0.52
Cauchy [11] |  84.55$\%$ |  0.82 |  4.89 |  0.68
Proposed |  90.91$\%$ |  0.88 |  8.70 |  0.78

In terms of True Positive (TP) values in Table III, the GMC based method shows
the best performance with 43.64$\%$. The TP of the proposed method is
41.82$\%$ which is only marginally lower than that of GMC, and higher than
that of Graziano et. al., TV, and Cauchy. As shown in Table III Cauchy and DT-
CWT based prior led to superior performance in terms of the ROC metrics
corresponding to True Negative (TN) and False Positive (FP), which are
48.18$\%$ and 5.45$\%$, respectively.

We also computed four quantitative metrics for evaluating the detection
performance, which are presented in Table III. In particular, for all four
performance metrics, the proposed method achieves the best results with 90.91%
detection accuracy. Compared to the results of the other four state-of-the-art
methods, the proposed method outperforms all by achieving around 6.5$\%$
accuracy gain compared to the Cauchy [11], and approximately 20$\%$ compared
to Graziano et. al. [17] and TV.

In order to illustrate the numerical performance analysis given above, for
visual analysis we also present detection results corresponding to an example
SAR image, which is depicted in Figure 3. When examining Figure 3, Graziano
et. al. [17] and the proposed method show remarkable detection results for the
example considered. However, TV, GMC and Cauchy based approaches have
relatively lower accuracy results (in particular for the detection of narrow-V
wakes).

Considering the proposed method in this paper is an improvement of the method
in [11], we can deduce that the addition of the DT-CWT based penalty to the
Cauchy-based penalty term led to a significant performance increase, in
particular for reducing false wake detections (FP values in Table III). Since
false detections correspond to noisy peak values in the reconstructed Radon
images, the DT-CWT penalty addition helps to reduce the noise in Radon domain
and turn these false detections into correct detections (TP values in Table
III).

(a) SAR image

(b) Graziano et. al. [17]

(c) TV

(d) GMC [8]

(e) Cauchy [11]

(f) Proposed method

Figure 3: Ship wake detection results for SAR Image 9. Yellow and green lines
refer to the detected turbulent and narrow-V wakes, respectively.

##  V Conclusions

In this paper, we proposed a novel regularisation strategy, which combines the
non-convex Cauchy based penalty function with an additional term based on the
DT-CWT. A proximal splitting optimisation algorithm was used to solve the
inverse problem and to enhance the information in the Radon domain.

The results of the performance analysis, which was based on 22 SAR images from
five different satellite platforms, demonstrates that the proposed approach
detects ship wakes with a high accuracy of around 91$\%$. It is obvious that
the proposed penalty function benefits from the advantages of DT-CWT
(improving the translation invariance and the directional feature selectivity
[18]) and removes thus noisy artefacts from the reconstructed images. The
proposed method achieves better detection performance than existing inverse
problem based ship wake detection methods. On the other hand, the proposed
method still offers potential for further improvement (e.g. to achieve higher
TP) and developing more approaches, based on other types of sparse
representations, is part of our current endeavours.

##  VI Acknowledgment

We are grateful to the UK Satellite Applications Catapult and to OceanMind Ltd
for providing the COSMO-SkyMed and the Sentinel-1 data sets, respectively.

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