<figure><img src="image_1.png" /></figure>

HAL
open science

# AntoNet: a small linear contrast-enhancing CNN for pixel-wise detection

Soggia Antonio, Lionel Fillatre, Deruaz-Pepin Laurent

<figure><img src="image_2.png" /></figure>

## To cite this version:

Soggia Antonio, Lionel Fillatre, Deruaz-Pepin Laurent. AntoNet: a small linear contrast-enhancing CNN for pixel-wise detection. 2024. hal-04673096

HAL Id: hal-04673096

https://hal.science/hal-04673096v1

Preprint submitted on 19 Aug 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_3.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# AntoNet: a small linear contrast-enhancing CNN for pixel-wise detection

1st Soggia Antonio

2nd Fillatre Lionel

3rd Deruaz-Pepin Laurent

Thales DMS, I3S

Université Côte d'Azur, CNRS, I3S

Thales DMS

Sophia-Antipolis, France

Sophia-Antipolis, France

Sophia-Antipolis, France

antonio.soggia@etu.univ-cotedazur.fr

lionel.fillatre@i3s.unice.fr

Laurent.Deruaz-Pepin@fr.thalesgroup.com

**Abstract**—In current broadband passive sonar systems, a human operator observes a 2D image that represents different energy levels in the bearing and time dimensions. The temporal depth of the image helps the operator to detect and identify the paths taken by different sources. However, detecting weak noise sources amid ambient sea noise and strong sources can be challenging. The purpose of this work is to increase the contrast of those images using AI techniques. The performance of AntoNet, our proposed architecture based on Unet, is evaluated for reducing ambient noise and enhancing the signal of interest. The dataset is generated by a fast sonar image simulator based on a simplified sonar beamforming model. We aim to improve pixel-wise detection through clearer 2D images with a better SNR. AntoNet leverages the temporal depth of data to produce 2D images with an enhanced SNR while having less than 1,500 parameters. Performance evaluation demonstrates AntoNet's effectiveness in single source environments, achieving higher detection performance with minimal artifacts.

**Index Terms**—CNN, detection, contrast enhancement, sonar

## I. INTRODUCTION

Passive sonar systems, primarily used for detecting underwater objects without emitting sound, display broadband signals to the operator as a 2D (bearing-time) image like the noised one in Fig 1. However, due to the noisy acoustic

environment, weak sources could be hidden [1]. Information loss due to external noises, such as marine biology and rain [2], obstructs a clear view for the operator and may lead to misinformed decisions.

Briefly speaking, we assume that the passive sonar system produces a family of 2D image whose horizontal, resp. vertical, axis is the arrival direction of the weak source wave, resp. the time. Usually, as shown on the top of Fig 1, a pixel-wise detector is applied on each pixel of the sonar image such that each couple (arrival direction, time) is treated independently of the other ones. In other terms, we want to classify each pixel as being part of a signal or noise. Our method aims to enhance the signal before applying the pixel-wise detection analysis (see Fig 1). The contrast enhancement is based on the whole image in order to use weak source trajectory information for increasing the Signal-to-Noise Ratio (SNR) of each individual pixel.

Conventional contrast-enhancing algorithms [3], [4], [5], [6] while useful, can produce artefacts, phantom sources or be badly calibrated. Beyond these conventional methods, wavelets [7], [8] have also been explored for contrast-enhancing. However, while they offer certain advantages, they come with their own drawbacks [9]. Wavelets suffer from many problems such as artifacts, over-smoothing and computational complexity. We then tried a gradient-based detection algorithm approach using Canny [10] and LSD [11]. Sadly, they performed poorly in presence of strong noise as shown in Fig. 2. In fact, the noise in the sonar images is too complex for this kind of algorithms as it presents high-frequency and Gaussian components. It makes the noise be falsely detected as a signal.

The advent of Machine Learning and the rise of Convolutional Neural Networks (CNNs), gave way to new image processing algorithms, [12], [13], [14], [15]. In particular, variational auto-encoder architectures such as Unet [16] become common for image processing. They have been used for active sonar imagery [17], [18] or do not meet yet the expected performance [19]. Their application in the domain of sonar imaging, particularly from an interpretability point of view, remains largely uncharted. Ofir et al. [20] reached our same conclusion for a segmentation purpose. We want to extend this framework into a contrast-enhancing framework.

We propose a custom CNN-based variational auto-encoder, named AntoNet, tailored for our specific contrast-enhancing

<figure><img src="image_4.png" /><figcaption>Fig. 1. Comparison between a traditional detection pipeline (up) and ours with image enhancement before the pixel-wise detection (down).</figcaption></figure>

Thanks to ANRT agency for funding.
<figure><img src="image_5.png" /><figcaption>Fig. 2. Canny and LSD applied on a noisy image. The signal isn't even detected, all detections are false alarms.</figcaption></figure>

requirements. AntoNet's design facilitates the distinction between noise and underwater signals, crucial in sonar image processing. Its mathematical operations, akin to traditional signal processing [21], contribute to the model's transparency. Furthermore, AntoNet's reduced parameter count, at least 2,000 times less than existing models, ensures faster and more accurate processing, making it ideal for real-time applications. Coupled with the low number of parameters, AntoNet has only locally linear operations, thus making it easy to interpret. Interestingly, AntoNet adopts a design devoid of traditional activation functions [22], [23], [24], a deliberate choice to preserve linearity and facilitate interpretability. This unconventional approach is designed to promote a more straightforward understanding of the network's transformation process, fostering a deeper insight into how the model discerns between signal and noise.

The main contributions of this work are manifold:

* A simplified physical model that shows that each sonar image pixel follows a Gaussian probability distribution. This model allows us to generate enough labeled data to train our proposed neural network.

* A novel, lightweight, and linear CNN together with a non-linear version. The linear CNN, although it contains no nonlinear operator, has an acceptable loss of performance with respect to the nonlinear one.

* A high (10 dB) SNR gain on the output compared to non-enhanced images that shows that our contrast enhancement increases the SNR of the processed sonar images.

* A comparison with bigger state-of-the-art models like ORCA and Unet.

This work aims to offer a comprehensive guide on CNN-based algorithms for sonar image, serving as a reliable resource in underwater acoustic analysis. By offering insights into the contrast-enhancing process, AntoNet not only improves the trustworthiness of the contrast-enhancing output but also provides valuable feedback that can be used to refine the model further. Indeed, a reliable pre-processed image reduces problems about tracking [25], [26], [27], and classification [28] algorithms in noisy environments.

This paper is organized as follows. Section II describes our simplified sonar image formation model. Section III presents our AntoNet encoder-decoder CNN. Section IV is dedicated to numerical results while section V concludes the paper.

## II. SIMPLIFIED SONAR IMAGE FORMATION

Due to the unavailability of Waterfall data and industrial constraints, we have opted to simulate our signals and the noise. The simulation method employed is computationally efficient, allowing for the generation of a sufficient amount of data within a reasonable time frame. Additionally, the simulated data is designed to have the same statistical distribution as the Waterfall data after broad band processing. This section presents our simplified model: first, the source and antenna model and, next, the measurement model.

### A. Source and antenna model

For simplicity, we assume that the source is unique and follows a uniform rectilinear movement over the sea. This source is emitting a monochromatic wave with an unknown frequency $\nu$. A sonar equipped with a linear antenna measures the planar wavefront arriving from a direction with an angle $\theta$, like in Fig. 3. Let $\delta_k(\theta_t)$ be the reception delay of the

<figure><img src="image_6.png" /><figcaption>Fig. 3. A linear antenna composed of K linearly spaced sensors receives a planar wavefront that arrives with an angle $\theta$.</figcaption></figure>

wavefront at sensor $k$ with respect to the first sensor, namely the sensor 0, with the arrival angle $\theta_t$ at time $t$. A short calculation shows that

$$ \delta_k(\theta_t) = (k - 1) \frac{d \cos(\theta_t)}{c}, \quad (1) $$

where $d$ is the constant distance between the sensors and $c$ is the wavefront celerity. Let

$$ h_t^+ = [1, \dots, e^{2i\pi\nu\delta_k(\theta_t)}, \dots, e^{2i\pi\nu\delta_K(\theta_t)}] \quad (2) $$

be the phase vector of the source in direction $\theta_t$ with respect to the linear antenna. Here, $h^+$ denotes the conjugate transpose of the vector $h$.

### B. Measurement model

At time $t$, we assume that we have $N$ stationary sonar measurements $s_{t,i} \in \mathbb{C}^K$ such that

$$ s_{t,i} = a_{t,i}h_t + \eta_{t,i}, \quad i = 1, \dots, N, \quad (3) $$

where $\eta_{t,i}$ is a sequence of independent circularly-symmetric central complex normal noise with zero mean and covariance matrix $\sigma_{\eta}^2 I_K$ and $I_K$ is the identity matrix of size $K$. The
signal amplitude $a_{t,i}$ is a sequence of complex normal variable with zero mean and variance $\sigma_a^2$.

As usual in sonar processing, we need to integrate the measurements to highlight the presence of the wave. For a faster integration, we assume that $\nu = \frac{c}{2d} = \nu_0$ whatever is the true value of the monochromatic wave frequency (the observed frequency band is supposed to be narrow). Let

$$g_j^+ = [1, \dots, e^{-i\pi k \frac{j}{K-1}}, \dots, e^{-i\pi (K-1) \frac{j}{K-1}}] \quad (4)$$

be the steering vector for the $j$th channel beamforming when $j = 0, \dots, K-1$. The integrated energy on the $j$th channel at time $t$ is computed by

$$x_{t,j} = \frac{1}{N} \sum_{i=1}^{N} |g_j^+ s_{t,i}|^2. \quad (5)$$

Since $N$ is large in practice, a short calculation gives the asymptotic distribution of the channel. Let us denote $H_0(t, j)$, resp. $H_1(t, j)$, the case when the acoustic source is not present, resp. is present, in the channel $j$ at time $t$. After an appropriate standardization of the channel (we do not change the notation to keep them simple), we get the following asymptotic distribution

$$x_{t,j} \sim_{N \to +\infty} \begin{cases} \mathcal{N}(0, 1) & \text{under } H_0(t, j), \\ \mathcal{N}\left(\varrho_{t,j}^2 \sqrt{N}, (1 + \varrho_{t,j}^2)^2\right) & \text{under } H_1(t, j). \end{cases} \quad (6)$$

The value $\varrho_{t,j}^2$ is the SNR defined by

$$\varrho_{t,j}^2 = \frac{\sigma_a^2}{K \sigma_{\eta}^2} \left( \frac{\sin(\frac{\pi}{2} K \phi_{t,j})}{\sin(\frac{\pi}{2} \phi_{t,j})} \right)^2, \quad (7)$$

where $\phi_{t,j}$ is the gap between the true delay and the pointing delay:

$$\phi_{t,j} = \nu \delta(\theta_t) - \frac{j}{K-1}. \quad (8)$$

The sonar image is then defined as the matrix

$$X = (x_{t,j})_{1 \le t \le T, 0 \le j \le K-1} \in \mathbb{R}^{T \times K} \quad (9)$$

where $T$ is the last measurement time. The label image that we want to predict is the SNR image given by

$$Y = (y_{t,j})_{1 \le t \le T, 0 \le j \le K-1} \in \mathbb{R}^{T \times K} \quad (10)$$

where

$$y_{t,j} = \begin{cases} 0 & \text{under } H_0(t, j), \\ \varrho_{t,j}^2 \sqrt{N} & \text{under } H_1(t, j). \end{cases} \quad (11)$$

In practice, the label image is of course extremely difficult to measure. This explains why a simulated dataset is so important to learn the contrast-enhancement method. When we generate a sonar image from this model, we choose first for each couple $(t, j)$ the ground truth, $H_0(t, j)$ or $H_1(t, j)$, to obtain a binary label $y_{t,j} \in \{0, 1\}$ and to form the whole label image $Y$. Then, knowing $Y$, we generate the associated sonar image $X$ whose pixel $x_{t,j}$ follows the asymptotic distribution in (6) under hypothesis $H_{y(t,j)}(t, j)$. From the two images in Fig. 4 we can see the sonar image with noise and the labeled image to obtain. We will use a -2dB signal example for reference in this case due to the greater difficulty.

<figure><img src="image_7.png" /><figcaption>Fig. 4. An example of generated sonar image $X$ (left) and its associated label image $Y$ (right) for a -2dB SNR source signal.</figcaption></figure>

### III. CONTRAST ENHANCEMENT WITH A CNN

We suppose to have a data set

$$\mathcal{D} = \left\{ (X^{(n)}, Y^{(n)}), n \in \{1, \dots, N\} \right\}, \quad (12)$$

consisting of sonar image pairs $(X^{(n)}, Y^{(n)})$ where $X^{(n)}$ is given in (9) and $Y^{(n)}$ is the associated labeled image (10).

#### A. Loss function

In our case, since we are faced with an estimation problem, we want a loss function capable of calculating a deviation. A first idea was to turn to the mean square error (MSE) that leads to the expected risk

$$\text{MLSE}(f) = \mathbb{E}_{(X,Y) \sim \mathcal{D}}[(Y - f(X))^2] \quad (13)$$

where $f$ denotes any function that processes $X$ to get an estimate of $Y$. Assuming ergodicity of the data, we can use the mean to estimate the expectation. We then have a training error defined as:

$$\widehat{\text{MLSE}}(f) = \frac{1}{NTK} \sum_{n=1}^{N} \sum_{t=1}^{T} \sum_{j=0}^{K-1} \left( f(x_{t,j}^{(n)}) - y_{t,j}^{(n)} \right)^2. \quad (14)$$

Although MSE gave good estimation performance, it was unable to detect weak sources effectively probably due to the weak error if they weren't detected. It was therefore decided to explore more suitable functions. Among these, Logarithmic Mean Square Error or MLSE proved the most effective. As its name suggests, it is defined as:

$$\widehat{\text{MLSE}}(f) = \frac{1}{NTK} \sum_{n=1}^{N} \sum_{t=1}^{T} \sum_{j=0}^{K-1} \left( \log \left( 1 + f(x_{t,j}^{(n)}) \right) - \log \left( 1 + y_{t,j}^{(n)} \right) \right)^2. \quad (15)$$

MLSE is slightly less efficient than MSE in the sonar image estimation (when we want to estimate each pixel value whatever its label is) but it is much better in contrast-enhancement. In fact, it gives a larger relative weight to weak signals to enhance them.

#### B. AntoNet: our encoder-decoder CNN

To enhance the contrast in the sonar images, we propose to use a function $f$ modeled as a CNN named AntoNet. We begin by setting up a classic version of the CNN and then we remove the non-linearities as summarized in Table I.
<table><caption>TABLE I
COMPARISON BETWEEN LINEAR AND NON-LINEAR ANTONET</caption><thead><tr><th></th><th>Activation function</th><th>Batch Normalisation</th><th>Pooling</th></tr></thead><tbody><tr><td>Non-Lin. AntoNet</td><td>SeLu</td><td>Yes</td><td>Max</td></tr><tr><td>Lin. AntoNet</td><td>None</td><td>No</td><td>Average</td></tr></tbody></table>

CNNs are proven to be resistant to noise due to many reasons [29]. The main one being their hierarchical feature learning (HFL). This aspect contributes to their robustness to noise in various tasks, including contrast-enhancing [30]. Thanks to HFL, CNNs can learn features at multiple levels of abstraction, which could help overcome noise. CNNs begin to learn with the initial layers. These layers, akin to low-pass filters in signal processing, primarily focus on extracting low-frequency features from the sonar images. Basically, we apply a smoothing filter that allows the network to capture general patterns, i.e. the source trajectory in sonar images, by attenuating high-frequency noise components and fine details. As the network progresses to deeper layers, the initially extracted smoothed features are combined and recombined to form more complex representations. This hierarchical structuring, starting from low-frequency, broad features to more detailed, high-frequency elements, allows CNNs to effectively interpret the data and enhance the source trajectory.

AntoNet is a small and interpretable variant of Unet [31], calibrated to better suit our problem. It is similar to other networks inspired by Unet, such as C-Unet [16] or DeepUnet [32] which are used for classification or segmentation. The AntoNet structure is described in Fig. 5. It belongs to the family of encoder-decoder models: its structure is composed of two parts, the encoding and the decoding. In practice, we have 4 encoding and 4 decoding layers. Each model component is formed by one convolution, an optional batch normalization (only in the non-linear version) followed by a MaxPooling (in the nonlinear case) or an Average Pooling (in the linear case). After many attempts to keep the model as simple as possible, we decided to make it such that its convolution kernel's depth is 1 when the input/output is a matrix or 3 when the input is a tensor. This depth makes visualisation still possible while giving us satisfactory performance.

Hyper-parameters have been added to customize the learning process and solve potential problems like overfitting. Two types of regularization have been considered: $L^1/L^2$ and weight limitation. The aim of these two methods is to force the weights to remain bounded, so as not to have high output variability for small changes in the model's input. $L^1/L^2$ regularization, also called Elastic-net [33], is a way of penalizing weights that are too high in the network. While $L^1/L^2$ regularization was an indirect regularization method, by adding a constraint on the kernel weights we work directly on regularization by "forcing" the weights to remain below a certain limit. This method, combined with the previous one enables faster learning and considerably reduces the risk of overlearning [34].

<figure><img src="image_8.png" /><figcaption>Fig. 5. AntoNet structure: 4 encoding layers consisting of one convolution followed by a pooling operation and a batch normalisation (in the non-linear case) each. They are followed by 4 decoding layers separated by oversampling and concatenated with the encoded layers of the same dimension.</figcaption></figure>

## IV. RESULTS

We evaluate our approach with different metrics: loss function error, inference time, ROC curves and precision. We test our approach on 2,000 test sonar images with a SNR varying from -6 to 9 dB. We do it for Unet, ORCA [19], nonlinear AntoNet and linear AntoNet. These architectures are trained on a training dataset consisting on 20,000 sonar images with SNR varying between -10 dB and 10 dB.

Table II compares the models in terms of the number of parameters, the memory they occupy and the training/inference time. Our model, running in a RTX 4090 GPU, takes less than

<table><caption>TABLE II
COMPARISON OF MODEL SIZES, TRAINING AND INFERENCE TIME</caption><thead><tr><th>Model</th><th>Unet</th><th>ORCA</th><th>Lin. AntoNet</th><th>Non-Lin. AntoNet</th></tr></thead><tbody><tr><td>#Params</td><td>1.94M</td><td>57.14M</td><td>1.19k</td><td>1.29k</td></tr><tr><td>Memory</td><td>373MB</td><td>686MB</td><td>155kB</td><td>237kB</td></tr><tr><td>Training</td><td>350s</td><td>724s</td><td>7s</td><td>9s</td></tr><tr><td>Testing</td><td>40s</td><td>58s</td><td>900ms</td><td>1100ms</td></tr></tbody></table>

one second to test 2,000 images while Unet, resp. ORCA, takes 40, resp. 58, seconds. Applied on the same image, nonlinear AntoNet and linear AntoNet give similar visual contrast enhancement as shown in Fig. 6. Contrast-enhancing is better shown when we compare an horizontal slice of the image before and after the enhancement as shown in Fig. 7. The blue curve is the slide before the processing. The acoustic wave signal looks like a bump on the left. It is dominated by
<figure><img src="image_9.png" /><figcaption>Fig. 6. Non-linear AntoNet's (left) enhancement vs linear (right) enhancement applied on the sonar image on the left in Fig. 4.</figcaption></figure>

the noise spikes. The orange curve is the same slice after the enhancement. It is now very easy to distinguish the acoustic wave bump. The noise is clearly reduced.

<figure><img src="image_10.png" /><figcaption>Fig. 7. Input slide (in blue) vs AntoNet processed slice (in orange). The acoustic signal peak is visible at the left in the orange curve while it is hidden by the noise in the blue curve.</figcaption></figure>

Let us now discuss the detection results. The pixel-wise detector is a threshold detector that is considered as robust and efficient in our context. It is a good approximation, for low SNR regime, of the optimal quadratic detector that can be derived from the statistical test based on the hypotheses described in (6). ROC curves measure the detection rate as a function of the false alarm rate. To compute the errors, we applied the pixel-wise detector to each pixel $x_{t,j}$ of a sonar image (enhanced or not). We then compare this decision to the ground truth $y_{t,j}$ of the pixel: we consider that the pixel $x_{t,j}$ contains a signal if $y_{t,j} \neq 0$ and it does not contain any signal if $y_{t,j} = 0$. By collecting all the errors, we easily get the detection rate and the false alarm rate.

Fig. 8 shows more than 10 dB of loss gain between the input and the output. For example, a -5 dB enhanced image is processed with the same performance as a 5 dB unprocessed one. In other words, the detector applied after our contrast-enhancement performs with the same detection rate and false alarm rate than the same detector applied on the unprocessed image with a signal 10 dB stronger. Fig. 9 shows the precision

<figure><img src="image_11.png" /><figcaption>Fig. 8. ROC curves comparing the noisy input with AntoNet's output using a pixel-wise detector performance. Pd is the detection rate and Pfa is the false alarm rate.</figcaption></figure>

metric for our contrast-enhancing task. The precision metric is also computed from the errors computed above. Although Unet performs better, AntoNet gives good results. The linear version is even better than the non-linear one.

<figure><img src="image_12.png" /><figcaption>Fig. 9. Precision for different SNR: higher is better.</figcaption></figure>

Interestingly, our model is robust for signal estimation. Fig. 10 shows that linear AntoNet obtains the lower estimation loss.

<figure><img src="image_13.png" /><figcaption>Fig. 10. Test loss for different SNR: lower is better.</figcaption></figure>

V. CONCLUSION

We showed AntoNet's advantages over existing models:

* The reduced number of parameters and lightweight architecture allow for quick training and inference, making it practical for real-time applications.
* The use of linear operations and reduced complexity provide greater transparency into the model's working, facilitating better understanding and trust in its outputs.

* While keeping its simplicity, AntoNet performs competitively in contrast-enhancing tasks, particularly excelling in scenarios with low SNR.

One of the standout features of AntoNet is its computational efficiency. With fewer than 1,500 parameters, the model requires significantly less memory and computational power compared to traditional models like Unet and ORCA. This efficiency translates to faster training and inference times, making AntoNet highly suitable for real-time applications in sonar systems. Additionally, by utilizing linear operations and a simplified architecture, AntoNet provides a more transparent view of how the sonar image is transformed. This transparency is crucial for applications where understanding the model's decision-making process is important.

## REFERENCES

[1] Jonathan P. Towle, David Herold, Ryan Johnson, and Harold Vincent, "Low-cost acoustic sensors for littoral anti-submarine warfare (asw)." *Sensors, and Command, Control, Communications, and Intelligence (C3I) Technologies for Homeland Security and Homeland Defense VI*, vol. 6538, pp. 653814, 2007.

[2] Iksu Seo, Seongweon Kim, Youngwoo Ryu, Jungyong Park, and Dong Seog Han, "Underwater moving target classification using multilayer processing of active sonar system," *Applied Sciences*, vol. 9, no. 21, 2019.

[3] D. Pillon, M. Solal, and S. Brasseur, "Simultaneous detection and target motion analysis from conventional passive beamforming outputs," in *Acoustics, Speech, and Signal Processing, IEEE International Conference on*, Los Alamitos, CA, USA, apr 1991, pp. 1321–1324, IEEE Computer Society.

[4] S Sitbon and P Blanc-Benon, "Intégration dynamique en sonar passif: détection et localisation simultannées de sources faibles," *TS. Traitement du signal*, vol. 13, no. 5, pp. 449–458, 1996.

[5] Jerome Berclaz, Francois Fleuret, Engin Turetken, and Pascal Fua, "Multiple object tracking using k-shortest paths optimization," *IEEE Transactions on Pattern Analysis and Machine Intelligence*, vol. 33, no. 9, pp. 1806–1819, 2011.

[6] Wei Tian, Zhe Chen, Jie Shen, Fengchen Huang, and Lizhong Xu, "Underwater sonar image denoising through nonconvex total variation regularization and generalized kullback-leibler fidelity," *Journal of Ambient Intelligence and Humanized Computing*, vol. 13, no. 11, pp. 5237–5251, Nov 2022.

[7] S.G. Mallat, "A theory for multiresolution signal decomposition: The wavelet representation," *IEEE Transactions on Pattern Analysis and Machine Intelligence*, vol. 11, no. 7, pp. 674–693, 1989.

[8] Stephane G. Mallat, "Multiresolution approximations and wavelet orthonormal bases of $l^2(\mathbb{R})$." *Transactions of the American Mathematical Society*, vol. 315, pp. 69–87, 1989.

[9] K. Nawres, H. Kamel, and E. Noureddine, "Image denoising using wavelets: A powerful tool to overcome some limitations in nuclear imaging," in *2006 2nd International Conference on Information Communication Technologies*, 2006, vol. 1, pp. 1114–1118.

[10] John Canny, "A computational approach to edge detection," *IEEE Transactions on Pattern Analysis and Machine Intelligence*, vol. PAMI-8, no. 6, pp. 679–698, 1986.

[11] Rafael Grompone Von Gioi, Jeremie Jakubowicz, Jean-Michel Morel, and Gregory Randall, "Lsd: A fast line segment detector with a false detection control," *IEEE Transactions on Pattern Analysis and Machine Intelligence*, vol. 32, no. 4, pp. 722–732, 2008.

[12] David Bellamy, Leo Celi, and Andrew L. Beam, "Evaluating progress on machine learning for longitudinal electronic healthcare data," 2020.

[13] Juhwan Kim, Seokyong Song, and Son-Cheol Yu, "Denoising autoencoder based image enhancement for high resolution sonar image," in *2017 IEEE Underwater Technology (UT)*, 2017, pp. 1–5.

[14] Alexandru Isar, Sorin Moga, and Dorina Isar, "A new denoising system for sonar images," *EURASIP Journal on Image and Video Processing*, vol. 2009, pp. 1–14, 2009.

[15] Xingmei Wang, Qiming Li, Jingwei Yin, Xiao Han, and Wenqian Hao, "An adaptive denoising and detection approach for underwater sonar image," *Remote Sensing*, vol. 11, no. 4, 2019.

[16] Le Dong, Ling He, Mengdie Mao, Gaipeng Kong, Xi Wu, Qianni Zhang, Xiaochun Cao, and Ebroul Izquierdo, "Cunet: A compact unsupervised network for image classification," *IEEE Transactions on Multimedia*, vol. 20, no. 8, pp. 2012–2021, 2018.

[17] Fei Yuan, Fengqi Xiao, Kaihan Zhang, Yifan Huang, and En Cheng, "Noise reduction for sonar images by statistical analysis and fields of experts," *Journal of Visual Communication and Image Representation*, vol. 74, pp. 102995, 2021.

[18] Avi Abu and Roee Diamant, "Robust image denoising for sonar imagery," in *2018 OCEANS - MTS/IEEE Kobe Techno-Oceans (OTO)*, 2018, pp. 1–5.

[19] Christian Bergler, Manuel Schmitt, Andreas Maier, Simeon Smeele, Volker Barth, and Elmar Nöth, "Orca-clean: A deep denoising toolkit for killer whale communication.," in *INTERSPEECH*, 2020, pp. 1136–1140.

[20] N. Ofir and Y. Keller, "Multi-scale processing of noisy images using edge preservation losses," in *2020 25th International Conference on Pattern Recognition (ICPR)*, Los Alamitos, CA, USA, jan 2021, pp. 1–8, IEEE Computer Society.

[21] Saad Albawi, Tareq Abed Mohammed, and Saad Al-Zawi, "Understanding of a convolutional neural network," in *2017 International Conference on Engineering and Technology (ICET)*, 2017, pp. 1–6.

[22] Warren S McCulloch and Walter Pitts, "A logical calculus of the ideas immanent in nervous activity," *The bulletin of mathematical biophysics*, vol. 5, no. 4, pp. 115–133, 1943.

[23] Vinod Nair and Geoffrey E Hinton, "Rectified linear units improve restricted boltzmann machines," *Proceedings of the 27th International Conference on Machine Learning (ICML-10)*, pp. 807–814, 2010.

[24] Shikha Singh and Shruti Gupta, "Activation functions in neural networks," *Trends in Computer Science and Information Technology*, vol. 5, no. 2, pp. 56–67, 2020.

[25] Wei Chu and Abeer Alwan, "Reducing f0 frame error of f0 tracking algorithms under noisy conditions with an unvoiced/voiced classification frontend," in *2009 IEEE International Conference on Acoustics, Speech and Signal Processing*, 2009, pp. 3969–3972.

[26] Alexander M. Latham, Robert Pilawa-Podgurski, Kofi M. Odame, and Charles R. Sullivan, "Analysis and optimization of maximum power point tracking algorithms in the presence of noise," *IEEE Transactions on Power Electronics*, vol. 28, no. 7, pp. 3479–3494, 2013.

[27] B. Liao, Z. G. Zhang, and S. C. Chan, "A new robust kalman filter-based subspace tracking algorithm in an impulsive noise environment," *IEEE Transactions on Circuits and Systems II: Express Briefs*, vol. 57, no. 9, pp. 740–744, 2010.

[28] Thomas Mahiout, Lionel Fillatre, and Laurent Deruaz-Pepin, "Learning-based scattering transform for explainable classification," in *2022 30th European Signal Processing Conference (EUSIPCO)*, 08 2022, pp. 1402–1406.

[29] Yutao Hu, Xiaolong Jiang, Xuhui Liu, Xiaoyan Luo, Yao Hu, Xianbin Cao, Baochang Zhang, and Jun Zhang, "Hierarchical self-distilled feature learning for fine-grained visual categorization," *IEEE Transactions on Neural Networks and Learning Systems*, vol. PP, 2021.

[30] Francesco Cagnetta, Alessandro Favero, and Matthieu Wyart, "What can be learnt with wide convolutional neural networks?," in *Proceedings of the 40th International Conference on Machine Learning*. 2023, ICML'23, JMLR.org.

[31] Olaf Ronneberger, Philipp Fischer, and Thomas Brox, "U-net: Convolutional networks for biomedical image segmentation," *CoRR*, vol. abs/1505.04597, 2015.

[32] Ruirui Li, Wenjie Liu, Lei Yang, Shihao Sun, Wei Hu, Fan Zhang, and Wei Li, "Deepunet: A deep fully convolutional network for pixel-level sea-land segmentation," *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, vol. 11, no. 11, pp. 3954–3962, 2018.

[33] Hui Zou and Trevor Hastie, "Regularization and Variable Selection Via the Elastic Net," *Journal of the Royal Statistical Society Series B: Statistical Methodology*, vol. 67, no. 2, pp. 301–320, 03 2005.

[34] Harouna Soumare, Alia Benkahla, and Nabil Gmati, "Deep learning regularization techniques to genomics data," *Array*, vol. 11, pp. 100068, 2021.