of processing on the sphere rather than via a mapping into 2D ERP or CMP images.

<figure><img src="image_43.png" /><figcaption>Figure 6.5 – Mollweide projection of the (a) equirectangular image denoised by the DRUNet that was re-trained on ERP images and the (b) spherical image denoised by our SphereDRUNet. AWGN of σ = 30/255 was added on the original image. The zoomed versions of the red and green rectangular regions are shown in the bottom rows.</figcaption></figure>

6.4.1 Spherical convolution: OSLO-based vs. graph-based (Deep-Sphere)

Our hypothesis that omnidirectional image denoising is more efficient when performed
on the sphere is verified. One popular approach to operate directly on irregular topology
relies on graph signal processing. For the sake of completeness, we also want to prove that
the OSLO-based spherical convolution that we have chosen is the best approach. For this
purpose, we consider a graph-based baseline: the DeepSphere architecture [126]. Deep-
Sphere is also defined on the HEALPix sampling, but uses max-pooling and graph-based
<figure><img src="image_44.png" /><figcaption>Figure 6.6 – Mollweide projection of the (a) equirectangular image denoised by the DRUNet that was re-trained on ERP images and the (b) spherical image denoised by our SphereDRUNet. AWGN of σ = 30/255 was added on the original image. The zoomed versions of the red and green rectangular regions are shown in the bottom rows.</figcaption></figure>

convolutions approximated by Chebyshev polynomial formulation. We re-train the same SphereDRUNet architecture with graph-based convolutions in the same training framework as our OSLO-based SphereDRUNet. For computational reasons, both networks are trained and evaluated on HEALPix images of resolution 8 (786,432 pixels) in this case.

Figure 6.9 shows denoising results of spherical images restored with the OSLO-based SphereDRUNet and the graph-based SphereDRUNet. We can see that the graph-based convolution approach fails to recover the clean image from the noisy measurement. This limitation arises from the inherent nature of the graph, where the lack of directional information restricts the learning process to a single weight per neighborhood. By contrast, our OSLO-based approach demonstrates superior performance, successfully recovering the clean image from the noisy measurement. This highlights the importance of anisotropic filters and the effectiveness of the directional information embedded in the OSLO-based convolution, leading to enhanced denoising capabilities.
<figure><img src="image_45.png" /><figcaption>Figure 6.7 – Mollweide projection of the (a) equirectangular image denoised by the DRUNet that was re-trained on ERP images and the (b) spherical image denoised by our SphereDRUNet. AWGN of $\sigma = 50/255$ was added on the original image. The zoomed versions of the green rectangular region is shown in the bottom row.</figcaption></figure>

## 6.4.2 Ablation study: training a denoiser on the equirectangular projection

Mapping-based solutions are a widely adopted approach when it comes to spherical image processing. The reason is simple: 2D processing tools can be easily applied to these projections. However, traditional 2D operations (such as CNN tools) can not effectively process omnidirectional images, because these mappings display a significant distortion. In this section, we further demonstrate the limits of mapping-based learning. Table 6.3

<table><caption>Table 6.1 – Quantitative results of denoising equirectangular, cubemap and spherical data in terms of WS-PSNR [dB].</caption><thead><tr><th rowspan="2"></th><th colspan="5">Noise level σ*255</th></tr><tr><th>10</th><th>20</th><th>30</th><th>40</th><th>50</th></tr></thead><tbody><tr><td>Sphere + SphereDRUNet</td><td><strong>40.78</strong></td><td><strong>37.33</strong></td><td><strong>35.43</strong></td><td><strong>34.14</strong></td><td><strong>33.16</strong></td></tr><tr><td>ERP + DRUNet (trained on ERP)</td><td>39.85</td><td>35.38</td><td>32.46</td><td>30.15</td><td>28.16</td></tr><tr><td>CMP + DRUNet (trained on ERP)</td><td>30.30</td><td>29.65</td><td>28.73</td><td>27.61</td><td>26.38</td></tr></tbody></table>
<table><caption>Table 6.2 – Quantitative results of denoising equirectangular, cubemap and spherical data in terms of S-PSNR [dB].</caption><thead><tr><th rowspan="2"></th><th colspan="5">Noise level σ*255</th></tr><tr><th>10</th><th>20</th><th>30</th><th>40</th><th>50</th></tr></thead><tbody><tr><td>Sphere + SphereDRUNet</td><td><b>22.28</b></td><td><b>20.47</b></td><td><b>19.49</b></td><td><b>18.83</b></td><td><b>18.32</b></td></tr><tr><td>ERP + DRUNet (trained on ERP)</td><td>21.65</td><td>19.78</td><td>18.73</td><td>17.98</td><td>17.40</td></tr><tr><td>CMP + DRUNet (trained on ERP)</td><td>17.43</td><td>17.08</td><td>16.78</td><td>16.50</td><td>16.25</td></tr></tbody></table>

<figure><img src="image_46.png" /><figcaption>Figure 6.8 – Zoomed versions of (a) a portion of an equirectangular image and (b) its corresponding re-conversion from cubemap projection (i.e. the ERP image is converted to CMP, and re-converted back directly to ERP). Distortion and shifting are produced by the simple conversion between the mappings.</figcaption></figure>

shows the results (in terms of WS-PSNR [dB]) of denoising ERP images with the initial DRUNet network trained to denoise 2D images and the DRUNet that we re-trained on equirectangular projections. We observe that the performance of the DRUNet network does not greatly enhance when we re-train it over spherical projections compared to the initial DRUNet that was trained to denoise 2D images. For instance, the gain is only 0.09 dB for $\sigma = 50/255$, vs. a gain of 5.09 dB when we consider a processing on the sphere. This comparison further confirms that two-dimensional convolutions are not able to successfully learn spherical features. Due to their inherent flat nature, they cannot capture the curvatures and non-planar characteristics of spherical surfaces. Thus, 2D learning tools do not generalize well on 360 mappings in the task of denoising.
<figure><img src="image_47.png" /><figcaption>Figure 6.9 – Denoising results of spherical images restored with the OSLO-based Sphere-DRUNet (proposed network) and graph-based SphereDRUNet. Results are in terms of (a) WS-PSNR and (b) S-PSNR [dB].</figcaption></figure>

<table><caption>Table 6.3 – Quantitative results of denoising equirectangular images with the initial DRUNet network trained to denoise perspective 2D images and the DRUNet that we re-trained on ERP inputs. Results are in terms of WS-PSNR [dB].</caption><thead><tr><th rowspan="2"></th><th colspan="5">Noise level σ*255</th></tr><tr><th>10</th><th>20</th><th>30</th><th>40</th><th>50</th></tr></thead><tbody><tr><td>ERP + DRUNet (trained on ERP)</td><td><strong>39.85</strong></td><td><strong>35.38</strong></td><td><strong>32.46</strong></td><td><strong>30.15</strong></td><td><strong>28.16</strong></td></tr><tr><td>ERP + DRUNet (trained on 2D images)</td><td>39.53</td><td>35.19</td><td>32.33</td><td>30.03</td><td>28.07</td></tr><tr><td>Gain</td><td>0.32</td><td>0.19</td><td>0.13</td><td>0.12</td><td>0.09</td></tr></tbody></table>

## 6.5 Conclusion

In this last chapter, we have addressed the task of omnidirectional image denoising. Our study investigates the effectiveness of performing denoising directly on the sphere, as opposed to employing a projection. To address this, we have proposed a novel spherical denoising CNN (SphereDRUNet), by transferring a state-of-the-art Gaussian denoiser to the sphere, using the HEALPix spherical sampling and the OSLO-based convolution. Our network has shown promising denoising results on the sphere for different noise levels.

Our research demonstrates that denoising the spherical image gives superior results compared to denoising a corresponding mapping such as the equirectangular or the cube-map projections. This can be explained by the fact that a network needs less weights to recover the same high frequency components on the sphere compared to a projection.
Ultimately, this work does not only advance the field of omnidirectional image de-
noising, but also opens doors for solving other inverse problems. In fact, our proposed
spherical regularizer can be leveraged in a Plug-and-play algorithm in order to solve a
variety of inverse problems.
CONCLUSION

The landscape of inverse problems solutions has undergone different transformations over time, notably pivoting with the advent of deep learning. In particular, research in regularization techniques has evolved to embrace more effective approaches to address the ill-posed nature of inverse problems. In this thesis, we focused on contributing to the research in inverse problem regularization through deep learning-based methodologies.

We began by giving a detailed review on inverse problems in imaging in Chapter 2. We first introduced the problem of interest and illustrated different examples that we have addressed in this thesis. Then, we elaborated on existing approaches that have been developed to solve inverse problems, and concentrated on the literature of deep learning-based regularization methods. In particular, regularizing inverse problems by leveraging the power of denoisers in a Bayesian framework is the core of this thesis.

At the beginning of this thesis, the Deep Image Prior [1] was getting popular for solving inverse problems. The DIP, distinguished by its untrained generative network which only leverages its architecture along with the degraded image to tackle inverse problems, ignited our curiosity. We were motivated to explore the potential advantages of combining its strengths with those of a generic regularizer trained in a supervised manner on a large image dataset.

Chapter 3 presented our first contribution, focusing on regularizing the estimate generated by the Deep Image Prior [1] using the DRUNet [2] within an ADMM framework. Our method yielded improved results compared to DIP and other techniques employing hand-crafted regularizations with the DIP. The complexity of the parameter-tuning caused by the ADMM and the additional hyper-parameter of the denoiser inspired us to transfer the prior implicitly defined by a denoiser to a regularizing network, that can be used in a simple gradient descent algorithm.

Chapter 4 exposed our second contribution on a novel approach to train a regularizer that can be used in a Plug-and-play gradient descent algorithm. Based on the assumption that a denoiser represents the proximal operator of an underlying differentiable regular-
izer, we proved that there exists a relation between the denoiser and the gradient of the
corresponding regularizer. We proposed to train a network modeling the gradient of a
regularizer by using a pre-trained deep denoiser. We jointly trained both networks by also
updating the denoiser, in order for it to be consistent with the definition of a proximal
operator. We tested the efficiency of our approach by using our network in a Plug-and-
play gradient descent algorithm, and observed that it outperforms state-of-the-art generic
approaches. We also showed that our regularizing network also serves as a pre-trained
network in unrolled gradient descent.

Chapter 5 marked the beginning of the second part of this thesis, which focused on omnidirectional images. We gave a short review on processing tools and challenges for omnidirectional images, and discussed the existing literature on inverse problems.

As we delved into this second part, we observed a lack of comprehensive research
on omnidirectional image denoising. Given the significant role denoisers play in modern
regularization strategies and the importance of image denoising in image processing, we
devoted Chapter 6 to omnidirectional image denoising. Our contribution in this chapter
was two-fold: we introduced a spherical denoiser network designed and trained specifically
for omnidirectional images, which gave promising denoising performance for different noise
levels. We also showed that image denoising gives better results when working directly
on the sphere rather than when processing their underlying projections.

The contributions presented in this thesis allow several improvements and open perspectives for future work. For inverse problems in perspective imagery, we suggest:

* It would be interesting to compare the behaviour of the Deep Image Prior regularized by our regularizing network (ReG in chapter 4) in a gradient descent algorithm with the denoiser-based regularization that we propose in Chapter 3.

* One limitation of our regularizing network presented in Chapter 4 is, that we do not enforce the network to have a symmetric Jacobian matrix. Therefore, our training strategy does not guarantee that the network can be interpreted as a conservative vector field. One possible direction is to study whether such a constraint could improve the Plug-and-play gradient descent without sacrificing the computational complexity. This would be possible by using a network directly modeling the regularizer and explicitly computing its gradient with back-propagation.

For solving inverse problems in omnidirectional imagery, the following directions are
possible:
* The spherical denoiser introduced in Chapter 6 is a first step towards an omnidirectional Plug-and-play on the sphere. Although this can be easily done for some applications such as pixel-wise inpainting, the tasks of super-resolution and deblurring require further implementations as they require to perform convolution on the sphere. In fact, the current implementation of convolution in OSLO supports 1-hop neighborhood, and uses iterative computation of the 1-hop convolution with a proper aggregation method in order to increase the size of kernels. This works well for training, as the weights of the filters are learned and not pre-determined. One initial step would involve re-implementing the convolution to access a pixel's secondary neighbors, enabling the use of filters with higher dimensions. Then, we could implement Plug-and-play ADMM on the sphere to solve different inverse problems using our SphereDRUNet.

* We also suggest to extend the work presented in Chapter 4 to the sphere. A network modeling the gradient of a regularizer can be trained using the spherical denoising network introduced in Chapter 6, and then be used in a Plug-and-play gradient descent algorithm on the sphere.
<figure><img src="image_48.png" /></figure>
# PARAMETER SETTINGS OF CHAPTER 4

## A.1 Plug-and-play ADMM: formulation and parameter settings

While the Plug-and-play ADMM can provide high quality image reconstruction, the parameter setting is not trivial and strongly influences the results. We detail in this section how we have parameterized the Plug-and-play ADMM method for our comparisons. First, let us remind the ADMM equations for solving linear inverse problems as detailed in 2.4.2.1:

$$ \mathbf{x}^{k+1} = \underset{\mathbf{x}}{\operatorname{argmin}} \|\mathbf{A}\mathbf{x} - \mathbf{y}\|_2^2 + \rho^k \left\|\mathbf{x} - \left(\mathbf{z}^k - \frac{\mathbf{l}^k}{\rho^k}\right)\right\|_2^2, \quad (A.1) $$

$$ \mathbf{z}^{k+1} = \mathcal{D}_s(\mathbf{x}^{k+1} + \mathbf{l}^k / \rho), \quad (A.2) $$

$$ \mathbf{l}^{k+1} = \mathbf{l}^k + \rho(\mathbf{x}^{k+1} - \mathbf{z}^{k+1}), \quad (A.3) $$

where $l$ is the dual variable (typically zero-initialized), $\rho$ is the penalty parameter and $s = \sqrt{\sigma^2/\rho} = \sigma/\sqrt{\rho}$.

Note that $\sigma$ is a parameter of the problem to be solved and should not depend on the algorithm. As discussed in 4.4.1.1, $\sigma$ must be equal to the true noise level $\sigma_n$ added in the degradation. However, in the noiseless scenario, using $\sigma = \sigma_n = 0$ removes the regularization term. Hence similarly to our method, we choose a very small non-zero value in this case. For our experiments, we have used $\sigma = \max(\sigma_n, 0.001/255)$.

Note that the PnP-ADMM may thus parameterize the denoiser with a different standard deviation $s$ than the noise level $\sigma$ by setting the penalty parameter $\rho \neq 1$. In practice, the results of PnP-ADMM strongly depend on the setting of $\rho$. A common strategy in ADMM is to increase the value of $\rho$ at each iteration with an update of the form $\rho^{k+1} = \rho^k \cdot \alpha$ for some fixed parameter $\alpha \ge 1$. Hence, this requires setting two parameters $\rho_0$ and $\alpha$. A more intuitive parameterization typically used in the Plug-and-play context (e.g. [2], [146]) considers instead the denoising standard deviations $s^0$ and $s^N$ respectively at the first and last iterations. Knowing that $s = \sigma/\sqrt{\rho^k}$ at each iteration,
$\rho^0$ and $\alpha$ can be set accordingly as:

$$ \rho^0 = \left(\frac{\sigma}{s^0}\right)^2 \quad \text{and} \quad \alpha = \left(\frac{s^0}{s^N}\right)^{2/N} \qquad (A.4) $$

Therefore, the main parameters that we need to set in PnP-ADMM are the number of iterations $N$ and the initial and final noise standard deviation of the denoiser (i.e. $s^0$ and $s^N$). For our experiments, we have tested two configurations: the first configuration is similar to [2] and [146] with $s$ varying between a sufficiently high value $s^0$ and the true noise standard deviation $s^N = \sigma_n$ (or a small value when $\sigma_n = 0$). In the second configuration, $s$ is kept constant (i.e. $s^0 = s^N$). For both configurations, we have tuned the parameters to obtain the best results on the Set5 dataset, and we kept the best configuration. The parameters are given in Table A.1.

<table><caption>Table A.1 – Parameters used for the PnP-ADMM with the denoiser of [2]. $\sigma_n$: standard deviation of the AWGN added on the degraded image, $s^0$ noise standard deviation of the denoiser at the first iteration, $s^N$: noise standard deviation of the denoiser at the last iteration, $N$: number of iterations.</caption><thead><tr><th></th><th></th><th>σ<sub>n</sub> * 255</th><th>s<sup>0</sup> * 255</th><th>s<sup>N</sup> * 255</th><th>N</th></tr></thead><tbody><tr><td rowspan="6">Super-Resolution x2</td><td rowspan="3">Bicubic</td><td>0</td><td>50</td><td>0.1</td><td>25</td></tr><tr><td>√2</td><td>20</td><td>20</td><td>30</td></tr><tr><td>2.55</td><td>30</td><td>30</td><td>30</td></tr><tr><td rowspan="3">Gaussian</td><td>0</td><td>50</td><td>0.1</td><td>25</td></tr><tr><td>√2</td><td>30</td><td>30</td><td>30</td></tr><tr><td>2.55</td><td>45</td><td>45</td><td>30</td></tr><tr><td rowspan="6">Super-Resolution x3</td><td rowspan="3">Bicubic</td><td>0</td><td>50</td><td>0.1</td><td>40</td></tr><tr><td>√2</td><td>60</td><td>60</td><td>30</td></tr><tr><td>2.55</td><td>100</td><td>100</td><td>30</td></tr><tr><td rowspan="3">Gaussian</td><td>0</td><td>50</td><td>0.1</td><td>25</td></tr><tr><td>√2</td><td>30</td><td>30</td><td>30</td></tr><tr><td>2.55</td><td>45</td><td>45</td><td>30</td></tr><tr><td rowspan="6">Deblurring</td><td rowspan="3">[Image: Black and white dots]</td><td>√2</td><td>25</td><td>25</td><td>20</td></tr><tr><td>2.55</td><td>30</td><td>30</td><td>20</td></tr><tr><td>7.65</td><td>35</td><td>35</td><td>20</td></tr><tr><td rowspan="3">[Image: Black and white dots]</td><td>√2</td><td>25</td><td>25</td><td>30</td></tr><tr><td>2.55</td><td>30</td><td>30</td><td>30</td></tr><tr><td>7.65</td><td>35</td><td>35</td><td>30</td></tr><tr><td rowspan="2">Pixel-wise inpainting</td><td>0.1</td><td>0</td><td>255</td><td>1</td><td>118</td></tr><tr><td>0.2</td><td>0</td><td>255</td><td>1</td><td>200</td></tr></tbody></table>
## A.2 Parameter settings for the other methods

<table><caption>Table A.2 – Parameters used for the projection operator (One-Net) [105]. $\sigma_n$: standard deviation of the AWGN added on the degraded image, $\rho$: penalty parameter of the ADMM, $n$: number of iterations</caption><thead><tr><th></th><th></th><th>σ<sub>n</sub> * 255</th><th>ρ</th><th>n</th></tr></thead><tbody><tr><td rowspan="6">Super-Resolution x2</td><td rowspan="3">Bicubic</td><td>0</td><td>0.05</td><td>1</td></tr><tr><td>√2</td><td>0.06</td><td>12</td></tr><tr><td>2.55</td><td>0.06</td><td>12</td></tr><tr><td rowspan="3">Gaussian</td><td>0</td><td>0.02</td><td>1</td></tr><tr><td>√2</td><td>0.06</td><td>10</td></tr><tr><td>2.55</td><td>0.06</td><td>10</td></tr><tr><td rowspan="6">Super-Resolution x3</td><td rowspan="3">Bicubic</td><td>0</td><td>0.01</td><td>5</td></tr><tr><td>√2</td><td>0.06</td><td>10</td></tr><tr><td>2.55</td><td>0.05</td><td>12</td></tr><tr><td rowspan="3">Gaussian</td><td>0</td><td>0.01</td><td>5</td></tr><tr><td>√2</td><td>0.07</td><td>10</td></tr><tr><td>2.55</td><td>0.07</td><td>10</td></tr><tr><td rowspan="3">Deblurring</td><td>0</td><td>0.01</td><td>15</td></tr><tr><td>√2</td><td>0.02</td><td>15</td></tr><tr><td>2.55</td><td>0.02</td><td>5</td></tr><tr><td rowspan="2">Pixel-wise inpainting</td><td>0.1</td><td>0</td><td>0.004</td><td>300</td></tr><tr><td>0.2</td><td>0</td><td>0.01</td><td>250</td></tr></tbody></table>
<table><caption>Table A.3 – Parameters used for the RED [4] in a Gradient Descent framework using DRUNet. $\sigma_n$: standard deviation of the AWGN added on the degraded image, $w$: weight of the regularization, $\sigma_f$: noise standard deviation of the denoiser, $\mu$: gradient step size.</caption><thead><tr><th></th><th>σ<sub>n</sub> * 255</th><th>w</th><th>σ<sub>f</sub> * 255</th><th>μ</th></tr></thead><tbody><tr><td rowspan="3">Super-Resolution x2<br>(Bicubic and Gaussian)</td><td>0</td><td>0.005</td><td>7</td><td>0.08</td></tr><tr><td>√2</td><td>0.03</td><td>7</td><td>0.08</td></tr><tr><td>2.55</td><td>0.07</td><td>7</td><td>0.08</td></tr><tr><td rowspan="3">Super-Resolution x3<br>(Bicubic and Gaussian)</td><td>0</td><td>0.005</td><td>10</td><td>0.08</td></tr><tr><td>√2</td><td>0.01</td><td>10</td><td>0.08</td></tr><tr><td>2.55</td><td>0.03</td><td>10</td><td>0.08</td></tr><tr><td rowspan="6">Deblurring</td><td rowspan="3">●</td><td>√2</td><td>0.01</td><td>10</td><td>0.01</td></tr><tr><td>2.55</td><td>0.03</td><td>16</td><td>0.01</td></tr><tr><td>7.65</td><td>0.03</td><td>16</td><td>0.01</td></tr><tr><td rowspan="3">● ● ●</td><td>√2</td><td>0.01</td><td>16</td><td>0.01</td></tr><tr><td>2.55</td><td>0.01</td><td>16</td><td>0.01</td></tr><tr><td>7.65</td><td>0.03</td><td>16</td><td>0.01</td></tr><tr><td rowspan="2">Pixel-wise inpainting</td><td>0.1</td><td>0</td><td>0.001</td><td>27</td><td>0.006</td></tr><tr><td>0.2</td><td>0</td><td>0.001</td><td>24</td><td>0.008</td></tr></tbody></table>
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**Titre :** Apprentissage profond pour les problèmes inverses et application à l'imagerie omnidirectionnelle

**Mot clés :** problèmes inverses, régularisation, deep learning, images omnidirectionnelles, plug-and-play, débruitage

**Résumé :** Cette thèse est consacrée à contribuer à des solutions d'apprentissage profond pour régulariser des problèmes inverses en imagerie perspective et omnidirectionnelle. Dans la première partie, nous nous concentrons sur les images 2D et nous commençons par proposer de régulariser le Deep Image Prior avec le débriteur DRUNet de pointe. Cette combinaison améliore les performances du DIP et se compare favorablement avec les méthodes qui ont été proposées précédemment pour le régulariser. Ensuite, nous proposons une nouvelle approche pour entraîner un réseau modélisant le gradient d'un régulateur en utilisant un débriteur appris. Nous

utilisons ce réseau dans un algorithme de descente de gradient Plug-and-play et montrons qu'il surpasse les méthodes génériques existantes. Dans la deuxième partie de cette thèse, nous nous concentrons sur les images omnidirectionnelles et nous abordons le problème du débruitage, comme première étape vers des méthodes de régularisation basées sur le débriteur. Nous introduisons un nouveau débriteur sphérique, en transférant le débriteur DRUNet de pointe sur la sphère. Nous montrons également que le débruitage d'images omnidirectionnelles est plus efficace lorsqu'il est effectué directement sur la sphère plutôt que débruiter sa projection.

**Title:** Deep learning for inverse problems and application to omnidirectional imaging

**Keywords:** Inverse problems, regularization, deep learning, omnidirectional images, plug-and-play, denoising

**Abstract:** This thesis is devoted to contribute to deep learning solutions to regularize inverse problems in perspective and omnidirectional imagery. In the first part of this manuscript, we focus on perspective imagery and we begin by proposing to regularize the Deep Image Prior with the state-of-the-art DRUNet denoiser. This combination enhances the performance of the DIP and compares favourably with the methods that have been proposed previously to regularize it. Then, we propose a novel approach to train a network modeling the gradient of a regularizer by using a deep denoiser. We use this

network in a Plug-and-play gradient descent algorithm and show that it outperforms existing generic methods. In the second part of this thesis, we focus on omnidirectional images and we address the problem of omnidirectional image denoising, as a first step towards denoiser-based regularization methods. We introduce a novel spherical denoiser, by transferring the state-of-the-art DRUNet denoiser into the sphere. We also show that omnidirectional image denoising is more efficient when performed directly on the sphere rather than a corresponding mapping.