Unite´ de recherche INRIA Lorraine, Technopoˆle de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LE` S NANCY Unite´ de recherche INRIA Rennes, Ir[r]

Abstract
Today’s robots are designed **for** humans, but are rarely deployed among humans. This thesis addresses problems of perception, planning, and safety that arise when deploying a mobile robot in human environments. A first key challenge is that of quickly navigating to a human-specified goal – one with known semantic type, but unknown coordinate – in a previously unseen world. This thesis formulates the con- textual scene understanding problem as an **image** translation problem, by learning to estimate the planning cost-to-go from aerial images of similar environments. The proposed perception algorithm is united with a motion planner to reduce the amount of exploration time before finding the goal. In dynamic human environments, pedes- trians also present several important technical challenges **for** the motion planning system. This thesis contributes a deep reinforcement learning-based (RL) formula- tion of the multiagent collision avoidance problem, with relaxed assumptions on the behavior model and number of agents in the environment. Benefits include strong performance among many nearby agents and the ability to accomplish long-term autonomy in pedestrian-rich environments. These and many other state-of-the-art robotics systems rely on Deep Neural Networks **for** perception and planning. How- ever, blindly applying deep learning in safety-critical domains, such as those involving humans, remains dangerous without formal guarantees on robustness. **For** example, small perturbations to sensor inputs are often enough to change network-based deci- sions. This thesis contributes an RL framework that is certified robust to uncertainties in the observation space.

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trix norms with p ≥ 1. The related convex constrained op- timization problems are solved through a novel epigraphical projection method. This formulation can be efficiently imple- mented thanks to the flexibility offered by recent primal-dual proximal **algorithms**. Experiments carried out **for** color images demonstrate the interest of considering a Non-Local Structure Tensor TV and show that the proposed epigraphical projec- tion method leads to significant improvements in terms of convergence speed over existing numerical solutions.

Stochastic approximation techniques have been used in var- ious contexts in data science. We propose a stochastic ver- sion of the forward-backward algorithm **for** minimizing the sum of two convex functions, one of which is not necessarily smooth. Our framework can handle stochastic approxima- tions of the gradient of the smooth function and allows **for** stochastic errors in the evaluation of the proximity operator of the nonsmooth function. The almost sure convergence of the iterates generated by the algorithm to a minimizer is es- tablished under relatively mild assumptions. We also propose a stochastic version of a popular primal-dual proximal split- ting algorithm, establish its convergence, and apply it to an online **image** **restoration** problem.

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ENL = µ 2 x /σ x 2 (6.24)
with µ 2
x the average intensity of the selected area and σ x 2 its variance. Larger ENL
values indicate stronger speckle rejection and, consequently, an improved ability to tell apart different gray levels. Tab. 6.4 reports the ENL values **for** the proposed and reference **algorithms** (we discard PPB and BM3D in the homomorphic context, by now). Results are quite consistent, indicating PPB 2 by far as the technique with the strongest speckle rejection ability, followed by MAP-S, SAR-BM3D and the oth- ers. On the other hand, this is immediately obvious by visual inspection of results, like those **for** the Rosen3 **image**, whose filtered versions are shown in Fig. 6.7 . The PPB **image** looks more pleasant than the others and is probably more helpful to gain a quick insight of the scene. On the downside, it presents widespread artifacts resembling watercolor strokes but, with neither the noiseless **image** nor an expert interpreter, it is difficult to decide whether this implies any loss of details. Some help comes from the analysis of ratio images obtained, as proposed in [ OQ04 ], as the pointwise ratio between the SAR original z and denoised ˆx images

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3. Numerical minimization. We focus in this section on the numerical approximation of a solution of problem ( 2.16 ). **For** that, we consider the associated discrete functional. As the functional is convex w.r.t. each variable separately, we use an alternating minimization algorithm. Each minimization step reduces to the minimization of a criterion composed of Poisson and regularizing terms. A lot of **algorithms** have been proposed in the literature **for** that purpose [ 18 , 36 , 23 , 17 , 44 , 11 ]. In this work, we use a scale gradient projection (SGP) algorithm proposed in [ 11 ] **for** marginal minimizations. The main advantage of this gradient descent algorithm is to propose a scaling strategy and a step-length updating rule deﬁned speciﬁcally **for** such a criterion in order to improve the speed of convergence. **For** the sake of compactness, we denote the global method SGPAM **for** scale gradient projection based alternating minimization. In this section, we ﬁrst introduce the discrete version of the considered problem and present the alternating minimization scheme. Then, we introduce the SGP method in its general form. Finally, we show how we can apply this algorithm to the **image** and PSF estimation problems.

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Contributions: The main contributions of this work are the following. We introduce three strategies to accelerate patch- based **image** **restoration** **algorithms** that use a GMM prior. We show that, when used jointly, they lead to a speed-up of the EPLL algorithm by two orders of magnitude. Compared to the popular BM3D algorithm, which represents the current state-of- the-art in terms of speed among CPU-based implementations, the proposed algorithm is almost an order of magnitude faster. The three strategies introduced in this work are general enough to be applied individually or in any combination to accelerate other related **algorithms**. **For** example, the random subsampling strategy is a general technique that could be reused in any algorithm that considers overlapping patches to process images; the flat tail spectrum approximation can accelerate any method that needs Gaussian log-likelihood or multiple Mahalanobis metric calculations; finally, the binary search tree **for** Gaussian matching can be included in any algorithm based on a GMM prior model and can be easily adapted **for** vector quantization techniques that use a dictionary.

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Given the plethora of **image** **restoration** **algorithms** available, the question arises as to their applicability **for** real ultrasound images. Assuming the shift-invariant convolution model is accurate, simulations provide a method of checking solutions against a ground truth. However, any limitations of the model may degrade real-life results. **For** instance, shift-variance of ℎ [6] may need to be accounted **for** by splitting the **image** into blocks, and even without shift-invariance, errors in estimating ℎ may lead to **restoration** errors. In addition, non-linear propagation and/or scattering will also invalidate the model, the latter often causing **image** degradation that is partially responsible **for** **image** clutter [7].

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In Section 2, we first review the proximal point method proposed in [16]. In Section 3, we apply the proximal point method presented in Section 2 to general ill-posed operator equations, that are particularized in Section 4 to several **image** **restoration** problems. Thus, we show that the proximal point method combined either with an a priori or with an a posteriori stopping rule provides stable approximation of the true **image**. In the introduction of Section 4, we briefly mention preliminary work and the related models in **image** processing that we consider in this paper. Furthermore, in Section 4.1, we present several **algorithms** **for** **image** deblurring with Gaussian, Laplace, or Poisson noise models with corresponding convex fidelity terms, and in Section 4.2 we extend the iterative idea to **image** **restoration** via cartoon + texture model. Finally, in Section 5, several numerical results are presented **for** each **image** **restoration** model. Comparisons with other methods of similar spirit or one-step gradient descent models are also presented.

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Nantes, France patrick.lecallet@univ-nantes.fr
Abstract—Learning-based black-box approaches have proven to be successful at several tasks in **image** and video processing domain. Many of these approaches depend on gradient-descent and back-propagation **algorithms** which requires to calculate the gradient of the loss function. However, many of the visual metrics are not differentiable, and despite their superior accuracy, they cannot be used to train neural networks **for** imaging tasks. Most of the **image** **restoration** neural networks rely on mean squared error to train. In this paper, we investigate visual system based metrics in order to provide perceptual loss functions that can replace mean squared error **for** gradient descent- based **algorithms**. We also share our preliminary results on the proposed approach.

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Given the plethora of **image** **restoration** **algorithms** available, the question arises as to their applicability **for** real ultrasound images. Assuming the shift-invariant convolution model is accurate, simulations provide a method of checking solutions against a ground truth. However, any limitations of the model may degrade real-life results. **For** instance, shift-variance of ℎ [6] may need to be accounted **for** by splitting the **image** into blocks, and even without shift-invariance, errors in estimating ℎ may lead to **restoration** errors. In addition, non-linear propagation and/or scattering will also invalidate the model, the latter often causing **image** degradation that is partially responsible **for** **image** clutter [7].

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This paper proposes accelerated subspace optimization methods in the context of **image** **restoration**. Subspace optimization methods belong to the class of iterative descent **algorithms** **for** unconstrained optimization. At each iteration of such methods, a stepsize vector allowing the best combination of several search directions is computed through a multi-dimensional search. It is usually obtained by an inner iterative second-order method ruled by a stopping criterion that guarantees the convergence of the outer algorithm. As an alternative, we propose an original multi-dimensional search strategy based on the majorize-minimize principle. It leads to a closed-form stepsize formula that ensures the convergence of the subspace algorithm whatever the number of inner iterations. The practical efficiency of the proposed scheme is illustrated in the context of edge-preserving **image** **restoration**.

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-functions and combination with the ARTUR algorithm [Charbonnier-96, Khoumri-97].
8.1 Principle
We assume as before an homogeneous regularization potential. We implement first the Metropolis updating schemes associated to respective posterior energies (51) and (52). Then we implement the generalized stochastic gradient **algorithms** as- sociated to stochastic equations (49) and (50). **For** each of the related stochastic gradient-like **algorithms**, we can attribute a different normalization constant (the Younes factor). However we choose the same one in all our experiments. We also decide that the convergence criterium should bring on hyperparameter only, since

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While our covariant re-fitting technique recovers the classical post re-fitting so- lution in specific cases, the proposed algorithm offers more stable solutions. Unlike the Lasso post re-fitting technique, ours does not require identifying a posteriori the support of the solution, i.e., the set of non-zero coefficients. In the same vein, it does not require identifying the jump locations of the aniso-TV solution. Since the Lasso or the aniso-TV solutions are usually obtained through iterative **algorithms**, stopped at a prescribed convergence accuracy, the support or jump numerical identification might be imprecise (all the more **for** ill-posed problems). Such erroneous support identifications lead to results that strongly deviate from the sought re-fitting. Our covariant re-fitting jointly estimates the re-enhanced solution during the iterations of the original algorithm and, as a by product, produces more robust solutions.

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It is well known that reducing bias is not always favorable in terms of mean square error because of the so-called bias-variance trade-off. It is important to highlight that a debiasing procedure is expected to re-inject part of the variance, therefore increasing the residual noise. Hence, the mean square error is not always expected to be improved by such techniques. Debiasing is nevertheless essential in applications where the **image** intensities have a physical sense and critical decisions are taken from their values. **For** instance, the authors of [7] suggest using **image** **restoration** techniques to estimate a temperature map within a tumor tissue **for** real time automatic surgical intervention. In such applications, it is so crucial that the estimated temperature is not biased. A remaining residual noise is indeed favorable compared to an uncontrolled bias.

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of the functional (5), the values of the parameters µ and ν are tuned by considering several numerical experiments [18, 30, 31]. It is often observed that the numerical **algorithms** are very sensitive to the chosen values of these parameters.
In this paper, we propose to address the **restoration**-segmentation problem without mixing these two objectives through arbitrary weights. To this end, a possible approach is to reformulate the problem as a Nash game. In [1], the authors consider the **image** **restoration** problem using Tikhonov regularization. This well-known method mixes two antagonistic objectives, the first one expresses the distance to the raw data, and the second one a penalized smoothing term. A Nash game approach was used which led to restored images -equilibria- much less sensitive to parameter tuning than the classical Tikhonov method is. Following the game terminology, we define two players: one is **restoration**, with the **image** intensity as strategy, and the other is segmentation with contours as strategy. Cost functions are the classical relevant ones **for** **restoration** and segmentation respectively. The two players play a static game with complete information, and we consider as solution to the game the so-called Nash Equilibrium.

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Figure 3.2 shows the resulting **image** u (`) computed by the HB [ 34 ] and the L- BFGS **algorithms** after ` = 25 iterations. The vertical stripes in the exemplar **image** v give rise to long-range dependencies between the wavelet coefficients, which require many iterations to be set up properly. L-BFGS is able to generate them within 25 iterations, while the original algorithm of [ 34 ] requires more iterations to create them. Let us finally note that while the L-BGFS approach brings some improvements with respect to the original HB algorithm, it suffers from the same drawback in term of texture modeling. Figure 3.3 shows two examples of syntheses obtained by the L- BFGS optimization method. This highlight the (well known) fact that methods based on first order statistics of multiscale decompositions are quite good at reproducing highly stochastic micro-textures (in the sense [ 27 ]) such as the exemplar on the left of the figure, but fail at generating geometric patterns, such as in the exemplar on the right. Synthesizing geometric features requires either to use higher order statistical constraints on the wavelet coefficients (see **for** instance [ 50 ]) or more complicated non-linear feature extractors, as exemplified in the next section.

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Considering applications such as compressive sensing, when the low sampling rate may lead to a bad initial guess of the weight function, some authors [37, 48] suggested to update the weights during the iterations to improve them. **For** exam- ple, Zhang, et al. [48] employed patches and up- dated the weights by recomputing the initial step (5) with the current estimate of the clean **image**. A similar update is used in [1, 37]. To trade off be- tween the accuracy and computational costs, the authors only reevaluate the weights every few steps [48] and in practical implementation, they only use the first few largest weights in a local search win- dow and set the rest of them to zero. Of course, because of these last ’tricks’, the **algorithms** may fail to find the optimal solution of the variational model which is not fully satisfactory. They however provide interesting **restoration** results.

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Index Terms— Bayesian inference, **image** **restoration**, Poisson
noise, split-and-augmented Gibbs sampler.
1. INTRODUCTION
Poisson noise can appear in a lot of **image** processing problems where observations are obtained through a count of particles (e.g. photons) emitted by the object of interest and arriving in the im- age plane where a detector (e.g. a charged coupled device (CCD) camera) is located [1]. **For** instance, this statistical property of the noise occurs in emission tomography (ET) [2], fluorescence mi- croscopy [3] or astronomy [4, 5]. In particular, a growing interest in restoring astronomical Poissonian images can be traced back at least to the Hubble Space Telescope optical aberration in the early 90’s. Methods to tackle such **image** **restoration** problems in these years were mainly based on Tikhonov-Miller inverse filter and maximum likelihood (ML) estimation via the expectation-maximization (EM) algorithm [2, 6, 7]. This is **for** instance the case **for** the classical Richardson-Lucy algorithm [8, 9] used and revisited in a lot of ap- plications [10, 11] since noise amplification tended to appear. We refer the interested reader to [12] **for** a review of Poissonian **image** **restoration** methods up to 2006. Since then, much research has been devoted to these problems and a lot of advances have been made in optimization-based methods. Among other works, [13] considered a forward-backward splitting algorithm using the Anscombe vari- ance stabilizing transform (VST) [14] while [15–17] used alternate minimization schemes in order to tackle the exact Poissonian like- lihood function. The optimization-based **algorithms** derived by the aforementioned authors appear to be efficient in various scenarios ranging from analysis to synthesis approaches [18] with different regularization functions (e.g. total variation (TV) [19] or ℓ1 norm) in low or high signal-to-noise ratio (SNR) cases.

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Figure 3.2 shows the resulting **image** u (ℓ) computed by the HB [34] and the L- BFGS **algorithms** after ℓ = 25 iterations. The vertical stripes in the exemplar **image** v give rise to long-range dependencies between the wavelet coefficients, which require many iterations to be set up properly. L-BFGS is able to generate them within 25 iterations, while the original algorithm of [34] requires more iterations to create them. Let us finally note that while the L-BGFS approach brings some improvements with respect to the original HB algorithm, it suffers from the same drawback in term of texture modeling. Figure 3.3 shows two examples of syntheses obtained by the L- BFGS optimization method. This highlight the (well known) fact that methods based on first order statistics of multiscale decompositions are quite good at reproducing highly stochastic micro-textures (in the sense [27]) such as the exemplar on the left of the figure, but fail at generating geometric patterns, such as in the exemplar on the right. Synthesizing geometric features requires either to use higher order statistical constraints on the wavelet coefficients (see **for** instance [50]) or more complicated non-linear feature extractors, as exemplified in the next section.

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