Section: Application Domains
Sparse signals & optimization
This research topic involves Geostat team and is used to set up an InnovationLab with I2S company
Sparsity can be used in many ways and there exist various sparse models in the literature; for instance minimizing the ${l}_{0}$ quasinorm is known to be an NPhard problem as one needs to try all the possible combinations of the signal's elements. The ${l}_{1}$ norm, which is the convex relation of the ${l}_{0}$ quasinorm results in a tractable optimization problem. The ${l}_{p}$ pseudonorms with $0<p<1$ are particularly interesting as they give closer approximation of ${l}_{0}$ but result in a nonconvex minimization problem. Thus, finding a global minimum for this kind of problem is not guaranteed. However, using a nonconvex penalty instead of the ${l}_{1}$ norm has been shown to improve significantly various sparsitybased applications. Nonconvexity has a lot of statistical implications in signal and image processing. Indeed, natural images tend to have a heavytailed (kurtotic) distribution in certain domains such as wavelets and gradients. Using the ${l}_{1}$ norm comes to consider a Laplacian distribution. More generally, the hyperLaplacian distribution is related to the ${l}_{p}$ pseudonorm ($0<p<1$) where the value of $p$ controls how the distribution is heavytailed. As the hyperLaplacian distribution for $0<p<1$ represents better the empirical distribution of the transformed images, it makes sense to use the ${l}_{p}$ pseudonorms instead of ${l}_{1}$. Other functions that better reflect heavytailed distributions of images have been used as well such as Studentt or Gaussian Scale Mixtures. The internal properties of natural images have helped researchers to push the sparsity principle further and develop highly efficient algorithms for restoration, representation and coding. Group sparsity is an extension of the sparsity principle where data is clustered into groups and each group is sparsified differently. More specifically, in many cases, it makes sense to follow a certain structure when sparsifying by forcing similar sets of points to be zeros or nonzeros simultaneously. This is typically true for natural images that represent coherent structures. The concept of group sparsity has been first used for simultaneously shrinking groups of wavelet coefficients because of the relations between wavelet basis elements. Lastly, there is a strong relationship between sparsity, nonpredictability and scale invariance.
We have shown that the two powerful concepts of sparsity and scale invariance can be exploited to design fast and efficient imaging algorithms. A general framework has been set up for using nonconvex sparsity by applying a firstorder approximation. When using a proximal solver to estimate a solution of a sparsitybased optimization problem, sparse terms are always separated in subproblems that take the form of a proximal operator. Estimating the proximal operator associated to a nonconvex term is thus the key component to use efficient solvers for nonconvex sparse optimization. Using this strategy, only the shrinkage operator changes and thus the solver has the same complexity for both the convex and nonconvex cases. While few previous works have also proposed to use nonconvex sparsity, their choice of the sparse penalty is rather limited to functions like the ${l}_{p}$ pseudonorm for certain values of $p\ge 0.5$ or the Minimax Concave (MC) penalty because they admit an analytical solution. Using a firstorder approximation only requires calculating the (super)gradient of the function, which makes it possible to use a wide range of penalties for sparse regularization. This is important in various applications where we need a flexible shrinkage function such as in edgeaware processing. Apart from nonconvexity, using a firstorder approximation makes it easier to verify the optimality condition of proximal operatorbased solvers via fixedpoint interpretation. Another problem that arises in various imaging applications but has attracted less works is the problem of multisparsity, when the minimization problem includes various sparse terms that can be nonconvex. This is typically the case when looking for a sparse solution in a certain domain while rejecting outliers in the datafitting term. By using one intermediate variable per sparse term, we show that proximalbased solvers can be efficient. We give a detailed study of the Alternating Direction Method of Multipliers (ADMM) solver for multisparsity and study its properties. The following subjects are addressed and receive new solutions:

Edge aware smoothing: given an input image $g$, one seeks a smooth image $u$ "close" to $g$ by minimizing:
$\underset{u}{\text{argmin}}\phantom{\rule{3.33333pt}{0ex}}{\displaystyle \frac{\lambda}{2}{\parallel ug\parallel}_{2}^{2}+\psi \left(\nabla u\right)}$where $\psi $ is a sparcityinducing nonconvex function and $\lambda $ a positive parameter. Splitting and alternate minimization lead to the subproblems:
$\begin{array}{ccc}\text{(sp1)}\hfill & :& {\displaystyle {v}^{(k+1)}\leftarrow \underset{v}{\text{argmin}}\phantom{\rule{3.33333pt}{0ex}}\psi \left(v\right)+\frac{\beta}{2}{\parallel \nabla {u}^{\left(k\right)}v\parallel}_{2}^{2}}\hfill \\ \text{(sp2)}\hfill & :& {\displaystyle {u}^{(k+1)}\leftarrow \underset{u}{\text{argmin}}{\phantom{\rule{3.33333pt}{0ex}}\lambda \parallel ug\parallel}_{2}^{2}+\beta {\parallel \nabla u{v}^{(k+1)}\parallel}_{2}^{2}.}\hfill \end{array}$We solve subproblem $\text{(sp2)}$ through deconvolution and efficient estimation via separable filters and warmstart initialization for fast GPU implementation, and subproblem $\text{(sp1)}$ through nonconvex proximal form.

Structuretexture separation: gesign of an efficient algorithm using nonconvex terms on both the datafitting and the prior. The resulting problem is solved via a combination of HalfQuadratic (HQ) and MaximizationMinimization (MM) methods. We extract challenging texture layers outperforming existing techniques while maintaining a low computational cost. Using spectral sparsity in the framework of lowrank estimation, we propose to use robust Principal Component Analysis (RPCA) to perform robust separation on multichannel images such as glare and artifacts removal of flash/noflash photographs. As in this case, the matrix to decompose has much less columns than lines, we propose to use a QR decomposition trick instead of a direct singular value decomposition (SVD) which makes the decomposition faster.

Robust integration: in many applications, we need to reconstruct an image from corrupted gradient fields. The corruption can take the form of outliers only when the vector field is the result of transformed gradient fields (lowlevel vision), or mixed outliers and noise when the field is estimated from corrupted measurements (surface reconstruction, gradient camera, Magnetic Resonance Imaging (MRI) compressed sensing, etc.). We use nonconvexity and multisparsity to build efficient integrability enforcement algorithms. We present two algorithms : 1) a local algorithm that uses sparsity in the gradient field as a prior together with a sparse datafitting term, 2) a nonlocal algorithm that uses sparsity in the spectral domain of nonlocal patches as a prior together with a sparse datafitting term. Both methods make use of a multisparse version of the HalfQuadratic solver. The proposed methods were the first in the literature to propose a sparse regularization to improve integration. Results produced with these methods significantly outperform previous works that use no regularization or simple ${l}_{1}$ minimization. Exact or nearexact recovery of surfaces is possible with the proposed methods from highly corrupted gradient fields with outliers.

Learning image denoising: deep convolutional networks that consist in extracting features by repeated convolutions with highpass filters and pooling/downsampling operators have shown to give nearhuman recognition rates. Training the filters of a multilayer network is costly and requires powerful machines. However, visualizing the first layers of the filters shows that they resemble wavelet filters, leading to sparse representations in each layer. We propose to use the concept of scale invariance of multifractals to extract invariant features on each sparse representation. We build a biLipschitz invariant descriptor based on the distribution of the singularities of the sparsified images in each layer. Combining the descriptors of each layer in one feature vector leads to a compact representation of a texture image that is invariant to various transformations. Using this descriptor that is efficient to calculate with learning techniques such as classifiers combination and artificially adding training data, we build a powerful texture recognition system that outperforms previous works on 3 challenging datasets. In fact, this system leads to quite close recognition rates compared to latest advanced deep nets while not requiring any filters training.