Category Archives: image processing

Axis 4: Sparse, non-local, patch-based approaches and harmonic analysis

This research axis is motivated by the problems of data analysis in high dimension (signals and images) organized in the form of vectors or matrices of large size.

The first part of this axis focuses on sparse-based approaches for the analysis of high-dimensional data when they can be well approximated in a space of low dimension. We are interested in the development of variational models which naturally promote this sparsity to a preselected representation (wavelets, gradient fields, etc.) or learned on an external dataset. This representation (the space of small size) can also be selected by the model itself, it is called dictionary learning which is usually expressed as a matrix factorization problem. For examples, we focus on approximation methods based on thresholding wavelets in image processing, as well as variable selection problems with the Lasso (Least Absolute Shrinkage and Selection Operator). Beyond the development of such models, we are seeking to understand their behavior. In particular, in the first example, we are interested in conditions ensuring that a signal can be recovered, or that its reconstruction error can be bounded (robustness). In the second example, we focus on the conditions ensuring the identification of good variables.

A second part concerns the analysis of repeating patterns in the case of images or multidimensional signals. These repetitions indeed provide key information for the processing or interpretation of the signal. In particular, we are interested in non-local approaches used in image processing that rely on the similarity of patterns using patches (typically small 8 by 8 rectangular windows). The term non-local means here that only the content of the patches is relevant, regardless of the spatial location. Part of our work involves the design and choice of metrics ensuring a robust matching of patterns as well as the implementation of efficient search algorithms. We are also interested in the analysis of these repetitions, typically on the connectivity graph. Finally, we aim at developing and studying new variational models taking into account these non-local information.

Axis 3: Guiding therapy imaging

These activities fall within the scope of interventional radiology, a booming discipline, for which France ranks in the forefront in the world. The possibility of depositing an energy locally and non-invasively opens new perspectives for more reliable therapeutic strategies of malignant tumors, less aggressive for the patients, which will allow a reduction of time and costs of hospitalization. In this context, an activity of the IOP team is oriented around image processing methods in real time, allowing the guiding imaging (ultrasound or MRI) of a mini or non-invasive treatment (thermal ablation or radiotherapy), also allowing local deposition of medications.

Axis 2: Optimal transport for image and signal processing

This axis focuses on the development of new methodologies for large data analysis such as histograms, images, or point clouds, based on concepts from the optimal transport theory. This methodology results in the use of non-Euclidean metric (such as Wasserstein distances) to extract the geometrical information in the presence of non-linear sources of variability in the data. In this context, a new method of Principal Component Analysis based on the Wasserstein distance has recently been proposed with applications to statistical analysis of histograms.

The use of optimal transport has also been proposed for various image processing problems. By generalizing transport distances by regularizing the associated transport plans, new image interpolation methods were developed for applications in oceanography. The Wasserstein distance was also considered to more traditional problems such as image segmentation or color transfer.

Axis 1: Variational methods and regularization

Variational methods are widely used in image processing. They allow us to propose models taking into account specificities of the tackled problems. They also enable the study of the properties of solutions. The proposed functional in image processing are not smooth (to account for the presence of interfaces) and not necessarily convex. This naturally raises questions of existence of solutions, uniqueness, and (fast) computation. The choice of the regularization is often based on an assumption of sparsity (eg, of the gradient, or in a transformed domain). Non-local interactions and the concept of patch are typically involved in the design of functional. Finally, all these approaches are based in principle on the weighting of the data fidelity term and the regularization term, which leads to the question of the reliable estimation of this parameter.

Estimation of Kullback-Leibler losses

We address the question of estimating Kullback-Leibler losses rather than squared losses in recovery problems where the noise is distributed within the exponential family. We exhibit conditions under which these losses can be unbiasedly estimated or estimated with a controlled bias. Simulations on parameter selection problems in image denoising applications with Gamma and Poisson noises illustrate the interest of Kullback-Leibler losses and the proposed estimators.

Preprint available here

Convergence of FISTA

Charles Dossal proved in 2015 with Antonin Chambolle the convergence of an extremely popular algorithm since its introduction in 2008 by Beck and Teboulle (for being extremely efficient in practice), the FISTA algorithm (Fast Iterative Shrinkage Algorithm). The convergence of the algorithm has remained an open problem in the communities of optimization/compressed sensing/image processing/machine learning for 7 years until Charles Dossal work.

Non parametric noise estimation

In order to provide a fully automatic denoising algorithm, we have developed an automatic noise estimation method that relies on the non-parametric detection of homogeneous areas. First, the homogeneous regions of the image are detected by computing Kendall’s rank correlation coefficient [1]. Computed on neighboring pixel sequences, it indicates the dependancy between neighbors, hence reflects the presence of structure inside an image block. This test is non-parametric, so the performance of the detection is independant of the noise statistical distribution. Once the homogeneous areas are detected, the noise level function, i.e., the function of the noise variance with respect to the image intensities, is estimated as a second order polynomial minimizing the \ell^1 error on the statistics of these regions.

Matlab implementation of the noise estimation algorithm

Related papers:

– C. Sutour, C.-A. Deledalle et J.-F. Aujol. Estimation of the noise level function based on a non-parametric detection of homogeneous image regions. Submitted to Siam Journal on Imaging Sciences, 2015.

– C. Sutour, C.-A. Deledalle et J.-F. Aujol. Estimation du niveau de bruit par la détection non paramétrique de zones homogènes. Submitted to Gretsi, 2015.

References

[1] Buades, A., Coll, B., and Morel, J.-M. (2005). A review of image denoising algorithms, with a new one. Multiscale Modeling and Simulation, 4(2): 490–530.

Adaptive regularization of NL-means

The denoising algorithm that has been developed is based on an adaptive regularization of the NL-means [1]. The proposed model is the following:

(1)   \begin{align*} u_{\text{TVNL}} &= \underset{u \in \mathbb{R}^N}{\operatorname{argmin}} \sum_{i \in \Omega} \lambda_i \left(u_i-u^{\text{NL}}_i\right)^2 + \text{TV}(u),\\ \lambda_i &= \gamma \left(\frac{\sigma_{\text{residual}}(i)}{\sigma_{\text{noise}}(i)}\right)^{-1} = \gamma \Big(\sum_j w_{i,j}^2\Big)^{-1/2}. \end{align*}

where u_{\NL} is the solution obtained with the NL-means algorithm, TV refers to the total variation of the image and w_{i,j} is the weight that measures the similarity between the patch of index i and the patch of index j in the NL-means algorithm. The ratio \left(\frac{\sigma_{\text{residual}}(i)}{\sigma_{\text{noise}}(i)}\right)^{-1} reflects the noise variance reduction performed by the NL-means. This formulation allows locally adaptive regularization of the NL-means solution u_{\NL}, thanks to a confidence index \lambda_i that reflects the quality of the denoising performed by the NL-means.

This model can be adapted to the different noise statistics belonging to the exponential family (Gaussian, Poisson, multiplicative…). It can also be adapted to video denoising thanks to the use of 3D patches combined to a spatio-temporal TV regularization.

Matlab implementation of RNL

Results of video denoising with R-NL and comparisons

Related papers:
1. C. Sutour, C.-A. Deledalle et J.-F. Aujol. Adaptive regularization of the NL-means : Application to image and video denoising. IEEE Transactions on image processing, vol. 23(8) : 3506-3521, 2014.

2. C. Sutour, J.-F. Aujol, C.-A. Deledalle et J.-P. Domenger. Adaptive regularization of the NL-means for video denoising. International Conference on Image Processing (ICIP), pages 2704–2708. IEEE, 2014.

3. C. Sutour, J.-F. Aujol et C.-A. Deledalle. TV-NL : Une coopération entre les NL-means et les méthodes variationnelles. Gretsi, 2013.

References

[1] Buades, A., Coll, B., and Morel, J.-M. (2005). A review of image denoising algorithms, with a new one. Multiscale Modeling and Simulation, 4(2): 490–530.