Title: The manifold structure of patches in images
Abstract: In this talk I will study the manifold structure of sets of patches extracted
from natural signals and images. For some simple datasets such as cartoon images or turbulent
textures, the local geometry of the signal can be described using a manifold parameterized
with a small number of parameters. I will begin my talk by a description of this manifold
structure for various ensembles suitable for natural signals, images and textures modeling.
I will consider two approaches to incorporate these local manifold constraints into a classical
signal and image processing pipeline, where the goal is to perform image restoration. The
first one imposes that the signal/image should be close to a curve/surface traced on the
feature manifold. This curve/surface should also match a given sets of measurements, in
order to solve some inverse problems. The second approach uses differential operators on
manifolds in order to regularize the signal/image viewed as a function defined on the feature
manifold. This point of view shed light on the success of non- local image denoising methods
such as NLmeans. This manifold analogy allows one to extend the NLmeans to a thresholding in
an orthogonal basis adapted to the image content.
I will show applications of these two approaches to denoising, inpainting, super-resolution,
compressive sampling reconstruction and texture synthesis.
11h30: Alessandro FOI (Department of Signal Processing, Tampere University of Technology, Finland)
Title: Removal of signal-dependent noise: the BM3D filter and optimized variance-stabilizing transformations
Abstract: While the vast majority of advanced denoising algorithms are designed, developed,
and optimized for the traditional additive i.i.d. Gaussian noise model, the noisy signals found
in applications very seldom can be described by such a simple model. Particularly in the case of
digital images, the noise is predominantly signal-dependent. Variance-stabilizing transformations
are commonly exploited in order to make exotic data easily tractable by standard methods. However,
for the most common families of distributions (e.g., Binomial, Poisson, etc.) exact stabilization
is not possible and even approximate stabilization turns out to be rather difficult to achieve. An
imprecise stabilization can seriously impair the effectiveness of the denoising. The talk is composed
of two parts. First, we review the Block-Matching and 3D filtering (BM3D) algorithm and its recent
modification with Shape-Adaptive Principal Component Analysis. This class of algorithms exploits
the nonlocal self-similarity of images by an adaptive 3D transform-domain shrinkage. It represents
the state-of-the-art for the removal of i.i.d. Gaussian noise. In the second and main part of the
talk, we consider the design of optimized nonparametric variance-stabilizing transformations. We
approach the variance stabilization problem as an explicit optimization problem and propose recursive
procedures to minimize a nonlinear stabilization functional that measures the discrepancy between
the standard deviation of the transformed variables and the fixed desired constant. We demonstrate
a number of optimized variance stabilizers for the most common distribution families. These stabilizers
are shown to outperform the existing stabilizers. In particular, variance-stabilizing transformations
optimized for low-count Poisson data are presented. The denoising of images from digital sensors is
used as an illustrative combined application of optimized transformations with the BM3D filter.
14h15: Julien MAIRAL (ENS/INRIA project-team WILLOW, Paris)
Title: Sparse learned representations for image restoration
Abstract: Sparse representations of signals have drawn considerable interest in recent years.
The assumption that natural signals, such as images, admit a sparse decomposition over a redundant
dictionary leads to efficient algorithms for handling such sources of data. In particular, the
design of well adapted dictionaries for images has been a major challenge. Recently, Elad and
Aharon have proposed to learn the dictionary for image denoising, which has shown to perform better
than using off-the-shelf ones. In this talk, we present the framework of dictionary learning
for various image and video processing tasks such has image and video inpainting or image
demosaicking as well as some recent achievements to make this task efficient in terms of memory
and computations.
