We consider the analysis operator and synthesis dictionary learning problems
based on the the $\ell_1$ regularized sparse representation model. We reveal
the internal relations between the $\ell_1$-based analysis model and synthesis
model. We then introduce an approach to learn both analysis operator and
synthesis dictionary simultaneously by using a unified framework of bi-level
optimization. Our aim is to learn a meaningful operator (dictionary) such that
the minimum energy solution of the analysis (synthesis)-prior based model is as
close as possible to the ground-truth. We solve the bi-level optimization
problem using the implicit differentiation technique. Moreover, we demonstrate
the effectiveness of our leaning approach by applying the learned analysis
operator (dictionary) to the image denoising task and comparing its performance
with state-of-the-art methods. Under this unified framework, we can compare the
performance of the two types of priors.

采用双层优化统一框架，同时学习分析算子和合成字典，以尽可能接近真实值的方式来优化乘积数学模型，并将学习到的分析算子应用于图像去噪任务。

使用双层优化学习基于 L1 分析和合成稀疏先验

Learning $\ell_1$-based analysis and synthesis sparsity priors using  bi-level optimization

We consider the problem of learning a low-dimensional signal model from a
collection of training samples. The mainstream approach would be to learn an
overcomplete dictionary to provide good approximations of the training samples
using sparse synthesis coefficients. This famous sparse model has a less well
known counterpart, in analysis form, called the cosparse analysis model. In
this new model, signals are characterised by their parsimony in a transformed
domain using an overcomplete (linear) analysis operator. We propose to learn an
analysis operator from a training corpus using a constrained optimisation
framework based on L1 optimisation. The reason for introducing a constraint in
the optimisation framework is to exclude trivial solutions. Although there is
no final answer here for which constraint is the most relevant constraint, we
investigate some conventional constraints in the model adaptation field and use
the uniformly normalised tight frame (UNTF) for this purpose. We then derive a
practical learning algorithm, based on projected subgradients and
Douglas-Rachford splitting technique, and demonstrate its ability to robustly
recover a ground truth analysis operator, when provided with a clean training
set, of sufficient size. We also find an analysis operator for images, using
some noisy cosparse signals, which is indeed a more realistic experiment. As
the derived optimisation problem is not a convex program, we often find a local
minimum using such variational methods. Some local optimality conditions are
derived for two different settings, providing preliminary theoretical support
for the well-posedness of the learning problem under appropriate conditions.

本研究提出了基于 L1 优化的受限极小化学习方法，用于从训练语料库中学习一个分析算子，通过投影次梯度和 Douglas-Rachford 分割技术，对足够大小的干净训练集进行稳健回收地真实分析算子，同时在一些噪声 cosparse 信号的帮助下，为图像找到一个分析算子。

基于约束过完备分析算子学习的余稀信号建模

Constrained Overcomplete Analysis Operator Learning for Cosparse Signal  Modelling

Exploiting a priori known structural information lies at the core of many
image reconstruction methods that can be stated as inverse problems. The
synthesis model, which assumes that images can be decomposed into a linear
combination of very few atoms of some dictionary, is now a well established
tool for the design of image reconstruction algorithms. An interesting
alternative is the analysis model, where the signal is multiplied by an
analysis operator and the outcome is assumed to be the sparse. This approach
has only recently gained increasing interest. The quality of reconstruction
methods based on an analysis model severely depends on the right choice of the
suitable operator.
In this work, we present an algorithm for learning an analysis operator from
training images. Our method is based on an $\ell_p$-norm minimization on the
set of full rank matrices with normalized columns. We carefully introduce the
employed conjugate gradient method on manifolds, and explain the underlying
geometry of the constraints. Moreover, we compare our approach to
state-of-the-art methods for image denoising, inpainting, and single image
super-resolution. Our numerical results show competitive performance of our
general approach in all presented applications compared to the specialized
state-of-the-art techniques.

本文介绍了从训练图像中学习分析算子的算法，并将其应用于图像去噪、修补和超分辨率的领域，相对于专业技术上的最新方法具有竞争性的综合性能。