The process of using one image to guide the filtering process of another one
is called Guided Image Filtering (GIF). The main challenge of GIF is the
structure inconsistency between the guidance image and the target image.
Besides, noise in the target image is also a challenging issue especially when
it is heavy. In this paper, we propose a general framework for Robust Guided
Image Filtering (RGIF), which contains a data term and a smoothness term, to
solve the two issues mentioned above. The data term makes our model
simultaneously denoise the target image and perform GIF which is robust against
the heavy noise. The smoothness term is able to make use of the property of
both the guidance image and the target image which is robust against the
structure inconsistency. While the resulting model is highly non-convex, it can
be solved through the proposed Iteratively Re-weighted Least Squares (IRLS) in
an efficient manner. For challenging applications such as guided depth map
upsampling, we further develop a data-driven parameter optimization scheme to
properly determine the parameter in our model. This optimization scheme can
help to preserve small structures and sharp depth edges even for a large
upsampling factor (8x for example). Moreover, the specially designed structure
of the data term and the smoothness term makes our model perform well in
edge-preserving smoothing for single-image tasks (i.e., the guidance image is
the target image itself). This paper is an extension of our previous work [1],
[2].

本文提出了一种基于数据项和光滑项的 Robust Guided Image Filtering（RGIF）通用框架，旨在解决引导图像与目标图像之间的结构不一致性以及目标图像中噪声的问题。该模型可同时去噪目标图像和执行对噪声具有鲁棒性的 GIF 操作，并具有良好的边缘保持平滑处理效果。通过提出的迭代加权最小二乘法（IRLS）方法来解决高度非凸问题，进一步开发基于数据驱动的参数优化方案，以适当确定模型中参数。

稳健引导滤波

Robust Guided Image Filtering

Iteratively Re-weighted Least Squares (IRLS) is a method for solving
minimization problems involving non-quadratic cost functions, perhaps
non-convex and non-smooth, which however can be described as the infimum over a
family of quadratic functions. This transformation suggests an algorithmic
scheme that solves a sequence of quadratic problems to be tackled efficiently
by tools of numerical linear algebra. Its general scope and its usually simple
implementation, transforming the initial non-convex and non-smooth minimization
problem into a more familiar and easily solvable quadratic optimization
problem, make it a versatile algorithm. However, despite its simplicity,
versatility, and elegant analysis, the complexity of IRLS strongly depends on
the way the solution of the successive quadratic optimizations is addressed.
For the important special case of $\textit{compressed sensing}$ and sparse
recovery problems in signal processing, we investigate theoretically and
numerically how accurately one needs to solve the quadratic problems by means
of the $\textit{conjugate gradient}$ (CG) method in each iteration in order to
guarantee convergence. The use of the CG method may significantly speed-up the
numerical solution of the quadratic subproblems, in particular, when fast
matrix-vector multiplication (exploiting for instance the FFT) is available for
the matrix involved. In addition, we study convergence rates. Our modified IRLS
method outperforms state of the art first order methods such as Iterative Hard
Thresholding (IHT) or Fast Iterative Soft-Thresholding Algorithm (FISTA) in
many situations, especially in large dimensions. Moreover, IRLS is often able
to recover sparse vectors from fewer measurements than required for IHT and
FISTA.

该研究通过探讨压缩感知和稀疏恢复问题等特定领域中的迭代算法，证明了使用共轭梯度法来解决二次优化问题可以在保证收敛的同时显著提高其复杂度，并发现 IRLS 方法在大维度情况下可以优于 IHT 和 FISTA 等一阶方法，并且在所需测量 fewer 的情况下仍可以恢复稀疏向量。

迭代重新加权最小二乘方法的共轭梯度加速

Conjugate gradient acceleration of iteratively re-weighted least squares  methods

We propose a method for nonparametric density estimation that exhibits
robustness to contamination of the training sample. This method achieves
robustness by combining a traditional kernel density estimator (KDE) with ideas
from classical $M$-estimation. We interpret the KDE based on a radial, positive
semi-definite kernel as a sample mean in the associated reproducing kernel
Hilbert space. Since the sample mean is sensitive to outliers, we estimate it
robustly via $M$-estimation, yielding a robust kernel density estimator (RKDE).
An RKDE can be computed efficiently via a kernelized iteratively re-weighted
least squares (IRWLS) algorithm. Necessary and sufficient conditions are given
for kernelized IRWLS to converge to the global minimizer of the $M$-estimator
objective function. The robustness of the RKDE is demonstrated with a
representer theorem, the influence function, and experimental results for
density estimation and anomaly detection.

提出一种非参数密度估计算法，通过将传统的核密度估计器（KDE）与经典 $M$-estimation 的思想相结合，以及利用核化迭代加权最小二乘（IRWLS）算法对样本均值进行鲁棒性估计从而获得鲁棒 Kernel 密度估计器 (RKDE)，并给出了它的相关理论性质和试验结果。