Side Window Filtering

发表在 CVPR 2019 Oral ^[1]

原文 pdf 链接

作者在知乎上的介绍

¶ Algorithm

give $I$ such that minimize cost function $E$.

where

$I = \frac{1}{N_n} \sum{\omega \cdot q}$
$E = || q - I ||_2^2$
possible windows: $S = {L, R, U, D, NW, NE, SW, SE}$
input of pixel: $q$, output of pixel: $I$
local window: $\Omega$, pixel in window $\Omega$: $q$, kernel_weight: $\omega$

¶ Background

¶ Edge-preserving filters

Two categories

¶ global optimization based algorithms

total variation (TV) algorithm
iterative shrinkage approach
relative total variation algorithm
weighted least squares algorithm

¶ local optimization based algorithms

bilateral filter 双边滤波器^[2]
bilateral filter accelerated versions
guided filter 导向滤波
guided filter extensions
rolling guidance filter
mutual structure joint filtering
curvature filter^[3]

¶ Filtering Fundamentals

¶ common assume

image is piecewise linear
approximate a pixel as the weighted average of its neighbor pixels over a local window
$$
I_i’ = \sum_{j \in \Omega_i} \omega_{ij} q_j
$$

¶ cost function

$$
E_i = { {|| I_i - I_i’ ||}_2 }^2
$$

$$
= (I_i - \sum_{j \in \Omega_i} \omega_{ij} q_j) ^ 2
$$

¶ trade off

manipulating the input image towards a desired target 去噪声
keeping it close to the original 保真

¶ Type of typical edges ^[4]

step edge

     ┌─────
     │
     │
     │
─────┘

ramp edge

          ┌─────
         /
       /
      /
─────┘

roof edge

    /\
   /  \
  /    \
 /      \
/        \

line edge

     ┌─┐
     │ │
     │ │
     │ │
─────┘ └─────

这些函数是连续但不可导的（考虑$|x|$）

¶ Anything

1.2. Problem and Motivation 部分用 $g(x, y)$ 和 Taylor expansion 来说明在边缘处，两侧点的取值差距较大。因此跳跃点两侧应当分开来进行考虑。

¶ Definition of side window

2. Side Window Filtering Technique 中对参数 $\varphi$ 的说明不足。如下图所示，论文中认为 $OD$ 是定长 $r$，矩形 $Q$ 是固定的；而 $OA$ 为变长 $\varphi$，矩形 $P$ 是可伸缩的
1
2
3
4
5
A┌─────┐B
│ P │
O├─────┤E
│ Q │
D└─────┘C
论文中认为 $OE$ 线条的宽度为 $1$ 像素，而其他线条宽度为 $0$

¶ I Don’t Understand

It is worth noting that optimization problem of the form similar
to eq. (2) is found in many applications including coloriza-
tion [14][22] and image segmentation [25][28], where the
weight functions are usually referred to as affinity functions.
Nonlinear approximation filtering such as median filtering
can also be formulated as a similar form of optimization
problem