DIP_review_2_2 - Image reconstruction & Restoration

1. Noise

sinusoidal noise(在频率域中以共轭冲激串的形式出现)
- $f(x,y) = Asin(u_0x+v_0y)$
- $F(u,v) = \frac{A\pi}{j}[\delta(u-u_0,v-v_0) - \delta(u+u_0,v+v_0)]$
cosine noise
$\cdots$
适合用频率域滤波器解决，如Bandpass filter & Notch filter

Arithmetic mean filter
- $\hat{f}(x,y) = \frac{1}{mn}\sum_{(s,t)\in S_{xy}}g(s,t)$
Geometric mean filter
- $\hat{f}(x,y) = (\prod_{(s,t)\in S_{xy}}g(s,t))^{\frac{1}{mn}}$
- 适用于解决高斯/均匀噪声
Harmonic mean filter(谐波均值滤波器)
- $\hat{f}(x,y) = \frac{mn}{\sum_{(s,t)\in S_{xy}}\frac{1}{g(s,t)}}$
Contraharmonic mean filter(逆谐波均值滤波器)
- $\hat{f}(x,y) = \frac{\sum_{(s,t)\in S_{xy}}g(s,t)^{Q+1}}{\sum_{(s,t)\in S_{xy}}g(s,t)^{Q}}$
- 当 $Q = -1$ , 此时的逆谐波均值滤波器等于谐波均值滤波器
- 当 $Q > 0$ , 适用于处理pepper noise即黑噪声
- 当 $Q < 0$ , 适用于处理salt noise即白噪声

Median filter
- $\hat{f}(x,y) = median_{(s,t)\in S_{xy}}{g(s,t)}$
- 适用于处理比例不太大， $<= 0.2$ 的salt and pepper noise
Max filter
- $\hat{f}(x,y) = max_{(s,t)\in S_{xy}}{g(s,t)}$
- 适用于处理pepper noise
Min filter
- $\hat{f}(x,y) = min_{(s,t)\in S_{xy}}{g(s,t)}$
- 适用于处理salt noise
Midpoint filter
- $\hat{f}(x,y) = \frac{1}{2}[min_{(s,t)\in S_{xy}}{g(s,t)}+max_{(s,t)\in S_{xy}}{g(s,t)}]$
Alpha-trimmed mean filter
- $\hat{f}(x,y) = \frac{1}{mn-d}\sum_{(s,t)\in S_{xy}}{g_r(s,t)}$
- 本质是舍弃patch中前后各 $\frac{d}{2}$ ,因此适用于处理椒盐噪声

Adaptive Median filter(自适应中值滤波器)
- 第一部分为寻找合适的patch以及对应的 $z_{med}$ 。当找到合适的 $z_{med}$ 在该patch中,那么进入第二部分。否则扩大patch的size直至阈值去寻找。若大于阈值时仍未找到，说明该部分无法improve，输出等于最大值或是最小值的 $z_{med}$
- 第二部分为寻找某一点的最好输出。若该点的original pixel value就处于正常范围内则直接输出，否则输出 $z_{med}$
Adaptive, Local Noise-reduction filter 局部噪声抑制滤波器
- Suppose we already know
  - $\hat{I}(x,y):$ 退化图像
  - $\sigma_y^2:$ 整个图像的噪声方差
  - $\hat{\mu}_L(x,y):$ 局部均值
  - $\hat{\sigma}_L(x,y):$ 局部方差
- Principle:
  - $\hat{\hat}(x,y) = \hat{I}(x,y) - \frac{\sigma_y^2}{\hat{\sigma}_L^2}(\hat{I}(x,y) - \hat{\mu}_L)$
  - If $\sigma_y^2 = 0$ , then $\hat{\hat}(x,y) = \hat{I}(x,y)$ . 说明噪声方差在全图为0，直接等于原图即可
  - If $ \frac{\sigma_y^2}{\hat{\sigma}_L^2} \approx 1$, then $\hat{\hat}(x,y) = \hat{\mu}_L(x,y)$ . 说明局部的方差与噪声相似，用均值替换来消除噪声
  - If $\sigma_y^2 <<\hat{\sigma}_L^2$ , then $\hat{\hat}(x,y) = \hat{I}(x,y)$ . 说明局部的方差大，特征更重要，比如edge，因此需要保留局部特征

step 1: Notch pass filter 扣出噪声
- $\eta(x,y) = F^{-1}\{H_{NP}(u,v)G(u,v)\}$
step 2: 再减去噪声*最优系数
- $\hat{f}(x,y) = g(x,y) - w(x,y)\eta(x,y)$
step 3: 估计local方差
- $\sigma^2(x,y) = \frac{1}{(2a+1)(2b+1)}\sum\sum[\hat{f}(x+s,y+t) - \overline{\hat{f}}(x,y)]$
step 4: 最小化方差，找到最优系数 $w(x,y)$ $w (x, y)$
- $\frac{\partial \sigma^2(x,y)}{\partial w(x,y)} = 0$
- $w(x,y) = \frac{\overline{g(x,y)\eta(x,y)} - \overline{g}(x,y) \overline{\eta}(x,y)}{\overline{\eta^2}(x,y)-\overline{\eta}^2(x,y)}$