SHARPENING (HIGHPASS) SPATIAL FILTERS

目录

• Laplacian
• UNSHARP MASKING AND HIGHBOOST FILTERING
• First-Order Derivatives
  • Roberts cross-gradient
  • Sobel operators

上一部分介绍的blur能够将图片模糊化, 这部分介绍的是突出图片的边缘的细节.

什么是边缘呢? 往往是像素点跳跃特别大的点, 这部分和梯度的概念是类似的, 可以如下定义图片的一阶导数而二阶导数:

$$\frac{\partial f}{\partial x} = f(x+1) - f(x), \\ \frac{\partial^2 f}{\partial x^2} = f(x+1) + f(x-1) - 2f(x).$$

注: 或许用差分来表述更为贴切.

如上图实例所示, 描述了密度值沿着$x$的变化, 一阶导数似乎能划分区域, 而二阶导数能够更好的“识别"边缘.

Laplacian

著名的laplacian算子:

$$\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2},$$

在digital image这里:

$$\Delta f = f(x+1， y) + f(x-1, y) + f(x, y+1) + f(x, y-1) - 4 f(x, y).$$

这个算子用kernel表示是下面的(a), 但是在实际中也有(b, c, d)的用法, (b, d)额外用到了对角的信息, 注意到这些kernels都满足

$$\sum_{ij}w_{ij} = 0.$$

最后

$$g(x, y) = f(x, y) + c[\nabla^2 f(x, y)],$$

$c=-1$, 如果a, b, $c=1$如果c, d.

kernel = -np.ones((3, 3))
kernel[1, 1] = 8
laps = cv2.filter2D(img, -1, kernel)
laps = (laps - laps.min()) / (laps.max() - laps.min()) * 255
img_pos = img + laps
img_neg = img - laps
fig, axes = plt.subplots(1, 4)
axes[0].imshow(img, cmap='gray')
axes[1].imshow(laps, cmap='gray')
axes[2].imshow(img_pos, cmap='gray')
axes[3].imshow(img_neg, cmap='gray')
plt.tight_layout()
plt.show()

kernel = np.ones((3, 3))
kernel[0, 0] = 0
kernel[0, 2] = 0
kernel[1, 1] = -4
kernel[2, 0] = 0
kernel[2, 2] = 0
laps = cv2.filter2D(img, -1, kernel)
laps = (laps - laps.min()) / (laps.max() - laps.min()) * 255
img_pos = img + laps
img_neg = img - laps
fig, axes = plt.subplots(1, 4)
axes[0].imshow(img, cmap='gray')
axes[1].imshow(laps, cmap='gray')
axes[2].imshow(img_pos, cmap='gray')
axes[3].imshow(img_neg, cmap='gray')
plt.tight_layout()
plt.show()

有点奇怪... 注意到我上面对laps进行标准化处理了, 如果没这个处理其实感觉是差不多的$c=1,-1$.

UNSHARP MASKING AND HIGHBOOST FILTERING

注意到, 之前的box kernel,

$$w_{box}(s, t) = \frac{1}{mn},$$

考虑$3 \times 3$的kernel size下:

$$w_{lap} = 9(E - \cdot w_{box}),$$

这里

$$E(s, t) =0, \forall s\not=2, t\not=2.$$

故假设

$$g_{mask} (x, y) = f(x, y) - \bar{f} (x, y),$$

其中$\bar{f}$是通过box filter 模糊的图像, 则

$$\Delta f = 9 \cdot g_{mask}.$$

故$g_{mask}$也反应了细节边缘信息.

进一步定义

$$g(x, y) = f(x, y) + k g_{mask}(x, y).$$

kernel = np.ones((3, 3)) / 9
img_mask = (img - cv2.filter2D(img, -1, kernel)) * 9
img_mask = (img_mask - img_mask.mean()) / (img_mask.max() - img_mask.min())
fig, ax = plt.subplots(1, 1)
ax.imshow(img_mask, cmap='gray')
plt.show()

First-Order Derivatives

最后再说说如何用一阶导数提取细节.

定义

$$M(x, y) = \|\nabla f\| = \sqrt{(\frac{\partial f}{\partial x})^2 + (\frac{\partial f}{\partial y})^2}.$$

注: 也常常用$M(x, y) = |\frac{\partial f}{\partial x}| + |\frac{\partial f}{\partial y}|$代替.

Roberts cross-gradient

把目标区域按照图(a)区分, Roberts cross-gradient采用如下方式定义:

$$\frac{\partial f}{\partial x} = z_9 - z_5, \: \frac{\partial f}{\partial y} = z_8 - z_6,$$

即右下角的对角之差. 所以相应的kernel变如图(b, c)所示(其余部分为0, $3 \times 3$).

注: 计算$M$需要两个kernel做两次卷积.

Sobel operators

Sobel operators 则是

$$\frac{\partial f}{\partial x} = (z_7 + 2z_8 + z_9) - (z_1 + 2z_2 + z_3) \\ \frac{\partial f}{\partial y} = (z_3 + 2z_6 + z_9) - (z_1 + 2z_4 + z_7),$$

即如图(d, e)所示.

kernel = np.zeros((3, 3))
kernel[1, 1] = -1
kernel[2, 2] = 1
part1 = cv2.filter2D(img, -1, kernel)
kernel = np.zeros((3, 3))
kernel[1, 2] = -1
kernel[2, 1] = 1
part2 = cv2.filter2D(img, -1, kernel)
img_roberts = np.sqrt(part1 ** 2 + part2 ** 2)

part1 = cv2.Sobel(img, -1, dx=1, dy=0, ksize=3)
part2 = cv2.Sobel(img, -1, dx=0, dy=1, ksize=3)
img_sobel = np.sqrt(part1 ** 2 + part2 ** 2)

fig, axes = plt.subplots(1, 2)
axes[0].imshow(img_roberts, cmap='gray')
axes[1].imshow(img_sobel, cmap='gray')