目录
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[
abla^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 imes 3)的kernel size下:
[w_{lap} = 9(E - cdot w_{box}),
]
这里
[E(s, t) =0, forall s
ot=2, t
ot=2.
]
故假设
[g_{mask} (x, y) = f(x, y) - ar{f} (x, y),
]
其中(ar{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) = |
abla 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 imes 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')
