JoshWindsor 发表于 2022-4-27 14:23

【图像处理】轻松搞懂图像锐化

图像锐化的目的是使模糊图像变清晰，方法可以大致分为两类：微分法、高频加重滤波法。其中微分法可以分为梯度法、Sobel算子法、Laplace算子法。
在做图像锐化时要注意，处理的图像必须有较高的信噪比，否则锐化会使噪声收到比信号还强的增强，一般先降噪后锐化。
微分法

微分运算是求信号的变化率，加强高频分量，使图像轮廓清晰。希望通过导数运算使边缘和轮廓变清晰，这就要求该导数运算是各向同性的（旋转不变性），可以证明偏导数的平方和运算具有此性质。

[*]梯度法
对于图像 https://www.zhihu.com/equation?tex=f%28x%2Cy%29 ,在点 https://www.zhihu.com/equation?tex=%28x%2Cy%29 处的梯度是一个矢量，定义为https://www.zhihu.com/equation?tex=+G%5Cleft%5B+f%5Cleft%28+x%2Cy+%5Cright%29+%5Cright%5D+%3D%5Cleft%5B+%5Cbegin%7Barray%7D%7Bc%7D+%09%5Cfrac%7B%5Cpartial+f%7D%7B%5Cpartial+x%7D%5C%5C+%09%5Cfrac%7B%5Cpartial+f%7D%7B%5Cpartial+y%7D%5C%5C+%5Cend%7Barray%7D+%5Cright%5D+++。
梯度的幅度表示为 https://www.zhihu.com/equation?tex=G%5Bf%28x%2Cy%29%5D%3D%5B%28%5Cfrac%7B%5Cpartial+f%7D%7B%5Cpartial+x%7D%29%5E2%2B%28%5Cfrac%7B%5Cpartial+f%7D%7B%5Cpartial+y%7D%29%5E2%5D%5E%7B1%2F2%7D
对于数字图像而言， https://www.zhihu.com/equation?tex=+G%5Cleft%5B+f%5Cleft%28+x%2Cy+%5Cright%29+%5Cright%5D+%3D%5Cleft%5C%7B+%5Cleft%5B+f%5Cleft%28+i%2Cj+%5Cright%29+-f%5Cleft%28+i%2B1%2Cj+%5Cright%29+%5Cright%5D+%5E2%2B%5Cleft%5B+f%5Cleft%28+i%2Cj+%5Cright%29+-f%5Cleft%28+i%2Cj%2B1+%5Cright%29+%5Cright%5D+%5E2+%5Cright%5C%7D+%5E%7B1%2F2%7D+ ，
该式可以简化成 https://www.zhihu.com/equation?tex=+G%5Cleft%5B+f%5Cleft%28+x%2Cy+%5Cright%29+%5Cright%5D+%3D%5Cleft%7C+f%5Cleft%28+i%2Cj+%5Cright%29+-f%5Cleft%28+i%2B1%2Cj+%5Cright%29+%5Cright%7C%2B%5Cleft%7C+f%5Cleft%28+i%2Cj+%5Cright%29+-f%5Cleft%28+i%2Cj%2B1+%5Cright%29+%5Cright%7C+



各个像素的位置

图像的最后一行（列）的梯度无法求得，一般用前一行（列）的梯度代替。
当梯度计算完之后，可以根据需要生成不同的梯度增强图像，
1）第一种， https://www.zhihu.com/equation?tex=+g%5Cleft%28+x%2Cy+%5Cright%29+%3DG%5Cleft%5B+f%5Cleft%28+x%2Cy+%5Cright%29+%5Cright%5D+++ ，只显示灰度变化大的边缘轮廓，灰度变化平缓的呈黑色。
2）第二种， https://www.zhihu.com/equation?tex=g%5Cleft%28+x%2Cy+%5Cright%29+%3D%5Cleft%5C%7B+%5Cbegin%7Barray%7D%7Bc%7D+%09G%5Cleft%5B+f%5Cleft%28+x%2Cy+%5Cright%29+%5Cright%5D+%26%2CG%5Cleft%5B+f%5Cleft%28+x%2Cy+%5Cright%29+%5Cright%5D+%5Cge+thresh%5C%5C+%09f%5Cleft%28+x%2Cy+%5Cright%29+%26%2C+otherwise%5C%5C+%5Cend%7Barray%7D+%5Cright.+
可以显示出非常明显的边缘轮廓，又不会破坏原灰度变化平缓的背景。
3）第三种， https://www.zhihu.com/equation?tex=+g%5Cleft%28+x%2Cy+%5Cright%29+%3D%5Cleft%5C%7B+%5Cbegin%7Bmatrix%7D+%09L_G%26%09%09%2CG%5Cleft%5B+f%5Cleft%28+x%2Cy+%5Cright%29+%5Cright%5D+%5Cge+thresh%5C%5C+%09f%5Cleft%28+x%2Cy+%5Cright%29%26%09%09%2Cotherwise%5C%5C+%5Cend%7Bmatrix%7D+%5Cright.+++
边缘轮廓用固定的灰度级 https://www.zhihu.com/equation?tex=L_G+ 表示。
4）第四种， https://www.zhihu.com/equation?tex=+g%5Cleft%28+x%2Cy+%5Cright%29+%3D%5Cleft%5C%7B+%5Cbegin%7Bmatrix%7D+%09G%5Cleft%5B+f%5Cleft%28+x%2Cy+%5Cright%29+%5Cright%5D%26%09%09%2CG%5Cleft%5B+f%5Cleft%28+x%2Cy+%5Cright%29+%5Cright%5D+%5Cge+thresh%5C%5C+%09L_G%26%09%09%2Cotherwise%5C%5C+%5Cend%7Bmatrix%7D+%5Cright.+++
背景用固定灰度级 https://www.zhihu.com/equation?tex=L_G 表示，便于研究边缘灰度的变化。
5）第五种， https://www.zhihu.com/equation?tex=+g%5Cleft%28+x%2Cy+%5Cright%29+%3D%5Cleft%5C%7B+%5Cbegin%7Bmatrix%7D+%09L_G%26%09%09%2CG%5Cleft%5B+f%5Cleft%28+x%2Cy+%5Cright%29+%5Cright%5D+%5Cge+thresh%5C%5C+%09L_B%26%09%09%2Cotherwise%5C%5C+%5Cend%7Bmatrix%7D+%5Cright.+++
将背景和边缘用二值图像表示，便于研究边缘所在位置。

2. Sobel算子
采用梯度微分锐化图像，会让噪声、条纹得到增强，Sobel算子在一定程度上解决了这个问题， https://www.zhihu.com/equation?tex=+S_x%3D%5Cleft%5B+f%5Cleft%28+i%2B1%2Cj-1+%5Cright%29+%2B2f%5Cleft%28+i%2B1%2Cj+%5Cright%29+%2Bf%5Cleft%28+i%2B1%2Cj%2B1+%5Cright%29+%5Cright%5D++%5C%5C+-%5Cleft%5B+f%5Cleft%28+i-1%2Cj-1+%5Cright%29+%2B2f%5Cleft%28+i-1%2Cj+%5Cright%29+%2Bf%5Cleft%28+i-1%2Cj%2B1+%5Cright%29+%5Cright%5D++%5C%5C+S_y%3D%5Cleft%5B+f%5Cleft%28+i-1%2Cj%2B1+%5Cright%29+%2B2f%5Cleft%28+i%2Cj%2B1+%5Cright%29+%2Bf%5Cleft%28+i%2B1%2Cj%2B1+%5Cright%29+%5Cright%5D++%5C%5C+-%5Cleft%5B+f%5Cleft%28+i-1%2Cj-1+%5Cright%29+%2B2f%5Cleft%28+i%2Cj-1+%5Cright%29+%2Bf%5Cleft%28+i%2B1%2Cj-1+%5Cright%29+%5Cright%5D++
从这个式子中，可以得到两个性质，

[*]Sobel引入了平均的因素，因此对噪声有一定的平滑作用
[*]Sobel算子的操作就是相隔两个行（列）的差分，所以边缘两侧元素的得到了增强，因此边缘显得粗而亮。
Sobel算子表示形式为：

https://www.zhihu.com/equation?tex=+H_1%3D%5Cleft%5B+%5Cbegin%7Bmatrix%7D+%09-1%26%09%090%26%09%091%5C%5C+%09-2%26%09%090%2A%26%09%092%5C%5C+%09-1%26%09%090%26%09%091%5C%5C+%5Cend%7Bmatrix%7D+%5Cright%5D+%5C%2C%5C%2C+++H_2%3D%5Cleft%5B+%5Cbegin%7Bmatrix%7D+%09-1%26%09%09-2%26%09%09-1%5C%5C+%090%26%09%090%2A%26%09%090%5C%5C+%091%26%09%092%26%09%091%5C%5C+%5Cend%7Bmatrix%7D+%5Cright%5D+

3. 拉普拉斯算子（二阶微分）
拉普拉斯运算也是各向同性的线性运算。拉普拉斯算子为： https://www.zhihu.com/equation?tex=+%5Cnabla+%5E2f%3D%5Cfrac%7B%5Cpartial+%5E2f%7D%7B%5Cpartial+x%5E2%7D%2B%5Cfrac%7B%5Cpartial+%5E2f%7D%7B%5Cpartial+y%5E2%7D++ ,锐化之后的图像 https://www.zhihu.com/equation?tex=g%3Df-k+%5Cnabla+%5E2f

https://www.zhihu.com/equation?tex=k 为扩散效应的系数。

https://www.zhihu.com/equation?tex=%5Cbegin%7Balign%2A%7D+++%5Cfrac%7B%5Cpartial+%5E2f%5Cleft%28+x%2Cy+%5Cright%29%7D%7B%5Cpartial+x%5E2%7D+%26%3D+%5Cnabla+_xf%5Cleft%28+i%2B1%2Cj+%5Cright%29+-%5Cnabla+_xf%5Cleft%28+i%2Cj+%5Cright%29+%5C%5C+%26%3D+%5Cleft%5B+f%5Cleft%28+i%2B1%2Cj+%5Cright%29+-f%5Cleft%28+i%2Cj+%5Cright%29+%5Cright%5D+-%5Cleft%5B+f%5Cleft%28+i%2Cj+%5Cright%29+-f%5Cleft%28+i-1%2Cj+%5Cright%29+%5Cright%5D+%5C%5C+%26%3D+f%5Cleft%28+i%2B1%2Cj+%5Cright%29+%2Bf%5Cleft%28+i-1%2Cj+%5Cright%29+-2f%5Cleft%28+i%2Cj+%5Cright%29+%5C%5C+%5Cfrac%7B%5Cpartial+%5E2f%5Cleft%28+x%2Cy+%5Cright%29%7D%7B%5Cpartial+y%5E2%7D%26%3Df%5Cleft%28+i%2Cj-1+%5Cright%29+%2Bf%5Cleft%28+i%2Cj%2B1+%5Cright%29+-2f%5Cleft%28+i%2Cj+%5Cright%29%5C%5C+%5Cend%7Balign%2A%7D

https://www.zhihu.com/equation?tex=%5Cbegin%7Balign%7D+%5Cnabla+%5E2f+%26%3D%5Cfrac%7B%5Cpartial+%5E2f%5Cleft%28+x%2Cy+%5Cright%29%7D%7B%5Cpartial+x%5E2%7D%2B%5Cfrac%7B%5Cpartial+%5E2f%5Cleft%28+x%2Cy+%5Cright%29%7D%7B%5Cpartial+y%5E2%7D+%5C%5C+%26%3D-5%5Cleft%5C%7B+f%5Cleft%28+i%2Cj+%5Cright%29+-%5Cfrac%7B1%7D%7B5%7D%5Cleft%5B+f%5Cleft%28+i%2B1%2Cj+%5Cright%29+%2Bf%5Cleft%28+i-1%2Cj+%5Cright%29+%2Bf%5Cleft%28+i-1%2Cj-1+%5Cright%29+%2Bf%5Cleft%28+i%2B1.j%2B1+%5Cright%29+%2Bf%5Cleft%28+i%2Cj+%5Cright%29+%5Cright%5D+%5Cright%5C%7D++%5Cend%7Balign%7D
由此式可知，数字图像在点的拉普拉斯算子，可以由该点的灰度值减去该点及其邻域四个点的平均灰度值求得。
拉普拉斯算子的模板图可表示为：

https://www.zhihu.com/equation?tex=+H%3D%5Cleft%5B+%5Cbegin%7Bmatrix%7D+%090%26%09%09-1%26%09%090%5C%5C+%09-1%26%09%094%26%09%09-1%5C%5C+%090%26%09%09-1%26%09%090%5C%5C+%5Cend%7Bmatrix%7D+%5Cright%5D+++
还有其他的常用拉普拉斯模板：

https://www.zhihu.com/equation?tex=+H%3D%5Cleft%5B+%5Cbegin%7Bmatrix%7D+%09-1%26%09%09-1%26%09%09-1%5C%5C+%09-1%26%09%098%26%09%09-1%5C%5C+%09-1%26%09%09-1%26%09%09-1%5C%5C+%5Cend%7Bmatrix%7D+%5Cright%5D+++

高通滤波

高通滤波法采用高通滤波器让高频分量通过，使图像的边缘和线条变得清楚，实现锐化。
高通滤波器而用空域法和频域法实现，在空间域使用卷积方法，模板如

https://www.zhihu.com/equation?tex=+H%3D%5Cleft%5B+%5Cbegin%7Bmatrix%7D+%09-1%26%09%09-1%26%09%09-1%5C%5C+%09-1%26%09%099%26%09%09-1%5C%5C+%09-1%26%09%09-1%26%09%09-1%5C%5C+%5Cend%7Bmatrix%7D+%5Cright%5D+++
可以看出和拉普拉斯算子基本相同。

个人理解：为了能够让高频通过，点与其四个邻域点应该满足可以过零的性质，因为导数跨零，原函数曲线不会是单调的，必然会在零点处改变单调性，产生高频效果。

代码

Sobel算子
import cv2
import matplotlib.pyplot as plt

plt.subplot(1, 2, 1)
img = cv2.imread("image.jpg", 0)# 0为灰度图
plt.imshow(img)

plt.subplot(1, 2, 2)
depth = cv2.CV_16S
#求X方向梯度（创建grad_x, grad_y矩阵）
grad_x = cv2.Sobel( img, depth, 1, 0 )
abs_grad_x = cv2.convertScaleAbs( grad_x )

#求Y方向梯度
grad_y = cv2.Sobel( img, depth, 0, 1 )
abs_grad_y = cv2.convertScaleAbs( grad_y )

#合并梯度
sobel_img = cv2.addWeighted( abs_grad_x, 0.5, abs_grad_y, 0.5, 0 )

plt.imshow(sobel_img)
plt.show()


灰度图 以及 Sobel算子

Laplace算子
import cv2
import matplotlib.pyplot as plt

plt.subplot(1, 2, 1)
img = cv2.imread("image.jpg", 0)# 0为灰度图
plt.imshow(img)

plt.subplot(1, 2, 2)
depth = cv2.CV_16S
kernel_size = (3, 3)# 模板大小
scale = 1# 扩散系数
laplace_img = cv2.Laplacian(img, depth, kernel_size, scale=scale)

plt.imshow(laplace_img)
plt.show()


灰度图 以及 Laplace算子

mypro334 发表于 2022-4-27 14:30

噢……我的天……高等数学
[思考]

kyuskoj 发表于 2022-4-27 14:37

请问大佬有什么边缘增强的综述性文章吗

super1 发表于 2022-4-27 14:41

emmm，不是很清楚，我是看的数字图像处理书。文章可以搜一搜百度学术和谷歌学术
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