图像处理之梯度及边缘检测算子

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一、sobel 算子

Sobel算子包含两组 3 ∗ 3的矩阵，分别为横向及纵向模板，将之与图像作平面卷积，即可分别得出横向及纵向的亮度差分近似值。下面是$G_x$ 和$G_y$的模板。
$$G_x=\left[\begin{array}{ccc}+1& 0& -1\\ +2& 0& -2\\ +1& 0& -1\end{array}\right]\ast A$$

$$G_y=\left[\begin{array}{ccc}+1& +2& +1\\ 0& 0& 0\\ -1& -2& -1\end{array}\right]\ast A$$
如上式，$G_x$与$G_y$分别表示对图像A进行横向和纵向梯度检测得到的结果。
取二者平方和即可得到图像上每一点的梯度值，即在该点同时计算$x$方向与$y$方向的梯度。
$$G=\sqrt{G_x^2+G_y^2}$$
该点的梯度方向可以通过取这两个值的比的反正切$arctan$得到：
$$\Theta =arctan\left(\frac{G_y}{G_x}\right)$$
实现代码如下：

def SobelX(img,threshold): height = img.shape[0] width = img.shape[1] G_x = np.array([[-1, 0, 1], [-2, 0, 2], [-1, 0, 1]]) result = np.zeros(img.shape) for i in range(0, width - 2): for j in range(0, height - 2): v = np.sum(G_x * img[i:i + 3, j:j + 3]) result[i,j] =v if(result[i,j]<threshold): result[i,j]=0 return result def SobelY(img,threshold): height = img.shape[0] width = img.shape[1] G_y = np.array([[-1, -2, -1], [0, 0, 0], [1, 2, 1]]) result = np.zeros(img.shape) for i in range(0, width - 2): for j in range(0, height - 2): h = np.sum(G_y * img[i:i + 3, j:j + 3]) result[i,j] =h if(result[i,j]<threshold): result[i,j]=0 return result def Sobel(img,threshold): height = img.shape[0] width = img.shape[1] G_x = np.array([[-1, 0, 1], [-2, 0, 2], [-1, 0, 1]]) G_y = np.array([[-1, -2, -1], [0, 0, 0], [1, 2, 1]]) result = np.zeros(img.shape) for i in range(0, width - 2): for j in range(0, height - 2): v = np.sum(G_x * img[i:i + 3, j:j + 3]) h = np.sum(G_y * img[i:i + 3, j:j + 3]) result[i,j] = np.sqrt((v  2) + (h  2)) if(result[i,j]<threshold): result[i,j]=0 return result 

二、Scharr算子

def Scharr(img,threshold): height = img.shape[0] width = img.shape[1] G_x = np.array([[-3, 0, 3], [-10, 0, 10], [-3, 0, 3]]) G_y = np.array([[-3, -10, -3], [0, 0, 0], [3, 10, 3]]) result = np.zeros(img.shape) for i in range(0, width - 2): for j in range(0, height - 2): v = np.sum(G_x * img[i:i + 3, j:j + 3]) h = np.sum(G_y * img[i:i + 3, j:j + 3]) result[i,j] = np.sqrt((v  2) + (h  2)) if(result[i,j]<threshold): result[i,j]=0 return result 

三、Roberts算子

$$\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$$
$$\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$$

def Roberts(img,threshold): height = img.shape[0] width = img.shape[1] G_x = np.array([[-1, 0], [0,1]]) G_y = np.array([[0, -1], [1,0]]) result = np.zeros(img.shape) for i in range(0, width - 1): for j in range(0, height - 1): v = np.sum(G_x * img[i:i + 2, j:j + 2]) h = np.sum(G_y * img[i:i + 2, j:j + 2]) result[i,j] = np.sqrt((v  2) + (h  2)) if(result[i,j]<threshold): result[i,j]=0 return result 

四、拉普拉斯算子

代码实现如下：

def Laplacian(img): temLaplacian = np.array([[0, 1, 0], [1, -4, 1], [0, 1, 0]]) height, width = img.shape[::-1] result = np.zeros(img.shape) for i in range(0, width - 2): for j in range(0, height - 2): result[i][j] = np.abs(np.sum(temLaplacian * img[i:i + 3, j:j + 3])) return result 

import numpy as np import cv2 import imgShow as iS def SobelX(img,threshold): height = img.shape[0] width = img.shape[1] G_x = np.array([[-1, 0, 1], [-2, 0, 2], [-1, 0, 1]]) result = np.zeros(img.shape) for i in range(0, width - 2): for j in range(0, height - 2): v = np.sum(G_x * img[i:i + 3, j:j + 3]) result[i,j] =v if(result[i,j]<threshold): result[i,j]=0 return result def SobelY(img,threshold): height = img.shape[0] width = img.shape[1] G_y = np.array([[-1, -2, -1], [0, 0, 0], [1, 2, 1]]) result = np.zeros(img.shape) for i in range(0, width - 2): for j in range(0, height - 2): h = np.sum(G_y * img[i:i + 3, j:j + 3]) result[i,j] =h if(result[i,j]<threshold): result[i,j]=0 return result def Sobel(img,threshold): height = img.shape[0] width = img.shape[1] G_x = np.array([[-1, 0, 1], [-2, 0, 2], [-1, 0, 1]]) G_y = np.array([[-1, -2, -1], [0, 0, 0], [1, 2, 1]]) result = np.zeros(img.shape) for i in range(0, width - 2): for j in range(0, height - 2): v = np.sum(G_x * img[i:i + 3, j:j + 3]) h = np.sum(G_y * img[i:i + 3, j:j + 3]) result[i,j] = np.sqrt((v  2) + (h  2)) if(result[i,j]<threshold): result[i,j]=0 return result def Sobel(img,threshold): height = img.shape[0] width = img.shape[1] G_x = np.array([[-1, 0, 1], [-2, 0, 2], [-1, 0, 1]]) G_y = np.array([[-1, -2, -1], [0, 0, 0], [1, 2, 1]]) result = np.zeros(img.shape) for i in range(0, width - 2): for j in range(0, height - 2): v = np.sum(G_x * img[i:i + 3, j:j + 3]) h = np.sum(G_y * img[i:i + 3, j:j + 3]) result[i,j] = np.sqrt((v  2) + (h  2)) if(result[i,j]<threshold): result[i,j]=0 return result def Scharr(img,threshold): height = img.shape[0] width = img.shape[1] G_x = np.array([[-3, 0, 3], [-10, 0, 10], [-3, 0, 3]]) G_y = np.array([[-3, -10, -3], [0, 0, 0], [3, 10, 3]]) result = np.zeros(img.shape) for i in range(0, width - 2): for j in range(0, height - 2): v = np.sum(G_x * img[i:i + 3, j:j + 3]) h = np.sum(G_y * img[i:i + 3, j:j + 3]) result[i,j] = np.sqrt((v  2) + (h  2)) if(result[i,j]<threshold): result[i,j]=0 return result def Roberts(img,threshold): height = img.shape[0] width = img.shape[1] G_x = np.array([[-1, 0], [0,1]]) G_y = np.array([[0, -1], [1,0]]) result = np.zeros(img.shape) for i in range(0, width - 1): for j in range(0, height - 1): v = np.sum(G_x * img[i:i + 2, j:j + 2]) h = np.sum(G_y * img[i:i + 2, j:j + 2]) result[i,j] = np.sqrt((v  2) + (h  2)) if(result[i,j]<threshold): result[i,j]=0 return result def Laplacian(img): temLaplacian = np.array([[0, 1, 0], [1, -4, 1], [0, 1, 0]]) height, width = img.shape[::-1] result = np.zeros(img.shape) for i in range(0, width - 2): for j in range(0, height - 2): result[i][j] = np.abs(np.sum(temLaplacian * img[i:i + 3, j:j + 3])) return result img=cv2.imread("./originImg/HorizontalAndVertical.jpg") img=cv2.cvtColor(img,cv2.COLOR_BGR2GRAY) sobelImg=Sobel(img,56) iS.showImagegray(sobelImg, img, 25, 15, 'sobelDetection', 'origin', './ProcessedImg/sobelDetection.jpg') imageList=[] origin_img=[img,'origin_img'] imageList.append(origin_img) sobelx=SobelX(img,0) sobel2=[sobelx,'Sobel_X'] imageList.append(sobel2) sobely=SobelY(img,0) sobel1=[sobely,'Sobel_Y'] imageList.append(sobel1) sobelImg=Sobel(img,56) sobel3=[sobelImg,'Sobel'] imageList.append(sobel3) iS.showMultipleimages(imageList,25,25,'./ProcessedImg/sobelEdge.jpg') img1=cv2.imread('./originImg/Goldhill.tif') img1=cv2.cvtColor(img1,cv2.COLOR_BGR2GRAY) LapImg=Laplacian(img1) iS.showImagegray(LapImg, img1, 25, 15, 'LapImg', 'origin', './ProcessedImg/lapImg.jpg') scharrImg=Scharr(img,56) iS.showImagegray(scharrImg, img, 25, 15, 'scharrDetection', 'origin', './ProcessedImg/scharrDetection.jpg') robertsImg=Roberts(img,56) iS.showImagegray(robertsImg, img, 25, 15, 'robertsDetection', 'origin', './ProcessedImg/robertsDetection.jpg') # cv2.imshow('sobely',sobely) # cv2.waitKey(0) # cv2.destroyAllWindows() 

画图代码：

import matplotlib.pyplot as plt import numpy as np import math #图像实际大小为 W*100 * H*100 像素 , def showImagegray(newImg,oldImg,W,H,newImgtitle,oldImgtitle,saveImgpath): plt.figure(figsize=(W,H)) plt.subplot(121) plt.title(oldImgtitle,fontsize=30) plt.axis('off') plt.imshow(oldImg, cmap='gray') plt.subplot(122) plt.title(newImgtitle,fontsize=30) plt.axis('off') plt.imshow(newImg, cmap='gray') # plt.tight_layout() # 调整整体空白 plt.savefig(saveImgpath) plt.show() def showMultipleimages(imageList,W,H,saveImgpath): imageLength=len(imageList) plt.rcParams['figure.figsize'] = (W,H) col=row=math.ceil(np.sqrt(imageLength)) fig, a = plt.subplots(col, row) m = 0 for i in range(col): for j in range(row): a[i][j].set_title(imageList[m][1]) a[i][j].imshow(imageList[m][0], cmap=plt.cm.gray) m += 1 #去掉边框和刻度 for ax in a.flat: ax.set_axis_off() fig.tight_layout() # 调整整体空白 plt.subplots_adjust(wspace=0.2, hspace=0.2) # 调整子图间距 plt.savefig(saveImgpath) plt.show()