目录
锐化（高通）空间滤波器使用一阶导数锐化图像－梯度锐化（高通）空间滤波器
平滑通过称为低通滤波类似于积分运算锐化通常称为高通滤波微分运算高过（负责细节的）高频，衰减或抑制低频使用一阶导数锐化图像－梯度
在图像处理中，一阶导数是用梯度幅度实现的，图像的梯度定义为二维列向量
∇f≡grad(f)=[gxgy]=[∂f/∂x∂f/∂y](3.57)\nabla f \equiv \text{grad}(f) = \begin{bmatrix} g_x \\ g_y \end{bmatrix} = \begin{bmatrix} \partial f /\partial x \\ \partial f /\partial y \ \end{bmatrix} \tag{3.57}∇f≡grad(f)=[gxgy]=[∂f/∂x∂f/∂y](3.57)
向量∇f\nabla f∇f的幅度表示为M(x,y)M(x, y)M(x,y)，也经常使用向量范数∥∇f∥\lVert\nabla f \rVert∥∇f∥
M(x,y)=∥f∥=mag(∇f)=gx2+gy2(3.58)M(x, y) = \lVert f \rVert = \text{mag}(\nabla f) = \sqrt{g_x^2 + g_y^2} \tag{3.58}M(x,y)=∥f∥=mag(∇f)=gx2+gy2(3.58)
是梯度向量方向的变化率在(x,y)(x, y)(x,y)处的值，是与原图像大小相同的图像，通常称为梯度图像
在某些实现中，使用绝对值来近似平方运算和平方根运算更合适：
M(x,y)≈∣gx∣+∣gy∣(3.59)M(x,y) \approx |g_x| + |g_y| \tag{3.59}M(x,y)≈∣gx∣+∣gy∣(3.59)
这个表达式通常会损失各向同性。
最简近似的一阶导数是gx=(z8−z5)g_x = (z_8 - z_5)gx=(z8−z5)和gy=(z6−z5)g_y = (z_6 - z_5)gy=(z6−z5)
罗伯特交叉梯度算子，早期的图像处理使用交叉差值
gx=(z9−z5)和gy=(z8−z6)(3.60)g_x = (z_9 - z_5)和g_y = (z_8 - z_6) \tag{3.60}gx=(z9−z5)和gy=(z8−z6)(3.60)
梯度图像计算为：
M(x,y)=[(z9−z5)2+(z8−z6)2]1/2(3.61)M(x, y) = \Big[(z_9 - z_5)^2 + (z_8 - z_6)^2 \Big]^{1/2} \tag{3.61}M(x,y)=[(z9−z5)2+(z8−z6)2]1/2(3.61)
M(x,y)≈∣z9−z5∣+∣z8−z6∣(3.62)M(x, y) \approx |z_9 - z_5| + |z_8 - z_6| \tag{3.62}M(x,y)≈∣z9−z5∣+∣z8−z6∣(3.62)
3×33\times 33×3的核
gx=∂f/∂x=(z7+2z8+z9)−(z1+2z2+z3)(3.63)g_x = \partial f/ \partial x = (z_7 + 2z_8 + z_9) - (z_1 +2z_2 + z_3) \tag{3.63}gx=∂f/∂x=(z7+2z8+z9)−(z1+2z2+z3)(3.63)
gy=∂f/∂y=(z3+2z6+z9)−(z1+2z4+z7)(3.64)g_y = \partial f/ \partial y = (z_3 + 2z_6 + z_9) - (z_1 +2z_4 + z_7) \tag{3.64}gy=∂f/∂y=(z3+2z6+z9)−(z1+2z4+z7)(3.64)
M(x,y)=[gx2+gy2]1/2=[[(z7+2z8+z9)−(z1+2z2+z3)]2+[(z3+2z6+z9)−(z1+2z4+z7)]2]1/2(3.65)M(x, y) = [g_x^2 + g_y^2]^{1/2} = \Big[[(z_7 + 2z_8 + z_9) - (z_1 +2z_2 + z_3)]^2 + [(z_3 + 2z_6 + z_9) - (z_1 +2z_4 + z_7)]^2\Big]^{1/2} \tag{3.65}M(x,y)=[gx2+gy2]1/2=[[(z7+2z8+z9)−(z1+2z2+z3)]2+[(z3+2z6+z9)−(z1+2z4+z7)]2]1/2(3.65)
def visualize_show_annot(img_show, img_annot, ax, string='img_annot'):"""add annotation to the image, values of each pixelparam: img: input imageparam: ax: axes of the matplotlib"""height, width = img_annot.shapeimg_show = img_show[:height, :width]ax.imshow(img_show, cmap='gray', vmin=0, vmax=255)thresh = 10 #img_show.max()/2.5for x in range(height):for y in range(width):if string == 'img_annot':ax.annotate(str(round(img_annot[x][y],2)), xy=(y,x),horizontalalignment='center',verticalalignment='center',color='white' if img_annot[x][y]>thresh else 'black')else:ax.annotate(string + str(x + y + 1), xy=(y,x),horizontalalignment='center',verticalalignment='center',color='white' if img_annot[x][y]>thresh else 'black')
# 一阶导数算子，罗伯特交叉梯度算子，Sobel算子height, width = 3, 3img_show = np.ones([height, width], dtype=np.uint8) * 250img_ori = np.zeros([height, width], dtype=np.uint8)fig = plt.figure(figsize=(8, 6))ax1 = fig.add_subplot(2, 3, 1)ax1.set_title('3x3 Region'), visualize_show_annot(img_show, img_ori, ax1, string='z'), plt.xticks([]), plt.yticks([])img_ori = np.zeros([2, 2], np.int)img_ori[0, 0] = -1img_ori[1, 1] = 1ax2 = fig.add_subplot(2, 3, 2)ax2.set_title('Robert operator'), visualize_show_annot(img_show, img_ori, ax2), plt.xticks([]), plt.yticks([])img_ori = np.zeros([2, 2], np.int)img_ori[0, 1] = -1img_ori[1, 0] = 1ax3 = fig.add_subplot(2, 3, 3)ax3.set_title('Robert operator'), visualize_show_annot(img_show, img_ori, ax3), plt.xticks([]), plt.yticks([])img_ori = np.zeros([3, 3], np.int)img_ori[0, :] = np.array([-1, -2, -1])img_ori[2, :] = np.array([1, 2, 1])ax4 = fig.add_subplot(2, 3, 4)ax4.set_title('Sobel operator'), visualize_show_annot(img_show, img_ori, ax4), plt.xticks([]), plt.yticks([])img_ori = np.zeros([3, 3], np.int)img_ori[:, 0] = np.array([-1, -2, -1])img_ori[:, 2] = np.array([1, 2, 1])ax5 = fig.add_subplot(2, 3, 5)ax5.set_title('Sobel operator'), visualize_show_annot(img_show, img_ori, ax5), plt.xticks([]), plt.yticks([])plt.tight_layout()plt.show()
# Sobel梯度增强边缘img_ori = cv2.imread("DIP_Figures/DIP3E_Original_Images_CH03/Fig0342(a)(contact_lens_original).tif", 0)sobel_x = np.zeros([3, 3], np.int)sobel_x[0, :] = np.array([-1, -2, -1])sobel_x[2, :] = np.array([1, 2, 1])sobel_y = np.zeros([3, 3], np.int)sobel_y[:, 0] = np.array([-1, -2, -1])sobel_y[:, 2] = np.array([1, 2, 1])# gx = separate_kernel_conv2D(img_ori, kernel=sobel_x)# gy = separate_kernel_conv2D(img_ori, kernel=sobel_y)gx = cv2.filter2D(img_ori, ddepth=-1, kernel=sobel_x)gy = cv2.filter2D(img_ori, ddepth=-1, kernel=sobel_y)gx = np.where(gx >= 100, gx, 0)gx = np.where(gx < 100, gx, 1)gy = np.where(gy >= 100, gy, 0)gy = np.where(gy < 100, gy, 1)# 先对gx gy做二值化处理再应用下面的公式# img_sobel = np.sqrt(gx**2 + gy**2) # 二值化后，平方根的效果与绝对值很接近img_sobel = abs(gx) + abs(gy)img_sobel = np.uint8(normalize(img_sobel) * 255)plt.figure(figsize=(15, 12))plt.subplot(1, 2, 1), plt.imshow(img_ori, 'gray', vmax=255), plt.title("Original"), plt.xticks([]), plt.yticks([])plt.subplot(1, 2, 2), plt.imshow(img_sobel, 'gray', vmax=255), plt.title("Sobel"), plt.xticks([]), plt.yticks([])plt.tight_layout()plt.show()
梯度还可用来突出灰度级图像中很难看到的小尺度图像（如异物、保护液中的气泡或镜片中的微小缺陷）。在平坦的灰度场中增强小的不连续的能力是梯度的呬个重要特征
第3章 Python 数字图像处理(DIP) - 灰度变换与空间滤波17 - 锐化高通滤波器 - 梯度图像（罗伯特 Sobel算子）
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