在做物体检测时，由于成本和应用场合的限制，不能够一味地增加相机的分辨率，或者已经用了分辨率很高的相机，但是视野范围很大，仍然无法实现很高的精度，这时就要考虑亚像素技术，亚像素技术就是在两个像素点之间进行进一步的细分，从而得到亚像素级别的边缘点的坐标（也就是float类型的坐标），一般来说，现有的技术可以做到2细分、4细分，甚至很牛的能做到更高，通过亚像素边缘检测技术的使用，可以节约成本，提高识别精度。
常见的亚像素技术包括灰度矩、Hu矩、空间矩、Zernike矩等，通过查阅多篇论文，综合比较精度、时间和实现难度，选择Zernike矩作为项目中的亚像素边缘检测算法。基于Zernike矩的边缘检测原理，下一篇文章详细再总结一下，这里大体把步骤说一下，并使用OpenCV实现。
大体步骤就是，首先确定使用的模板大小，然后根据Zernike矩的公式求出各个模板系数，例如我选择的是7*7模板（模板越大精度越高，但是运算时间越长），要计算M00，M11R，M11I，M20，M31R，M31I，M40七个Zernike模板。第二步是对待检测图像进行预处理，包括滤波二值化等，也可以在进行一次Canny边缘检测。第三步是将预处理的图像跟7个Zernike矩模板分别进行卷积，得到7个Zernike矩。第四步是把上一步的矩乘上角度校正系数（这是因为利用Zernike的旋转不变性，Zernike模型把边缘都简化成了x=n的直线，这里要调整回来）。第五步是计算距离参数l和灰度差参数k，根据k和l判断该点是否为边缘点。以下是基于OpenCV的实现。
理论部分可以参考：(12条消息) Zernike矩之边缘检测（附源码）_xiaoluo91的专栏-CSDN博客_zernike矩边缘检测
以下部分为python实现的代码段，仅供参考。（以上理论部分为原始的Zernike矩，代码为改进的Zernike矩，参考文章为：基于Zernike正交矩的图像亚像素边缘检测算法改进。高世一）
import cv2import numpy as npimport matplotlib.pyplot as pltimport timeg_N = 7M00 = np.array([0, 0.0287, 0.0686, 0.0807, 0.0686, 0.0287, 0,0.0287, 0.0815, 0.0816, 0.0816, 0.0816, 0.0815, 0.0287,0.0686, 0.0816, 0.0816, 0.0816, 0.0816, 0.0816, 0.0686,0.0807, 0.0816, 0.0816, 0.0816, 0.0816, 0.0816, 0.0807,0.0686, 0.0816, 0.0816, 0.0816, 0.0816, 0.0816, 0.0686,0.0287, 0.0815, 0.0816, 0.0816, 0.0816, 0.0815, 0.0287,0, 0.0287, 0.0686, 0.0807, 0.0686, 0.0287, 0]).reshape((7, 7))M11R = np.array([0, -0.015, -0.019, 0, 0.019, 0.015, 0,-0.0224, -0.0466, -0.0233, 0, 0.0233, 0.0466, 0.0224,-0.0573, -0.0466, -0.0233, 0, 0.0233, 0.0466, 0.0573,-0.069, -0.0466, -0.0233, 0, 0.0233, 0.0466, 0.069,-0.0573, -0.0466, -0.0233, 0, 0.0233, 0.0466, 0.0573,-0.0224, -0.0466, -0.0233, 0, 0.0233, 0.0466, 0.0224,0, -0.015, -0.019, 0, 0.019, 0.015, 0]).reshape((7, 7))M11I = np.array([0, -0.0224, -0.0573, -0.069, -0.0573, -0.0224, 0,-0.015, -0.0466, -0.0466, -0.0466, -0.0466, -0.0466, -0.015,-0.019, -0.0233, -0.0233, -0.0233, -0.0233, -0.0233, -0.019,0, 0, 0, 0, 0, 0, 0,0.019, 0.0233, 0.0233, 0.0233, 0.0233, 0.0233, 0.019,0.015, 0.0466, 0.0466, 0.0466, 0.0466, 0.0466, 0.015,0, 0.0224, 0.0573, 0.069, 0.0573, 0.0224, 0]).reshape((7, 7))M20 = np.array([0, 0.0225, 0.0394, 0.0396, 0.0394, 0.0225, 0,0.0225, 0.0271, -0.0128, -0.0261, -0.0128, 0.0271, 0.0225,0.0394, -0.0128, -0.0528, -0.0661, -0.0528, -0.0128, 0.0394,0.0396, -0.0261, -0.0661, -0.0794, -0.0661, -0.0261, 0.0396,0.0394, -0.0128, -0.0528, -0.0661, -0.0528, -0.0128, 0.0394,0.0225, 0.0271, -0.0128, -0.0261, -0.0128, 0.0271, 0.0225,0, 0.0225, 0.0394, 0.0396, 0.0394, 0.0225, 0]).reshape((7, 7))M31R = np.array([0, -0.0103, -0.0073, 0, 0.0073, 0.0103, 0,-0.0153, -0.0018, 0.0162, 0, -0.0162, 0.0018, 0.0153,-0.0223, 0.0324, 0.0333, 0, -0.0333, -0.0324, 0.0223,-0.0190, 0.0438, 0.0390, 0, -0.0390, -0.0438, 0.0190,-0.0223, 0.0324, 0.0333, 0, -0.0333, -0.0324, 0.0223,-0.0153, -0.0018, 0.0162, 0, -0.0162, 0.0018, 0.0153,0, -0.0103, -0.0073, 0, 0.0073, 0.0103, 0]).reshape(7, 7)M31I = np.array([0, -0.0153, -0.0223, -0.019, -0.0223, -0.0153, 0,-0.0103, -0.0018, 0.0324, 0.0438, 0.0324, -0.0018, -0.0103,-0.0073, 0.0162, 0.0333, 0.039, 0.0333, 0.0162, -0.0073,0, 0, 0, 0, 0, 0, 0,0.0073, -0.0162, -0.0333, -0.039, -0.0333, -0.0162, 0.0073,0.0103, 0.0018, -0.0324, -0.0438, -0.0324, 0.0018, 0.0103,0, 0.0153, 0.0223, 0.0190, 0.0223, 0.0153, 0]).reshape(7, 7)M40 = np.array([0, 0.013, 0.0056, -0.0018, 0.0056, 0.013, 0,0.0130, -0.0186, -0.0323, -0.0239, -0.0323, -0.0186, 0.0130,0.0056, -0.0323, 0.0125, 0.0406, 0.0125, -0.0323, 0.0056,-0.0018, -0.0239, 0.0406, 0.0751, 0.0406, -0.0239, -0.0018,0.0056, -0.0323, 0.0125, 0.0406, 0.0125, -0.0323, 0.0056,0.0130, -0.0186, -0.0323, -0.0239, -0.0323, -0.0186, 0.0130,0, 0.013, 0.0056, -0.0018, 0.0056, 0.013, 0]).reshape(7, 7)def zernike_detection(path):img = cv2.imread(path)img = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY)blur_img = cv2.medianBlur(img, 13)c_img = cv2.adaptiveThreshold(blur_img, 255, cv2.ADAPTIVE_THRESH_GAUSSIAN_C, cv2.THRESH_BINARY_INV, 7, 4)ZerImgM00 = cv2.filter2D(c_img, cv2.CV_64F, M00)ZerImgM11R = cv2.filter2D(c_img, cv2.CV_64F, M11R)ZerImgM11I = cv2.filter2D(c_img, cv2.CV_64F, M11I)ZerImgM20 = cv2.filter2D(c_img, cv2.CV_64F, M20)ZerImgM31R = cv2.filter2D(c_img, cv2.CV_64F, M31R)ZerImgM31I = cv2.filter2D(c_img, cv2.CV_64F, M31I)ZerImgM40 = cv2.filter2D(c_img, cv2.CV_64F, M40)point_temporary_x = []point_temporary_y = []scatter_arr = cv2.findNonZero(ZerImgM00).reshape(-1, 2)for idx in scatter_arr:j, i = idxtheta_temporary = np.arctan2(ZerImgM31I[i][j], ZerImgM31R[i][j])rotated_z11 = np.sin(theta_temporary) * ZerImgM11I[i][j] + np.cos(theta_temporary) * ZerImgM11R[i][j]rotated_z31 = np.sin(theta_temporary) * ZerImgM31I[i][j] + np.cos(theta_temporary) * ZerImgM31R[i][j]l_method1 = np.sqrt((5 * ZerImgM40[i][j] + 3 * ZerImgM20[i][j]) / (8 * ZerImgM20[i][j]))l_method2 = np.sqrt((5 * rotated_z31 + rotated_z11)/(6 * rotated_z11))l = (l_method1 + l_method2)/2k = 3 * rotated_z11 / (2 * (1 - l_method2 ** 2)**1.5)# h = (ZerImgM00[i][j] - k * np.pi / 2 + k * np.arcsin(l_method2) + k * l_method2 * (1 - l_method2 ** 2) ** 0.5)# / np.pik_value = 20.0l_value = 2**0.5 / g_Nabsl = np.abs(l_method2 - l_method1)if k >= k_value and absl <= l_value:y = i + g_N * l * np.sin(theta_temporary) / 2x = j + g_N * l * np.cos(theta_temporary) / 2point_temporary_x.append(x)point_temporary_y.append(y)else:continuereturn point_temporary_x, point_temporary_ypath = 'E:/program/image/lo.jpg' # 此处需要修改为自己的路径！！！time1 = time.time()point_temporary_x, point_temporary_y = zernike_detection(path)time2 = time.time()print(time2-time1)
文章中的point_temporary_x, point_temporary_y即为所存亚像素点。
由于本人是菜鸟一枚，不知道怎么样在图像中显示亚像素（希望有大佬会的话，可以教教我）。。就用散点图画了一个。如下：
原图：
亚像素散点图为：
其中会产生双层边缘，如下：
产生的原因是由于边缘线与两端背景都有像素差，（猜测一下）如果是二值图象直接检测的话，应该就不会有双层边缘了。
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经大佬指点，成功在图像中显示亚像素！以下为效果图：
下图为右眼眼角的边缘显示：
下面为完整代码：
import cv2import numpy as npimport matplotlib.pyplot as pltimport timeg_N = 7M00 = np.array([0, 0.0287, 0.0686, 0.0807, 0.0686, 0.0287, 0,0.0287, 0.0815, 0.0816, 0.0816, 0.0816, 0.0815, 0.0287,0.0686, 0.0816, 0.0816, 0.0816, 0.0816, 0.0816, 0.0686,0.0807, 0.0816, 0.0816, 0.0816, 0.0816, 0.0816, 0.0807,0.0686, 0.0816, 0.0816, 0.0816, 0.0816, 0.0816, 0.0686,0.0287, 0.0815, 0.0816, 0.0816, 0.0816, 0.0815, 0.0287,0, 0.0287, 0.0686, 0.0807, 0.0686, 0.0287, 0]).reshape((7, 7))M11R = np.array([0, -0.015, -0.019, 0, 0.019, 0.015, 0,-0.0224, -0.0466, -0.0233, 0, 0.0233, 0.0466, 0.0224,-0.0573, -0.0466, -0.0233, 0, 0.0233, 0.0466, 0.0573,-0.069, -0.0466, -0.0233, 0, 0.0233, 0.0466, 0.069,-0.0573, -0.0466, -0.0233, 0, 0.0233, 0.0466, 0.0573,-0.0224, -0.0466, -0.0233, 0, 0.0233, 0.0466, 0.0224,0, -0.015, -0.019, 0, 0.019, 0.015, 0]).reshape((7, 7))M11I = np.array([0, -0.0224, -0.0573, -0.069, -0.0573, -0.0224, 0,-0.015, -0.0466, -0.0466, -0.0466, -0.0466, -0.0466, -0.015,-0.019, -0.0233, -0.0233, -0.0233, -0.0233, -0.0233, -0.019,0, 0, 0, 0, 0, 0, 0,0.019, 0.0233, 0.0233, 0.0233, 0.0233, 0.0233, 0.019,0.015, 0.0466, 0.0466, 0.0466, 0.0466, 0.0466, 0.015,0, 0.0224, 0.0573, 0.069, 0.0573, 0.0224, 0]).reshape((7, 7))M20 = np.array([0, 0.0225, 0.0394, 0.0396, 0.0394, 0.0225, 0,0.0225, 0.0271, -0.0128, -0.0261, -0.0128, 0.0271, 0.0225,0.0394, -0.0128, -0.0528, -0.0661, -0.0528, -0.0128, 0.0394,0.0396, -0.0261, -0.0661, -0.0794, -0.0661, -0.0261, 0.0396,0.0394, -0.0128, -0.0528, -0.0661, -0.0528, -0.0128, 0.0394,0.0225, 0.0271, -0.0128, -0.0261, -0.0128, 0.0271, 0.0225,0, 0.0225, 0.0394, 0.0396, 0.0394, 0.0225, 0]).reshape((7, 7))M31R = np.array([0, -0.0103, -0.0073, 0, 0.0073, 0.0103, 0,-0.0153, -0.0018, 0.0162, 0, -0.0162, 0.0018, 0.0153,-0.0223, 0.0324, 0.0333, 0, -0.0333, -0.0324, 0.0223,-0.0190, 0.0438, 0.0390, 0, -0.0390, -0.0438, 0.0190,-0.0223, 0.0324, 0.0333, 0, -0.0333, -0.0324, 0.0223,-0.0153, -0.0018, 0.0162, 0, -0.0162, 0.0018, 0.0153,0, -0.0103, -0.0073, 0, 0.0073, 0.0103, 0]).reshape(7, 7)M31I = np.array([0, -0.0153, -0.0223, -0.019, -0.0223, -0.0153, 0,-0.0103, -0.0018, 0.0324, 0.0438, 0.0324, -0.0018, -0.0103,-0.0073, 0.0162, 0.0333, 0.039, 0.0333, 0.0162, -0.0073,0, 0, 0, 0, 0, 0, 0,0.0073, -0.0162, -0.0333, -0.039, -0.0333, -0.0162, 0.0073,0.0103, 0.0018, -0.0324, -0.0438, -0.0324, 0.0018, 0.0103,0, 0.0153, 0.0223, 0.0190, 0.0223, 0.0153, 0]).reshape(7, 7)M40 = np.array([0, 0.013, 0.0056, -0.0018, 0.0056, 0.013, 0,0.0130, -0.0186, -0.0323, -0.0239, -0.0323, -0.0186, 0.0130,0.0056, -0.0323, 0.0125, 0.0406, 0.0125, -0.0323, 0.0056,-0.0018, -0.0239, 0.0406, 0.0751, 0.0406, -0.0239, -0.0018,0.0056, -0.0323, 0.0125, 0.0406, 0.0125, -0.0323, 0.0056,0.0130, -0.0186, -0.0323, -0.0239, -0.0323, -0.0186, 0.0130,0, 0.013, 0.0056, -0.0018, 0.0056, 0.013, 0]).reshape(7, 7)def zernike_detection(path):img = cv2.imread(path)img = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY)blur_img = cv2.medianBlur(img, 13)c_img = cv2.adaptiveThreshold(blur_img, 255, cv2.ADAPTIVE_THRESH_GAUSSIAN_C, cv2.THRESH_BINARY_INV, 7, 4)ZerImgM00 = cv2.filter2D(c_img, cv2.CV_64F, M00)ZerImgM11R = cv2.filter2D(c_img, cv2.CV_64F, M11R)ZerImgM11I = cv2.filter2D(c_img, cv2.CV_64F, M11I)ZerImgM20 = cv2.filter2D(c_img, cv2.CV_64F, M20)ZerImgM31R = cv2.filter2D(c_img, cv2.CV_64F, M31R)ZerImgM31I = cv2.filter2D(c_img, cv2.CV_64F, M31I)ZerImgM40 = cv2.filter2D(c_img, cv2.CV_64F, M40)point_temporary_x = []point_temporary_y = []scatter_arr = cv2.findNonZero(ZerImgM00).reshape(-1, 2)for idx in scatter_arr:j, i = idxtheta_temporary = np.arctan2(ZerImgM31I[i][j], ZerImgM31R[i][j])rotated_z11 = np.sin(theta_temporary) * ZerImgM11I[i][j] + np.cos(theta_temporary) * ZerImgM11R[i][j]rotated_z31 = np.sin(theta_temporary) * ZerImgM31I[i][j] + np.cos(theta_temporary) * ZerImgM31R[i][j]l_method1 = np.sqrt((5 * ZerImgM40[i][j] + 3 * ZerImgM20[i][j]) / (8 * ZerImgM20[i][j]))l_method2 = np.sqrt((5 * rotated_z31 + rotated_z11) / (6 * rotated_z11))l = (l_method1 + l_method2) / 2k = 3 * rotated_z11 / (2 * (1 - l_method2 ** 2) ** 1.5)# h = (ZerImgM00[i][j] - k * np.pi / 2 + k * np.arcsin(l_method2) + k * l_method2 * (1 - l_method2 ** 2) ** 0.5)# / np.pik_value = 20.0l_value = 2 ** 0.5 / g_Nabsl = np.abs(l_method2 - l_method1)if k >= k_value and absl <= l_value:y = i + g_N * l * np.sin(theta_temporary) / 2x = j + g_N * l * np.cos(theta_temporary) / 2point_temporary_x.append(x)point_temporary_y.append(y)else:continuereturn point_temporary_x, point_temporary_ypath = 'E:/program/image/lo.jpg' # 此处需要修改为自己的路径！！！time1 = time.time()point_temporary_x, point_temporary_y = zernike_detection(path)time2 = time.time()print(time2 - time1)# gray : 进行检测的图像gray = cv2.imread(path, 0)plt.imshow(gray, cmap="gray")# point检测出的亚像素点point = np.array([point_temporary_x, point_temporary_y])# s:调整显示点的大小plt.scatter(point[0, :], point[1, :], s=10, marker="*")plt.show()
参考&改编自：OpenCV实现基于Zernike矩的亚像素边缘检测_zilanpotou182的博客-CSDN博客_opencv亚像素边缘检测
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