一。原理
參考：
camera calibration using opencv
相機標定原理、步驟
opencv-python 攝像頭標定

生成黑白棋盤標定圖和單目相機標定
相機外參估計

坐標系

坐標系名稱介紹圖形

世界坐標系	world coordinate，測量坐標系，三維直角坐標系，以其為基準可以描述相機和待測物體的空間位置。世界坐標系的位置可以根據實際情況自由確定
相機坐標系	camera coordinate，三維直角坐標系，原點位于鏡頭光心處，x，y軸分別與相面的兩邊平行，z軸為鏡頭光軸，與像平面垂直
像素坐標系	pixel coordinate，二維直角坐標系，反映相機CCD/CMOS芯片中像素的排列情況。原點位于左上角，u，v分別于像面的兩邊平行，單位是像素（整數）。但是它不利于坐標轉換
圖像坐標系	XOY,單位通常是毫米，原點是相機光軸與相面的交點（也叫主點）

4大坐標系理解圖：

坐標系之間轉換：

轉換

世界坐標系-》相機坐標系
圖像坐標系-》像素坐標系
相機坐標系-》圖像坐標系
世界坐標系-》像素坐標系

過程理解：
要找到3D點在圖像平面上的投影，我們首先需要使用外參（旋轉和平移）將該點從世界坐標系統轉換成相機坐標系，接下來，使用相機的內參將點投影到圖像平面上。

相機標定3+要素（內參，外參，畸變參數等）：
0.先看下總公式：

P:3x4投影矩陣
K：內參 3x3矩陣
R：旋轉矩陣 3x3
t：平移矩陣 3x1

1.內參K

fx，fy：x，y焦距，通常值一樣
cx，cy：圖像平面光學中心的x，y坐標，一般近似圖像中心點
gamma：是軸之間的傾斜度，一般是0

2.外參
Rt
R:旋轉矩陣 3x3 但opencv標定時3x1矩陣？？！！
t: 平移矩陣 3x1

3.畸變參數

分徑向畸變（k1，k2，k3）和切向畸變（p1，p2）

畸變類型內容示例

徑向畸變	來自于透鏡形狀,包括枕型畸變、桶型畸變，光學中心畸變為0，越往邊緣移動，畸變也嚴重
切向畸變	透鏡制造上的缺陷使得透鏡本身與圖像平面不平行而產生的，包括薄透鏡畸變、離心畸變
二。相機標定步驟
1、打印一張棋盤格，把它貼在一個平面上，作為標定物。
2、通過調整標定物或攝像機的方向，為標定物拍攝一些不同方向的照片。
3、從照片中提取棋盤格角點。
4、估算理想無畸變的情況下，五個內參和六個外參。
5、應用最小二乘法估算實際存在徑向畸變下的畸變系數。
6、極大似然法，優化估計，提升估計精度。

三。代碼注釋
內參，畸變參數的求解：

#!/usr/bin/env pythonimport cv2 import numpy as np import os import glob# Defining the dimensions of checkerboard CHECKERBOARD = (6,9) criteria = (cv2.TERM_CRITERIA_EPS + cv2.TERM_CRITERIA_MAX_ITER, 30, 0.001)# Creating vector to store vectors of 3D points for each checkerboard image objpoints = [] # Creating vector to store vectors of 2D points for each checkerboard image imgpoints = [] # Defining the world coordinates for 3D points objp = np.zeros((1, CHECKERBOARD[0]*CHECKERBOARD[1], 3), np.float32) objp[0,:,:2] = np.mgrid[0:CHECKERBOARD[0], 0:CHECKERBOARD[1]].T.reshape(-1, 2) prev_img_shape = None# Extracting path of individual image stored in a given directory images = glob.glob('./images/*.jpg') for fname in images:img = cv2.imread(fname)gray = cv2.cvtColor(img,cv2.COLOR_BGR2GRAY)# Find the chess board corners# If desired number of corners are found in the image then ret = trueret, corners = cv2.findChessboardCorners(gray, CHECKERBOARD, cv2.CALIB_CB_ADAPTIVE_THRESH+cv2.CALIB_CB_FAST_CHECK+cv2.CALIB_CB_NORMALIZE_IMAGE)"""If desired number of corner are detected,we refine the pixel coordinates and display them on the images of checker board"""if ret == True:objpoints.append(objp)# refining pixel coordinates for given 2d points.corners2 = cv2.cornerSubPix(gray,corners,(11,11),(-1,-1),criteria)imgpoints.append(corners2)# Draw and display the cornersimg = cv2.drawChessboardCorners(img, CHECKERBOARD, corners2,ret)cv2.imshow('img',img)cv2.waitKey(10)cv2.destroyAllWindows()h,w = img.shape[:2]""" Performing camera calibration by passing the value of known 3D points (objpoints) and corresponding pixel coordinates of the detected corners (imgpoints) """ ret, mtx, dist, rvecs, tvecs = cv2.calibrateCamera(objpoints, imgpoints, gray.shape[::-1],None,None)print("Camera matrix {}: \n".format(mtx.shape)) print(mtx) print("dist {}: \n".format(dist.shape)) print(dist) print("rvecs {}: \n".format(np.array(rvecs).shape)) print(rvecs) print("tvecs {}: \n".format(np.array(tvecs).shape)) print(tvecs)# Using the derived camera parameters to undistort the imageimg = cv2.imread(images[0]) # Refining the camera matrix using parameters obtained by calibration newcameramtx, roi = cv2.getOptimalNewCameraMatrix(mtx, dist, (w, h), 1, (w, h))# Method 1 to undistort the image dst = cv2.undistort(img, mtx, dist, None, newcameramtx)# Method 2 to undistort the image mapx,mapy=cv2.initUndistortRectifyMap(mtx, dist, None, newcameramtx, (w, h), 5)dst = cv2.remap(img, mapx, mapy, cv2.INTER_LINEAR) print("dst shape is ", dst.shape)x, y, w, h = roi cropped_frame = dst[y:y+h, x:x+w] print("cropped_frame shape is ", cropped_frame.shape)# Displaying the undistorted image cv2.imshow("distorted image", img) cv2.imshow("undistorted image", dst) cv2.imshow("cropped image", cropped_frame) cv2.waitKey(0)#計算誤差 mean_error = 0 for i in range(len(objpoints)):imgpoints2, _ = cv2.projectPoints(objpoints[i], rvecs[i], tvecs[i], mtx, dist)error = cv2.norm(imgpoints[i], imgpoints2, cv2.NORM_L2)/len(imgpoints2)mean_error += errorprint("mean error: ", mean_error/len(objpoints))

外參的求解：

import cv2 import numpy import numpy as nptwod_point = "1189 254 1216 254 1242 255 1268 255 1294 256 1322 254 1347 255 1375 253 1187 279 1214 " \"278 1243 281 1269 279 1296 280 1321 279 1346 281 1374 280" twod_point = np.array(twod_point.split( ),dtype=np.float) print(len(twod_point)//2) twod_point = twod_point.reshape((16,2)) print(twod_point.shape)threed_point = np.zeros(shape=(16,3)) for i in range(len(twod_point)):x = twod_point[i][0]y = twod_point[i][1]threed_point[i][0] = (x -1189)/27 *4threed_point[i][1] = (y - 254)/25 *4 newcameramtx=[[1.67276660e+03,0.00000000e+00,6.22925147e+02],[0.00000000e+00,1.67104004e+03,3.44706162e+02],[0.00000000e+00,0.00000000e+00,1.00000000e+00]] dist=[[-6.11401280e-01 ,4.08011840e-01 ,-5.83043580e-05 ,-1.16998902e-03,-7.99593012e-01]] newcameramtx = np.array(newcameramtx,dtype=np.uint8) dist = np.array(dist,dtype=np.uint8) _,r,t = cv2.solvePnP(threed_point,twod_point,newcameramtx,dist) np.set_printoptions(suppress=True)def Pix2World(point2D, rVec, tVec, cameraMat, height):"""Function used to convert given 2D points back to real-world 3D pointspoint2D : An array of 2D pointsrVec : Rotation vectortVec : Translation vectorcameraMat: Camera Matrix used in solvePnPheight : Height in real-world 3D spaceReturn : output_array: Output array of 3D points"""point3D = []point2D = (np.array(point2D, dtype='float32')).reshape(-1, 2)numPts = point2D.shape[0]point2D_op = np.hstack((point2D, np.ones((numPts, 1))))rMat = cv2.Rodrigues(rVec)[0]# print(rMat)rMat_inv = np.linalg.inv(rMat)kMat_inv = np.linalg.inv(cameraMat)for point in range(numPts):uvPoint = point2D_op[point, :].reshape(3, 1)tempMat = np.matmul(rMat_inv, kMat_inv)tempMat1 = np.matmul(tempMat, uvPoint)tempMat2 = np.matmul(rMat_inv, tVec)s = (height + tempMat2[2]) / tempMat1[2]p = tempMat1 * s - tempMat2point3D.append(p)point3D = (np.array(point3D, dtype='float32')).reshape([-1, 1, 3])return point3Dframe = cv2.imread("source.jpg") # cap = cv2.VideoCapture("rtsp://admin:MLLaaa100848@192.168.1.64:554/h264/ch1/main/av_stream") # ret, frame = cap.read() drawing = False #鼠標按下為真 mode = True #如果為真，畫矩形，按m切換為曲線 px,py,ix,iy=-1,-1,-1,-1 def draw_circle(event,x,y,flags,param):global ix, iy, drawing, mode,px,py,sourceif event == cv2.EVENT_LBUTTONDOWN:drawing = Trueix, iy = x, yp3d = Pix2World([ix, iy], r, t, newcameramtx, 0)source = p3delif event == cv2.EVENT_LBUTTONUP:if drawing == True:if mode == True:px,py = x,ycv2.line(frame, (ix, iy), (x, y), (0, 255, 0), 10)p3d2 = Pix2World([px, py], r, t, newcameramtx, 0)coord = p3d2-sourcelength = round(np.sqrt(np.sum((p3d2-source) ** 2))*10,3)cv2.putText(frame,"L:"+str(length)+'mm',(int((px+ix)/2)-10,int((py+iy)/2)-10),cv2.FONT_HERSHEY_SIMPLEX, 0.5, (0, 0, 255), 2)print("-"*20,'\n',"實際長度",length,'\n','-'*20,)cv2.imshow("frame",frame)cv2.waitKey(0)cv2.namedWindow('frame') cv2.setMouseCallback("frame",draw_circle) cv2.imshow("frame", frame) k = cv2.waitKey(0)# while ret: # ret, frame = cap.read() # cv2.imshow("frame",frame) # k = cv2.waitKey(0) # # if cv2.waitKey(1) & 0xFF == ord('q'): # # break # if k == ord('s'): # break cv2.destroyAllWindows() # cap.release()

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