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### How to use camera parameters to calculate world distance in cm between pixels?

I calibrated my camera using, separately, ROS, OpenCV and Matlab. People say that I need extrinsic parameters to calculate real distance between pixels in cm from the image. ROS does not provide extrinsic parameters explicitly, it provides a (4,3) projection matrix which is the output of multiplied intrinsic and extrinsic parameters.

That is why I again calibrated my camera using OpenCV and Matlab to get extrinsic parameters. Although I searched how can I calculate real distance in cm between pixels (i.e from (x1,y1) to (x2,y2)), I could not figure out how to calculate the real distance. Moreover, I did not understand that which parameters to use for distance calculation. I want to use OpenCV to calculate the distance between pixels and write the output as a txt file so that I can use this txt file to move my robot. For example, here is arrays of pixel output sample for the path,

array([[  4.484375  , 799.515625  ],
[ 44.484375  , 487.        ],
[255.296875  , 476.68261719],
[267.99707031, 453.578125  ],
[272.484375  , 306.        ],
[403.484375  , 300.515625  ],
[539.484375  , 296.515625  ],
[589.421875  , 270.00292969],
[801.109375  , 275.18554688],
[819.        , 467.515625  ]])


I want to find the real distance in cm between these pixels in the same order. OpenCv Code that calculating parameters (taken from the official documentation of opencv)

import numpy as np
import cv2
import glob

# termination criteria
criteria = (cv2.TERM_CRITERIA_EPS + cv2.TERM_CRITERIA_MAX_ITER, 30, 0.001)

cbrow = 6
cbcol = 9

# prepare object points, like (0,0,0), (1,0,0), (2,0,0) ....,(6,5,0)
objp = np.zeros((cbrow*cbcol,3), np.float32)
objp[:,:2] = np.mgrid[0:cbcol,0:cbrow].T.reshape(-1,2)

# Arrays to store object points and image points from all the images.
objpoints = [] # 3d point in real world space
imgpoints = [] # 2d points in image plane.

images = glob.glob('C:\Users\Ender\Desktop\CalibrationPhotos\*.jpg')

for fname in images:

gray = cv2.cvtColor(img,cv2.COLOR_BGR2GRAY)

# Find the chess board corners
ret, corners = cv2.findChessboardCorners(gray, (cbcol,cbrow),None)

# If found, add object points, image points (after refining them)
if ret == True:
print "%s: success" % fname
objpoints.append(objp)

cv2.cornerSubPix(gray,corners,(11,11),(-1,-1),criteria)
imgpoints.append(corners)

# Draw and display the corners
cv2.drawChessboardCorners(img, (cbcol,cbrow), corners,ret)
cv2.imshow('img',img)
cv2.waitKey(150)

else:
print "%s: failed" % fname
cv2.imshow('img',img)
cv2.waitKey(1)

cv2.destroyAllWindows()

ret, mtx, dist, rvecs, tvecs = cv2.calibrateCamera(objpoints, imgpoints, gray.shape[::-1],None,None)

print "mtx"
print mtx
print "dist"
print dist
#print "rvecs"
#print rvecs
#print "tvecs"
#print tvecs
np.savez("CalibData.npz" ,mtx=mtx, dist=dist, rvecs=rvecs, tvecs=tvecs)

#UNDISTROTION
img = cv2.imread('C:\Users\Ender\Desktop\Maze\Maze Images\Partial Maze Images-Raw\Raw7.jpg')
h,  w = img.shape[:2]
newcameramtx, roi=cv2.getOptimalNewCameraMatrix(mtx,dist,(w,h),1,(w,h))

dst = cv2.undistort(img, mtx, dist, None, newcameramtx)

# crop the image
x,y,w,h = roi
dst = dst[y:y+h, x:x+w]
cv2.imwrite('calibresult.jpg',dst)

#Re-projection Errors
total_error = 0
for i in xrange(len(objpoints)):
imgpoints2, _ = cv2.projectPoints(objpoints[i], rvecs[i], tvecs[i], mtx, dist)
error = cv2.norm(imgpoints[i],imgpoints2, cv2.NORM_L2)/len(imgpoints2)
total_error += error

print "total error: ", total_error/len(objpoints)


All in all, how can I calculate the distance in cm between pixels. Which parameters should I use to do this ? and how?