1 Camera Calibration {#tutorial_py_calibration} 2 ================== 3 4 Goal 5 ---- 6 7 In this section, 8 - We will learn about distortions in camera, intrinsic and extrinsic parameters of camera etc. 9 - We will learn to find these parameters, undistort images etc. 10 11 Basics 12 ------ 13 14 Today's cheap pinhole cameras introduces a lot of distortion to images. Two major distortions are 15 radial distortion and tangential distortion. 16 17 Due to radial distortion, straight lines will appear curved. Its effect is more as we move away from 18 the center of image. For example, one image is shown below, where two edges of a chess board are 19 marked with red lines. But you can see that border is not a straight line and doesn't match with the 20 red line. All the expected straight lines are bulged out. Visit [Distortion 21 (optics)](http://en.wikipedia.org/wiki/Distortion_%28optics%29) for more details. 22 23 ![image](images/calib_radial.jpg) 24 25 This distortion is represented as follows: 26 27 \f[x_{distorted} = x( 1 + k_1 r^2 + k_2 r^4 + k_3 r^6) \\ 28 y_{distorted} = y( 1 + k_1 r^2 + k_2 r^4 + k_3 r^6)\f] 29 30 Similarly, another distortion is the tangential distortion which occurs because image taking lense 31 is not aligned perfectly parallel to the imaging plane. So some areas in image may look nearer than 32 expected. It is represented as below: 33 34 \f[x_{distorted} = x + [ 2p_1xy + p_2(r^2+2x^2)] \\ 35 y_{distorted} = y + [ p_1(r^2+ 2y^2)+ 2p_2xy]\f] 36 37 In short, we need to find five parameters, known as distortion coefficients given by: 38 39 \f[Distortion \; coefficients=(k_1 \hspace{10pt} k_2 \hspace{10pt} p_1 \hspace{10pt} p_2 \hspace{10pt} k_3)\f] 40 41 In addition to this, we need to find a few more information, like intrinsic and extrinsic parameters 42 of a camera. Intrinsic parameters are specific to a camera. It includes information like focal 43 length (\f$f_x,f_y\f$), optical centers (\f$c_x, c_y\f$) etc. It is also called camera matrix. It depends on 44 the camera only, so once calculated, it can be stored for future purposes. It is expressed as a 3x3 45 matrix: 46 47 \f[camera \; matrix = \left [ \begin{matrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{matrix} \right ]\f] 48 49 Extrinsic parameters corresponds to rotation and translation vectors which translates a coordinates 50 of a 3D point to a coordinate system. 51 52 For stereo applications, these distortions need to be corrected first. To find all these parameters, 53 what we have to do is to provide some sample images of a well defined pattern (eg, chess board). We 54 find some specific points in it ( square corners in chess board). We know its coordinates in real 55 world space and we know its coordinates in image. With these data, some mathematical problem is 56 solved in background to get the distortion coefficients. That is the summary of the whole story. For 57 better results, we need atleast 10 test patterns. 58 59 Code 60 ---- 61 62 As mentioned above, we need atleast 10 test patterns for camera calibration. OpenCV comes with some 63 images of chess board (see samples/cpp/left01.jpg -- left14.jpg), so we will utilize it. For sake of 64 understanding, consider just one image of a chess board. Important input datas needed for camera 65 calibration is a set of 3D real world points and its corresponding 2D image points. 2D image points 66 are OK which we can easily find from the image. (These image points are locations where two black 67 squares touch each other in chess boards) 68 69 What about the 3D points from real world space? Those images are taken from a static camera and 70 chess boards are placed at different locations and orientations. So we need to know \f$(X,Y,Z)\f$ 71 values. But for simplicity, we can say chess board was kept stationary at XY plane, (so Z=0 always) 72 and camera was moved accordingly. This consideration helps us to find only X,Y values. Now for X,Y 73 values, we can simply pass the points as (0,0), (1,0), (2,0), ... which denotes the location of 74 points. In this case, the results we get will be in the scale of size of chess board square. But if 75 we know the square size, (say 30 mm), and we can pass the values as (0,0),(30,0),(60,0),..., we get 76 the results in mm. (In this case, we don't know square size since we didn't take those images, so we 77 pass in terms of square size). 78 79 3D points are called **object points** and 2D image points are called **image points.** 80 81 ### Setup 82 83 So to find pattern in chess board, we use the function, **cv2.findChessboardCorners()**. We also 84 need to pass what kind of pattern we are looking, like 8x8 grid, 5x5 grid etc. In this example, we 85 use 7x6 grid. (Normally a chess board has 8x8 squares and 7x7 internal corners). It returns the 86 corner points and retval which will be True if pattern is obtained. These corners will be placed in 87 an order (from left-to-right, top-to-bottom) 88 89 @sa This function may not be able to find the required pattern in all the images. So one good option 90 is to write the code such that, it starts the camera and check each frame for required pattern. Once 91 pattern is obtained, find the corners and store it in a list. Also provides some interval before 92 reading next frame so that we can adjust our chess board in different direction. Continue this 93 process until required number of good patterns are obtained. Even in the example provided here, we 94 are not sure out of 14 images given, how many are good. So we read all the images and take the good 95 ones. 96 97 @sa Instead of chess board, we can use some circular grid, but then use the function 98 **cv2.findCirclesGrid()** to find the pattern. It is said that less number of images are enough when 99 using circular grid. 100 101 Once we find the corners, we can increase their accuracy using **cv2.cornerSubPix()**. We can also 102 draw the pattern using **cv2.drawChessboardCorners()**. All these steps are included in below code: 103 104 @code{.py} 105 import numpy as np 106 import cv2 107 import glob 108 109 # termination criteria 110 criteria = (cv2.TERM_CRITERIA_EPS + cv2.TERM_CRITERIA_MAX_ITER, 30, 0.001) 111 112 # prepare object points, like (0,0,0), (1,0,0), (2,0,0) ....,(6,5,0) 113 objp = np.zeros((6*7,3), np.float32) 114 objp[:,:2] = np.mgrid[0:7,0:6].T.reshape(-1,2) 115 116 # Arrays to store object points and image points from all the images. 117 objpoints = [] # 3d point in real world space 118 imgpoints = [] # 2d points in image plane. 119 120 images = glob.glob('*.jpg') 121 122 for fname in images: 123 img = cv2.imread(fname) 124 gray = cv2.cvtColor(img,cv2.COLOR_BGR2GRAY) 125 126 # Find the chess board corners 127 ret, corners = cv2.findChessboardCorners(gray, (7,6),None) 128 129 # If found, add object points, image points (after refining them) 130 if ret == True: 131 objpoints.append(objp) 132 133 cv2.cornerSubPix(gray,corners,(11,11),(-1,-1),criteria) 134 imgpoints.append(corners) 135 136 # Draw and display the corners 137 cv2.drawChessboardCorners(img, (7,6), corners2,ret) 138 cv2.imshow('img',img) 139 cv2.waitKey(500) 140 141 cv2.destroyAllWindows() 142 @endcode 143 One image with pattern drawn on it is shown below: 144 145 ![image](images/calib_pattern.jpg) 146 147 ### Calibration 148 149 So now we have our object points and image points we are ready to go for calibration. For that we 150 use the function, **cv2.calibrateCamera()**. It returns the camera matrix, distortion coefficients, 151 rotation and translation vectors etc. 152 @code{.py} 153 ret, mtx, dist, rvecs, tvecs = cv2.calibrateCamera(objpoints, imgpoints, gray.shape[::-1],None,None) 154 @endcode 155 ### Undistortion 156 157 We have got what we were trying. Now we can take an image and undistort it. OpenCV comes with two 158 methods, we will see both. But before that, we can refine the camera matrix based on a free scaling 159 parameter using **cv2.getOptimalNewCameraMatrix()**. If the scaling parameter alpha=0, it returns 160 undistorted image with minimum unwanted pixels. So it may even remove some pixels at image corners. 161 If alpha=1, all pixels are retained with some extra black images. It also returns an image ROI which 162 can be used to crop the result. 163 164 So we take a new image (left12.jpg in this case. That is the first image in this chapter) 165 @code{.py} 166 img = cv2.imread('left12.jpg') 167 h, w = img.shape[:2] 168 newcameramtx, roi=cv2.getOptimalNewCameraMatrix(mtx,dist,(w,h),1,(w,h)) 169 @endcode 170 #### 1. Using **cv2.undistort()** 171 172 This is the shortest path. Just call the function and use ROI obtained above to crop the result. 173 @code{.py} 174 # undistort 175 dst = cv2.undistort(img, mtx, dist, None, newcameramtx) 176 177 # crop the image 178 x,y,w,h = roi 179 dst = dst[y:y+h, x:x+w] 180 cv2.imwrite('calibresult.png',dst) 181 @endcode 182 #### 2. Using **remapping** 183 184 This is curved path. First find a mapping function from distorted image to undistorted image. Then 185 use the remap function. 186 @code{.py} 187 # undistort 188 mapx,mapy = cv2.initUndistortRectifyMap(mtx,dist,None,newcameramtx,(w,h),5) 189 dst = cv2.remap(img,mapx,mapy,cv2.INTER_LINEAR) 190 191 # crop the image 192 x,y,w,h = roi 193 dst = dst[y:y+h, x:x+w] 194 cv2.imwrite('calibresult.png',dst) 195 @endcode 196 Both the methods give the same result. See the result below: 197 198 ![image](images/calib_result.jpg) 199 200 You can see in the result that all the edges are straight. 201 202 Now you can store the camera matrix and distortion coefficients using write functions in Numpy 203 (np.savez, np.savetxt etc) for future uses. 204 205 Re-projection Error 206 ------------------- 207 208 Re-projection error gives a good estimation of just how exact is the found parameters. This should 209 be as close to zero as possible. Given the intrinsic, distortion, rotation and translation matrices, 210 we first transform the object point to image point using **cv2.projectPoints()**. Then we calculate 211 the absolute norm between what we got with our transformation and the corner finding algorithm. To 212 find the average error we calculate the arithmetical mean of the errors calculate for all the 213 calibration images. 214 @code{.py} 215 mean_error = 0 216 for i in xrange(len(objpoints)): 217 imgpoints2, _ = cv2.projectPoints(objpoints[i], rvecs[i], tvecs[i], mtx, dist) 218 error = cv2.norm(imgpoints[i],imgpoints2, cv2.NORM_L2)/len(imgpoints2) 219 tot_error += error 220 221 print "total error: ", mean_error/len(objpoints) 222 @endcode 223 Additional Resources 224 -------------------- 225 226 Exercises 227 --------- 228 229 -# Try camera calibration with circular grid. 230