ME6406-Homework 3 Solved

1.      Camera Model and Calibration 
a)   Camera Model. Write a program (CameraModel.m) to transform the image ‘*’ (represented by 20 feature points in table 1) from the 3D world coordinate (XwYwZw) to the 2D undistorted image coordinate (uv) (Fig. 1). Use [Rx(135°)], T=[3 3.5 7.5]T, f=1.3 to illustrate your solutions. Determine and show these 20 feature points in the uv plane. Save the (Xw, Yw) and (u, v) values in camera_calibration_data.mat for b). 

b)  Camera Calibration. Write a program (CameraCalibration.m) to calibrate the camera. Using the above data saved in camera_calibration_data.mat. Compute f, [R], T. 

                    

Xw 
−2 
−1 
0 
1 
2 
Yw 
0 
0 
0 
0 
0 
Zw 
0 
0 
0 
0 
0 
Xw 
3 
−2 
−1 
0 
1 
Yw 
0 
1 
1 
1 
1 
Zw 
0 
0 
0 
0 
0 
Xw 
2 
3 
−1 
−2 
−1 
Yw 
1 
1 
2 
2 
3 
Zw 
0 
0 
0 
0 
0 
Xw 
−2  
−1 
−2 
−1 
−2 
Yw 
3 
4 
4 
5 
5 
Zw 
0 
0 
0 
0 
0 
             Table 1 Camera calibration points 


 

                                          Fig. 1 Camera model and calibration            

2.      Robot Eye-on-Hand Calibration 
Fig. 2a shows the setup for performing an eye-on-hand calibration where a stationary planar calibration board is viewed at 3 different locations by a camera mounted on a robot gripper. Fig. 2b shows the images in three camera image planes. The transformation matrices from CW to Ci can be determined by the camera calibration ([Hci] where i=1, 2, 3). The rigid body transformations of the robot gripper from Station 1 to 2 and 2 to 3 ([Hg12] and [Hg23]) are given by the robot controller. Write a MATLAB program for the eye-an-hand calibration. Using the given ([Hc1] [Hc2] [Hc3]) data in ‘robot_hand_eye_data.mat’ to illustrate your solutions:  

a)   Compute ([Rc12], Tc12) and ([Rc23], Tc23). 

b)  Obtain the equivalent angle-axis representation (n, θ) for each of the rotation matrixes; [Rc12], [Rc23], [Rg12] and [Rg23].  

c)   Compute Pc12, Pc23, Pg12 and Pg23. Check your solutions by computing [Rg12] and [Rg23] using Equations (8) and (10) in [2] and comparing with those given in the data file ‘robot_hand_eye_data.mat’. 

d)  Use the procedure in [2] to compute Pcg, [Rcg] and Tcg. 

  

3.      Ellipse-Circle Correspondence 
A circle captured by a camera (with focal length f=0.8690cm) in the image plane has the following general ellipse equation, Au2 +2Buv Cv+ 2 +2Du+2Ev F+ =0 . The coefficients are given in file ‘coef2021.mat’, and the circle radius r = 6.5cm. Where is the center of the circle with respect to the camera frame? Find the plane equation (with respect to the camera frame) that contains the circle. (Without additional information, multiple solutions are possible. Find all valid solutions). 

  

4.      Morphology 
Use the following steps and the structure element in Fig.3b to denoise the ‘Fingerprint.jpg’ image.  

•       AӨB 

•      (AӨB)B 

•      [(AӨB)B] B 

•      {[(AӨB)B] B}ӨB 

  

Show the corresponding images obtained after all 4 operations.