CMSC5711 Revision (1) (v.7.b) revised 17. 04.20 Q1: A 3D point X is at (in meters) in the world coordinate system. The parameters of a camera are shown below: The world coordinate system and the camera coordinate system are the same. The focal length F=6mm. Horizontal pixel width sx= Vertical pixel width sy The CCD sensor size = 10mm x 10mm. The resolution of the image captured by the camera is 500x500. The image centre is at (4.8 mm, 5.1 mm) from the right bottom corner of the image plane. The origin (1,1) of the image is at the right bottom corner. The x-coordinate is increasing from right to left, and y-coordinate is increasing from bottom to top. Find sx and sy in meters. Find the image centre (Ox, Oy) in pixels. Find the focal length in pixels. Find the 2D image position of the point X in pixels. CMSC5711 revision 1 v.8b

Q2: A 3D point is at [0.5, 0.6, 1.2]T meters from the world centre. (a) The 3D point X is rotated (by R1) around the world centre and then translated (by T=[Tx,Ty,Tz]T ) to a new position X’, where   The rotation angles are (1=1.2, 2=0, 3=0.57) (in degrees) rotating around X,Y,Z axis respectively, and the translation T is meters. Assume the world coordinate system and the camera coordinate system are the same. Calculate the position of the 3D point X’ in meters. (Hint: The entries of R are in radian, you need to convert angles to radian before use.) (b) If the rotation R1 is not rotating around the world centre but around a 3D point Y instead ( where Y is [0.45, 0.25, 1.05]T in meters from the world centre). All other parameters remain unchanged, calculate the position of the 3D point X’ in meters again. CMSC5711 revision 1 v.8b

Q3: An image X1 and a mask mask1 are shown below Find the convolution result of X1 and mask1, the result should include partial overlapping cases. Describe how to find the edge image of a gray level input image using convolution and Sobel kernel masks. CMSC5711 revision 1 v.8b

r(k) N(k) Pr(r(k)) S(k) Round off (S(k)) r(0) 135 r(1) 278 r(2) 4521 Q4: An original gray level image has resolution M=100 row and N=100 columns. The gray level resolution (L) of each pixel is 6. R(k) is the gray level of index k, N(k) is the number of pixels that have level R(k). Pr(R(k)) is the probability of the pixels in the image having gray level R(k). After histogram normalization, S(k) is the normalized gray level of index k. Discuss why histogram normalization is important in image processing. Based on the following table (you may copy it to your answer book first), fill in the blanks. Discuss the term “histogram back projection” and its applications. Find the histogram back projection of a pixel with gray level 3 in the original image r(k) N(k) Pr(r(k)) S(k) Round off (S(k)) r(0) 135   r(1) 278 r(2) 4521 r(3) 244 r(4) 3987 r(5) 1352 CMSC5711 revision 1 v.8b

Q5: In a stereo system, a 3-D Point P1=[X1,Y1,Z1]T is in the left (reference) camera coordinate system and its projection is q1 in the left image. And this point becomes P2 =[X2,Y2,Z2]T in the right camera coordinate system and its projection is q2 in the right image. q1 and q2 are in homogenous coordinates. P1 and P2 are related by P2=R*P1 +T, where R and T are the rotation and translation of the right camera respectively. Draw the diagram depicting the two cameras, the point P1 in 3D, image projections of P1 in both images, the epipoles, the epipolar lines in the right image of two left image feature points ‘a’ and ‘b’ (you are free to select any two image points). Write the Essential matrix E in term of R and T. Write the relation of q1, q2, T and R. Discuss the relation between essential matrix E and fundamental matrix F. Use formulas to illustrate your answer. Describe the algorithm to find F. CMSC5711 revision 1 v.8b

CMSC5711 revision 1 v.8b

Q7: Find the edge image of Image (I) if we use Prewitt masks and the threshold is set to be 0.5. Show your calculation steps. (Consider cases only when the mask and image are fully overlapped).   CMSC5711 revision 1 v.8b

The initial window is centred at (x,y)=(4,5) as shown above. Q8: As shown above, an object has pixels of gray levels 1 or above. The empty cells have gray level 0. A mean shift algorithm is applied to find the position of the object of size around 5x5 with gray level 1 or above. That means you need to find the centre of a 5x5 window covering pixels of the object.  The initial window is centred at (x,y)=(4,5) as shown above. Describe the procedure of finding the object by a mean-shift algorithm. Find the location of the centre of the 5x5 box at each step of the mean-shift algorithm. Round off numbers to integers during your calculations. CMSC5711 revision 1 v.8b