Problem 3 (30 marks) In this question, the target image involved in the landmark registration is a scaled version of the source image as shown in Fig. 2. The scaling factors in the vertical and horizontal dimensions are the same. (Hints: Thus, no translation and rotational operations are involved). N landmarks Pr (n = 1, 2, ..., N) in the source image and corresponding landmarks p. (n = 1, 2, ...., N) in the target image have been identified (e.g., black '+' landmarks in Fig. 2). The 2D coordinates of these two sets of landmarks are denoted by: in in n Pn = 1 and [ in N The goal of this problem is to find out the optimal scaling operation based on the landmarks to register the source image with the target image. You can express your answers in terms of the following constants: Si = En in S; = EN-LN S, = im S: = ENiinin Sije = EN1 inih Soje = Eindh Siv; = EN=1%jn Source image Target image n=1 'n S; = Su = ENTRE Sjj = N2 =1 =1 Fig. 2 a. (10 marks) The cost function C quantifies the mean-squared distance between the target landmarks and the transformed source landmarks. Express C for this problem in terms of the coordinates of target landmarks, (2) the source landmarks, (inin), and the scaling factor required for the registration, denoted by k. b. (20 marks) Find k that minimizes derived in Parta of this question. Problem 3 (30 marks) In this question, the target image involved in the landmark registration is a scaled version of the source image as shown in Fig. 2. The scaling factors in the vertical and horizontal dimensions are the same. (Hints: Thus, no translation and rotational operations are involved). N landmarks Pr (n = 1, 2, ..., N) in the source image and corresponding landmarks p. (n = 1, 2, ...., N) in the target image have been identified (e.g., black '+' landmarks in Fig. 2). The 2D coordinates of these two sets of landmarks are denoted by: in in n Pn = 1 and [ in N The goal of this problem is to find out the optimal scaling operation based on the landmarks to register the source image with the target image. You can express your answers in terms of the following constants: Si = En in S; = EN-LN S, = im S: = ENiinin Sije = EN1 inih Soje = Eindh Siv; = EN=1%jn Source image Target image n=1 'n S; = Su = ENTRE Sjj = N2 =1 =1 Fig. 2 a. (10 marks) The cost function C quantifies the mean-squared distance between the target landmarks and the transformed source landmarks. Express C for this problem in terms of the coordinates of target landmarks, (2) the source landmarks, (inin), and the scaling factor required for the registration, denoted by k. b. (20 marks) Find k that minimizes derived in Parta of this