hw1 CSE 252a homework solutions with correct answers|100% correct

hw1 October 27, 2017 1 CSE 252A Computer Vision I Fall 2017 1.1 Assigment 1 This assignment contains theoretical and programming exercises. If you plan to submit hand written answers for theoretical exercises, please be sure your writing is readable and merge those in order with the final pdf you create out of this notebook. 1.2 Problem 1: Perspective Projection [5 pts] Consider a perspective projection where a point P = [x y z] T is projected onto an image plane Π′ represented by k = f ′ 0 as shown in the following figure. The first second and third coordinate axes are denoted by i,j,k respectively. Consider the projection of a ray in the world coordinate system Q = [2 -3 0] + t[0 2 1] where −∞ ≤ t ≤ −1. Calculate the coordinates of the endpoints of the projection of the ray onto the image plane. 1.3 Solution : P ′ = [ f ′ x z , f ′ y z ] T Thus, for Q = [2 -3 0] + t[0 2 1] = [2, -3 + 2t, t] Q ′ = [ f ′ 2 t , f ′ −3 + 2t t ] T where −∞ ≤ t ≤ −1 1 1.4 Problem 2: Thin Lens Equation [5 pts] An illuminated arrow forms a real inverted image of itself at a distance of w = 90cm, measured along the optical axis of a convex thin lens as shown above. The image is half the size of the object 1. How far from the object must the lens be placed? 2. What is the focal length of the lens? 3. Suppose the arrow moves x cm to the right while the lens and image plane remain fixed. This will result in an out of focus image; what is the radius of the corresponding blur circle formed from the tip of the arrow on the image plane assuming the diameter of the lens is d? 1.5 Solution 1. since the image is real, inverted, and half of the object