CS/RBE 549 Computer Vision Fall 2020 Name: Due: Tue 08 Sep 2020 HW #1 1. Prove that for a thin lens, the image is in focus when 1 − + 1 = 1 Reason as follows: A ray leaving the object at ⃗ = (, ) parallel to the axis passes through the lens, then bends to pass through the focal point (0, ) before hitting the image plane at ⃗ = (, ). If the image is in focus, then similarly, a ray leaving the object at ⃗passing through the negative focal point (0, −) will be bent parallel to the z axis and hit the image plane at the same point ⃗. Hint: As discussed in class, consider similar triangles from the lens to the focal point and the focal point to the image plane. There are 2 pairs of similar triangles, one each for the positive and negative focal points. Then show that − + = 2. Suppose that, in the imaging geometry above, the image plane is located distance ′ = + ∆ from the lens, so that the image is out of focus. Show that the blur circle has diameter = |∆| , where d is the lens diameter. Hint: Consider rays coming from the top and bottom of the lens that would be in focus at . What happens when they hit the image plane at ′ ? 3. A typical human eyeball is 2.4 cm in diameter and contains roughly 150,000,000 receptors. Ignoring the fovea and blind spot, assume that the receptors are uniformly distributed (they actually aren’t) across a hemisphere (it is actually closer to 160°, subtending a solid angle of 1.6 steradian rather than 2 steradian for a true hemisphere). a. How many receptors are there per mm2? ⃗ ⃗ z Positive focal point (0, ) Negative focal point (0, −) b. You observe Gompei the Goat1 on a hill 1km away. Assuming a 1m spherical goat, and using a value of f equal to the eye’s diameter, on how many receptors does the image of Gompei fall? 4. Show that a ray in the world projects to a line segment in the image as follows: Define world ray = {| = + , 0 ≤ ≤ ∞}. Show that it projects to image line segment = {| = (1 − ) + } where is the projection of onto the image plane and is the projection of ray in the limit as → ∞. You should find that ranges from 0 to 1 and is related non-linearly to . 1 Gompei Kuwada, WPI class of 1893, cared for the school’s mascot, which was a goat. WPI students and faculty continue to refer to the goat as Gompei, incorrectly it turns out. I.P.
