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.