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.