CS549-Homework 1 Solved

In addition to the similarities and difference from Class 01, list 5 other ways in which eyes and cameras are similar. List 5 other ways in which they are different
Prove that for a thin lens, the image is in focus when
passes through the lens, then bends to pass through focal point (0, 𝑓𝑓)  before hitting the image plane at 𝑋𝑋⃗𝑐𝑐 = (𝑦𝑦𝑐𝑐, 𝑧𝑧𝑐𝑐). If the image is in focus, then similarly, a ray leaving the object at 𝑋𝑋⃗𝑤𝑤passing through the negative focal point (0, −𝑓𝑓) will be bent parallel to the z axis and hit the image plane at the same point 𝑋𝑋⃗𝑐𝑐.

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 for the positive and negative focal point. Then show that

−𝑧𝑧𝑐𝑐𝑓𝑓 + 𝑧𝑧𝑤𝑤𝑓𝑓 = 𝑧𝑧𝑐𝑐𝑧𝑧𝑤𝑤

 

Suppose that, in the imaging geometry above, the image plane is located distance 𝑧𝑧𝑐𝑐′ =
𝑧𝑧𝑐𝑐 + ∆𝑧𝑧 from the lens, so that the image is out of focus. Show that the blur circle has

|∆𝑧𝑧| diameter 𝐷𝐷 = 𝑑𝑑 𝑧𝑧𝑐𝑐 , where d is the lens diameter. 

Hint: Consider rays coming from the top and bottom of the lens that would be in focus at 𝑧𝑧𝑐𝑐.

What happens when they hit the image plane at 𝑧𝑧𝑐𝑐′?

 

A typical human eyeball is 2.4 cm in diameter and contains roughly 150,000,000 receptors.
Ignoring the fovea, assume that the receptors are uniformly distributed across a hemisphere (it is actually closer to 160°).

How many receptors are there per mm2?
Mars has a diameter of 8,000 km and an average distance from Earth of 225,000,000 km. Using a value of f equal to the eye’s diameter, on how many receptors does the image of Mars fall?
Show that a ray in the world projects to a line segment in the image as follows:
Define world ray 𝑅𝑅𝑤𝑤 = 𝑥𝑥⃗𝑤𝑤|𝑥𝑥⃗𝑤𝑤 = 𝑠𝑠⃗𝑤𝑤 + 𝛼𝛼𝑡𝑡⃗𝑤𝑤, 0 ≤ 𝛼𝛼 ≤ ∞. Show that it projects to camera line segment 𝐿𝐿𝑐𝑐 = 𝑥𝑥⃗𝑐𝑐|𝑥𝑥⃗𝑐𝑐 = (1 −𝛽𝛽)𝑠𝑠⃗𝑐𝑐 + 𝛽𝛽𝑡𝑡⃗𝑐𝑐 where 𝑠𝑠⃗𝑐𝑐 is the projection of 𝑠𝑠⃗𝑤𝑤 onto the image plane and 𝑡𝑡⃗𝑐𝑐 is the projection of ray 𝑅𝑅𝑤𝑤in the limit as 𝛼𝛼 → ∞. You should find that 𝛽𝛽 ranges from 0 to 1 and is related non-linearly to 𝛼𝛼.