CS 549 -Computer Vision- HW #1 Solved

Description

 

 

1. 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.

 

2. 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 𝑋𝑋⃗^𝑐𝑐 = (𝑦𝑦^𝑐𝑐, 𝑧𝑧^𝑐𝑐). If the image is in focus, then similarly, a ray leaving the object at 𝑋𝑋⃗^𝑤𝑤passing through the negative focal point (0, −𝑓𝑓) will be bent parallel to the z axis and hit the image plane at the same point 𝑋𝑋⃗^𝑐𝑐.

 

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

−𝑧𝑧^𝑐𝑐𝑓𝑓 + 𝑧𝑧^𝑤𝑤𝑓𝑓 = 𝑧𝑧^𝑐𝑐𝑧𝑧^𝑤𝑤

 

3. Suppose that, in the imaging geometry above, the image plane is located distance 𝑧𝑧^𝑐𝑐′ =

𝑧𝑧^𝑐𝑐 + ∆𝑧𝑧 from the lens, so that the image is out of focus. Show that the blur circle has

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

 

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

What happens when they hit the image plane at 𝑧𝑧^𝑐𝑐′?

 

4. 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°).

1. How many receptors are there per mm²?
2. 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?

 

 

5. Show that a ray in the world projects to a line segment in the image as follows:

Define world ray 𝑅𝑅^𝑤𝑤 = 𝑥𝑥⃗^𝑤𝑤|𝑥𝑥⃗^𝑤𝑤 = 𝑠𝑠⃗^𝑤𝑤 + 𝛼𝛼𝑡𝑡⃗^𝑤𝑤, 0 ≤ 𝛼𝛼 ≤ ∞. Show that it projects to camera line segment 𝐿𝐿^𝑐𝑐 = 𝑥𝑥⃗^𝑐𝑐|𝑥𝑥⃗^𝑐𝑐 = (1 −𝛽𝛽)𝑠𝑠⃗^𝑐𝑐 + 𝛽𝛽𝑡𝑡⃗^𝑐𝑐 where 𝑠𝑠⃗^𝑐𝑐 is the projection of 𝑠𝑠⃗^𝑤𝑤 onto the image plane and 𝑡𝑡⃗^𝑐𝑐 is the projection of ray 𝑅𝑅^𝑤𝑤in the limit as 𝛼𝛼 → ∞. You should find that 𝛽𝛽 ranges from 0 to 1 and is related non-linearly to 𝛼𝛼.