A short quiz on computational stereo

Adrian F. Clark

1. In a camera, the incoming light is focussed onto:

   • the optical axis

   • the back focal plane

   • the lens
   • the front focal plane

   The place where a camera forms its image is known as the back focal plane.

2. What is disparity?

   • The foreshortening of an object with distance
   • The relative size of the same object when it is nearby and far away

   • The discrepancy in the apparent position of an object in the left and right images

   • The distance to an object in an image

   Disparity, which is essentially the same as parallax, is the `shift' in appearance between the left and right eyes' views.

3. If the diameter of a circle is measured as (2.0 +/- 0.2) m, what is the percentage error in the circle's circumference?

   • 20%

   • 10%

   • 1%
   • 30%

   The percentage error in the measurement is 100 x (0.2 / 2.0) = 10%. As the circumference is given by 2 pi r = pi d (where d is the diameter), the percentage error in the circumference is also 10%. Although pi is an irrational number and hence has an infinite number of decimal places, it is known to >10^9 decimal places which is well beyond the accuracy of any measurement we are likely to make.

4. If the diameter of a circle is measured as (2.0 +/- 0.2) m, what is the percentage error in the circle's area?

   • 20%

   • 1%
   • 10%
   • 30%

   The percentage error in the measurement is 100 x (0.2 / 2.0) = 10%. As the area is given by pi r^2 = pi d^2/4 (where d is the diameter), we are calculating the product of two quantities with errors. In this case, the fractional (percentage) errors add, so we have 10 + 10 = 20% error in the area.

5. Which of the following equations determines the distance to an object $Z$ given its disparity $D$, baseline $B$ and focal length $f$?

   • $Z = B / fD$

   • $Z = fB / D$

   • $Z = fD / B$
   • $Z = D / fB$

   This is an equation that you need to be very familiar with; if you got this wrong, make sure you learn the equation and are able to derive it!

6. Why is a pin-hole camera rarely used in practice?

   • Exposure times are long

   • The pin-hole is always too large for an image to be made
   • Only distant objects are in focus
   • Only nearby objects are in focus

   A pin-hole lets very little light into the camera, so exposure times are long.

7. How does disparity vary with distance?

   • disparity is inversely proportional to the square of distance
   • disparity is proportional to distance
   • disparity is proportional to the square of distance

   • disparity is inversely proportional to distance

   Disparity D and distance Z are related by Z = fB / D so Z is inversely proportional to D.

8. A particular measurement of the height of a building is repeated many times. The mean and standard deviation of the measurements are calculated and found to be (50.0 +/- 0.5) m. Assuming the uncertainties are Gaussian-distributed, which of the following statements is true?

   • With 95% confidence, you can say that the height of the building is in the range 49.5--50.5 m

   • With 95% confidence, you can say that the height of the building is in the range 49.0--51.0 m

   • With 95% confidence, you can say that the height of the building is in the range 48.5--51.5 m
   • With 95% confidence, you can say that the height of the building is in the range 49.75--50.25 m

   We obtain 95% confidence with 2 standard deviations (SDs) from the mean. As the SD is 0.5 m, 2 x SD = 1 m, and that corresponds to 49--51 m.

9. If the length of the sides of a brick is (20.00 +/- 0.02) cm, the length of two bricks placed side by side is

   • The result is (40.02 +/- 0.04) cm
   • The result is (40.04 +/- 0.02) cm
   • The result is (40.00 +/- 0.02) cm

   • The result is (40.00 +/- 0.04) cm

   When we add quantities, we must add their errors.

10. If the diameter of a sphere is measured as (2.0 +/- 0.2) m, what is the percentage error in the sphere's volume?

    • 10%
    • 1%
    • 20%

    • 30%

    The percentage error in the measurement is 100 x (0.2 / 2.0) = 10%. As the volume is given by 4/3 pi r^3 = pi d^3/6 (where d is the diameter), we are calculating the product of three quantities with errors. Hence, their fractional (percentage) errors add, so we have 3 x 10 = 30% error in the volume.