A short quiz on computational stereo

Adrian F. Clark

1. When dividing two quantities that have associated errors, which of the following procedures is correct?

   • We add the absolute errors

   • We add the fractional errors

   • We subtract the fractional errors
   • We subtract the absolute errors

   When we multiply or divide quantities, we must add their fractional errors.

2. 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 (40.04 +/- 0.02) cm

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

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

   When we add quantities, we must add their errors.

3. 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 49.75--50.25 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 49.0--51.0 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.

4. Which of the following statements is generally true regarding uncertainties?

   • Uncertainties cannot be thought of as following a Gaussian distribution
   • About 67% of measurements should lie within two standard deviations
   • About 67% of measurements should lie within three standard deviations

   • About 67% of measurements should lie within one standard deviation

   About 67% of values lie within $+/- 1$ standard deviation of the mean of a Gaussian distribution, which is the distribution that most experimental uncertainties follow.

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

   • $Z = fD / B$
   • $Z = B / fD$

   • $Z = fB / D$

   • $Z = D / fB$

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

   • 10%

   • 20%
   • 30%
   • 1%

   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.

7. How does disparity vary with distance?

   • disparity is proportional to the square of distance

   • disparity is inversely proportional to distance

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

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

8. As the baseline, the distance between the cameras, increases, what happens to the disparity?

   • the disparity increases

   • the disparity stays the same
   • the product of disparity and focal length stays the same
   • the disparity decreases

   The disparity must increase as the baseline increases. You can easily show this by pointing at something and moving your head from side to side.

9. Which of the following is not an assumption we have made in considering computational stereo?

   • the optical axes of the cameras are parallel
   • the images are captured by both cameras simultaneously

   • the object is stationary

   • the cameras are identical

   The one thing we haven't assumed is that the object is stationary: this would make motion capture practically impossible!

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

    • Only nearby objects are in focus

    • Exposure times are long

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

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