Depth Estimation from Stereo Disparity and Astronomical Parallax

Overview

A. Disparity Map Computation from Stereo Images

This section focuses on computing disparity maps from random dot stereograms using stereo image pairs (left1/right1 and left2/right2).

Method:

• Maximum disparity: D = 15
• Local context window size: C = 10
• Total margin: D + C = 25
• Disparity computed by:
  • Normalizing row vectors from stereo images.
  • Finding the dot product to determine best-matching disparity.
  • Recording disparity with the highest dot product.
• MATLAB’s imagesc() function visualizes the results (lighter pixels = higher disparity).

Results:

• Average disparity (top 10% of highest values):
  • Set 1: 10.3035
  • Set 2: 10.3275
• Estimated distances using the parallax depth formula ( z = \frac{f(d - D)}{D} ):
  • Focal length: f = 2 cm, Interocular distance: d = 10 cm
  • Pixel size: p = 0.025 cm
  • Converted disparity: D ≈ 0.25 cm

  Set	Distance (d ≫ D)	Distance (d − D)
  1	77.6432 cm	75.6432 cm
  2	77.4631 cm	75.4631 cm

Note: Both distance methods are acceptable as discussed in class and Piazza.

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B. Star Distance Estimation Using Astronomical Parallax

This section uses parallax and known imaging parameters to estimate the distance to a star.

Parameters:

• Focal length: f = 2 m
• Interocular distance (Earth’s orbit diameter): d = 80,000,000 miles = 1.2875×10¹¹ m
• Measured disparity: D = 6.2×10⁻⁶ m

Calculations:

• [ z = \frac{f \cdot d}{D} = \frac{2 \cdot 1.2875 \times 10^{11}}{6.2 \times 10^{-6}} = 4.1531 \times 10^{16} \text{ m} ]
• Converted to miles: [ z \approx 2.5806 \times 10^{13} \text{ miles} ]
• Converted to light-years: [ z \approx \frac{2.5806 \times 10^{13}}{5.879 \times 10^{12}} \approx 4.39 \text{ light-years} ]

Interpretation:

• The estimated distance corresponds closely to Alpha Centauri (either Rigil Kentaurus or Toliman).
• Their known distance: 4.3441 ± 0.0022 light-years.
• These stars form a binary system, often treated as one in early measurements.