Least Squares Fitting

Overview

A1. Best Fit Line Using Least Squares

This task explores two approaches for fitting a line to data points from line data 2.txt:

• Non-Homogenous Least Squares
  • Minimizes vertical distance to points
  • Solved using MATLAB with slope-intercept form: y = mx + b
  • Implemented using matrix formulation Ux = y and solved with x = U \ y
• Homogenous Least Squares
  • Minimizes perpendicular distance to points
  • Solves for line ax + by = d using eigen decomposition on centered data
  • Eigenvector with smallest eigenvalue determines line orientation

Observations:

• Homogenous least squares produced a visually better fit for noisy data.
• RMSE was calculated with respect to both vertical and perpendicular distances:
  • Non-homogenous performed better for vertical RMSE (as expected)
  • Homogenous performed slightly better for perpendicular RMSE
• Close RMSE values raise uncertainty about the precision of perpendicular error calculations
• Notably, the non-homogenous perpendicular RMSE was slightly lower than its vertical RMSE, possibly due to the line being nearly horizontal or minor computational error.

B1. Perspective Analysis: building.jpeg

Lines were drawn over building.jpeg to assess vanishing points:

• Some sets of parallel lines converged as expected
• Others, especially those in the same plane, did not produce colinear vanishing points
• Inconsistencies:
  • Floor heights vary unnaturally
  • Suspicious object placement and reflections
• Conclusion: Image is not in true perspective and may be AI-generated or enhanced

B2. Perspective Analysis: chandelier.tiff

A symmetrical chandelier was analyzed using reference lines:

• No valid vanishing points found
• Asymmetry observed in:
  • Arm elevations
  • Relative distances from the central body
  • Misaligned central point at the chandelier’s tip
• Anomalies (e.g., a rod visually interweaving with a flat structure) are physically implausible
• Conclusion: Image is not symmetrical or in perspective and is likely AI-generated or manipulated