Camera Calibration and Lambertian Rendering

Overview

A. Finding the Camera Matrix

Using homogeneous least squares, a 3×4 camera matrix M was estimated. The matrix was derived by:

1. Constructing matrix P,
2. Computing U = PᵗP,
3. Extracting the smallest eigenvector,
4. Reshaping it into matrix M.

The accuracy of M was evaluated by comparing projected image points to ground truth.

Results:

• RMSE of M: 3.68
• RMSE of HW04 matrices:
  • Camera 1: 23.9
  • Camera 2: 40.7

This confirms that our matrix M gives the most accurate projection among all.

Note: The calibration minimizes Pm = 0 with |m| = 1 constraint and not the direct re-projection error.

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B. Rendering a Lambertian Sphere

Using the computed camera matrix M, we rendered a Lambertian surface sphere in 3D space.

Setup:

• Camera position: (9, 14, 11)
• Light position: (33, 29, 44)
• Sphere model: Generated via nested loop mesh
• Shading: Based on surface normals and dot product with light vector
• Blue channel: Encodes luminance, scaled by albedo (set to 4)

Alternate Lighting Scenario

• Light moved to (-30, 0, 0)
• Sphere shading adjusts to reflect new light direction
• Realistic darkening of shadowed regions

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Conclusion

This assignment demonstrates a practical approach to camera matrix estimation and realistic rendering using Lambertian reflectance. The computed camera matrix significantly improves projection accuracy, and the rendering logic reflects correct lighting behavior.