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Unlocking the Interreflection Puzzle: Matrix Formulation of Shape Estimation

Delve into the complex world of interreflections in computer graphics vision, exploring how to estimate shapes of concave surfaces amidst interreflection effects. Understand the relationship between shape and interreflections, solving the inverse problem using Radiance and the Interreflection Kernel. Learn about pseudo-shape and reflectance from Photometric Stereo and iterative refinement techniques. Discover key assumptions and observations that guide the shape estimation process.

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Unlocking the Interreflection Puzzle: Matrix Formulation of Shape Estimation

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  1. Interreflections : The Inverse Problem Lecture #9 Thanks to Shree Nayar, Seitz et al, Levoy et al, David Kriegman

  2. Email introduction section by tomorrow (Thurs. 22) at 11:59pm • Still exploring ideas? Take a look at Ramesh Raskar’s idea hexagon: https://www.slideshare.net/cameraculture/raskar-ideahexagonapr2010

  3. Graphics

  4. Vision: Estimating Shape of Concave Surfaces Shape from Photometric Stereo Actual Shape Need to account for Interreflections!!

  5. Shape and Interreflections: Chicken and Egg • If we remove the effects of interreflections, • we know how to compute shape. • But, interreflections depend on the shape!! • So, which comes first?

  6. Linear System of Radiosity Equations - RECAP Known Known Unknown • Matrix Inversion to Solve for Radiosities.

  7. Use Radiance instead of Radiosity • Vision shape-from-intensity algorithms work on Radiance. • Assume image pixel covers infinitesimal scene patch area . • Form factor is called “Interreflection Kernel”, K (better name). Radiance of a facet is given by the linear combination of radiances from other facets. Loosely, we can say, this weighted averaging in the direction of concave curvature.

  8. Why do concavities appear shallow? Radiance of a facet is given by the linear combination of radiances from other facets. Loosely, we can say this weighted averaging in the direction of concave curvature.

  9. Matrix Form of Interreflection Equation

  10. Pseudo Shape and Reflectance • Apply any shape from intensity algorithm • ignoring interreflections! Pseudo-shape from Photometric Stereo Actual Shape • KEY IDEA: The pseudo shape and reflectance (albedo) is • related to the actual shape and reflectance.

  11. Pseudo Shape/Reflectance from Photometric Stereo Facet Matrix : Source direction Three Source Directions :

  12. Pseudo Shape/Reflectance from Photometric Stereo Three Source Directions : Pseudo Shape :

  13. Key Observations • Pseudo Shape and Albedos are independent of source • direction! This allows us to reconstruct actual shape. • Pseudo Facets: Lambertian! • “Smoothed” versions of actual facets (shallow) • Pseudo albedos may be greater than 1.

  14. Iterative Refinement of Shape and Albedos • Start with pseudo shape and albedos as initial guesses. • Compute Interreflection Kernel K, and Albedo matrix P. • Iterate until convergence.

  15. Important Assumptions and Observations • Any shape-from-intensity method can be used. • Assumes shape is continuous (for integrability). • All facets contributing to interreflections must be • visible to sensor. • Facets are infinitesimal lambertian patches. • Complexity: • Convergence shown for 2 facets. • Does not always converge to the right facet • for large tilt angles (> 70). (M Iterations, n facets)

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