Constraint Propagation and Least-Squares Reconstruction

The interior of a face carries its own constraint

Every constraint is one linear equation

Solving in the least-squares sense

Checking the answer by re-rendering

The reconstruction stacks far more equations than there are unknown pixels, and a few edge classifications in the map are wrong. Why is the overdetermined formulation an advantage here rather than a problem?

A pixel lies strictly inside a flat face of a block, away from every edge. Under the formulation of this lesson, how many equations does this pixel contribute to the system, and what are they?

At some pixel the reconstructed heights are $Y(x-1, y) = 1$, $Y(x, y) = 3$ and $Y(x+1, y) = 5$. Compute the finite-difference estimate $\frac{\partial^2 Y}{\partial x^2} \approx 2Y(x, y) - Y(x+1, y) - Y(x-1, y)$ and write the number. A perfectly planar solution would give 0.