Homework: Due Week 2

1.

$$x \in [0,1]$$ B₁(x) = 1 (1)

$$x \in [1,4]$$ B₂(x) = 1 (2)

$$= {\rm linear} x \in [0,1]$$ B₃(x) (3)

$$= {\rm linear} x \in [0,4]$$ B₄(x) (4)

$$= {\rm linear} x \in [1,4]$$ B₅(x) (5)

Where ${\rm linear}$ is the linear interpolation basis over the three points, 0,1,4. Project f(x) = x² and g(x) = x³ - 5x² + 4x onto B_i. Which sets of basis function B_1,2 or B_3,4,5 work better. What does better mean?

2.

Look at your Monte Carlo solutions for the Cornell box. What sort of basis functions would capture the radiance function well? Why?