(“Screened Poisson smoothing”) Suppose we sample a function f(x) at n positions x1, x2, . . . , xn,...

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(“Screened Poisson smoothing”) Suppose we sample a function f(x) at n positions x₁, x₂, . . . , x_n, yielding a point ≡ (f(x₁), f(x₂), . . . , f(x_n)) ∈ R ⁿ. Our measurements might be noisy, however, so a common task in graphics and statistics is to smooth these values to obtain a new vector ∈ R ⁿ.

(a) Provide least-squares energy terms measuring the following: (i) The similarity of and . (ii) The smoothness of . Hint: We expect f(xi₊₁) − f(x_i) to be small for smooth f.

(b) Propose an optimization problem for using the terms above to obtain , and argue that it can be solved using linear techniques.

(c) Suppose n is very large. What properties of the matrix in 4.13b might be relevant in choosing an effective algorithm to solve the linear system?

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(“Screened Poisson smoothing”) Suppose we sample a function f(x) at n positions x1, x2, . . . , xn, yielding a point  ≡ (f(x1), f(x2), . . . , f(xn)) ∈ R n. Our measurements might be noisy, however, so a common task in graphics and statistics is to smooth these values to obtain a new vector  ∈ R n.

(“Screened Poisson smoothing”) Suppose we sample a function f(x) at n positions x 1 , x 2 , . . . , x n , yielding a point ≡ (f(x 1 ), f(x 2 ), . . . , f(x n )) ∈ R n . Our measurements might be...

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