Using Taylor series in 1D Grayscale Image

Question

Could someone point me in the direction of how to solve this?

I = [I1, . . . , In] is a 1D grayscale image and D = [D1, . . . , Dn] represents the second derivative of I. I am given the four pixel intensities I1, I2, In−1, In] and the second derivative values D3, . . . , Dn−2. How would I compute the rest of I’s intensities?

What I have tried so far: Each Ix intensity can be approximated via Taylor series: Ix = I(0) + x(dI(o)/dx) + (1/2)x^2(d^2I(x)/dx^2).

I am sure the trick is in using the sliding window algorithm for fitting a 2nd degree polynomial, and that a matrix is involved in solving. However, I am unsure how big the sliding window should be (3, 5 pixels?), etc.

Answer 1

I am sure these are finite differences, so the first "derivatives" are just

$B_i := I_{i+1} - I_i$

and the second "derivatives" are the differences of those:

$D_i := B_{i+1} - B_i$

Write it out and solve for $I_i$.