Using Taylor series in 1D Grayscale Image

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.

• Hint: What's the definition of the second derivative? Note that you can use LaTeX here to typeset mathematics in a more readable way. See here for a short introduction.
– D.W.
Feb 28 '16 at 4:16

$B_i := I_{i+1} - I_i$
$D_i := B_{i+1} - B_i$
Write it out and solve for $I_i$.