Iterative Optimization in Inverse Problems

Background 11

Fourier coefficients will reconstruct g(x) on the interval [−2L, 2L]; this

DFT estimate is

g

DFT

(x)=

M



m=−M

a

m

e

imπx/2L

, (1.15)

for |x|≤2L. This will give us a reconstruction of f(x)itselfoverthe

interval [−L, L], but will also give us a reconstruction of the rest of g(x),

which we already know to be zero. So we are wasting the additional data

by reconstructing g(x) instead of f (x). We need to use our prior knowledge

that g(x)=0forL<|x|≤2L.

We want to use the prior knowledge that g(x)=0forL<|x|≤2L

to improve our reconstruction. Suppose that we take as our reconstruction

the modified DFT (MDFT) [40]:

f

MDFT

(x)=

M



j=−M

b

j

e

ijπx/2L

, (1.16)

for |x|≤L,