In most modern medical imaging systems, images are measured at discrete grid poi...

In most modern medical imaging systems, images are measured at discrete grid points. Interpolation is needed when we want to treat the images as functions defined on a continuous domain. For 1-D signals, linear interpolation is one of the simplest. It can easily be extended to bilinear interpolation in 2-D and trilinear interpolation in 3-D. Linear interpolation is defined as

In 2-D, bilinear interpolation for f (p) from f (A), f (B), f (C), and f (D) is done in the following steps (see Figure P3.3):

• Using linear interpolation to get f (E) from f (A) and f (B).

• Using linear interpolation to get f (F) from f (C) and f (D).

• Using linear interpolation to get f (P) from f (E) and f (F).

(a) Derive an explicit expression for f (P) using f (A), f (B), f (C), and f (D).

(b) Prove that the results are the same whether we get f (P) from f (E) and f (F), or from f (G) and f (H).

(c) In Problem 3.25, if we take measurements on rectangular grids in the image plane at are integers, how should we interpolate the value for in the physical domain?

Reference: Problem 3.25

A medical imaging system has a geometric distortion, which is well modeled as

Where x and y are the coordinates in the image plane and ξ and η are coordinates in the physical domain.

(a) If we take measurements on rectangular grids in the image plane, find a way to correct the geometric distortion.

(b) If we want to take measurements on rectangular grids in the physical domain, on what points should we take measurements in the image domain?