How to Ensure a Voxel Contains Only Type B Material in Spin-Echo Imaging?

[stosw]

Homework Statement

I have an image similar to the one given here with two types of material inside of it and I need to have at least one voxel in the image is entirely type B material (The smaller inside material). I am given all the dimensions on the material A and B but not told where material B is located inside of material A. I need to design a voxel size deltaX, deltaY, deltaZ and a scanning geometry that will guarantee getting a voxel of purely tissue B.

hint: This part is mostly about geometry and the sampling theorem.

Homework Equations

Area of a cylinder?
The cylindrical structure has its long axis parallel to zhat and has a length of 5mm.

The Attempt at a Solution

I understand how the pulses create the spin-echo effects, but no idea how to set up sampling or geometry for this sort of thing. Any hints would be greatly appreciated.

 

[member 553083]

The area of a cylinder is given by A = πr^2h where r is the radius and h is the height. In this case, the height is 5mm and the radius can be determined by using the ratio of the two types of material. Assuming that for every unit of length along the zhat axis, there are equal amounts of material A and B, the ratio of their respective lengths must be 1:1. This means the radius of material B is half the radius of material A. The sampling theorem states that if we sample a signal at a frequency greater than twice the highest frequency of the signal, then it can be accurately reconstructed. Therefore, we need to choose a voxel size deltaX, deltaY, deltaZ such that it samples the signal at least twice its highest frequency. The highest frequency of the signal is determined by its sharpest edges, which in this case is the boundary between the two materials. Since the boundary has an abrupt change in signal, it has a high frequency. By choosing a voxel size deltaX, deltaY, deltaZ such that it samples the signal at least twice its highest frequency, we can guarantee that at least one voxel will contain purely type B material.