Lecture 11: Coordinate Systems And The General Equation For X-ray CT

Question 1

Why is a general equation needed for x-ray CT?

Answer 1

An equation needs to be calculated that works at any angle, as the system rotates.

Question 2

What is the general equation for a projection?

Answer 2

Question 3

The path of each ray through the density function in a CT scanner is represented by the _____ ______ _________.

Answer 3

Dirac delta function

Question 4

What is the equation for the projection of an image in the rotated plane, using the intensity values calculated by the x-ray CT scanner?

Answer 4

I(S) = intensity
I_0 = initial intensity
P = projection
φ = rotation angle
r = radius

Question 5

Define back projection reconstruction

Answer 5

The standard method of reconstructing CT slices by converting the data from projection space to image space. This is done by inverting the general equation for the projection to find µ(x, y).

Question 6

What is the central section theorem?

Answer 6

“The 1D Fourier transform of a projection through an object is the same as a section through the 2D Fourier transform of the object itself along the Fourier conjugate of the projection.”

Question 7

What is the purpose of the central section theorem?

Answer 7

To prove that there is enough information in a projection to reconstruct µ(x, y).