Calculate parallel projection given function f(x,y)

[kmoh111]

Homework Statement

Calculate the parallel projection on an infinite object defined by:

f(x,y) = cos(2pi(2x+y)) from the angle phi = 45 degrees.

Hint: Use the Central Slice Theorem and Fourier Transform (FT) of f(x,y).
2nd Hint: On a 2D image in Fourier space, delta functions are only points. See whether these points are measured by the 45-degree line and then take FT inverse.

Homework Equations

The Attempt at a Solution

My attempt at an answer:
From Fourier Transfer (FT) pairs we know that the FT of cos(2pix) is a dirac delta function in k-space. That is, as a function of kx and ky. The parallel projection will be the inverse FT of the dirac delta function - but this function should be in terms of r, and phi.

I'm not sure how to get the delta function as a function of kx and ky into a delta function in terms of r and phi.

Another approach is to note that cos (2pi(2x+y)) is a rotated and scaled function of cos (2pi x). The Fourier Transform will also be a rotated and scaled function of cos (2pi x) - but I'm not sure what FT properties are needed to derive the FT.

I've also attached a PDF with my attempt to solve the problem.

Thank you.

 

[mark726]

Thank you for your post. Your approach using the Central Slice Theorem and the Fourier Transform is correct. However, to obtain the parallel projection, you need to use the inverse Fourier Transform, not the inverse of the delta function.

The inverse Fourier Transform of a delta function in k-space will give you a constant function in r-space. This means that the parallel projection will be a constant value for all points along the 45-degree line in the image.

To obtain the value of this constant, you can use the FT pair for cos(2pi x) and its inverse. Then, using the properties of the Fourier Transform, you can rotate and scale the function to match the given function cos(2pi(2x+y)). Finally, taking the inverse Fourier Transform will give you the parallel projection on the 45-degree line.

I hope this helps. Please let me know if you have any further questions.