# I Fourier transform sum of two images

1. Apr 28, 2016

### BobP

The FT decomposes images into its individual frequency components
In its absolute crudest form, would the sum of these two images (R) give the L image?

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2. Apr 28, 2016

### blue_leaf77

The sum should also have the vertical grey arm. But I just don't see how those pictures connect to the subject of FT.

3. Apr 28, 2016

### BobP

I thought the FT was about decomposing images into different frequencies. So I was showing the inverse of what I thought the FT and asking if it was correct as it was easier for me to show it this way.
Is what I showed not related?

4. Apr 28, 2016

### blue_leaf77

You can use those pictures as an analogy to an FT, because FT is about summing functions of the form $e^{ik_xx}$ (for 1D) and $e^{ik_xx}e^{ik_yy}$ (for 2D) having certain amplitude distribution.

5. Apr 28, 2016

### pixel

What are R and L?

6. Apr 28, 2016

### Khashishi

I'm not exactly sure what you are asking,. But, the Fourier transform is a linear map. So F{f(x)+g(x)} = F{f(x)} +F{g(x)}.

7. Apr 29, 2016

### BobP

Right and left (referrring to the images)

8. Apr 29, 2016

### BobP

Ok, in addition then which of the two examples shown correctly represents the transformation between the spatial domain (right) and domain (left)

I am only learning the FT to gain a very crude understanding of how image reconstruction is done. As I do not have a physics background the course organiers are not teaching the maths

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9. Apr 29, 2016

### blue_leaf77

To me, the bottom picture seems to correspond to a pair of Fourier transform conjugates (can you see why?).

10. Apr 29, 2016

### BobP

Presumably because it is symmetrical about the centre of k-space.
But as the image only displays a single frequency I wasn't sure if I only needed to display two conugate pairs or lots

By your answer am I correct in assuming you think the bottom image is correct then?

11. Apr 29, 2016

### blue_leaf77

A way of justifying the bottom images is to do the math. The right picture of the bottom pair looks like it being composed of three delta functions. Mathematically, it reads
$$f(x,y) = \delta(y) (\delta(x+a) + \delta(x) + \delta(x-a))$$
Now Fourier transform $f(x,y)$ and see if you will get something that resembles the left picture.

12. Apr 29, 2016

### BobP

Hi. Like I said I don't know any of the maths so I am learning everything conceptually.
So I don't really know what you mean by 3 delta functions :(

13. Apr 29, 2016

### blue_leaf77

FT is one subject of math, there is no other way to learn FT except by learning the maths.
Delta functions is a mathematical object usually used to represent a zero-dimensional point. In reality obviously there is no such object, however if a pinhole is much smaller compared to the wavelength of light illuminating it (in the case of the diffraction of light), it can be modeled as a delta function.

14. Apr 29, 2016

### BobP

I see. well thank you for the confirmation but as I have a very limited knowledge of calculus it seems like it'll be a long time before I can prove why the bottom image is correct.

Thanks again for your help though :)

15. Apr 29, 2016