[FRIAM] Fourier image space, duality, filtering and compression

[Angel Edward]

Sorry I missed the discussion. I was in Vancouver all week for SIGGRAPH.

A few comments after looking at the two websites. I taught Fourier transforms for many years both through digital signal processing and though optics as part of a course on hybrid processing at the Institute of Optics at Rochester.
 
The main problem with these sites is that they fail to distinguish between the continuous Fourier transform and the discrete Fourier transform, which have major differences that affect what you see in the examples. In particular, you need to talk about sampling and reconstruction to talk about the DFT. The Nyqist sampling theorem is key to understanding the effects of windowing in space or frequency, the uncertainty principle for the transforms, the limits of displays and the artifacts due the windows. In the transforms in the second site, I’m pretty certain that all the dominant frequencies in the transforms are window artifacts and not due to the data. At least the first site gives you the option of applying a spatial window to the data although it doesn’t give you a way to understand some of the displays you can generate with window applied to band limited data.

There is an Appendix in my books about the FT and I have some notes from a summer short course at the Institute of Optics that cover these issues.

Ed
__________

Ed Angel

Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab)
Professor Emeritus of Computer Science, University of New Mexico

1017 Sierra Pinon
Santa Fe, NM 87501
505-984-0136 (home)		 	edward.angel at gmail.com
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> On Aug 12, 2022, at 2:09 PM, Stephen Guerin <stephen.guerin at simtable.com> wrote:
> 
> Jon, Mike and I were talking about fourier transforms on images, JPEG compression and filtering at this morning's FRIAM.
> 
> Here's a nice site that lets you load an image, and its fourier transform (amplitude spectrum) in cartesian or log-polar.
>      https://www.djmannion.net/img_freq_web/ <https://www.djmannion.net/img_freq_web/>
> 
>  A menu allows you to pull up sample images, simple sine waves in various frequencies and orientation and a live webcam image to see their point dualities in fourier space.
> 
> The lower panel lets you filter high and low cutoffs (much like jpeg compression) to see the impact on the image.
> 
> links related to compression:
>   https://blog.demofox.org/2020/11/04/frequency-domain-image-compression-and-filtering/ <https://blog.demofox.org/2020/11/04/frequency-domain-image-compression-and-filtering/>
>   https://micro.magnet.fsu.edu/primer/java/digitalimaging/processing/fouriertransform/ <https://micro.magnet.fsu.edu/primer/java/digitalimaging/processing/fouriertransform/>
> 
> different but related use of fourier transform for finding epicycles in a drawing:
> https://www.jezzamon.com/fourier/ <https://www.jezzamon.com/fourier/>  
> https://www.myfourierepicycles.com/ <https://www.myfourierepicycles.com/>
> 
> -S
> 
> 
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