[HN Gopher] Introduction to the Fourier transform for image proc...
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Introduction to the Fourier transform for image processing (2001)
Author : todsacerdoti
Score : 192 points
Date : 2023-05-15 08:28 UTC (2 days ago)
(HTM) web link (www.cs.unm.edu)
(TXT) w3m dump (www.cs.unm.edu)
| revskill wrote:
| I always think most of mathematicians is genius. They can hold
| all concepts and relationships between thems in their mind.
|
| The result is the way they explain concepts to broader audience
| mostly failed. Abstract nonsense for no purpose.
|
| I'm fine with it... until i see the application to learn more
| about theory.
| COGlory wrote:
| Are you asking what the practical application of 2D Fourier
| analysis is?
|
| I use it every single day, to correct aberrations in electron
| microscope images that otherwise obliterates high spatial
| frequency features and information.
| gtpedrosa wrote:
| Tangent since it is not exclusive to image processing, but When I
| was studying Fourier Transforms at university I recall that the
| book "Fourier Series" by Gerogi P. Tolstov helped me a lot. Just
| throwing it out there in case it helps anyone, including to
| refresh the definitions the article starts with.
| capn_duck wrote:
| I read this book too and it really drove things home. I
| remember after feeling a little bit unsure about the relevance
| of orthogonal systems, feeling the thrill of beginning to see
| and anticipate where the argument was going. It felt like I was
| discovering it for myself. I think there are lot of good math
| and physics books from the Soviet era.
| peter_d_sherman wrote:
| >"MAGNITUDE VS. PHASE:
|
| [...]
|
| Note that f(x,y) is the image and is REAL, but F(u,v) (abbreviate
| as F) is the FT and is, in general, COMPLEX.
|
| Generally, F is represented by its MAGNITUDE and PHASE rather
| that its REAL and IMAGINARY parts, where: MAGNITUDE(F) = SQRT(
| REAL(F)^2+IMAGINARY(F)^2 ) PHASE(F) = ATAN( IMAGINARY(F)/REAL(F)
| )
|
| Briefly, the MAGNITUDE tells "how much" of a certain frequency
| component is present and the PHASE tells "where" the frequency
| component is in the image."
|
| This is interesting... I've never thought of FT's in terms of
| _Magnitude_ and _Phase_ before.... I wonder if these two aspects
| of FT equations could have any correspondences with other
| "aspect pairs" (for lack of a better term) in Physics, for
| example, Amperage and Voltage, Speed and Acceleration, Space and
| Time, Wavelength and Frequency, etc., etc. (in other words, a
| thorough comparison would need to be done between Magnitude and
| Phase and other "aspect pairs" in Physics, and see what's the
| same, see what's different, etc.
|
| There might be something to discover there... or at least
| (re)understand a little bit better...
|
| Anyway, excellent article (I learned stuff I didn't know about
| the Fourier Transform) -- thanks to the author for writing it,
| upvoted and favorited!
| dreamcompiler wrote:
| Multiplying Mag/Phase representations is easy; adding them is
| hard. The opposite is true for the Re/Im representation.
|
| So some of your representation choice depends on what you're
| doing. If you're multiplying two FTs (i.e. to perform
| convolution in the spatial or time domain) then Mag/Phase is
| easier. If you're adding signals together, Re/Im is easier.
| peter_d_sherman wrote:
| Interesting! Sort of like Mag/Phase is the opposite of Re/Im
| -- where adding and multiplication are reversed in complexity
| in both systems...
| jetunsaure wrote:
| Nice explanation, but do we still need to use Playboy for
| reference images?
| throwthisawayhn wrote:
| [flagged]
| neonate wrote:
| We don't still, but people used to. The article is over 20 yrs
| old.
| https://web.archive.org/web/20010725011144/https://www.cs.un...
| [deleted]
| ArtWomb wrote:
| Conversely, where is the actual Lena reference image? 8-bit
| pixel depth, 512x512, uncompressed tiff format preferably. Is
| there a canonical persistent URI someplace?
| ska wrote:
| Only sort of. Since the image has been in use since before
| web services etc., more than one different copy was used as
| "canonical" so it's sort of unsolvable if you want to
| reproduce work. There has been more recent efforts to pick a
| reference one, as linked in this thread.
| ArtWomb wrote:
| Reply within a hour, thnx hn ;)
|
| Intrigue. Looks like it was removed from the SIPI db?
|
| https://sipi.usc.edu/database/database.php?volume=misc
|
| World needs a new 600 DPI reference scan. To the Playboy
| Archives!
| zamadatix wrote:
| https://web.archive.org/web/20160322020730/https://sipi.usc..
| ..
|
| At this point archive services are the best bet for this kind
| of image.
| throwthisawayhn wrote:
| Yes, the "Lenna Story" website: http://www.lenna.org/
|
| A permanent link to the TIFF image itself is:
| http://www.lenna.org/lena_std.tif
| yazzku wrote:
| It's just a nude, grow up. And the image used in compression
| benchmarks only shows her face.
| jetunsaure wrote:
| It's interesting that there's so much resistance to stop
| using this particular image when there are billions of
| alternatives out there. Lena Forsen herself has asked that
| people stop using it, which seems like a pretty reasonable
| request.
| akhayam wrote:
| Every picture in this article is worth a million words. Well
| done!
|
| I also used a lot of pictures when teaching Discrete Cosine
| Transform in a grad course back in the day:
|
| http://rmarsh.cs.und.edu/CLASS/CS446/DiscreteCosineTransform...
|
| Students loved it and could instantly relate to the theory
| because of these pictures. Embarrassingly, this continues to be
| my most cited publication to date :)
| fartsucker69 wrote:
| seems like a really bad introduction if you don't already know
| what a fourier transform is
| midjji wrote:
| A better way to eliminate image border effects isnt windowing,
| but mirrored periodic repetition.
| enriquto wrote:
| But then this is equivalent to the cosine transform, isn't it?
| A much better trick than mirrored repetition seems to be the
| "periodic + smooth" decomposition:
| https://sbrisard.github.io/posts/20180212-on_the_periodic-pl...
| midjji wrote:
| Yeah, and its good at minimizing high frequencies, which is
| why the dct is used in image compression. There is no good
| general windowing, understanding is key, and where smoothing
| or windowing will hide the effect of the border, thus
| inhibiting understanding, mirroring helps understanding.
| [deleted]
| laszlokorte wrote:
| Fourier Transform is one of my favourite topic in maths and
| computer science. I am anways happy to stumble upon new resources
| about it.
|
| While taking courses in signal processing at university I built a
| collection of interactive visualiztations to provide a
| experimental and intuitive approach without much technical
| explanations.[1]
|
| I already submitted a few of them to HN and received very
| motivating feedback.[2] I am sure some of you who like the OP
| article might also enjoy these.
|
| The latest visualization is dedicated to the trade-off between
| time and frequency resolution/uncertainty priciple.
|
| [1]: https://tools.laszlokorte.de/ [2]:
| https://news.ycombinator.com/item?id=29455894 [3]:
| https://static.laszlokorte.de/time-frequency/
| 1-6 wrote:
| This is a bit out of context but nonetheless, it's inspiring:
| https://www.jezzamon.com/fourier/
|
| Spectrograms and fourier transforms can also be applied to making
| music through diffuse AI. https://www.riffusion.com/about
| tails4e wrote:
| Beautifully done. God I wish our fourier series lecturer in
| university motivated the class even a little with illustrations
| like this. It was 100% derivation and hard maths, no intuition
| at all
| pantulis wrote:
| Fourier series are kinde of more graspable if they are taught
| in the context of signal processing --as a nerd I could
| understand what the frequency domain was, after all, and the
| maths made sense to me. But when it comes to Laplace
| Transform in control systems everything became impossible.
| dreamcompiler wrote:
| The day I realized Laplace transforms converted difficult
| linear differential equations into simple algebra problems
| was the day I began to appreciate Laplace transforms.
| pizza wrote:
| It's definitely worth persisting with the Laplace
| transform.. you get a ton of stuff you can do when you have
| H(s)=Y(s)/X(s)
| scrlk wrote:
| Brian Douglas produced a decent set of lectures on control
| systems, I found them to be quite useful during my degree:
| https://www.youtube.com/@ControlLectures
| danwills wrote:
| I love image FFT! Quite some time ago I implemented it (just a
| wrapped FFTW) as a plugin for the 'Shake' compositing software
| that we were using at the time at RSP. It was tricky but
| incredible fun and like a mini-superpower! People were hand-
| painting full-colour bokeh images for very fancy defocus effects,
| among other things. Then Apple bought Shake, and we swapped to
| Nuke, which also has an FFT built-in which is cool, although last
| time I looked it didn't do the complex-to-magnitude/phase
| transformation-and-back which is very handy when working with FTs
| of pixels.
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