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I don't understand how frequencies are defined in images/photographs. As far as I understand it now, high frequencies are like sharp things in images, like edges or so, and low frequencies are kind of the opposite?

I also would like to understand the outcome of Discrete Fourier Transformations, like how to read them properly.

It would be cool if somebody could explain to me the following:

  1. What are frequencies in pictures and how are they defined?

  2. How do you read the outcome of a Discrete Fourier Transformation?

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Related: photo.stackexchange.com/questions/21762/… –  coneslayer Jun 25 '13 at 16:25
    
thanks, I already read this, it helped me, but im still a little clueless. –  Jakob Abfalter Jun 25 '13 at 16:29
    
The second part of your question is covered by the link @coneslayer provides. It could probably use better answers, but it's still the same question. :) –  mattdm Jun 25 '13 at 16:30

2 Answers 2

up vote 11 down vote accepted

I will only answer the first question: What are frequencies in images?

Fourier Transform is a mathematical technique where the same image information is represented not for each pixel separately but rather for each frequency. Think about it this way. The sea has waves some of which are very slow moving (like tides), others are medium in size and still some others are tiny like the ripples formed from a gust. You can think of them as three separate waves but at each point on the surface of the sea and a moment in time, you get just one height of water.

The same applies to images. You can think of the image being made up of various waves or frequencies. To create your image, start with the average colour (actually thinking of gray scale images is easier). Then add waves of different wave lengths and strength to slowly build up details in the picture.

Source Image: Source Image

First Frequency (Average): Average

The second frequency along the vertical dimension is a wave starting at zero at the bottom of the image, rising, becoming zero again along the centred horizon and falling below zero to finally become zero at the top of the image. (I described a Fourier Series without phase shift, but the analogy still holds.)

Here you can see the second frequency along the horizontal, vertical and diagonal. Notice that you can make out where the mountain will be (dark) and where the sky and lake will be (lighter).

Second Frequency: First Component

Each additional wave or frequency brings along more ripples and as such, more detail. To get different images, the wave height/amplitude can be changed as well as the starting point of the wave, also called the Phase.

Third Frequency: Third

Interestingly, the information amount is the same in this representation and one can go back and forth between normal images (spatial domain) and Fourier Transformed images (frequency domain). In the frequency domain we need to keep information of all frequencies along with the amplitude and the phase information.

Here it is using 50% of the frequencies: 50%

There are variants of all this, with distinctions to be made among Fourier Series, Fourier Transform and Discrete Fourier Transform and Discrete Cosine Transform(DCT).

One interesting application is in the use of compression algorithms like JPEG. Here the DCT is used to save more of the important parts of the image (the low frequencies) and less of the high frequencies.

I wrote this in the hope that novice readers can get a basic understanding of the idea of Fourier Transforms. For that I made some simplifications that I hope the more advanced readers will forgive me.

Animated

Sorry, the GIF is 5MB large so I can't include it here. Here is the link or run the code to generate it yourself.


Usage of Frequencies in Post-Processing

There are numerous methods that rely on frequencies for post processing, mostly because we never look at single pixels individually. Many algorithms work on frequency because it is more natural to think about them this way. But also because the Fourier Transform contains the same information we can express any mathematical operation (or post processing step) in the frequency and the spatial domains! Sometimes the pixel-wise description is better but often the frequency description is better. (Better primarily means faster in this context.)

One technique I would like to point for no particular reason except that it is artists working directly with frequencies and that is *frequency separation *. I am not going to describe it but you can see how it works on YouTube for both Photoshop and GIMP.

You create two layers one with the low frequencies and one with the high frequencies. For portraits you can do skin smoothing on the high frequencies without affecting the skin tones in the low frequencies.

Code

This is some code to generate the above examples. It can be run as a simple Python program.

from PIL import Image
from numpy.fft import rfft2, irfft2
import numpy as np

def save_dims(ft, low, high, name):
    ft2 = np.zeros_like(ft)
    # copy the frequencies from low to high but all others stay zero.
    ft2[low:high, low:high] = ft[low:high, low:high]
    save(ft2, name)

def save(ft, name):
    rft = irfft2(ft)
    img = Image.fromarray(rft)
    img = img.convert('L')
    img.save(name)

def main():
    # Convert input into grayscale and save.
    img = Image.open("input.jpg")
    img = img.convert('L')
    img.save('input_gray.png')
    # Do Fourier Transform on image.
    ft = rfft2(img)
    # Take only zeroth frequency and do Inverse FT and save.
    save_dims(ft, 0, 1, 'output_0.png')
    # Take first two frequencies in both directions.
    save_dims(ft, 0, 2, 'output_1.png')
    save_dims(ft, 0, 3, 'output_2.png')
    # Take first 50% of frequencies.
    x = min(ft.shape)
    save_dims(ft, 0, x/2, 'output_50p.png')

def generateGif():
    ''' Generates images to be later converted to a gif.
    This requires ImageMagick:
    convert -delay 100 -loop 0 output_*.png animation.gif
    '''
    # Requires images2gif from code.google.com/p/visvis/source/browse/vvmovie/images2gif.py 
    # from images2gif import writeGif

    img = Image.open('input.jpg')
    img = img.convert('L')
    # Resize image before any calculation.
    size = (640,480)
    img.thumbnail(size, Image.ANTIALIAS)
    ft = rfft2(img)

    images = []
    for x in range(0, max(ft.shape)):
        ft2 = np.zeros_like(ft)
        ft2[0:x, 0:x] = ft[0:x,0:x]
        rft = irfft2(ft2)
        img_out = Image.fromarray(rft).convert('L')
        fname = 'animation/output_%05d.jpg' %(x, )
        img_out.save(fname, quality=60, optimize=True)

    #writeGif('animation.gif', images, duration=0.2)


if __name__=='__main__':
    main()
    #generateGif()
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1  
If the aim is for novice readers to understand, some visual guides might be useful. Perhapse generate some "wavelets" in separate images, then blend them together to demonstrate what the convolved "image" might look like. (As it stands now, the wall of text is probably likely to be a bit off-putting for "novice" readers.) –  jrista Jun 26 '13 at 0:05
    
Agree, I'll do it once I get near a computer. –  Unapiedra Jun 26 '13 at 10:14
    
Done. If someone can figure out how to compress the GIF that would be swell. –  Unapiedra Jun 26 '13 at 18:12
    
It should be clarified that, while theoretically we could, assuming we had infinite knowledge of the image at hand, decompose it to component frequencies and recompose it with no loss...in the real world we can't. Convolution of a real world image, which occurs at each and every "interface" along the optical pipeline, is effectively an irreversible process. We can't ever know all convolution factors, and therefor reconstruction of an FFT back into an image is difficult, and extreme modifications usually result in artifacts and data loss. –  jrista Jun 26 '13 at 18:21
2  
jrista's comment is misleading in that FT is blamed for information loss. Of course, photography is a lossy process and so is post-processing. If I convert a discrete image to Fourier Space, do some lossy processing there, and then convert back, of course I loose information. But it happens in the processing step and not in the conversion step. True, because of machine precision every mathematical operation looses information but if we are talking about 8 bit per channel images, we won't notice machine precision errors. –  Unapiedra Jun 27 '13 at 18:05

Reading the outcome of a bidimensional FFT is... Well, tricky. Basically because the output is in the complex numbers domain.

I cannot give you a very detailed description because I have forgotten most of it myself, but basically you have a decomposition expressed as the sum of a series of exponentials of imaginary numbers.

There are frequency transforms that are more useful for image processing than the DFT, like the Cosine Transform (used in JPEG encoding for example, which has only real coefficients) and the different kinds of Wavelets.

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