What does multiplying two Fourier transforms do in image processing?
Asked 3/27/2012
3 views
2 answers
0
I'm learning image processing and saw a reference to creating effects by multiplying two FFT results, but it didn't explain what that multiplication means. In practical terms, what does multiplying two Fourier transforms do, and how is it used in image processing? For example, how does this relate to effects like blurring or other common operations?
Originally by Photography Stack Exchange contributor. Source · Licensed CC BY-SA 4.0
Photography Stack Exchange contributor
14y ago
2 Answers
6
You should add more details with respect to the specific special effect.
In general, given two function f(x) and g(x) and their Fourier transforms F(k), G(k), the product of the two transform F(k)G(k) when brought back to the coordinate space (that is, when anti-transformed) is equal to the convolution of the original f and g.
Suppose that you are convoluting a function f with a gaussian kernel (to achieve a gaussian blur). Instead of doing the convolution integral you could find easier to multiply the Fourier transforms of the f and of the kernel, and then transform back.
Here you find a nice description of blurring and unblurring which showcases the idea without burdening the explanation with excessive mathematical notation .
Originally by user5032. Source · Licensed CC BY-SA 4.0
user5032
14y ago
0
Generated from our catalog & community — verify before relying on it.
Multiplying two Fourier transforms is useful because of the convolution theorem: if you take two functions/images, transform them to the frequency domain, multiply those transforms, and then inverse-transform the result, you get the convolution of the originals in normal image space.
In practice, this means you can apply many filters efficiently in the frequency domain.
Example: Gaussian blur.
- Start with the image.
- Take its FFT.
- Take the FFT of a Gaussian kernel.
- Multiply the two FFTs.
- Inverse FFT the result.
That gives the same result as convolving the image with the Gaussian kernel in spatial space, often more efficiently for large kernels.
So the multiplication itself usually represents applying a filter. Depending on the second transform, this can be used for blurring, and in some contexts for deblurring/unblurring as well. The exact “special effect” depends on what the second function or kernel is.
Recommended products
UniqueBot
AI14y ago
Your Answer
Related Questions
What do low and high frequencies mean in an image, and how do you read a discrete Fourier transform?
How can I align a remake photo to match an old photo’s crop, scale, and rotation in GIMP?
How can I learn photo post-processing techniques using GIMP?
What do Amount, Radius, and Threshold do in Unsharp Mask?
How does wavelet sharpening in RegiStax work, and what do the layer controls do?