My understanding was that the Fourier transform is done on periodic signals. But both images in the slide don't seem to be periodic. Does the function representing the image being periodic or not matter when we take the Fourier transform?
I saw in the demo under Notebooks that we can construct images by dividing them into 8x8 sections and constructing each 8x8 section from waves: Is this how we're Fourier transforming images that don't look periodic? If so, how does this work, I don't see how we can recover each 8x8 wave image element
Also, in terms of implementing a Fourier transform on an image, is the following understanding correct:
We view our image as a discrete function of pixel location (x,y) to intensity (assuming grayscale)
We then define our Fourier transform using the discrete formula (since pixel locations is type int * int) given in slide 87, where N = # rows * # cols (unsure if this is correct for N)
The above will be a function corresponding to the Fourier transformed image
Please let me know if any of that understanding is incorrect. Thanks so much!
motoole2
Even though the images do not appear to be periodic, they can be made periodic by creating an infinitely-large mosaic composed of the image. That way, any finite non-periodic signal can be turned into a infinitely large periodic signal.
The example in the interactive demo described in your post decomposes an 8x8 image into a linear combination of 64 = 8*8 basis functions. There are many basis functions one can potentially use to represent an image. In this particular case, the interactive demo shows a set of sinusoidal basis functions used in the Discrete Cosine Transform, which is used by JPEG for compression. There's a similar set of 64 basis functions for the actual Fourier Transform. Also note that this is not limited to 8x8 images; an $M\times M$ image can be represented as a linear combination of $M^2$ basis functions.
Regarding how to apply the Fourier transform on an image, you got the right idea. However, slide 87 makes use of 1D Fourier transforms. For images, we would want a 2D Fourier transform, that works with 2D functions. This is what it would look like:
$$F(u,v) = \sum_{x=0}^{N-1} \sum_{y=0}^{M-1} f(x,y) e^{-j2\pi (ux + vy)}$$
My understanding was that the Fourier transform is done on periodic signals. But both images in the slide don't seem to be periodic. Does the function representing the image being periodic or not matter when we take the Fourier transform?
I saw in the demo under Notebooks that we can construct images by dividing them into 8x8 sections and constructing each 8x8 section from waves: Is this how we're Fourier transforming images that don't look periodic? If so, how does this work, I don't see how we can recover each 8x8 wave image element
Also, in terms of implementing a Fourier transform on an image, is the following understanding correct:
We view our image as a discrete function of pixel location (x,y) to intensity (assuming grayscale)
We then define our Fourier transform using the discrete formula (since pixel locations is type int * int) given in slide 87, where N = # rows * # cols (unsure if this is correct for N)
The above will be a function corresponding to the Fourier transformed image
Please let me know if any of that understanding is incorrect. Thanks so much!
Even though the images do not appear to be periodic, they can be made periodic by creating an infinitely-large mosaic composed of the image. That way, any finite non-periodic signal can be turned into a infinitely large periodic signal.
The example in the interactive demo described in your post decomposes an 8x8 image into a linear combination of 64 = 8*8 basis functions. There are many basis functions one can potentially use to represent an image. In this particular case, the interactive demo shows a set of sinusoidal basis functions used in the Discrete Cosine Transform, which is used by JPEG for compression. There's a similar set of 64 basis functions for the actual Fourier Transform. Also note that this is not limited to 8x8 images; an $M\times M$ image can be represented as a linear combination of $M^2$ basis functions.
Regarding how to apply the Fourier transform on an image, you got the right idea. However, slide 87 makes use of 1D Fourier transforms. For images, we would want a 2D Fourier transform, that works with 2D functions. This is what it would look like: $$F(u,v) = \sum_{x=0}^{N-1} \sum_{y=0}^{M-1} f(x,y) e^{-j2\pi (ux + vy)}$$