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I am reading "Image Processing and Analysis" by Chan and Shen, c 2005 SIAM. They introduce some notation I'm not 100% sure how to interpret:

$$ u_0(x)=u(x)+n(x), x=(x_1,x_2) \in \Omega $$

They state $u_0$ represents a noisy image, $u$ is the clean image and $n$ is Gaussian white noise. I assume that $ x=(x_1,x_2) $ is meant to convey that x is a two-dimensional vector indexing both image height and width. I don't know how to interpret $\Omega$. Is it meant to denote the entire image index range?

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  • $\begingroup$ $\Omega$ is just the image space. So $\Omega = [0,X -1] \times [0,Y - 1]$ assuming your image has size $X\times Y$. $\endgroup$ Mar 14, 2023 at 2:39

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It sounds exactly like that!

  • $\Omega$ is the image domain, it can be either discrete or continuous. No idea how it is defined in that particular book.
  • $x=(x_1, x_2)$ are the 2D image coordinates
  • $u_0, u, n$ are functions that return intensity values at the given location $x$.
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