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The Wikipedia entry on Radon transform shows an equation like the following: $$ \int_L f(\mathbf{x})|\mathbf{dx}|. $$ What does absolute value around $$|\mathbf{dx}|$$ mean in an integral?

I understand that for the Radon transform we want to integrate the function along the line $L$, but I don't understand the notation in the equation I listed above.

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    $\begingroup$ Their link to line integral takes you right to its definition. How would you calculate the length of the curve $L$? $\endgroup$
    – Kurt G.
    Commented Sep 15, 2023 at 16:47
  • $\begingroup$ @KurtG. I think the |dx| can be (at least loosely) interpreted as a small length along the curve L. It's not clear to me why the absolute value signs are needed. $\endgroup$
    – user3731622
    Commented Sep 15, 2023 at 17:00
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    $\begingroup$ There is no looseness whatsoever in the definition for which Wikipedia and I provided a link. A good reason for the absolute value signs is that there is another type of line integral that integrates a vector along a curve ("work done"). Please google that and get familiar with it. In short: its proper notation is $\int_L\mathbf{F}\cdot d\mathbf{x}\,.$ $\endgroup$
    – Kurt G.
    Commented Sep 15, 2023 at 17:05
  • $\begingroup$ For one thing, $|dz|$ is the natural notation in complex analysis when one wants to distinguish it from the regular $dz$ line integrals. $\endgroup$
    – Ninad Munshi
    Commented Sep 15, 2023 at 18:25
  • $\begingroup$ @KurtG. I think I understand the line integral you described for "work done," but I still don't understand why the absolute value is used in Radon transform. $\endgroup$
    – user3731622
    Commented Sep 16, 2023 at 1:01

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