Talk:Radon transform

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 Field: Analysis


This page lacks any explanation[edit]

At the moment this page provides little more than a classification, definitions and a bit on applications, but no explanation whatsoever. A description of the transform, preferable in the context of CT reconstruction, would go a great length to making this a proper Wikipedia page. — Preceding unsigned comment added by 192.16.186.53 (talk) 09:55, 22 August 2012 (UTC)

Filter back projection[edit]

I know people want to keep things abstract, but the vast majority of people referencing this page will want to use the 2D(and rarely 3D) version of the dual transform. They should be given explicitly(in addition to the N dimensional variant). —Preceding unsigned comment added by 142.90.103.110 (talk) 20:56, 20 March 2010 (UTC)

Reference to GT Herman book[edit]

I came to look at this at the request of Sławomir Biały. The first edition of Herman's book is a seminal text in the field and certainly would earn a place in a bibliography (along with Natterer's) book. I have not seen the second edition, but in the 1980s when I first learnt about Tomography herman's book was the first place to go. In as far as I have edited the article it is fair to say that is one of the sources I used. That said, we might want to separate a bibliography, as a guide to where people should start to read about this stuff, as oppose to references which specifically support theorems quoted.Billlion (talk) 18:26, 2 January 2010 (UTC)

Ok. I've added the book back into the references. Sławomir Biały (talk) 22:36, 2 January 2010 (UTC)

Definition[edit]

I'm not 100% sure about this, but it looks to me like the statement: "α is the angle L makes with the x axis" is not quite correct, since α is the angle between the x axis and the normal vector to the line L.

--Klubi 16:08, 29 January 2010 —Preceding unsigned comment added by 131.234.89.201 (talk)

You are right. This error was corrected by 108.3.68.169 a day later.

--Keilandreas (talk) 01:07, 20 March 2010 (UTC)

New Illustration of the Radon Transform[edit]

I created a new illustration of the Radon transform as vector graphic (PDF) and uploaded it to Wikimedia Commons at [[1]]. I would appreciate comments on the content of this image (if it could be used as a replacement) and how to link it here. --Keilandreas (talk) 10:40, 20 March 2010 (UTC)

I have trouble rendering that image in my browser. I don't know if this is an issue with how Wikimedia generates thumbnails of PDF files or if it is an issue on my end. The best supported file format is scalable vector graphics (SVG). I don't know IPE well, but I believe it supports export to SVG. I would encourage you to do this. Sławomir Biały (talk) 14:15, 20 March 2010 (UTC)
Ipe can only save in its internal format (.ipe), .pdf and .esp. I tried Inkscape to convert the pdf or eps to svg without luck (fonts are not rendered but replaced by some other font). Maybe someone else can help? --Keilandreas (talk) 05:32, 26 October 2010 (UTC)

Can we get this done? The diagram currently in use doesn't match the math terminology used in the article and makes it difficult to use. — Preceding unsigned comment added by 165.123.243.100 (talk) 19:35, 13 December 2011 (UTC)

Dual/Adjoint of the Radon Transform[edit]

Can somebody please clarify the definition of the dual transform? It seems that the two definitions given are not equivalent. Should the normalization factor \frac{1}{2\pi} really be part of the dual? Keilandreas (talk) 13:06, 20 March 2010 (UTC)

A probability measure assigns length one to the circle. The 1/2π is necessary to get a total length of one, so the two definitions are equivalent. I can attest that this is precisely how Helgason defines the dual Radon transform, but there may be other normalization conventions in the literature (I don't know). Sławomir Biały (talk) 14:15, 20 March 2010 (UTC)

d sigma?[edit]

Is d sigma just length along the line? Should the equation

Rf(L) = \int_L f(x)\,d\sigma(x)

just read

Rf(L) = \int_L f(x)\, \|dx\|

or perhaps

Rf(L) = \int_L f(\mathbf{x})\, \|d\mathbf{x}\|

or am I missing something? —Ben FrantzDale (talk) 15:32, 30 March 2010 (UTC)

The meaning is actually explained in the text of the definition – sigma is arclength measure. The notation you suggest is sometimes used, but is far from universal. I would suggest leaving it as it is. Hanche (talk) 16:33, 30 March 2010 (UTC)
I saw that in the text, but I was half expecting the integral to be weighted additionally by the radius (integrating over a wedge as in an area integral in polar coordinates rather than a line). I'll be bold and try to add some clarity. In particular, I see d\sigma(x) as implying a dependence on x, which made me think an additional weighting may be going on. —Ben FrantzDale (talk) 19:46, 30 March 2010 (UTC)
|dx| seems fine to me, but you might want to explain the notation. E.g., "where |dx| is the measure of arclength along the line", or something. Sławomir Biały (talk) 20:50, 30 March 2010 (UTC)

More figures?[edit]

I think that there are some images that are very common to the discussion of radon transforms. In particular, transforms and inverse transforms of the Shepp-Logan phantom. I think that a figure showing an image, its sinogram, and then it's inverse radon transform, would be very useful. If you agree, I will go ahead and run the calculations. If no one cares enough to respond, then I will add it. hovden(talk) 19:27, 21 June 2010 (UTC)

In case you find it useful, see this article: http://www.aapm.org/meetings/99AM/pdf/2806-57576.pdf Sławomir Biały (talk) 11:53, 23 June 2010 (UTC)
Right. The figures in the link you provided are the types of figures I was planning to add. In fact, almost identical. hovden (talk) 1:06, 02 July 2010 (ET)
Ok. One thing I like are the artifacts caused by insufficient angular sampling as a good illustration of the inverse transform. Sławomir Biały (talk) 13:26, 2 July 2010 (UTC)
Another important addition would be a graph of the ramp function used in the filtered backprojection. For two dimensions, this is generally shaped like a wide gaussian subtracted from a narrow gaussian. --Aflafla1 (talk) 04:00, 16 April 2012 (UTC)
A diagram showing what exactly "the space of straight lines L in R2" is, is needed. Additionally, a diagram or better description of what the "parametrization of any straight line L with respect to arc length t" means is also needed. I have 2 degrees in Physics and I can't understand this article. If this is an article in mathematics, then the mathematics is poorly explained. If this is a general description, than the there is too much mathematics. — Preceding unsigned comment added by 152.77.240.235 (talk) 10:14, 11 June 2014 (UTC)
Are these things not both already explained in the article? The arc length parameterization of a line is given explicitly, and coordinates are given on the space of straight lines. It's hard to be more explicit than that. Sławomir Biały (talk) 10:50, 11 June 2014 (UTC)
I am not sure if you noticed but I posted this comment under the "More figures?" section and not the "This page lacks any explanation" section. I agree the arc length parameterization of a line is given explicitly though I am (or rather was) at a loss for what exactly that means: 'arc length of what?', 'where is this arc length relative to the object function f(x)?' and 'why a parameterization with respect to arc length?'. I would agree that the answer to these questions would be more apparent if I understood what 'all lines in R2' looked like. If you look at figure 1 in the article (the only figure in the article explaining any of the mathematics) why would it not be feasible to believe that 'all lines in R2' represent all lines parallel to AA' spanning R2? You said that the coordinates are given on the space of straight lines, which is correct however this is clearly after "the space of straight lines L in R2" is first mentioned so if nothing else the wording or order of the article lacks clarity. I would like to reiterate that this article is not written for someone who knows little about the radon transform, which is the type of person that would be researching on wiki this subject, but instead is written for someone who is fairly familiar with the subject and would not be reading the article for any particularly good reason. I would like to see more diagrams explaining the mathematics of the radon transform added to the article. — Preceding unsigned comment added by 152.77.240.235 (talk) 13:03, 12 June 2014 (UTC)