Find the Radon transform of a circular cylinder with radius...

Find the Radon transform of a circular cylinder with radius 6 and height 3 located at the origin. The cylinder is characterized by $$ f(x, y)=\left\{\begin{array}{l} 3 \text { for } x^2+y^2 \leq 6^2 \\ 0 \text { otherwise } \end{array}\right. $$ From the transform obtained, and using the Radon transform properties, find the Radon transform of (i) $$ f(x, y)=\left\{\begin{array}{l} 3 \text { for } x^2+y^2 \leq 3^2 \\ 0 \text { otherwise } \end{array}\right. $$ (ii) $$ f(x, y)=\left\{\begin{array}{l} 3 \text { for }(x-1)^2+(y-2)^2 \leq 6^2 \\ 0 \text { otherwise } \end{array}\right. $$ (iii) $$ f(x, y)=\left\{\begin{array}{l} 3 \text { for }(3 x)^2+(3 y)^2 \leq 6^2 \\ 0 \text { otherwise } \end{array}\right. $$

Find the Radon transform of a circular cylinder with radius 6 and height 3 located at the origin.
The cylinder is characterized by
$$
f(x, y)=\left\{\begin{array}{l}
3 \text { for } x^2+y^2 \leq 6^2 \\
0 \text { otherwise }
\end{array}\right.
$$
From the transform obtained, and using the Radon transform properties, find the Radon transform of
(i)
$$
f(x, y)=\left\{\begin{array}{l}
3 \text { for } x^2+y^2 \leq 3^2 \\
0 \text { otherwise }
\end{array}\right.
$$
(ii)
$$
f(x, y)=\left\{\begin{array}{l}
3 \text { for }(x-1)^2+(y-2)^2 \leq 6^2 \\
0 \text { otherwise }
\end{array}\right.
$$
(iii)
$$
f(x, y)=\left\{\begin{array}{l}
3 \text { for }(3 x)^2+(3 y)^2 \leq 6^2 \\
0 \text { otherwise }
\end{array}\right.
$$

Key Concepts

Radon Transform Definition

The Radon transform converts a two?dimensional function into its set of line integrals over all possible lines. In imaging and tomography, this transform is used to obtain projection data that can later be inverted to reconstruct the original function. Its parameters typically include the distance from the origin to the line and the angle of the line’s normal.

Projection Geometry for Characteristic Functions

When dealing with functions defined by characteristic functions (i.e., functions that take constant values over specific regions and zero elsewhere), the Radon transform reduces to computing the length of the intersection of the integration line with the region. For circular regions, this intersection often involves simple geometric expressions that depend on the distance from the center of the circle to the integration line.

Scaling Property of the Radon Transform

The scaling property describes how the Radon transform behaves when the spatial coordinates of the function are scaled. If a function is dilated or contracted, the corresponding Radon transform will reflect this change by stretching or compressing the projection data. This property is crucial when comparing objects of different sizes, as it allows one to relate the integral projections of a scaled object to those of the original.

Translation Property of the Radon Transform

The translation property of the Radon transform explains how shifting the function in the spatial domain affects its projections. A translation in the original function will result in a corresponding shift in the projection data. This property enables one to derive the transform for objects that are not centered at the origin by appropriately adjusting the projection parameters.

Utilization of Symmetry in Radon Transforms

For functions with circular symmetry, such as disks or cylinders (in their cross-sectional view), the Radon transform simplifies considerably. The inherent symmetry allows one to express the transform primarily in terms of the distance from the center, making the integration and subsequent analysis more manageable. This concept is used to derive transforms for different configurations of circular objects by exploiting rotational invariance.

Transcript

00:03 So in this problem, we're trying to find the volume that is under the plane and inside the cylinder.

00:11 And although they didn't say, we're going to assume they meant above the x, y plane.

00:16 Okay, so you can see that volume.

00:21 Here's the top of it.

00:23 And then here's the rest of it right there.

00:27 Okay.

00:27 So we know that it's going to be the top, which is the plane, minus the bottom, which is zero.

00:36 R -d -r -d -theta.

00:40 So the plane, we have to solve that for z.

00:43 So you get 6 minus x minus 2y all over 3.

00:51 Oops, i forgot i was in polar, so i'm going to have to change that to polar.

00:56 Okay, i'll do that on the next step.

00:58 So now what i need to know is what does this look like down in the x, y plane.

01:04 So there's its bottom of it.

01:10 I don't need a picture of that.

01:11 Okay, so what i need to do now is change this, which is what it looks like in the xx by plane, into polar.

01:21 So i get r squared cosine squared theta plus four r squared sine squared equals four.

01:32 So i'm going to factor an r squared out, and then i'll have to take the square root to get the end of it.

01:48 Okay.

01:51 I don't know what happened there...

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