Problem
Write the matrix which implements the forward projection and its transpose.
A simple case would be to consider a 2-D object made up of only 4 pixels and one projection. After that think about an object with many pixels and many projections.
Answer
I will use the convention used by the radon transform in Matlab in setting up the coordinates system which is as shown below (diagram from Matlab documentation page).
In our case, we need to perform the following projection, which is at angle as follows
The equation for the above mapping is , hence we write
Hence
But from the line integral at the above projection and , hence the above 2 equations becomes
By comparing coeﬃcients on the LHS and RHS for each of the above equations, we see that for the ﬁrst equation we obtain
For the second equation we obtain
Hence the matrix is
Taking the transpose
Hence if we apply operator onto the image , we obtain back a image, which is written as
Hence . In other words, the image is a 4 pixels
by smearing each pixel value over the plane along the line of sight as illustrated below
We repeat the above for
The equation for the above mapping is , hence we write
Therefore
We see from projection diagram that and , hence the above 3 equations become
By comparing coeﬃcients, we obtain from the ﬁrst equation and from the second equation and from the last equation . Hence the matrix is
Using to project the image we obtain
Hence , hence the back projection plane is
This also can be interpreted as back projecting the image on a onto a plane by smearing each pixel value on each pixel along its line of sight as illustrated below