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## Finding image forward projection and its transpose matrix

California State University, Fullerton, Summer 2008 page compiled on July 1, 2015 at 9:15pm

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.

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

can now be viewed as back projecting the image into a plane

by smearing each pixel value over the plane along the line of sight as illustrated below

### 1 Case of

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