What is the name of this transform similar to the Radon transform? I have a 2D image (obtained using computed tomography) that I'm "transforming" for image segmentation purposes, and I'm looking for a formal way of describing the transformation I'm doing.

The transformation I'm doing defines a centre point $(x_0,y_0)$, angles $\theta_i$ from 0 to 360 degrees, and a set of centrifugal lines defined by $x_0 + L\cos(\theta_i)$, $y_0 + L\sin(\theta_i)$. For simplicity, we can imagine that the original image is circular and thus $L$ is simply the radius of the circle.

The result of the transform is simply the 2D "stacking" of the intensity function $I(x,y)$ on each line defined above. This is essentially "unrolling" a circular image into a 2D image.

It looks superficially similar to the Radon transform, but there's not integral involved - I'm simply copying the intensity values. 

Does this have a formal name?
This is a simple Cartesian to Polar coordinates transformation. Instead of expressing the position of a pixel using $(x,y)$ you are expressing it using $(r, θ)$ where $r$ is distance from the centre $(x_0,y_0)$ and $θ$ is angle.

The images are exactly the same, wherever they are definable, but the "unrolling" results from still using $(x,y)$ to depict $(r,θ)$.

For more information, please see this and this.