The relevant question that everyone wants to know the answer to is:
What is your reason for needing to know how to solve this problem without using interp2 in MATLAB?
I think if you answer this question, it will help everyone understand what exactly your requirements are for whatever it is you are doing. Unless there is some compelling reason why you don't want to use interp2, everyone is going to suggest using interp2 because you are unlikely to write a faster or more memory efficient 2-D interpolation routine yourself, especially if you are using a recent version of MATLAB.
I have answered a variation of this question in the past on how to do log-polar transformations of images, in your case it looks like you want a polar->cartesian transformation, but the idea is the same. However, in my answer, I used interp2.