Maximum circular area sum

Given an $n \times n$ grid of integers, I would like to compute the circular area which gives the maximum sum of integers within the area. For example:

In this picture, the integers 1, 0, 1, 2, 3, 1, 2, 1, -3, -2, 2, -3, 1, 3, -2, 0, 2, -1 etc are all within the circular area.

If we restrict the circular area to be centered at one of the grid points, is there an efficient algorithm for this problem?

The input is just the grid of integers. The task is to find the center and radius. If the circular area includes part outside the grid that part contributes zero to the sum.

There are $n^4$ possible $(x, y, r^2)$ tuple, each describing a circle centered at $(x, y)$ with radius $r$. Since each circle is its smaller version added with $O(1)$ additional points, this yields a time complexity of $O(n^4)$. We may find the additional points by sorting every $r^2 = a^2 + b^2$ pair.

StackExchange Computer Science Q#164698, answer score: 2