I have shown how an integral along a path in a sampled space can be formulated as a weighted sum of the input samples. The form of the weighting function and the effort needed to calculate it depends on the way in which the input data is interpolated onto the path.
If an integral is approximated by taking equally spaced samples in space there may be problems associated with ``operator aliasing''. The nearest neighbor and/or linear interpolators that I have described can be used to reduce these problems. However these methods will not overcome problems related to aliasing in the sampled data. The assumption that a linear or bilinear interpolation of the data is a good approximation to the true data is violated in this case. Putting it another way, you can integrate along a path steeper than the aliasing limit if you use an interpolating integral, but this will not save you if the data itself is aliased at that dip.
The same method outlined here can be extended to treat a surface integral in a 3-D space, e.g. 3-D Kirchoff migration. It is left as an exercise for the reader.