# FFT in different coordinate systems

Is there a transformation that will enable one to calculate the FFT in an arbitrary coordinate system? What I am interested is the following two cases:

1. The space is Euclidean and infinitely differentiable everywhere
2. The space forms a curved Riemannian manifold with metric: $$ds^2 = \sum_{i, j} g_{ij} dx^i dx^j$$
• Hi there! Interesting question! Please give more details and clarify: the coordinate system you refer is a spatial construct to measure space coordinates of points? That can makes sense for image analsysis in non cartesian or non curvilinear coordinate systems... – Fat32 May 27 '16 at 0:14
• Well, the coordinate system does not matter from a mathematical point of view. Usually we have samples on a grid i.e. Rectangular coordinates. What if the samples are in some arbitrary coordinate system of dimension N. – user110971 May 27 '16 at 0:19
• ok then please make that arbitrary coordinate system more specific; i.e. a warped space? nonorthogonal? or just nonuniform samples from a rectengular or polar grid? In case of nonuniform samples, it can (possibly) be converted to uniform ones and usual DFT can be obtained then. – Fat32 May 27 '16 at 0:30
• A manifold? Hmm lets see who has taken a differential geometry course here :) – Fat32 May 27 '16 at 0:47
• Well case 1 is eucledian but the samples are not uniform in some manner. So if it is in polar coordinates you will get more samples near the origin. What if the coordinate system is not polar but an arbitrary one? – user110971 May 27 '16 at 0:50