The solutions to many 2D signal processing problems rely on the identification of patterns (e.g. molecules in biological imagery) or local structures (e.g. edges/ridges in general) that could appear with arbitrary orientations.
Our method of study is to consider an optimal linear approximation problem, where we simultaneously approximate a given square-integrable function together with all of its rotations by linear combinations of functions from a fixed set, say . That is to say, we are interested in minimising the following average approximation error
where computes the orthogonal projection onto the span of the basis. This minimisation leads to the following rotation-covariant linear expansion, which we name Fourier-Argand representation,
Each term of this expansion is rotation-covariant, in the sense that rotating each basis function amounts to multiplying it by a (complex) scalar, i.e., .
A few terms are sufficient to form a very accurate approximation to heavily elongated (high directional selectivity) functions while being readily steerable. See the figure below: (left) an elongated Gaussian (aspect ratio 1:10), (right) 10-term Fourier-Argand approximation.
Elongated Gaussian | 10-Term Fourier-Argand Approximation |
The representation allows us to efficiently detection image local structures. See the figure below for an example on ridge-like blood vessel detection.
Retinal Image | Filter Response | Detection Result |
We also show how the Fourier-Argand representation of arbitrary patterns can be obtained using the Radon transformation:
[1] Zhao, T. & Blu, T.,"The Fourier-Argand representation: an optimal basis of steerable patterns", IEEE Transactions on Image Processing, Vol. 29 (1), pp. 6357–6371, December 2020. |