Efficient steerable circular harmonic functions

• Very efficient Circular Harmonic Function (CHF) steerable filter technique presented.
• CHF implementation several times faster than direct convolution and FFT (conditionally).
• Technique provides exact results as direct convolution.
• Comparisons presented for pattern matching, orientation analysis, and denoising.

Two-dimensional (2D) filtering and pattern matching applications often involve processing of images using a set, Sf, which includes several transformed (e.g. rotated, scaled, warped) versions of the same 2D function, f. Depending on the application, f may correspond to a filter kernel or a pattern. In general, image processing using all functions in Sf poses a computational burden. Steerability offers a solution to the problem. Specifically, each function in Sf can be exactly expressed or closely approximated by a linear combination of a few basis or steering functions. In such cases, processing images by employing steering functions and obtaining linear combinations of the associated responses may be preferable over utilizing all functions in Sf. However, employing the steering functions is still a computationally expensive process. This paper presents a method to significantly improve the computational efficiency of circular harmonic functions, which are the steering functions when rotation is the transformation of interest. The method is presented mainly in the context of image filtering, yet 2D pattern matching is also discussed. The advantages of the proposed technique in relation to other techniques are presented, and its efficiency is demonstrated in terms of execution time and number of operations. An extension of the technique to more general cases where Sf includes rotated, but also otherwise transformed versions of f, is also discussed.