*Article* **A Framework for Circular Multilevel Systems in the Frequency Domain**

**Guomin Sun 1,\*,†, Jinsong Leng 1,† and Carlo Cattani 2,\*,†**


Received: 9 February 2018; Accepted: 31 March 2018; Published: 8 April 2018

**Abstract:** In this paper, we will construct a new multilevel system in the Fourier domain using the harmonic wavelet. The main advantages of harmonic wavelet are that its frequency spectrum is confined exactly to an octave band, and its simple definition just as Haar wavelet. The constructed multilevel system has the circular shape, which forms a partition of the frequency domain by shifting and scaling the basic wavelet functions. To possess the circular shape, a new type of sampling grid, the circular-polar grid (*CPG*), is defined and also the corresponding modified Fourier transform. The *CPG* consists of equal space along rays, where different rays are equally angled. The main difference between the classic polar grid and *CPG* is the even sampling on polar coordinates. Another obvious difference is that the modified Fourier transform has a circular shape in the frequency domain while the polar transform has a square shape. The proposed sampling grid and the new defined Fourier transform constitute a completely Fourier transform system, more importantly, the harmonic wavelet based multilevel system defined on the proposed sampling grid is more suitable for the distribution of general images in the Fourier domain.

**Keywords:** harmonic wavelet; filtering; multilevel system
