**1. Introduction**

Wavelet multiresolution representations are one of the effective techniques for analyzing signals and images. The wavelet multiresolution analysis (MRA) technology has been widely used in signal and image processing. It was first given by Mallat [1], and the authors study the difference of information between approximation of a signal at the resolution 2*j*+<sup>1</sup> and 2*j*, by decomposing this signal on a wavelet basis of *<sup>L</sup>*<sup>2</sup>(*R*). The 2D general MRA technique possesses a square shape in the frequency domain [2–4]. To design the filter of circular-shape in the Fourier domain, the classical polar Fourier transformation is considered. However, the classical polar Fourier transform retains the same shape as in the space domain, so new approaches are investigated. One way is to redefine the sampling grid in the Fourier domain. In [5], the authors introduce a pseudo-polar Fourier transform that samples the Fourier transform on the pseudo-polar grid, also known as the concentric squares grid. We will give more details in Section 3. In addition, [6] samples on points that are equally spaced on an arbitrary arc of the unit circle, which brings about the Fractional Fourier transform; and, in [7], the sampling is on spirals of the form *AWk*, with *A*, *W* ∈ *C*. Using this type of sampling, the authors develops a computation algorithm for numerically evaluating the *z* − *trans f orm*. Our goal is to obtain the sampling grid in a circular shape; therefore, we hope to design a new type of sampling that ensures the sampling points concentrated in a circular region. Then, the sampling grid has a circular shape in the Fourier domain. Inspired by the pseudo-polar Fourier transform in [5], we will also redefine the Fourier transform on circular sampling grid.

In recent years, many kinds of directional wavelets filters have been designed, in order to further efficiently capture the details of signals. The most widely used directional multilevel system includes curvelets [8], contourlets [9] and shearlets [10,11]. What these wavelets have in common is that they have compact support multiscale structure in the space domain. In the Fourier domain, the support of a multilevel system constitutes a high redundant partition. To reduce the redundancy, we consider designing the multilevel system in the frequency domain directly. We also must ensure that the multilevel system constructs the basis of *L*<sup>2</sup>(*R*) in the space domain.

To design the multilevel system in the Fourier domain, wavelets with compact support in frequency are needed. According to the definition of the harmonic wavelet [12–15], it is suitable to construct a directional multilevel structure with harmonic wavelets whose Fourier transforms are compact and are constructed from simple functions like Haar wavelets [16] in the space domain. We will review the basic definition and property of harmonic wavelet in Section 2.

In this work, by defining the circular-shape Fourier transform (CFT), we will construct the circular-shape directional multilevel system (CMS) in the Fourier domain due to the compact support of harmonic wavelets [12–15,17]. The specific structure is totally different from the general Cartesian system. By introducing the *CFT*, we plan to give a parallel analogy with the general classical Descartes Fourier transform, and the corresponding circular-shape directional multilevel system is constructed naturally, which is suitable for the circular shape of images in the Fourier domain. More details will be given in Section 4.

This paper is organized as follows: Section 2 reviews the basic definition and property of harmonic wavelets. The design of *CFT* is given in Section 3. Then, in Section 4, the multilevel system in the frequency domain based harmonic wavelet is constructed. The quantitative test measures and test results are displayed in Sections 5 and 6.
