The ellipselet transform

Background: A fair amount of important objects in natural images have circular and elliptical shapes. For example, the nucleus of most of the biological cells is circular, and a number of parasites such as Oxyuris have elliptical shapes in microscopic images. Hence, atomic representations by two-dimensional (2D) basis functions based on circle and ellipse can be useful for processing these images. The first researches have been done in this domain by introducing circlet transform. Methods: The main goal of this article is expanding the circlet to a new one with elliptical basis functions. Results: In this article, we first introduce a new transform called ellipselet and then compare it with other X-let transforms including 2D-discrete wavelet transform, dual-tree complex wavelet, curvelet, contourlet, steerable pyramid, and circlet transform in the application of image denoising. Conclusion: Experimental results show that for noises under 30, the ellipselet is better than other geometrical X-lets in terms of Peak Signal to Noise Ratio, especially for Lena which contains more circular structures. However, for Barbara which has fine structures in its texture, it has worse results than dual-tree complex wavelet and steerable pyramid.