Image representation is an essential problem in image processing. The most effective image representation method is multiscale geometric analysis (MGA). However, the current representative MGA method has some shortcomings, such as the fixed division of the scale spectrum and the direction spectrum, which cannot achieve well the sparsest representation of the image; the lack of rotation invariance and translation invariance make it impossible to achieve better results in the application. In this paper, a new circular flexible MGA method is proposed to solve this problem. First, the McClellan method was used to design a circular flexible multiscale decomposition, which had translation invariance and rotation invariance due to the circular spectral shape. Moreover, the spectral division of the new multiscale decomposition method could be flexibly changed according to the image, rather than being fixed at π/2 in the traditional manner. Second, new linear phase directional filter banks were proposed to divide the direction flexibly, which had selectivity in any direction. Finally, the two steps above were combined to construct a new MGA method, which maintained translation invariance, low redundancy, and flexibility in both scale and direction, resulting in better frequency localization and more sparse representation. The results of our denoising experiment show that the method proposed in this paper not only achieves a perfect reconstruction, but also performs well in image denoising compared to other state-of-the-art MGA methods.
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