Abstract Fractal analysis has been widely used in computer vision, especially in texture image processing and texture analysis. The key concept of fractal-based image model is the fractal dimension, which is invariant to bi-Lipschitz transformation of image, and thus capable of representing intrinsic structural information of image robustly. However, the invariance of fractal dimension generally does not hold after fifiltering, which limits the application of fractal-based image model. In this paper, we propose a novel fractal dimension invariant fifiltering (FDIF) method, extending the invariance of fractal dimension to fifiltering operations. Utilizing the notion of local self-similarity, we fifirst develop a local fractal model for images. By adding a nonlinear postprocessing step behind anisotropic fifilter banks, we demonstrate that the proposed fifiltering method is capable of preserving the local invariance of the fractal dimension of image. Meanwhile, we show that the FDIF method can be reinstantiated approximately via a CNN-based architecture, where the convolution layer extracts anisotropic structure of image and the nonlinear layer enhances the structure via preserving local fractal dimension of image. The proposed fifiltering method provides us with a novel geometric interpretation of CNN-based image model. Focusing on a challenging image processing task — detecting complicated curves from the texture-like images, the proposed method obtains superior results to the state-of-art approaches