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Fractal analysis has been widely used in computer vision, especially intexture image processing and texture analysis. The key concept of fractal-basedimage model is the fractal dimension, which is invariant to bi-Lipschitztransformation of image, and thus capable of representing intrinsic structuralinformation of image robustly. However, the invariance of fractal dimensiongenerally does not hold after filtering, which limits the application offractal-based image model. In this paper, we propose a novel fractal dimensioninvariant filtering (FDIF) method, extending the invariance of fractaldimension to filtering operations. Utilizing the notion of localself-similarity, we first develop a local fractal model for images. By adding anonlinear post-processing step behind anisotropic filter banks, we demonstratethat the proposed filtering method is capable of preserving the localinvariance of the fractal dimension of image. Meanwhile, we show that the FDIFmethod can be re-instantiated approximately via a CNN-based architecture, wherethe convolution layer extracts anisotropic structure of image and the nonlinearlayer enhances the structure via preserving local fractal dimension of image.The proposed filtering method provides us with a novel geometric interpretationof CNN-based image model. Focusing on a challenging image processing task ---detecting complicated curves from the texture-like images, the proposed methodobtains superior results to the state-of-art approaches.

Fractal Dimension Invariant Filtering and Its CNN-based Implementation