Fractal Dimension Invariant Filtering and Its CNN-based Implementation

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 filtering, which limits the application of fractal-based image model. In this paper, we propose a novel fractal dimension invariant filtering (FDIF) method, extending the invariance of fractal dimension to filtering operations. Utilizing the notion of local self-similarity, we first develop a local fractal model for images. By adding a nonlinear post-processing step behind anisotropic filter banks, we demonstrate that the proposed filtering method is capable of preserving the local invariance of the fractal dimension of image. Meanwhile, we show that the FDIF method can be re-instantiated 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 filtering 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.