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.
Fractal-based image model has been widely used to solve many problems of computer vision, including, texture analysis @cite_27 , bio-medical image processing @cite_40 , and image quality assessment @cite_19 . The local fractal analysis method in @cite_26 and the spectrum of fractal dimension in @cite_33 @cite_16 take advantage of the bi-Lipschitz invariance property of fractal dimension for texture classification, whose features are very robust to the deformation and scale changing of textures. Because the local self-similarity of image is often ubiquitous both within and across scales @cite_28 @cite_11 , natural images can also be modeled as fractals locally @cite_15 @cite_44 . Recently, the fractal model of natural image is applied to image super-resolution @cite_48 @cite_32 , where the local fractal analysis is used to enhance image gradient adaptively. @cite_40 , a fracal-based dissimilarity measurement is proposed to analyze MRI images. However, because the invariance of fractal dimension does not hold after filtering, it is difficult to merge fractal analysis into other image processing methods.
@cite_26 @cite_33 @cite_28 @cite_48 @cite_32 @cite_44 @cite_19 @cite_40 @cite_27 @cite_15 @cite_16 @cite_11
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In this paper, we propose a single image super-resolution and enhancement algorithm using local fractal analysis. If we treat the pixels of a natural image as a fractal set, the image gradient can then be regarded as a measure of the fractal set. According to the scale invariance (a special case of bi-Lipschitz invariance) feature of fractal dimension, we will be able to estimate the gradient of a high-resolution image from that of a low-resolution one. Moreover, the high-resolution image can be further enhanced by preserving the local fractal length of gradient during the up-sampling process. We show that a regularization term based on the scale invariance of fractal dimension and length can be effective in recovering details of the high-resolution image. Analysis is provided on the relation and difference among the proposed approach and some other state of the art interpolation methods. Experimental results show that the proposed method has superior super-resolution and enhancement results as compared to other competitors. We address the problem of developing discriminative, yet invariant, features for texture classification. Texture variations due to changes in scale are amongst the hardest to handle. One of the most successful methods of dealing with such variations is based on choosing interest points and selecting their characteristic scales [ PAMI 2005]. However, selecting a characteristic scale can be unstable for many textures. Furthermore, the reliance on an interest point detector and the inability to evaluate features densely can be serious limitations. Fractals present a mathematically well founded alternative to dealing with the problem of scale. However, they have not become popular as texture features due to their lack of discriminative power. This is primarily because: (a) fractal based classification methods have avoided statistical characterisations of textures (which is essential for accurate analysis) by using global features; and (b) fractal dimension features are unable to distinguish between key texture primitives such as edges, corners and uniform regions. In this paper, we overcome these drawbacks and develop local fractal features that are evaluated densely. The features are robust as they do not depend on choosing interest points or characteristic scales. Furthermore, it is shown that the local fractal dimension is invariant to local bi-Lipschitz transformations whereas its extension is able to correctly distinguish between fundamental texture primitives. Textures are characterised statistically by modelling the full joint PDF of these features. This allows us to develop a texture classification framework which is discriminative, robust and achieves state-of-the-art performance as compared to affine invariant and fractal based methods. Fractals have been shown to be useful in characterizing texture in a variety of contexts. Use of this methodology normally involves measurement of a parameter H, which is directly related to fractal dimension. In this work the basic theory of fractional Brownian motion is extended to the discrete case. It is shown that the power spectral density of such a discrete process is only approximately proportional to |f|a instead of in direct proportion as in the continuous case. An asymptotic Cramer-Rao bound is derived for the variance of an estimate of H. Subsequently, a maximum likelihood estimator (MLE) is developed to estimate H. It is shown that the variance of this estimator nearly achieves the minimum bound. A generation algorithm for discrete fractional motion is presented and used to demonstrate the capabilities of the MLE when the discrete fractional Brownian process is contaminated with additive Gaussian noise. The results show that even at signal-to-noise ratios of 30 dB, significant errors in estimation of H can result when noise is present. The MLE is then applied to X-ray images of the human calcaneus to demonstrate how the line-to-line formulation can be applied to the two-dimensional case. These results indicate that it has strong potential for quantifying texture. This paper addresses the problems of 1) representing natural shapes such as mountains, trees, and clouds, and 2) computing their description from image data. To solve these problems, we must be able to relate natural surfaces to their images; this requires a good model of natural surface shapes. Fractal functions are a good choice for modeling 3-D natural surfaces because 1) many physical processes produce a fractal surface shape, 2) fractals are widely used as a graphics tool for generating natural-looking shapes, and 3) a survey of natural imagery has shown that the 3-D fractal surface model, transformed by the image formation process, furnishes an accurate description of both textured and shaded image regions. The 3-D fractal model provides a characterization of 3-D surfaces and their images for which the appropriateness of the model is verifiable. Furthermore, this characterization is stable over transformations of scale and linear transforms of intensity. The 3-D fractal model has been successfully applied to the problems of 1) texture segmentation and classification, 2) estimation of 3-D shape information, and 3) distinguishing between perceptually smooth'' and perceptually textured'' surfaces in the scene.