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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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. Methods for super-resolution can be broadly classified into two families of methods: (i) The classical multi-image super-resolution (combining images obtained at subpixel misalignments), and (ii) Example-Based super-resolution (learning correspondence between low and high resolution image patches from a database). In this paper we propose a unified framework for combining these two families of methods. We further show how this combined approach can be applied to obtain super resolution from as little as a single image (with no database or prior examples). Our approach is based on the observation that patches in a natural image tend to redundantly recur many times inside the image, both within the same scale, as well as across different scales. Recurrence of patches within the same image scale (at subpixel misalignments) gives rise to the classical super-resolution, whereas recurrence of patches across different scales of the same image gives rise to example-based super-resolution. Our approach attempts to recover at each pixel its best possible resolution increase based on its patch redundancy within and across scales. 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. 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. "...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature Image texture provides a rich visual description of the surfaces in the scene. Many texture signatures based on various statistical descriptions and various local measurements have been developed. Existing signatures, in general, are not invariant to 3D geometric transformations, which is a serious limitation for many applications. In this paper we introduce a new texture signature, called the multifractal spectrum (MFS). The MFS is invariant under the bi-Lipschitz map, which includes view-point changes and non-rigid deformations of the texture surface, as well as local affine illumination changes. It provides an efficient framework combining global spatial invariance and local robust measurements. Intuitively, the MFS could be viewed as a "better histogram" with greater robustness to various environmental changes and the advantage of capturing some geometrical distribution information encoded in the texture. Experiments demonstrate that the MFS codes the essential structure of textures with very low dimension, and thus represents an useful tool for texture classification. We propose a new high-quality and efficient single-image upscaling technique that extends existing example-based super-resolution frameworks. In our approach we do not rely on an external example database or use the whole input image as a source for example patches. Instead, we follow a local self-similarity assumption on natural images and extract patches from extremely localized regions in the input image. This allows us to reduce considerably the nearest-patch search time without compromising quality in most images. Tests, that we perform and report, show that the local self-similarity assumption holds better for small scaling factors where there are more example patches of greater relevance. We implement these small scalings using dedicated novel nondyadic filter banks, that we derive based on principles that model the upscaling process. Moreover, the new filters are nearly biorthogonal and hence produce high-resolution images that are highly consistent with the input image without solving implicit back-projection equations. The local and explicit nature of our algorithm makes it simple, efficient, and allows a trivial parallel implementation on a GPU. We demonstrate the new method ability to produce high-quality resolution enhancement, its application to video sequences with no algorithmic modification, and its efficiency to perform real-time enhancement of low-resolution video standard into recent high-definition formats.