This work presents a novel descriptor for texture images based on fractal geometry and its application to image analysis. The descriptors are provided by estimating the triangular prism fractal dimension under different scales with a weight exponential parameter, followed by dimensionality reduction using Karhunen-Lo e ve transform. The efficiency of the proposed descriptors is tested on two well-known texture data sets, that is, Brodatz and Vistex, both for classification and image retrieval. The novel method is also tested concerning invariances in situations when the textures are rotated or affected by Gaussian noise. The obtained results outperform other classical and state-of-the-art descriptors in the literature and demonstrate the power of the triangular descriptors in these tasks, suggesting their use in practical applications of image analysis based on texture features.
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.  This article reports on the application of the recently introduced concept of multiscale fractal dimension (MFD) as a resource for quantifying three-dimensional gene expression patterns in embryonic development. While traditional fractal dimensions provide interesting possibilities for quantifying pattern complexity, as defined by the intensity in which the pattern interacts with its surrounding space, those approaches fail to take into account the important fact that natural objects are not perfectly self-similar. By expressing the fractal behavior explicitly in terms of the spatial scale, the MFD provides a more comprehensive and objective characterization of the complexity of natural data such as gene expression profiles. After presenting the MFD concept as well as a technique for its numerical estimation, the potential of this measure for objectively quantifying gene expression is discussed, and a complete example is provided regarding the three-dimensional expression of the myogenic marker gene Myf5 along successive somites in a mouse embryo. In this specific case, the adopted technique proved itself a useful means for identifying spatial variations of gene expression intensity. Abstract A new method is proposed to calculate the fractional (fractal) dimension of topographic surfaces. The method uses a three-dimensional geometric equivalent of the “walking dividers” method in two dimensions. that is it takes elevation values at the corners of squares, interpolates a center value, divides the square into four triangles, and then computes the surface areas of the tops of the prisms which result from raising the triangles to their given elevations. By repeating this calculation for different size squares, the relationship between the total area of the topographic surface and the spacing of the squares is established, and used to estimate the fractal dimension. A computer program in the C programming language is presented to read digital terrain matrices and compute the fractal dimension of the surface using the method. Some test data sets are provided along with their computed fractal dimensions. Finally, the advantages and disadvantages of the method are discussed.