The Study On Multifractal Theory And Application In Image Processing

Since recent more than ten years , as a core branch of fractal geometry for researching irregular geometrical figure and describing the nature, Multifractal theory plays an important role in actual application such as finance, physics, network traffic, biomedical engineering, .image processing, transmission, communication ,pattern recognition and so on. It also is a very active branch of non-linear field in the world. Because its potential value in theory and application, it has received much attention paid by related research fellow. Especially, it plays a very important role in modern image processing and analysis.In this dissertation, theoretic methods and application of Multifractal in image processing and analysis are mainly researched. The mechanism of fractal image processing is recommended, and at the same time, the mathematical and related theoretical fundament of fractal and Multifractal image processing are carefully analyzed and discussed. Some new algorithm and methods in image processing and analysis are proposed, which include application researches on image denoising, image compression, image texture analysis, and so on. Experiment results show that these algorithms and methods are useful and valid.The main contributions of this dissertation are as follows:1, After some recalls on fractal and multifractal theory, by defining the new partition function for the moments of the wavelet coefficients employing the wavelet transform modulus maxima method(WTMM), a new estimator for multifractal spectrum form a finite-length is presented in this dissertation. The estimators are based on a log-linear relationship between averages of the moments of wavelet coefficients and scales suggesting a linear regression approach. Based on the q-th sample moment of the wavelet coefficients at the given scale., using a regression procedure with potentially corrected error structure, the estimator first computes the partition function, Then Performing finite differencing on the partition function,we estimate α , finally can obtain an estimate of the spectrum f(α),The advantage of this algorithm is that it can calculate the multifractal spectrum effectively and simply. In constrast to previouse works on multifractal spectrum estimation, as a direct estimation strategy for α and f(α) it can avoids some of the practical difficulties when performing the Legendre transform,2, After 2-microlocal analysis introduced, 2-microlocal spaces were defined, and characterized by a wavelet analysis. We propose an algorithm which uses the new characterization to estimate the 2-microlocal frontier. The computation of both frontier.exponents(s,s') is performed using directly the values of the function. This method does not lose information by integrating or smoothing the data. Moreover one can extract from the frontier the iregularity information, i.e. the Holder exponents, with a better accuracy. Furthermore, the improved robust estimators of both the pointwise and the local exponents are obtained by 2-microlocal frontier.Based on 2-microlocal analysis and estimate of 2-microlocal frontier, a new approach for image denoising by singularity structures analysis and multifractal is proposed. Multifractal niose model is constructed. The method does not make assumption the type of noise or global smoothness of original data, and the image is characterized by its multifractal spectrum. This approach processes Hausdorff exponent of the image by defining a new transformation operationbased on 2-microlocal analysis and wavelet, so that the most points in denoised image lie in smooth regions while preserving the relative strength of the singularities in the images.Experiment results explain that the method leads to a smooth image and keeps the information of the original images.3^ The image texture segmentation method based on multifractal analysis defined for a sequence of capacities is presented. A multifractal analysis first is defined for sequences of Choquet capacities with respect to a general class of measures, and some core results are proposed concerning multifractal analysis about a sequence of capacities. In particular, we show how to construct the sequence of capacities used to image processing. In order to realize image texture segmentation, a novel feature vectors is defined with the singularity exponents and spectrum for sum, max, min, iso and Lp capacities of the image.Then, the fuzzy clustering Kohonen network (FKCN) has been discussed. An adaptive FKCN model is presented. Comparing with the classical Kohonen network, AFKN is nonsequential, supervised, and effective update neighborhoods which highly improves the ability of discrimination and make the detection algorithm more accurate and more robust. The proposed method can fully extract the texture information and provide accurate classification results for the different texture.The new method is essentially different from current approaches, It deals with 2D grey level Image from view point of capacities, not assumes that the 2D grey level Image can be considered as a spatial coordinate on the z-axis, since the scaling properties of the grey levels are generally different from those of the space coordinates. Instead, the grey levels were considered as capacities. This method has been extended to color image segmentation. The experimental result indicate that the proposed algorithm has better segmentation quality and improves convergence4, A generalized class of affine two-dimensional fractal-wave let transforms has been introduced. Iterated function system with grey level maps is discussed. Based on generalized two-dimensional fractal-wavelet transform, a new hybrid Wavelet-fractal image coding algorithm is presented. On the one hand, the paper proposes a novel Smooth symmetric biorthogonal wavelet basis selection algorithm by optimization of discrete finite variation (DFV) measure to get ride of the blocking effect in result image. On the other hand, in order to improve reconstructed image quality, an adaptive wavelet sub-trees partition algorithm is given which can split the wavelet subtree into children subtrees according to local image complexity. Then, the new approach is extended to 3-dimension color space for color image encoding Experimental results for gray image and color image prove that the new algorithm can obtain much better coding performance when the compression ratio is the same.5 > Based on multifractal measure, a fast method of adaptive wavelet packet image compression is presented to obtain the best reconstructed image in term of a higher PSNR, at the lowest bit rate. First we developed a new fast two-dimension convolution-decimation algorithm with factorized non-separable 2-D filter. A novel extending Shannon cost function is proposed. Wavelet packet bases can be obtained with the new cost function. Then the conception of the relative multifractal spectrum is introduced. Using the relative multifractal dimension spectrum, a new objective measure of image quality corresponding to the human visual impresion is developed. An approach to identify regions of complexity disparity, as Region of Interest (ROI), in an image is presented based on the relative multifractal dimension spectrum objective measure. A study of the effect of enhancing regions has been implemented. Later, a wavelet packet scheme of imagecompression through zerotree coding, the Region of Interest coding, and adaptive arithmetic coding is constructedThis technology solves the difference between the image quality and the compress rate, The experimental results reveal that the image quality of the regions of interest are better than those by the oringinal embedded zertree algorithm, and the proposed image quality measure corresponds to the human visual impresion.