Inverse problem for two-dimensional fractal sets using the wavelet transform and the moment method

Iterated function system's (IFSs) can achieve a very high compression ratio for a fractal image. It iteratively generates a fractal image by the use of, for example, the random iteration algorithm. However, the original sizes and coordinates of fractal images are usually different and they can be measured only from the decoded results. This is a drawback since the decoded fractal image shown on a display of fixed size may not be with an adequate scale and located in a desired position. It should make each decoded fractal image which can be generated with a desired size. In this paper, we propose a fixed-point-searching algorithm which can automatically determine the original size and the coordinates of a fractal image directly from its IFS code. With the proposed modification on the translation parameters in IFS code, we can resize and relocate the decoded image to become a desired one. The computation load is low because only the calculation of certain fixed points are required. Finally, the computer simulation for the fractal fern is performed to demonstrate the decoded fractal image can be displayed with a desired size.

Chaotic orbits and bifurcation from a fixed point generated by an iterated function system

Compression of images process is a step in the image processing. It is concerned with the transmission and storage of digitally created images. Fractal coding is a potential image and video compression approach with good reconstruction fidelity and relatively large compression ratios, because of its simplicity and great performance; fractal image compression (FIC) is a particularly popular approach in image compression applications. However, it has a significant disadvantage in the form of a long encoding time. This is because encoding any small bit necessitates a massive similarity search in the original data As a result; the FIC search time is reduced while the quality of the reconstructed images is maintained acceptable level in many introduced paper and other still a study topic in progress. Fractal images are images that are self-similar in that each individual part is the same as the total. This paper will discusses many attempts for more author that working on image and video compression using fractal compression technique based on various approach and with each discuss focuses on the main parameter of compression such compression ratio (CR), peak signal to noise ratio (PSNR) and encoding time, as well as the details of data set the used for testing also writing with each technique to creating fractal video and image compression. Index Terms— FIC, Video compression, parallel processing, Iteration Function System (IFS), Image processing.

Many effective methods have been proposed to solve the problem of image compression. Fractal image code can compress an image in a higher compress ratio level, the fundamental idea of fractal image compression is based on the iteration function system (IFS). In this paper, we proposed a novel method based on IFS with probabilities can be used in image compression and image retrieval, we obtain enlightenment from fractal no search methods, our method also no need to carried out searching stage when compress the image, so the consuming time with image compression is less than traditional methods. We utilized the image code as index file, and apply our method in image retrieval, experiment results indicate that our method is effective in image retrieval.

A novel optical architecture based on the random iteration algorithm to decode iterated function system (IFS) codes is presented. The optical part of this system performs the affine transformations of IFS. The code values in the affine transformations of IFS are first translated into the modulation voltages of the electro-optic modulators (EOMS), with a specific probability. Then, the multiplication and addition operations can easily be performed. With the O/E and A/D converters, the light intensity is converted into an electrical signal to address a specific pixel at the display device. A fractal image is then progressively decoded pixel by pixel and superimposed on the display device. The computer simulation and some experimental results for the proposed architecture are provided for verification. The decoding speed is fast, the image quality is high, and the resolution of the generated fractals depends on the resolution of the display device itself.

One of the main disadvantages of fractal image data compression is a loss time in the process of image compression (encoding) and conversion into a system of iterated functions (IFS). In this paper, the idea of the inverse problem of fixed point is introduced. This inverse problem is based on collage theorem which is the cornerstone of the mathematical idea of fractal image compression. Then this idea is applied by iterated function system, iterative system functions and grayscale iterated function system down to general transformation. Mathematical formulation form is also provided on the digital image space, which deals with the computer. Next, this process has been revised to reduce the time required for image compression by excluding some parts of the image that have a specific milestone. The neural network algorithms have been applied on the process of compression (encryption). The experimental results are presented and the performance of the proposed algorithm is discussed. Finally, the comparison between filtered ranges method and self-organizing method is introduced.

All images have two basic characters: geometry and texture. Basing the two characters, we present Markov Iterated Function System (MIFS) model, consisting of Iterated Function System (IFS) and Markov Random Field (MRF). The fractal images have self-similarities in statistics, especially in geometric topological structure. IFS is useful for discribing the self- similarities. In the meantime, we consider a texture to be a stochastic field and usually it is anisotropic. We choose MRF as the model of the texture. It is convenient to control the geometric characters and texture through changing the parameters of MIFS model. We explored a fast iterated algorithm to realize a MIFS sample and an algorithm to estimate the parameters of MIFS.

The Hutchinson metric is a natural measure of the discrepancy between two images for use in fractal image processing. A neural network is described which can quickly calculate this metric. By combining this with the architecture previously described by the author for implementing the Markov operator of an iterated function system (IFS) on a neural network, a fast method is obtained for determining the distance between a target image and the invariant measure of a trial IFS. This has obvious applications to Barnsley's fractal image compression scheme.

The hierarchical IFS coding proposed by the authors is applied to monochrome images. The present method is a fusion of the hierarchical procedure and fractal coding based on the iterated function system (IFS) theory. By using a hierarchical structure based on the Gaussian pyramid, the image is divided into frequency subbands so that the information is compressed. By using the information between the bands, the coding can effectively capture the similarity between the range and the domain. Further, an application of this method to color images is discussed. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 83(7): 71–78, 2000

Bounding the attractor of an IFS

Iterated function systems (IFSs) are capable of effectively describing complex shapes and textures by fractals. The interscale properties of such fractals are analyzed with the aid of the wavelet transform and general multiresolution analysis. The result is a formulation for homogeneous IFSs in general scale space which leads to a direct solution of the inverse problem of finding the IFS which best represents a given function. Multiscale techniques that are used in the analysis are discussed. Some previous results from the IFS literature are introduced.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Fractal image compression using the iterative function system (IFS) is based on the collage theorem proposed by Barnsley et al. (1986). In this conventional method, the errors between the reconstructed image and original image may be greater than the errors between the collage and original image because the collage theorem does not guarantee the former errors to be smaller than the latter errors. This paper proposes an extended collage theorem. An IFS algorithm based on this theorem determines the parameters after iterations of the contraction mappings. An image reconstructed according to the new theorem has higher quality than one based on the existing collage theorem. The reconstructed image can be gotten by fewer iterations than by the conventional IFS.

Approximation of an image by the attractor evolved through iterations of a set of contractive maps is usually known as fractal image compression. The set of maps is called iterated function system (IFS). Several algorithms, with different motivations, have been suggested towards the solution of this problem. But, so far, the theory of IFS with probabilities, in the context of image compression, has not been explored much. In the present article we have proposed a new technique of fractal image compression using the theory of IFS and probabilities. In our proposed algorithm, we have used a multiscaling division of the given image up to a predetermined level or up to that level at which no further division is required. At each level, the maps and the corresponding probabilities are computed using the gray value information contained in that image level and in the image level higher to that level. A fine tuning of the algorithm is still to be done. But, the most interesting part of the proposed technique is its extreme fastness in image encoding. It can be looked upon as one of the solutions to the problem of huge computational cost for obtaining fractal code of images.

Measures generated by Iterated Function Systems can be used in place of atomic measures in Gaussian integration. A stable algorithm for the numerical solution of the related approximation problem – an inverse problem in fractal construction – is proposed.

In this paper, the fractal function is used for the image representation and compression. By introducmg a searching strategyinto the fractal-function-based image coding, the image representation accuracy is improved by 10%. A regional adaptiveresolution-based fractal block coding is formed by combining the encoding and the decoding processes together with aquadtree structure to perform image compression. It is shown by the experiments that the image compression ratio isincreased up to 1 2 without a great loss of visual quality.Keywords: Image coding; image compression; fractal function. 1. INTRODUCTION AND THEORETICAL FOUNDATION Fractal has been stimulating great interests since John Hutchinson introduced the theory of iterated function system(IFS) [1] to model collections of contractive transformations in a metric space as dynamical systems. After the work of M.Barnsley and A. E. Jacquin [2]-[3] in using the partitioned iterated function system (PIFS) to do fractal-based imagecompression, many works have been done either to improve the compression ratio or to reduce the encoding time. F.Dudbridge[4]-[5] developed a method which is quite different from the traditional fractal (TF) image coding. He proposedusing a new fractal function (FF) for the image representation.Before the concept of fractal function is introduced the following preliminaries are needed: (1) w1 ,