Fractal Coding Research Articles

Adaptively hybrid fractal image coding

AbstractAn adaptively hybrid method was proposed to improve the performance of fractal coding methods. First, it is found that the range blocks with large variances (RBLVs) play a crucial role in degrading decoded images, and the effect of the remaining range blocks with small variances (RBSVs) can be ignored. Then, an adaptive method was proposed to divide the range blocks into the above two categories: RBLVs and RBSVs. Second, RBLVs were designed to be encoded in an extended domain block pool (EDBP). Then, better block‐matching effect can be obtained, which will result in better decoded image quality. Further, the no‐search fractal encoding method is adopted for RBSVs to achieve faster encoding speed and fewer bits per pixel (bpp). Finally, four fractal coding methods were adopted to assess the performance of the proposed method. Experimental results show that compared with the previous methods, the PSNR quality of decoded images in the proposed method can be improved by about 0.15–0.4 dB, about 20%–35% of the total computations in encoding process can be saved, and about 0.2 bpp can be saved. Moreover, under the same decoding time, the proposed method can achieve comparable or smaller deviations regarding the decoded image.

Image Encryption Technology Based on Fractal Image Compression Algorithm

As the most commonly used information transmission method, digital images often store a large amount of personal information. To prevent information leakage, encrypting images is essential. Common image encryption techniques suffer from certain limitations, such as overly simple encryption methods and long encryption times. In response to the above issues, this study proposes the Frobenius canonical form image encryption scheme. It calculates the fractal code through the fractal compression algorithm and to encrypt the image, it adjusts the brightness coefficient in the fractal code. To address unsatisfactory correlation coefficients in encrypted images, the Frobenius canonical form image encryption is improved by introducing the Arnold transformation encryption, which combines the two methods to reduce correlation coefficients. Finally, the knight tour algorithm is put forward. In response to the long image scrambling time in the knight tour algorithm, the Tetragonal theorem is combined with the scheme to encrypt the image. It is then re-encrypted using the Frobenius canonical form. The experimental findings illustrate that when using Frobenius canonical form, Arnold transformation combined with Frobenius canonical form, and the tetragonal algorithm combined with knight tour algorithm to encrypt Lena images, the three decryption methods correspond to image similarity of over 70%, over 80%, and over 90%, respectively. Combining the tetragonal algorithm and the knight tour algorithm can significantly increase the security of image encryption.

MULTIPARENT FRACTAL IMAGE CODING-BASED METHODS FOR SALT-AND-PEPPER NOISE REMOVAL

Salt-and-pepper noise consists of outlier pixel values which significantly impair image structure and quality. Multiparent fractal image coding (MFIC) methods substantially exploit image redundancy by utilizing multiple domain blocks to approximate the range block, partially compensating for the information loss caused by noise. Motivated by this, we propose two novel image restoration methods based on MFIC to remove salt-and-pepper noise. The first method integrates Huber M-estimation into MFIC, resulting in an improved anti-salt-and-pepper noise robust fractal coding approach. The second method incorporates MFIC into a total variation (TV) regularization model, including a data fidelity term, an MFIC term and a TV regularization term. An alternative iterative method based on proximity operator is developed to effectively solve the proposed model. Experimental results demonstrate that these two proposed approaches achieve significantly enhanced performance compared to traditional fractal coding methods.

The application of fractal and chaos theory in photo processing

Nowadays, more and more people are looking for a faster and safer way of encryption. Fractal coding and chaos encryption fit those two points. This article discussed the usage of Fractal and Chaos Theory in photo processing. For example, Fractals and chaos theory are usually used in picture encrypting and picture processing to make the object in the target picture much more clearly. And currently, the method become advanced. The current method can provide great defense from hacking. Meanwhile, in terms of compression performance, it achieves a higher compression ratio than other solutions while meeting the reconstruction visual quality. In this article, the writer analyses the future of Fractal coding by talking about the advantages and disadvantages of this method which came from different articles. Furthermore, in this paper, the author discusses the definition of Fractal and the properties of Fractal. The Iterated Function System (IFS) is also a key point in photo processing.

Review on Fractal Video and Image Compression Techniques

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.

A cellular automaton decoder for a noise-bias tailored color code

Self-correcting quantum memories demonstrate robust properties that can be exploited to improve active quantum error-correction protocols. Here we propose a cellular automaton decoder for a variation of the color code where the bases of the physical qubits are locally rotated, which we call the XYZ color code. The local transformation means our decoder demonstrates key properties of a two-dimensional fractal code if the noise acting on the system is infinitely biased towards dephasing, namely, no string-like logical operators. As such, in the high-bias limit, our local decoder reproduces the behavior of a partially self-correcting memory. At low error rates, our simulations show that the memory time diverges polynomially with system size without intervention from a global decoder, up to some critical system size that grows as the error rate is lowered. Furthermore, although we find that we cannot reproduce partially self-correcting behavior at finite bias, our numerics demonstrate improved memory times at realistic noise biases. Our results therefore motivate the design of tailored cellular automaton decoders that help to reduce the bandwidth demands of global decoding for realistic noise models.

Image compression and encryption algorithm based on uniform non-degeneracy chaotic system and fractal coding

Image compression and encryption algorithm based on uniform non-degeneracy chaotic system and fractal coding

Digital Video Compression Using DCT-Based Iterated Function System (IFS)

Large video files processing involves a huge volume of data. The codec, storage systems and network needs resource utilization, so it becomes important to minimize the used memory space and time to distribute these videos over the Internet using compression techniques. Fractal image and video compression falls under the category of lossy compression. It gives best results when used for natural images.
 This paper presents an efficient method to compress an AVI (Audio Video Interleaved) file with fractal video compression(FVC). The video first is separated into a sequence of frames that are a color bitmap images, then images are transformed from RGB color space to Luminance/Chrominance components (YIQ) color space; each of these components is compressed alone with Enhanced Partition Iterated Function System (EPIFS), then fractal codes are saved.
 The classical IFS suffers from a very long encoding time that needed to find the best matching for each range block when compared with the domain image blocks. In this work, the (FVC) is enhanced by enhancing the IFS of the fractal image compression using a classification scheme based on the Discrete Cosine Transform (DCT). Experimentation is performed by considering different block sizes and jump steps to reduce number of the tested domain blocks. Results shows a significant reduction in the encoding time with good quality and high compression ratio for different video files.

Fabric Image Retrieval Algorithm based on Fractal Coding and Zernike Moment under Wavelet Transform

Abstract: To help the workers retrieve the same or similar in textile companies, a fabric image retrieval algorithm based on fractal coding and Zernike moment under wavelet transform is proposed, which can quickly and accurately retrieve images similar to the fabric image to be retrieved from the fabric image database. Firstly, the low-frequency component is obtained by wavelet transform, and the transformed low-frequency sub-image is fractal encoded to obtain the coding parameters. Then, the Zernike moment of the low-frequency sub-image is calculated. The fractal coding parameters and Zernike moment under wavelet transform are combined as the fabric image retrieval characteristic. The algorithm overcomes the problems of low retrieval accuracy and time-consuming of direct feature extraction under single feature. Compared with the basic fractal algorithm (BFIC), the joint orthogonal fractal parameters with the improved Hu invariant moment algorithm (HVKF) and the sparse fractal image compression

Activation Functions Effect on Fractal Coding Using Neural Networks

Activation functions play an essential role in converting the output of the artificial neural network into nonlinear results, since without this nonlinearity, the results of the network will be less accurate. Nonlinearity is the mission of all nonlinear functions, except for polynomials. The activation function must be differentiable for backpropagation learning. This study’s objective is to determine the best activation functions for the approximation of each fractal image. Different results have been attained using Matlab and Visual Basic programs, which indicate that the bounded function is more helpful than other functions. The non-linearity of the activation function is important when using neural networks for coding fractal images because the coefficients of the Iterated Function System are different according to the different types of fractals. The most commonly chosen activation function is the sigmoidal function, which produces a positive value. Other functions, such as tansh or arctan, whose values can be positive or negative depending on the network input, tend to train neural networks faster. The coding speed of the fractal image is different depending on the appropriate activation function chosen for each fractal shape. In this paper, we have provided the appropriate activation functions for each type of system of iterated functions that help the network to identify the transactions of the system.

Fabric image retrieval algorithm based on fractal coding and Zernike moment under the wavelet transform

The predecessor of China Optics was China Optics and Applied Optics Digest, which was founded in 1985. At that time, it was the only retrieval journal in the field of optics in China. At the end of 2008, China Optics and Applied Optics Digest was renamed

Topological Order, Quantum Codes, and Quantum Computation on Fractal Geometries

We investigate topological order on fractal geometries embedded in $n$ dimensions. In particular, we diagnose the existence of the topological order through the lens of quantum information and geometry, i.e., via its equivalence to a quantum error-correcting code with a macroscopic code distance or the presence of macroscopic systoles in systolic geometry. We first prove a no-go theorem that $\mathbb{Z}_N$ topological order cannot survive on any fractal embedded in 2D. For fractal lattice models embedded in 3D or higher spatial dimensions, $\mathbb{Z}_N$ topological order survives if the boundaries of the interior holes condense only loop or membrane excitations. Moreover, for a class of models containing only loop or membrane excitations, and are hence self-correcting on an $n$-dimensional manifold, we prove that topological order survives on a large class of fractal geometries independent of the type of hole boundaries. We further construct fault-tolerant logical gates using their connection to global and higher-form topological symmetries. In particular, we have discovered a logical CCZ gate corresponding to a global symmetry in a class of fractal codes embedded in 3D with Hausdorff dimension asymptotically approaching $D_H=2+\epsilon$ for arbitrarily small $\epsilon$, which hence only requires a space-overhead $\Omega(d^{2+\epsilon})$ with $d$ being the code distance. This in turn leads to the surprising discovery of certain exotic gapped boundaries that only condense the combination of loop excitations and gapped domain walls. We further obtain logical $\text{C}^{p}\text{Z}$ gates with $p\le n-1$ on fractal codes embedded in $n$D. In particular, for the logical $\text{C}^{n-1}\text{Z}$ in the $n^\text{th}$ level of Clifford hierarchy, we can reduce the space overhead to $\Omega(d^{n-1+\epsilon})$. Mathematically, our findings correspond to macroscopic relative systoles in fractals.

A Double Random Matrix Design Model for Fractal Art Patterns Based on Visual Characteristics

This paper adopts the method of visual characteristics of the double random matrix to conduct in-depth research and analysis on the design of fractal art patterns. For the practical application needs in fractal graphic design, a method is proposed to automatically extract the core base pattern based on fractal graphic content and generate a four-sided continuous pattern. The method first uses the Canny operator for edge detection to analyze the area of the main pattern. Then, it uses a grayscale cogeneration matrix to extract and analyze the graphic texture features, based on which the best splicing method is selected to splice the extracted pattern, and then, it achieves two splicing methods of flat row and staggered four-sided continuous pattern. The method has good practicality with low complexity and high versatility under the premise of ensuring the beauty of the generated four-sided continuous pattern. It can assist designers to design patterns, improve efficiency, and save design costs. In this paper, we improve the existing image segmentation methods, adopt two segmentation methods, namely quadratic tree segmentation and HV segmentation, propose a new local codebook selection strategy, and study the degree of self-adaptation of different methods to images in terms of segmentation methods and local codebook selection strategies. It makes the network unable to train efficiently on long text prediction problems. Finally, the improved algorithm of this paper is tested on the standard and living image libraries, and the experimental results show that the use of local codebooks makes the image coding speed significantly improved compared with the fixed fractal. In the face of the image to be retrieved, it is only necessary to perform the coding operation to obtain the fractal code to perform similarity matching, which can meet the requirement of real-time retrieval. Applying the improved distance formula, the search accuracy obtained on the test gallery is significantly better than that of the grayscale histogram algorithm.

A chaos-based image compression and encryption scheme using fractal coding and adaptive-thresholding sparsification

With the wide application of digital images in the network, the problems of information security and transmission efficiency need to be solved urgently. Fractal encoding as an efficient compression method, its security is not guaranteed. To solve these problems, this paper proposes a new image compression and encryption scheme based on chaotic map and frequency division joint compression coding, which uses the combination of fractal encoding and adaptive-thresholding sparsification to compress images. Arnold scrambling and chaos-based plane scrambling diffusion are embedded in two compression processes respectively. In addition, an efficient encryption algorithm is designed based on the coupling-enhanced chaotic map, which encrypt the compressed data as a whole through simultaneous scrambling of parent blocks and sub-blocks and zigzag round-trip diffusion. The simulation results demonstrate that the proposed image compression and encryption scheme is suitable for gray and color images, and can achieve higher compression ratio and improve the efficiency of fractal compression. The designed encryption algorithm is reliable, and more efficient than several advanced image encryption schemes.

High Synthetic Audio Compression Model Based on Fractal Audio Coding and Error-Compensation

This study presented a model for improving audio files quality using fractal coding specifically when a high compression ratio is required. The proposed high synthetic audio compression model which can be called (HSACM) is based on conventional fractal coding and lifting wavelet transform. Various lifting wavelet transform families and levels are used and their effects on the reconstructed audio files are discussed as well. Audio files from GTZAN dataset and standard measurements for data compression are used in the evaluation of the proposed model. The results reveal that using block length 50 samples which is the worst case, PSNR is increased, on average, from 34.1 to 44.8 dB and from 34.1 to 40.5 dB using lifting wavelet transform with 3 and 2 levels, respectively. Thus, the PSNR is improved by 10 and 5 dB with slightly reducing the compression ratio by 6.2 and 12.5%, respectively. Moreover, it can be noticed that adopting lifting wavelet transform with basis Haar, db1, db4, db5, cdf1.1 and cdf2.2 provide higher audio quality while db6, db8, sym7 and sym8 give the worst audio quality. Furthermore, the performance of HSACM is compared with that of existing work to highlight its performance.

Six Fractal Codes of Biological Life Unifying ATOMS, WAVES and INFORMATION: Perspectives in Exobiology, Cancers Basic Research and Artificial Intelligence Biomimetism Decisions Making

In this theoretical discovery of a law of Life, there is MATHEMATICS (Geometry, Bits and Numbers) that UNIFY 3 universes as complementary as ATOMIC MASS, WAVES, and INFORMATION (DNA, RNA and Amino Acids). The discovery of a simple numerical formula for the projection of all the atomic mass of life-sustaining CONHSP bioatoms leads to the emergence of a set of Nested CODES unifying all the biological, genetic and genomic components by unifying them from bioatoms up to 'to whole genomes. In particular, we demonstrate the existence of a digital meta-code common to the three languages ​​of biology that are RNA, DNA and amino acid sequences. Through this meta-code, genomic and proteomic images appear almost analogous and correlated. The analysis of the textures of these images then reveals a binary code as well as an undulatory code whose analysis on the human genome makes it possible to predict the alternating bands constituting the cariotypes of the chromosomes. The application of these codes to perspectives in astrobiology, cancer, and specifically in INFORMATION THEORY with the emergence of binary codes and regions of local stability (voting process), whose fractal nature we demonstrate, is illustrated. PREFACE by Professor Luc Montagnier Addendum by Robert Friedman M.D After the discovery of the DNA double helix structure allowing both the stable storage of genetic information and its transfer through messenger RNA to protein synthesis organelles themselves structured by RNA most abundant in cells, the ribosomal. This wonder of nature exists in ALL living beings from the virus to humans and is based on two codes, the linear sequence of nucleotides and that derived from codons where three nucleotides allow with a certain flexibility - synonymous codons - the choice in the twenty amino acids. But we are missing a third CODE the one governing at multicellular beings from the rotifer to human, the stabilized modulation of gene expression in a nutshell the differentiation of cells from the single cell of the fertilized egg. It is logical to think that this program which begins as soon as fertilization is written in the DNA. We are also prone to associate it with non-coding DNA sequences although they control gene expression. I introduce here the notion developed by Jean-Claude Pérez of mathematical harmony, a higher order present in all living beings and whose existence it finds in genomes, including those of viruses. Thus the natural evolution of variants of the genome of coronavirus Covid 19 tends towards increasingly long Fibonacci series. It remains to determine the Who, the How and the Why of such developments. I will bet with my mathematician colleague that waves and fractals play a role. Luc Montagnier ADDENDUM Jean-claude has given scientists a strong new direction for research. He has identified a unified field of science guided by the Golden Ratio and Fibonacci Sequence. By identifying an overall guiding principle that makes possible fractal-like nesting at all levels of biological manifestation, future researchers can begin with the "whole" instead of the "parts". If we know that complex systems are organized at varying levels by the Golden Ratio and Fibonacci Sequence, we can look for those universal patterns first and then fill in the gaps with small details to complete the picture. It's like having an overall view of a crossword puzzle before beginning to assemble the individual pieces. Without an overarching vision and guiding principle, completing the puzzle is infinitely more difficult. Once scientists and researchers realize and begin using this "SECRET IN HIDDEN IN PLAIN SIGHT," their discoveries will be orders of magnitude more fruitful. Robert Friedman M.D

Face Recognition Based on Fractal Code and Deep Belief Networks

An enhanced algorithm to recognize the human face using bi-dimensional fractal codes and deep belief networks is presented in this work. The proposed method is experimentally robust against variations in the appearance of human face images, despite different disturbances affecting the measurements and the acquisition process such as occlusion, changes in lighting, pose, and expression or the presence or absence of structural components. That is mainly based on fractal codes (IFS) and bi-dimensional subspaces for features extraction and space reduction, combined with a deep belief network (DBN) classifier. The evaluation is performed through comparisons using probabilistic neural network (PNN) and nearest neighbours (KNN) approaches on three well-known databases (FERET, ORL, and FEI). The results suggest the effectiveness and robustness of the proposed approach.

A NONLOCAL LOW-RANK REGULARIZATION METHOD FOR FRACTAL IMAGE CODING

Fractal coding has been widely used as an image compression technique in many image processing problems in the past few decades. On the other hand side, most of the natural images have the characteristic of nonlocal self-similarity that motivates low-rank representations of them. We would employ both the fractal image coding and the nonlocal self-similarity priors to achieve image compression in image denoising problems. Specifically, we propose a new image denoising model consisting of three terms: a patch-based nonlocal low-rank prior, a data-fidelity term describing the closeness of the underlying image to the given noisy image, and a quadratic term measuring the closeness of the underlying image to a fractal image. Numerical results demonstrate the superior performance of the proposed model in terms of peak-signal-to-noise ratio, structural similarity index and mean absolute error.

Image Compression Using Fractal Functions

The rapid growth of geographic information technologies in the field of processing and analysis of spatial data has led to a significant increase in the role of geographic information systems in various fields of human activity. However, solving complex problems requires the use of large amounts of spatial data, efficient storage of data on on-board recording media and their transmission via communication channels. This leads to the need to create new effective methods of compression and data transmission of remote sensing of the Earth. The possibility of using fractal functions for image processing, which were transmitted via the satellite radio channel of a spacecraft, is considered. The information obtained by such a system is presented in the form of aerospace images that need to be processed and analyzed in order to obtain information about the objects that are displayed. An algorithm for constructing image encoding–decoding using a class of continuous functions that depend on a finite set of parameters and have fractal properties is investigated. The mathematical model used in fractal image compression is called a system of iterative functions. The encoding process is time consuming because it performs a large number of transformations and mathematical calculations. However, due to this, a high degree of image compression is achieved. This class of functions has an interesting property—knowing the initial sets of numbers, we can easily calculate the value of the function, but when the values of the function are known, it is very difficult to return the initial set of values, because there are a huge number of such combinations. Therefore, in order to de-encode the image, it is necessary to know fractal codes that will help to restore the raster image.

FASHE: A FrActal Based Strategy for Head Pose Estimation.

Head pose estimation (HPE) represents a topic central to many relevant research fields and characterized by a wide application range. In particular, HPE performed using a singular RGB frame is particular suitable to be applied at best-frame-selection problems. This explains a growing interest witnessed by a large number of contributions, most of which exploit deep learning architectures and require extensive training sessions to achieve accuracy and robustness in estimating head rotations on three axes. However, methods alternative to machine learning approaches could be capable of similar if not better performance. To this regard, we present FASHE, an approach based on partitioned iterated function systems (PIFS) to represent auto-similarities within face image through a contractive affine function transforming the domain blocks extracted only once by a single frontal reference image, in a good approximation of the range blocks which the target image has been partitioned into. Pose estimation is achieved by finding the closest match between fractal code of target image and a reference array by means of Hamming distance. The results of experiments conducted exceed the state of the art on both Biwi and Ponting'04 datasets as well as approaching those of the best performing methods on the challenging AFLW2000 database. In addition, the applications to GOTCHA Video Dataset demonstrate that FASHE successfully operates in-the-wild.