Search results for: Medical image analysis

Authors: J. Alamelu Mangai, Satej Wagle, V. Santhosh Kumar

Abstract:

The exponential increase in the volume of medical image database has imposed new challenges to clinical routine in maintaining patient history, diagnosis, treatment and monitoring. With the advent of data mining and machine learning techniques it is possible to automate and/or assist physicians in clinical diagnosis. In this research a medical image classification framework using data mining techniques is proposed. It involves feature extraction, feature selection, feature discretization and classification. In the classification phase, the performance of the traditional kNN k nearest neighbor classifier is improved using a feature weighting scheme and a distance weighted voting instead of simple majority voting. Feature weights are calculated using the interestingness measures used in association rule mining. Experiments on the retinal fundus images show that the proposed framework improves the classification accuracy of traditional kNN from 78.57 % to 92.85 %.

Keywords: Medical Image Mining, Data Mining, Feature Weighting, Association Rule Mining, k nearest neighbor classifier.

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Authors: Negar Riazifar, Mehran Yazdi

Abstract:

Discrete Wavelet Transform (DWT) has demonstrated far superior to previous Discrete Cosine Transform (DCT) and standard JPEG in natural as well as medical image compression. Due to its localization properties both in special and transform domain, the quantization error introduced in DWT does not propagate globally as in DCT. Moreover, DWT is a global approach that avoids block artifacts as in the JPEG. However, recent reports on natural image compression have shown the superior performance of contourlet transform, a new extension to the wavelet transform in two dimensions using nonseparable and directional filter banks, compared to DWT. It is mostly due to the optimality of contourlet in representing the edges when they are smooth curves. In this work, we investigate this fact for medical images, especially for CT images, which has not been reported yet. To do that, we propose a compression scheme in transform domain and compare the performance of both DWT and contourlet transform in PSNR for different compression ratios (CR) using this scheme. The results obtained using different type of computed tomography images show that the DWT has still good performance at lower CR but contourlet transform performs better at higher CR.

Keywords: Computed Tomography (CT), DWT, Discrete Contourlet Transform, Image Compression.

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Authors: C. Rajan, K. Geetha, S. Geetha

Abstract:

The motivation of image compression technique is to reduce the irrelevance and redundancy of the image data in order to store or pass data in an efficient way from one place to another place. There are several types of compression methods available. Without the help of compression technique, the file size is knowingly larger, usually several megabytes, but by doing the compression technique, it is possible to reduce file size up to 10% as of the original without noticeable loss in quality. Image compression can be lossless or lossy. The compression technique can be applied to images, audio, video and text data. This research work mainly concentrates on methods of encoding, DCT, compression methods, security, etc. Different methodologies and network simulations have been analyzed here. Various methods of compression methodologies and its performance metrics has been investigated and presented in a table manner.

Keywords: Image compression techniques, encoding, DCT, lossy compression, lossless compression, JPEG.

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Authors: Sheema Shuja Khattak, Gule Saman, Imran Khan, Abdus Salam

Abstract:

Image segmentation plays an important role in medical imaging applications. Therefore, accurate methods are needed for the successful segmentation of medical images for diagnosis and detection of various diseases. In this paper, we have used maximum entropy to achieve image segmentation. Maximum entropy has been calculated using Shannon, Renyi and Tsallis entropies. This work has novelty based on the detection of skin lesion caused by the bite of a parasite called Sand Fly causing the disease is called Cutaneous Leishmaniasis.

Keywords: Shannon, Maximum entropy, Renyi, Tsallis entropy.

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Authors: Shayan Rezvanigilkolaei, Shayesteh Vefaghnematollahi

Abstract:

Electrical impedance tomography is a non-invasive medical imaging technique suitable for medical applications. This paper describes an electrical impedance tomography device with the ability to navigate a robotic arm to manipulate a target object. The design of the device includes various hardware and software sections to perform medical imaging and control the robotic arm. In its hardware section an image is formed by 16 electrodes which are located around a container. This image is used to navigate a 3DOF robotic arm to reach the exact location of the target object. The data set to form the impedance imaging is obtained by having repeated current injections and voltage measurements between all electrode pairs. After performing the necessary calculations to obtain the impedance, information is transmitted to the computer. This data is fed and then executed in MATLAB which is interfaced with EIDORS (Electrical Impedance Tomography Reconstruction Software) to reconstruct the image based on the acquired data. In the next step, the coordinates of the center of the target object are calculated by image processing toolbox of MATLAB (IPT). Finally, these coordinates are used to calculate the angles of each joint of the robotic arm. The robotic arm moves to the desired tissue with the user command.

Keywords: Electrical impedance tomography, EIT, Surgeon robot, image processing of Electrical impedance tomography.

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Authors: Chutimon Thitipornvanid, Siripun Sanguansintukul

Abstract:

Choosing the right metadata is a critical, as good information (metadata) attached to an image will facilitate its visibility from a pile of other images. The image-s value is enhanced not only by the quality of attached metadata but also by the technique of the search. This study proposes a technique that is simple but efficient to predict a single human image from a website using the basic image data and the embedded metadata of the image-s content appearing on web pages. The result is very encouraging with the prediction accuracy of 95%. This technique may become a great assist to librarians, researchers and many others for automatically and efficiently identifying a set of human images out of a greater set of images.

Keywords: Metadata, Prediction, Multi-layer perceptron, Human facial image, Image mining.

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Authors: Alessandro Bevilacqua, Alessandro Gherardi, Ludovico Carozza, Filippo Piccinini

Abstract:

The last years have seen an increasing use of image analysis techniques in the field of biomedical imaging, in particular in microscopic imaging. The basic step for most of the image analysis techniques relies on a background image free of objects of interest, whether they are cells or histological samples, to perform further analysis, such as segmentation or mosaicing. Commonly, this image consists of an empty field acquired in advance. However, many times achieving an empty field could not be feasible. Or else, this could be different from the background region of the sample really being studied, because of the interaction with the organic matter. At last, it could be expensive, for instance in case of live cell analyses. We propose a non parametric and general purpose approach where the background is built automatically stemming from a sequence of images containing even objects of interest. The amount of area, in each image, free of objects just affects the overall speed to obtain the background. Experiments with different kinds of microscopic images prove the effectiveness of our approach.

Keywords: Microscopy, flat field correction, background estimation, image segmentation.

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Authors: Susmitha Vekkot, Pancham Shukla

Abstract:

In this paper, we focus on the fusion of images from different sources using multiresolution wavelet transforms. Based on reviews of popular image fusion techniques used in data analysis, different pixel and energy based methods are experimented. A novel architecture with a hybrid algorithm is proposed which applies pixel based maximum selection rule to low frequency approximations and filter mask based fusion to high frequency details of wavelet decomposition. The key feature of hybrid architecture is the combination of advantages of pixel and region based fusion in a single image which can help the development of sophisticated algorithms enhancing the edges and structural details. A Graphical User Interface is developed for image fusion to make the research outcomes available to the end user. To utilize GUI capabilities for medical, industrial and commercial activities without MATLAB installation, a standalone executable application is also developed using Matlab Compiler Runtime.

Keywords: Filter mask, GUI, hybrid architecture, image fusion, Matlab Compiler Runtime, wavelet transform.

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Authors: Osamah M. Al-Qershi, Khoo Bee Ee

Abstract:

In recent years image watermarking has become an important research area in data security, confidentiality and image integrity. Many watermarking techniques were proposed for medical images. However, medical images, unlike most of images, require extreme care when embedding additional data within them because the additional information must not affect the image quality and readability. Also the medical records, electronic or not, are linked to the medical secrecy, for that reason, the records must be confidential. To fulfill those requirements, this paper presents a lossless watermarking scheme for DICOM images. The proposed a fragile scheme combines two reversible techniques based on difference expansion for patient's data hiding and protecting the region of interest (ROI) with tamper detection and recovery capability. Patient's data are embedded into ROI, while recovery data are embedded into region of non-interest (RONI). The experimental results show that the original image can be exactly extracted from the watermarked one in case of no tampering. In case of tampered ROI, tampered area can be localized and recovered with a high quality version of the original area.

Keywords: DICOM, reversible, ROI-based, watermarking.

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Authors: K. Amudha, C. Nelson Kennedy Babu, S. Balu

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The security of the medical images and its related data is the major research area which is to be concentrated in today’s era. Security in the medical image indicates that the physician may hide patients’ related data in the medical image and transfer it safely to a defined location using reversible watermarking. Many reversible watermarking methods had proposed over the decade. This paper enhances the security level in brain tumor images to hide the patient’s detail, which has to be conferred with other physician’s suggestions. The details or the information will be hidden in Non-ROI area of the image by using the block cipher algorithm. The block cipher uses different keys to extract the details that are difficult for the intruder to detect all the keys and to spot the details, which are the key advantage of this method. The ROI is the tumor area and Non-ROI is the area rest of ROI. The Non-ROI should not be spoiled in any cause and the details in the Non-ROI should be extracted correctly. The reversible watermarking method proposed in this paper performs well when compared to existing methods in the process of extraction of an original image and providing information security.

Keywords: Brain tumor images, Block Cipher, Reversible watermarking, ROI.

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Authors: Yasser F. Hassan

Abstract:

The reduction or removal of noise in a color image is an essential part of image processing, whether the final information is used for human perception or for an automatic inspection and analysis. This paper describes the modeling system based on the rough neural network model to adaptive cellular automata for various image processing tasks and noise remover. In this paper, we consider the problem of object processing in colored image using rough neural networks to help deriving the rules which will be used in cellular automata for noise image. The proposed method is compared with some classical and recent methods. The results demonstrate that the new model is capable of being trained to perform many different tasks, and that the quality of these results is comparable or better than established specialized algorithms.

Keywords: Rough Sets, Rough Neural Networks, Cellular Automata, Image Processing.

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Authors: Roza Afarin, Saeed Mozaffari

Abstract:

The aim of this paper is image encryption using Genetic Algorithm (GA). The proposed encryption method consists of two phases. In modification phase, pixels locations are altered to reduce correlation among adjacent pixels. Then, pixels values are changed in the diffusion phase to encrypt the input image. Both phases are performed by GA with binary chromosomes. For modification phase, these binary patterns are generated by Local Binary Pattern (LBP) operator while for diffusion phase binary chromosomes are obtained by Bit Plane Slicing (BPS). Initial population in GA includes rows and columns of the input image. Instead of subjective selection of parents from this initial population, a random generator with predefined key is utilized. It is necessary to decrypt the coded image and reconstruct the initial input image. Fitness function is defined as average of transition from 0 to 1 in LBP image and histogram uniformity in modification and diffusion phases, respectively. Randomness of the encrypted image is measured by entropy, correlation coefficients and histogram analysis. Experimental results show that the proposed method is fast enough and can be used effectively for image encryption.

Keywords: Correlation coefficients, Genetic algorithm, Image encryption, Image entropy.

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Authors: R. M. Farouk

Abstract:

In this paper, we present an analytical analysis of the representation of images as the magnitudes of their transform with the discrete wavelets. Such a representation plays as a model for complex cells in the early stage of visual processing and of high technical usefulness for image understanding, because it makes the representation insensitive to small local shifts. We found that if the signals are band limited and of zero mean, then reconstruction from the magnitudes is unique up to the sign for almost all signals. We also present an iterative reconstruction algorithm which yields very good reconstruction up to the sign minor numerical errors in the very low frequencies.

Keywords: Wavelets, Image processing signal processing, Image reconstruction

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Authors: Yi Zhang, Kai Lu, Ying-hui Gao, Mo Wang

Abstract:

Histogram plays an important statistical role in digital image processing. However, the existing quantum image models are deficient to do this kind of image statistical processing because different gray scales are not distinguishable. In this paper, a novel quantum image representation model is proposed firstly in which the pixels with different gray scales can be distinguished and operated simultaneously. Based on the new model, a fast quantum algorithm of constructing histogram for quantum image is designed. Performance comparison reveals that the new quantum algorithm could achieve an approximately quadratic speedup than the classical counterpart. The proposed quantum model and algorithm have significant meanings for the future researches of quantum image processing.

Keywords: Quantum Image Representation, Quantum Algorithm, Image Histogram.

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Authors: S. Zith Dey Babu, S. Kour, S. Verma, C. Verma, V. Pathania, A. Agrawal, V. Chaudhary, A. Manoj Puthur, R. Goyal, A. Pal, T. Danti Dey, A. Kumar, K. Wadhwa, O. Ved

Abstract:

Background: Skin cancer is now is the buzzing button in the field of medical science. The cyst's pandemic is drastically calibrating the body and well-being of the global village. Methods: The extracted image of the skin tumor cannot be used in one way for diagnosis. The stored image contains anarchies like the center. This approach will locate the forepart of an extracted appearance of skin. Partitioning image models has been presented to sort out the disturbance in the picture. Results: After completing partitioning, feature extraction has been formed by using genetic algorithm and finally, classification can be performed between the trained and test data to evaluate a large scale of an image that helps the doctors for the right prediction. To bring the improvisation of the existing system, we have set our objectives with an analysis. The efficiency of the natural selection process and the enriching histogram is essential in that respect. To reduce the false-positive rate or output, GA is performed with its accuracy. Conclusions: The objective of this task is to bring improvisation of effectiveness. GA is accomplishing its task with perfection to bring down the invalid-positive rate or outcome. The paper's mergeable portion conflicts with the composition of deep learning and medical image processing, which provides superior accuracy. Proportional types of handling create the reusability without any errors.

Keywords: Computer-aided system, detection, image segmentation, morphology.

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Authors: K.Somasundaram, S.Domnic

Abstract:

A low bit rate still image compression scheme by compressing the indices of Vector Quantization (VQ) and generating residual codebook is proposed. The indices of VQ are compressed by exploiting correlation among image blocks, which reduces the bit per index. A residual codebook similar to VQ codebook is generated that represents the distortion produced in VQ. Using this residual codebook the distortion in the reconstructed image is removed, thereby increasing the image quality. Our scheme combines these two methods. Experimental results on standard image Lena show that our scheme can give a reconstructed image with a PSNR value of 31.6 db at 0.396 bits per pixel. Our scheme is also faster than the existing VQ variants.

Keywords: Image compression, Vector Quantization, Residual Codebook.

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Authors: G. N. Srinivasan, G. Shobha

Abstract:

This paper presents an overview of the methodologies and algorithms for statistical texture analysis of 2D images. Methods for digital-image texture analysis are reviewed based on available literature and research work either carried out or supervised by the authors.

Keywords: Image Texture, Texture Analysis, Statistical Approaches, Structural approaches, spectral approaches, Morphological approaches, Fractals, Fourier Transforms, Gabor Filters, Wavelet transforms.

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Authors: Yong-Gyu Lee, Gilwon Yoon

Abstract:

Image processing for capsule endoscopy requires large memory and it takes hours for diagnosis since operation time is normally more than 8 hours. A real-time analysis algorithm of capsule images can be clinically very useful. It can differentiate abnormal tissue from health structure and provide with correlation information among the images. Bleeding is our interest in this regard and we propose a method of detecting frames with potential bleeding in real-time. Our detection algorithm is based on statistical analysis and the shapes of bleeding spots. We tested our algorithm with 30 cases of capsule endoscopy in the digestive track. Results were excellent where a sensitivity of 99% and a specificity of 97% were achieved in detecting the image frames with bleeding spots.

Keywords: bleeding, capsule endoscopy, image processing, real time analysis

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Authors: Samia Boucherkha, Mohamed Benmohamed

Abstract:

In this paper we investigate the watermarking authentication when applied to medical imagery field. We first give an overview of watermarking technology by paying attention to fragile watermarking since it is the usual scheme for authentication.We then analyze the requirements for image authentication and integrity in medical imagery, and we show finally that invertible schemes are the best suited for this particular field. A well known authentication method is studied. This technique is then adapted here for interleaving patient information and message authentication code with medical images in a reversible manner, that is using lossless compression. The resulting scheme enables on a side the exact recovery of the original image that can be unambiguously authenticated, and on the other side, the patient information to be saved or transmitted in a confidential way. To ensure greater security the patient information is encrypted before being embedded into images.

Keywords: Medical Imaging, Invertible Watermarking, Authentication, Integrity.

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Authors: Faruq A. Al-Omari, Nabeel I. Al-Fayoumi

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In this paper, we present a new algorithm for clustering data in large datasets using image processing approaches. First the dataset is mapped into a binary image plane. The synthesized image is then processed utilizing efficient image processing techniques to cluster the data in the dataset. Henceforth, the algorithm avoids exhaustive search to identify clusters. The algorithm considers only a small set of the data that contains critical boundary information sufficient to identify contained clusters. Compared to available data clustering techniques, the proposed algorithm produces similar quality results and outperforms them in execution time and storage requirements.

Keywords: Data clustering, Data mining, Image-mapping, Pattern discovery, Predictive analysis.

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Authors: R. C. Joshi, Shashikala Tapaswi

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The paper proposes an approach using genetic algorithm for computing the region based image similarity. The image is denoted using a set of segmented regions reflecting color and texture properties of an image. An image is associated with a family of image features corresponding to the regions. The resemblance of two images is then defined as the overall similarity between two families of image features, and quantified by a similarity measure, which integrates properties of all the regions in the images. A genetic algorithm is applied to decide the most plausible matching. The performance of the proposed method is illustrated using examples from an image database of general-purpose images, and is shown to produce good results.

Keywords: Image Features, color descriptor, segmented classes, texture descriptors, genetic algorithm.

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Authors: Amnach Khawne, Kazuhiko Hamamoto, Orachat Chitsobhuk

Abstract:

Most of the image watermarking methods, using the properties of the human visual system (HVS), have been proposed in literature. The component of the visual threshold is usually related to either the spatial contrast sensitivity function (CSF) or the visual masking. Especially on the contrast masking, most methods have not mention to the effect near to the edge region. Since the HVS is sensitive what happens on the edge area. This paper proposes ultrasound image watermarking using the visual threshold corresponding to the HVS in which the coefficients in a DCT-block have been classified based on the texture, edge, and plain area. This classification method enables not only useful for imperceptibility when the watermark is insert into an image but also achievable a robustness of watermark detection. A comparison of the proposed method with other methods has been carried out which shown that the proposed method robusts to blockwise memoryless manipulations, and also robust against noise addition.

Keywords: Medical image watermarking, Human Visual System, Image Adaptive Watermark

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Authors: Panos Kotsas

Abstract:

This paper presents the application of a signal intensity independent registration criterion for non-rigid body registration of medical images. The criterion is defined as the weighted ratio image of two images. The ratio is computed on a voxel per voxel basis and weighting is performed by setting the ratios between signal and background voxels to a standard high value. The mean squared value of the weighted ratio is computed over the union of the signal areas of the two images and it is minimized using the Chebyshev polynomial approximation. The geometric transformation model adopted is a local cubic B-splines based model.

Keywords: Medical image, non-rigid, registration.

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Authors: Mahinda P. Pathegama, Özdemir Göl

Abstract:

Extraction of edge-end-pixels is an important step for the edge linking process to achieve edge-based image segmentation. This paper presents an algorithm to extract edge-end pixels together with their directional sensitivities as an augmentation to the currently available mathematical models. The algorithm is implemented in the Java environment because of its inherent compatibility with web interfaces since its main use is envisaged to be for remote image analysis on a virtual instrumentation platform.

Keywords: edge-end pixels, image processing, imagesegmentation, pixel extraction

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Authors: C. G. Jinimole, A. Harsha

Abstract:

One of the key problems facing in the analysis of Computed Tomography (CT) images is the poor contrast of the images. Image enhancement can be used to improve the visual clarity and quality of the images or to provide a better transformation representation for further processing. Contrast enhancement of images is one of the acceptable methods used for image enhancement in various applications in the medical field. This will be helpful to visualize and extract details of brain infarctions, tumors, and cancers from the CT image. This paper presents a comparison study of five contrast enhancement techniques suitable for the contrast enhancement of CT images. The types of techniques include Power Law Transformation, Logarithmic Transformation, Histogram Equalization, Contrast Stretching, and Laplacian Transformation. All these techniques are compared with each other to find out which enhancement provides better contrast of CT image. For the comparison of the techniques, the parameters Peak Signal to Noise Ratio (PSNR) and Mean Square Error (MSE) are used. Logarithmic Transformation provided the clearer and best quality image compared to all other techniques studied and has got the highest value of PSNR. Comparison concludes with better approach for its future research especially for mapping abnormalities from CT images resulting from Brain Injuries.

Keywords: Computed tomography, enhancement techniques, increasing contrast, PSNR and MSE.

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Authors: Bing-Fei Wu, Yen-Lin Chen, Chung-Cheng Chiu

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In this study, a new criterion for determining the number of classes an image should be segmented is proposed. This criterion is based on discriminant analysis for measuring the separability among the segmented classes of pixels. Based on the new discriminant criterion, two algorithms for recursively segmenting the image into determined number of classes are proposed. The proposed methods can automatically and correctly segment objects with various illuminations into separated images for further processing. Experiments on the extraction of text strings from complex document images demonstrate the effectiveness of the proposed methods.1

Keywords: image segmentation, multilevel thresholding, clustering, discriminant analysis

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Authors: Sumathi Poobal, G. Ravindran

Abstract:

Medical imaging uses the advantage of digital technology in imaging and teleradiology. In teleradiology systems large amount of data is acquired, stored and transmitted. A major technology that may help to solve the problems associated with the massive data storage and data transfer capacity is data compression and decompression. There are many methods of image compression available. They are classified as lossless and lossy compression methods. In lossy compression method the decompressed image contains some distortion. Fractal image compression (FIC) is a lossy compression method. In fractal image compression an image is coded as a set of contractive transformations in a complete metric space. The set of contractive transformations is guaranteed to produce an approximation to the original image. In this paper FIC is achieved by PIFS using quadtree partitioning. PIFS is applied on different images like , Ultrasound, CT Scan, Angiogram, X-ray, Mammograms. In each modality approximately twenty images are considered and the average values of compression ratio and PSNR values are arrived. In this method of fractal encoding, the parameter, tolerance factor Tmax, is varied from 1 to 10, keeping the other standard parameters constant. For all modalities of images the compression ratio and Peak Signal to Noise Ratio (PSNR) are computed and studied. The quality of the decompressed image is arrived by PSNR values. From the results it is observed that the compression ratio increases with the tolerance factor and mammogram has the highest compression ratio. The quality of the image is not degraded upto an optimum value of tolerance factor, Tmax, equal to 8, because of the properties of fractal compression.

Keywords: Fractal image compression, IFS, PIFS, PSNR, Quadtree partitioning.

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Authors: Maha Sharkas, Dahlia R. ElShafie, Nadder Hamdy

Abstract:

Image watermarking has proven to be quite an efficient tool for the purpose of copyright protection and authentication over the last few years. In this paper, a novel image watermarking technique in the wavelet domain is suggested and tested. To achieve more security and robustness, the proposed techniques relies on using two nested watermarks that are embedded into the image to be watermarked. A primary watermark in form of a PN sequence is first embedded into an image (the secondary watermark) before being embedded into the host image. The technique is implemented using Daubechies mother wavelets where an arbitrary embedding factor α is introduced to improve the invisibility and robustness. The proposed technique has been applied on several gray scale images where a PSNR of about 60 dB was achieved.

Keywords: Image watermarking, Multimedia Security, Wavelets, Image Processing.

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Authors: Ratchakit Sakuldee, Somkait Udomhunsakul

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Measurement of the quality of image compression is important for image processing application. In this paper, we propose an objective image quality assessment to measure the quality of gray scale compressed image, which is correlation well with subjective quality measurement (MOS) and least time taken. The new objective image quality measurement is developed from a few fundamental of objective measurements to evaluate the compressed image quality based on JPEG and JPEG2000. The reliability between each fundamental objective measurement and subjective measurement (MOS) is found. From the experimental results, we found that the Maximum Difference measurement (MD) and a new proposed measurement, Structural Content Laplacian Mean Square Error (SCLMSE), are the suitable measurements that can be used to evaluate the quality of JPEG200 and JPEG compressed image, respectively. In addition, MD and SCLMSE measurements are scaled to make them equivalent to MOS, given the rate of compressed image quality from 1 to 5 (unacceptable to excellent quality).

Keywords: JPEG, JPEG2000, objective image quality measurement, subjective image quality measurement, correlation coefficients.

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Authors: Xueting Liu, Daojin Song, Chuandai Dong, Hongkui Li

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From a set of shifted, blurred, and decimated image , super-resolution image reconstruction can get a high-resolution image. So it has become an active research branch in the field of image restoration. In general, super-resolution image restoration is an ill-posed problem. Prior knowledge about the image can be combined to make the problem well-posed, which contributes to some regularization methods. In the regularization methods at present, however, regularization parameter was selected by experience in some cases and other techniques have too heavy computation cost for computing the parameter. In this paper, we construct a new super-resolution algorithm by transforming the solving of the System stem Є=An into the solving of the equations X+A*X-1A=I , and propose an inverse iterative method.

Keywords: High-resolution MAP image, Reconstruction, Image interpolation, Motion Estimation, Hermitian positive definite solutions.

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