Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

With the explosion in the number of digital images taken every day, the demand for more accurate and visually pleasing images is increasing. However, the images captured by modern cameras are inevitably degraded by noise, which leads to deteriorated visual image quality. Therefore, work is required to reduce noise without losing image features (edges, corners, and other sharp structures). So far, researchers have already proposed various methods for decreasing noise. Each method has its own advantages and disadvantages. In this paper, we summarize some important research in the field of image denoising. First, we give the formulation of the image denoising problem, and then we present several image denoising techniques. In addition, we discuss the characteristics of these techniques. Finally, we provide several promising directions for future research.

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Brief review of image denoising techniques

In recent years, significant progress has been made in developing more accurate and efficient machine learning algorithms for segmentation of medical and natural images. In this review article, we highlight the imperative role of machine learning algorithms in enabling efficient and accurate segmentation in the field of medical imaging. We specifically focus on several key studies pertaining to the application of machine learning methods to biomedical image segmentation. We review classical machine learning algorithms such as Markov random fields, k-means clustering, random forest, etc. Although such classical learning models are often less accurate compared to the deep learning techniques, they are often more sample efficient and have a less complex structure. We also review different deep learning architectures, such as the artificial neural networks (ANNs), the convolutional neural networks (CNNs), and the recurrent neural networks (RNNs), and present the segmentation results attained by those learning models that were published in the past three years. We highlight the successes and limitations of each machine learning paradigm. In addition, we discuss several challenges related to the training of different machine learning models, and we present some heuristics to address those challenges.

Machine Learning Techniques for Biomedical Image Segmentation: An Overview of Technical Aspects and Introduction to State-of-Art Applications

To recover the corrupted pixels, traditional inpainting methods based on low-rank priors generally need to solve a convex optimization problem by an iterative singular value shrinkage algorithm. In this paper, we propose a simple method for image inpainting using low rank approximation, which avoids the time-consuming iterative shrinkage. Specifically, if similar patches of a corrupted image are identified and reshaped as vectors, then a patch matrix can be constructed by collecting these similar patch-vectors. Due to its columns being highly linearly correlated, this patch matrix is low-rank. Instead of using an iterative singular value shrinkage scheme, the proposed method utilizes low rank approximation with truncated singular values to derive a closed-form estimate for each patch matrix. Depending upon an observation that there exists a distinct gap in the singular spectrum of patch matrix, the rank of each patch matrix is empirically determined by a heuristic procedure. Inspired by the inpainting algorithms with component decomposition, a two-stage low rank approximation (TSLRA) scheme is designed to recover image structures and refine texture details of corrupted images. Experimental results on various inpainting tasks demonstrate that the proposed method is comparable and even superior to some state-of-the-art inpainting algorithms.

Patch-Based Image Inpainting via Two-Stage Low Rank Approximation

Image segmentation methodology is a part of nearly all computer schemes as a pre-processing phase to excerpt more meaningful and useful information for analysing the objects within an image. Segmentation of an image is one of the most conjoint scientific matter, essential technology and critical constraint for image investigation and dispensation. There has been a lot of research work conceded in several emerging algorithms and approaches for segmentation, but even at present, no solitary standard technique has been proposed. The methodologies present are broadly classified among two classes i.e. traditional approaches and Soft computing approaches or Computational Intelligence (CI) approaches. In this article, our emphasis is to focus on Soft Computing (SC) techniques which has been adopted for segmenting an image. Nowadays, it is quite often seen that SC or CI is cast-off frequently in Information Technology and Computer Technology. However, Soft Computing approaches working synergistically provides in anyway, malleable information processing competence to manipulate real-life enigmatic circumstances. The impetus of these methodologies is to feat the lenience for ambiguity, roughness, imprecise acumen and partial veracity for the sake to attain compliance, sturdiness and economical results. Neural Networks (NNs), Fuzzy Logic (FL), and Genetic Algorithm (GA) are the fundamental approaches of SC regulation. SC approaches has been broadly implemented and studied in the number of applications including scientific analysis, medical, engineering, management, humanities etc. The paper focuses on introducing the various SC methodologies and presenting numerous applications in image segmentation. The acumen is to corroborate the probabilities of smearing computational intelligence to segmentation of an image. The available articles about usage of SC in segmentation are investigated, especially focusing on the core approaches like FL, NN and GA and efforts has been also made for collaborating new techniques like Fuzzy C-Means from FL family and Deep Neural Network or Convolutional Neural Network from NN family. The impression behind this work is to simulate core Soft Computing methodologies, along with encapsulating various terminologies like evaluation parameters, tools, databases, noises etc. which can be advantageous for researchers. This study also identifies approaches of SC being used, often collectively to resolve the distinctive dilemma of image segmentation, concluding with a general discussion about methodologies, applications followed by proposed work.

https://link.springer.com/content/pdf/10.1007%2Fs11831-018-9257-4.pdf

Image Segmentation Using Computational Intelligence Techniques: Review

As one of the most popular image oversegmentations, superpixel has been commonly used as supporting regions for primitives to reduce computations in various computer vision tasks. In this paper, we propose a novel superpixel segmentation approach based on a distance function that is designed to balance among boundary adherence, intensity homogeneity, and compactness (COM) characteristics of the resulting superpixels. Given an expected number of superpixels, our method begins with initializing the superpixel seed positions to obtain the initial labels of pixels. Then, we optimize the superpixels iteratively based on the defined distance measurement. We update the positions and intensities of superpixel seeds based on the three-sigma rule. The experimental results demonstrate that our algorithm is more effective and accurate than previous superpixel methods and achieves a comparable tradeoff between superpixel COM and adherence to object boundaries.

A Simple Algorithm of Superpixel Segmentation With Boundary Constraint

To get better segmentation results, local information and global information should be taken into consideration together In this paper, we propose a new energy functional which combines a local intensity fitting term and an auxiliary global intensity fitting term, and we also give the method to adjust the weight of auxiliary global fitting term dynamically by using local contrast of the image The combination of the two terms improves the accuracy of segmentation results obviously while reduces dependence on location of initial contour The experiment results proved the effectiveness of our method

Active contour method combining local fitting energy and global fitting energy dynamically

In quantitative brain image analysis, accurate brain tissue segmentation from brain magnetic resonance image (MRI) is a critical step It is considered to be the most important and difficult issue in the field of medical image processing The quality of MR images is influenced by partial volume effect, noise, and intensity inhomogeneity, which render the segmentation task extremely challenging We present a novel fuzzy c-means algorithm (RCLFCM) for segmentation and bias field correction of brain MR images We employ a new gray-difference coefficient and design a new impact factor to measure the effect of neighbor pixels, so that the robustness of anti-noise can be enhanced Moreover, we redefine the objective function of FCM (fuzzy c-means) by adding the bias field estimation model to overcome the intensity inhomogeneity in the image and segment the brain MR images simultaneously We also construct a new spatial function by combining pixel gray value dissimilarity with its membership, and make full use of the space information between pixels to update the membership Compared with other state-of-the-art approaches by using similarity accuracy on synthetic MR images with different levels of noise and intensity inhomogeneity, the proposed algorithm generates the results with high accuracy and robustness to noise

A Modified Fuzzy C-Means Algorithm for Brain MR Image Segmentation and Bias Field Correction

Image super-resolution (SR) for a single low-resolution image is an important and challenging task in image processing. In this study, the authors propose a novel non-local feature back-projection method for image SR, which can effectively reduce jaggy and ringing artefacts common, in general, iterative back-projection (IBP) method. In their method, the objective high-resolution (HR) image is obtained by projecting reconstructed errors back to HR image iteratively. To optimise the initial HR image and constrain anisotropic errors propagation during IBP process, an efficient non-local feature interpolation algorithm is designed. Specially, edge information is used as constraints to make the interpolation surface preserve better shape. Furthermore, as post-processing, non-local similarities are utilised to remove noise and irregularities induced by errors propagation. Experimental results show that their method achieves better performance than state-of-the-art methods in terms of both quantitative metrics and visual qualities.

https://ietresearch.onlinelibrary.wiley.com/doi/pdf/10.1049/iet-ipr.2015.0467

Non-local feature back-projection for image super-resolution

Natural image priors play an important role in image denoising, and various prior-based methods have been widely proposed for noise removal. However, these methods tend to smooth the fine image textures while suppressing noise, degrading the image visual quality. To address this problem, in this paper, we propose an adaptive texture-preserving denoising method. In contrast to most existing prior-based denoising methods, two types of priors [gradient histogram matching priors and nonlocal self-similarity (NSS) priors] are proposed, and their combination is used for image denoising. We introduce a hyper-Laplacian distribution of the gradient histogram matching prior, which enforces the gradient histogram of the denoised image to be as close as possible to the estimated reference histogram from the original image. Meanwhile, the proposed model obtained by introducing the NSS priors effectively preserves fine image details and generates sharp image edges. To improve the accuracy of the method, a content-adaptive parameter selection scheme based on edge detection filters is proposed. Moreover, the optimization problem with two types of priors and the content-adaptive parameter added into the objective function becomes a challenging non-convex optimization problem. To effectively solve this problem, we have developed a new numerical solution based on augmented Lagrangian multipliers and alternating minimization scheme. The experimental results demonstrate that the proposed method effectively preserves the texture features of the denoised images and outperforms several variational methods and other state-of-the-art methods in terms of various evaluation indices and visual quality, especially at medium and high noise levels.

Xuemei Li

Papers

A Simple Algorithm of Superpixel Segmentation With Boundary Constraint

Active contour method combining local fitting energy and global fitting energy dynamically

A Modified Fuzzy C-Means Algorithm for Brain MR Image Segmentation and Bias Field Correction

Non-local feature back-projection for image super-resolution

Adaptive Texture-Preserving Denoising Method Using Gradient Histogram and Nonlocal Self-Similarity Priors