Developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined. Particular attention is paid to common estimation structures and to practical problems not properly solved yet. The problem of image reconstruction and restoration is first formulated. Some of the current regularization approaches used to solve the problem are then described. The concepts of a priori information and compound criterion are introduced. A Bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated. The practical aspects of computing the solution, first when the hyperparameters are known and second when they must be estimated, are then considered. Conclusions are drawn, and points that still need to be investigated are outlined. >

Image reconstruction and restoration: overview of common estimation structures and problems

Some aspects of statistical inference for a class of spatial-interaction models for finite images are presented: primarily the simultaneous autoregressive (SAR) models and conditional Markov (CM) models. Each of these models is characterized by a set of neighbors, a set of coefficients, and a noise sequence of specified characteristics. We are concerned with two problems: the estimation of the unknown parameters in both SAR and CM models and the choice of an appropriate model from a class of such competing models. Assuming Gaussian-distributed variables, we discuss maximum likelihood (ML) estimation methods. In general, the ML scheme leads to nonlinear optimization problems. To avoid excessive computation, an iterative scheme is given for SAR models, which gives approximate ML estimates in the Gaussian case and reasonably good estimates in some non-Gaussian situations as well. Likewise, for CM models, an easily computable consistent estimate is given. The asymptotic mean-squared error (mse) of this estimate for a four-neighbor CM model is shown tn be substantially less than the mse of the popular coding estimate. Asymptotically consistent decision rules are given for choosing an appropriate SAR or CM model. The usefulness of the estimation scheme and the decision rule for the choice of neighbors is illustrated by using synthetic patterns. Synthetic patterns obeying known SAR and CM models are generated, and the models corresponding to true and several competing neighbor sets are fitted. The estimation scheme yields estimates close to the parameters of the true models, and the decision rule for the choice of neighbors picks up the true model from the class of competing models.

Estimation and choice of neighbors in spatial-interaction models of images

Graphs are a central tool in machine learning and information processing as they allow to conveniently capture the structure of complex datasets. In this context, it is of high importance to develop flexible models of signals defined over graphs or networks. In this paper, we generalize the traditional concept of wide sense stationarity to signals defined over the vertices of arbitrary weighted undirected graphs. We show that stationarity is expressed through the graph localization operator reminiscent of translation. We prove that stationary graph signals are characterized by a well-defined power spectral density that can be efficiently estimated even for large graphs. We leverage this new concept to derive Wiener-type estimation procedures of noisy and partially observed signals and illustrate the performance of this new model for denoising and regression.

Stationary Signal Processing on Graphs

The purpose of this paper is to illustrate the usefulness of two dimensional noncausal autoregressive (NCAR) models for the synthesis of textures. These models characterize the gray level at a pixel as a linear combination of gray levels at nearby locations in all directions and an additive white noise variable. We first show that the class of NCAR models is capable of generating a wide variety of image patterns posessing the local replication attribute, an essential ingredient of many natural textures. It is also shown that the theoretical variograms of many NCAR models possess an oscillatory behavior, a characteristic of the variograms of many natural textures. Next, we give experimental results of synthesis of 64 × 64 textures resembling several real textures in the Brodatz album. The synthetic textures generated by 16 parameter NCAR models retain most the visual characteristics of the original textures. The variograms of the original and synthetic textures are also similar.

http://ieeexplore.ieee.org/iel6/29/26189/01164507.pdf

Texture synthesis using 2-D noncausal autoregressive models

Deep convolutional neural networks (DCNN) have been successfully used in many low-level vision problems such as image denoising. Recent studies on the mathematical foundation of DCNN has revealed that the forward propagation of DCNN corresponds to a dynamic system, which can be described by an ordinary differential equation (ODE) and solved by the optimal control method. However, most of these methods employ integer-order differential equation, which has local connectivity in time space and cannot describe the long-term memory of the system. Inspired by the fact that the fractional-order differential equation has long-term memory, in this paper we develop an advanced image denoising network, namely FOCNet, by solving a fractional optimal control (FOC) problem. Specifically, the network structure is designed based on the discretization of a fractional-order differential equation, which enjoys long-term memory in both forward and backward passes. Besides, multi-scale feature interactions are introduced into the FOCNet to strengthen the control of the dynamic system. Extensive experiments demonstrate the leading performance of the proposed FOCNet on image denoising. Code will be made available.

/pdf/focnet-a-fractional-optimal-control-network-for-image-35jc7fqln3.pdf

FOCNet: A Fractional Optimal Control Network for Image Denoising

Application of Partial Differential Equation (PDE) models for restoration of noisy images is considered. The hyperbolic, parabolic, and elliptic classes of PDE's yield recursive, semirecursive, and nonrecursive filtering algorithms. The two-dimensional recursive filter is equivalent to solving two sets of filtering equations, one along the horizontal direction and other along the vertical direction. The semirecursive filter can be implemented by first transforming the image data along one of its dimensions, say Column, and then recursive filtering along each row independently. The nonrecursive filter leads to Fourier domain Wiener filtering type transform domain algorithm. Comparisons of the different PDE model filters are made by implementing them on actual image data. Performances of these filters are also compared with Fourier Wiener filtering and spatial averaging methods. Performance bounds based on PDE model theory are calculated and implementation tradeoffs of different algorithms are discussed.

Partial differential equations and finite difference methods in image processing--Part II: Image restoration

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

J. Jain

Author Tools

Chat about Author

Papers

Partial differential equations and finite difference methods in image processing--Part II: Image restoration