Chapter 16: Image Restoration by Non-standard Diffusion

The presence of noise in images is unavoidable.It may be formed by image formation process,image recording,image transmission etc.These random distortions make it difficult to perform any required image processing task.High quality noise free remote sensing images are necessary for various applications.Therefore, removal of noise is necessary. The purpose of image denoising is to preserve edges as far as possible while removing noise, making the resulting images approximate the ideal image.This paper proposes a method for denoising remote sensing images using a combination of second order and fourth order partial differential equations.The advantages of both second order and fourth order partial differential equations are utilized here.The image is denoised using second order partial differential equations , fourth order partial differential equations and the combination of both.A comparison of the three methods is also presented.The proposed algorithm smooth out more noise and conserve more detail in the denoising process.

In this paper, we propose a local and contextual controlled (LCC) fourth order partial differential equation (PDE) method for noise removal. First, a region based fourth order PDE is proposed, which conductance coefficients are chosen adaptively in terms of the domain type. Thus, the proposed method can preserve the advantages of forth order PDE and avoid leaving isolated black and white speckles. Furthermore, the region based fourth order PDE and two discontinuity measures are incorporated into a LCC fourth order PDE, where the joint use of the two discontinuity measures leads to a complementary effect for edge preservation. The proposed LCC fourth order PDE is tested on two typical images, including texture image and natural scene image, which show the superiority of the proposed method.

Surface representation as a boundary-valued problem of partial differential equations (PDE) is an important topic of computer graphics and computer aided design. In existing references, various free-from surfaces were created with a fourth order PDE which is only able to meet the tangential conditions at the surface boundaries. The need for a sixth order PDE in surface modeling arises in two situations: one is to generate surfaces with curvature continuity; and the other is to use curvature values as a user handle for surface shape manipulation. In this chapter, we introduce such a sixth order PDE for free-form surface generation and develop a finite difference method to solve this PDE. We also investigate the effects of boundary curvature and the vector-valued shape parameters on the surface shape.

Visual information transmitted in the form of digital images is becoming a major method of communication in the modern age, but the image obtained after transmission is often corrupted with noise. The received image needs processing before it can be used in the applications. Image denoising involves the manipulation of the image data to produce a visually high quality image. This thesis reviews the existing denoising algorithms, such as filtering approach, wavelet based approach, and multifractal approach, and performs their comparative study. Different noise models including additive and multiplicative types are used. They include Gaussian noise, salt and pepper noise, speckle noise and Brownian noise. Selection of the denoising algorithm is application dependent. Hence, it is necessary to have knowledge about the noise present in the image so as to select the appropriate denoising algorithm. The filtering approach has been proved to be the best when the image is corrupted with salt and pepper noise. The Partial differential equation approach finds applications in denoising images corrupted with Gaussian noise. In the case where the noise characteristics are complex, the multi fractal approach can be used. A quantitative measure of comparison is provided by the signal to noise ratio of the image. Aim of the project is to reduce the speckle noise from the medical images by retaining most of the information, using fourth order partial differential equations and comparing the results with second order partial differential equation method, that is best suitable for a medical practitioner or a doctor to diagnose well by comparing this denoised image with the obtained.

Isogeometric analysis of insoluble surfactant spreading on a thin film

A Bernstein property of certain types of fourth order partial differential equations

Digital Image Restoration is a technique used to recover original image information from degraded images. Digital images lose information because of factors like noise, blurring and camera misfocus. Image restoration plays a significant role in improving the quality of pictures obtained from unmanned aircrafts, poor quality video tape of a crime scene and blurred satellite images. In this paper we review how the field of image restoration has evolved over the years, and how the techniques of image restoration are making inroads into various applications, thus improving users perception. The article also portrays recent advances and patents in the field of image restoration. Keywords: Image deblurring, image restoration, classical approaches to image restoration, adaptive approaches, median filter, image inpainting, inverse filtering, patents on image restoration applications

Due to effects of atmosphere and optical subsystem,detector and electronic subsystem in imaging system,there is degradation in imaging process.The degradation of imaging system induces imagery blur.The ability to acquire information from blurred image is debased The technique of image restoration can improve imagery quality,so it can enhance the ability to acquire information from restored image.According to the requirement of image restoration technique,high precision modulate transfer function(MTF) of imaging system is the base of high-quality imagery restored.Since there are differences of MTF between different remote sensors and at different period of same remote sensor,it is important to acquire MTF of imaging system in-flight.The paper presents systematically a in-flight measurement technique of remote sensor MTF,which is called knife edge technique.In the technique,we first analyze its principle and its requirement on targets in images,then introduce in detail several key techniques including sampling method line by line in image,registering line pixels with shifted knife method and fitting method to acquired knife edge.After introducing the technique of image restoration based on imaging system' MTF,the paper shows the improved image on the bases of experiment.The result appears imagery quality is improved obviously.

An Improved PDE Based Super-Resolution Reconstruction Algorithm

The non-linear diffusion and the fourth order partial differential equations (PDEs) are used to decompose a turbulent flow field into coherent and incoherent fields. The methods are applied against the velocity fields for both forced and decaying homogeneous isotropic turbulence. The aim of this paper is to examine the ability of the partial differential equations in extracting coherent vortices from a turbulent flow field in the spatial domain without transforming the turbulent velocity field into frequency domain. The comparison against the standard methods such as the wavelet and Fourier decompositions are also considered. The three dimensional velocity fields with a resolution of 1283 are generated using the Lattice Boltzmann method (LBM) where the Taylor micro-scale Reynolds numbers are 72 and 29 for forced and decaying fields, respectively. Results show that the coherent field and the random incoherent part contribute to all scales in the inertial range. The two filtering methods approximately identify the coherent fields without any loss of the geometrical structure of the vortices. The statistical properties such as flatness, skewness and spectrum of the extracted fields (coherent and incoherent parts) are also investigated for each filtering method.

Image Denoising and Image Segmentation are the two major areas of the medical image processing. The main objective of this paper is to develop a robust segmentation algorithm inorder to detect tumor in 2D MRI brain images. Here we use image denoising as the preprocessing step as noise plays an important role incase of accuracy of affected area of the image, especially in medical diagnostics. To denoise the image, fourth order partial differential equation is employed. A seeded region growing segmentation is used to detect the tumor in MRI brain image. Also skull removal procedure is employed using morphological operators to increase the accuracy of brain tumor detection. This method detects the tumor in the brain image efficiently and also tested for several brain tumor images.

Underwater computer vision has attracted increasing attention in the research community due to the recent advances in underwater platforms such as of rovers, gliders, autonomous underwater vehicles (AUVs) , and the like, that now make possible the acquisition of vast amounts of imagery and video for applications such as biodiversity assessment, environmental monitoring, and search and rescue. Despite growing interest, underwater computer vision is still a relatively under-researched area, where the attention in the literature has been paid to the use of computer vision techniques for image restoration and reconstruction, where image formation models and image processing methods are used to recover colour corrected or enhanced images. This is due to the notion that these methods can be used to achieve photometric invariants to perform higher-level vision tasks such as shape recovery and recognition under the challenging and widely varying imaging conditions that apply to underwater scenes. In this paper, we review underwater computer vision techniques for image reconstruction, restoration, recognition, depth, and shape recovery. Further, we review current applications such as biodiversity assessment, management and protection, infrastructure inspection and AUVs navigation, amongst others. We also delve upon the current trends in the field and examine the challenges and opportunities in the area.

The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space H with a self-adjoint positive definite operator A is considered. A stable three-step difference scheme for the approximate solution of the problem is presented. The main theorem on stability of this difference scheme is established. In applications, the stability estimates for the solution of difference schemes of the approximate solution of three nonlocal boundary value problems for third order partial differential equations are obtained. Numerical results for oneand two-dimensional third order partial differential equations are provided.

Image restoration poses a formidable challenge in the field of computer vision, endeavoring to restore high-quality images from degraded or corrupted versions. This research paper conducts a comprehensive comparison of three prominent image restoration methodologies: GFP GAN, DeOldify, and MIRNet. GFP GAN, featuring a specialized GAN architecture designed for image restoration tasks, introduces an AI-centric approach. DeOldify, a deep learning-based method, focuses on colorizing and restoring old images using advanced AI techniques, while MIRNet offers a lightweight network specifically crafted for image restoration within an AI framework. The comparative analysis involves training and testing each method on a diverse dataset comprising both degraded and ground truth images. Employing a confusion matrix, precision, accuracy, recall, and other evaluation metrics are computed to comprehensively assess the performance of these AI-based methods. The matrix affords insights into the strengths and weaknesses of each AI-driven approach, providing a nuanced understanding of their respective performances. Experimental evaluations divulge the relative effectiveness of GFP GAN, DeOldify, and MIRNet in addressing image restoration challenges, encompassing issues such as noise, blur, compression artifacts, and other degradations, within the context of AI methodologies. The results not only illuminate the advantages and limitations of each AI-infused method but also serve as a valuable resource for researchers and practitioners in selecting the most suitable AI-driven approach for their unique image restoration requirements. In conclusion, this paper offers a thorough comparison of GFP GAN, DeOldify, and MIRNet in the domain of AI-driven image restoration, leveraging a confusion matrix to analyze precision, accuracy, and other pertinent parameters. Through a meticulous consideration of these AI-powered evaluation metrics, this study furnishes invaluable insights into the nuanced performance of these methodologies, facilitating informed decision-making in various AI-driven image restoration applications

Partial differential equations of the second- and higher-order are encountered in a number of areas of importance to engineering and mathematical physics. As the processes that are accounted for in the development of the governing differential equations increase in complexity, the differential equations themselves tend to acquire a higher-order. The theory of second-order partial differential equations has found extensive applications in the study of problems in fluid mechanics, flow in porous media, heat conduction in solids, diffusive transport of chemicals in porous media, wave propagation in strings and membranes, and in mechanics of solids. Higher-order partial differential equations have been developed particularly in relation to studies in mechanics of solids. These include the formulation of the biharmonic equation for the solution of problems in plane elasticity and in the flow of viscous fluids, the fourth-order partial differential equations associated with the flexural mechanics of prismatic plates, the sixth-order partial differential equations used to describe the mechanics of shell structures and other fourth-order partial differential equations which describe the mechanics of plastic solids. In this chapter we shall restrict attention to the examination of second-order partial differential equations.