Image Enhancement Techniques In Process Tomography

This thesis is concerned with reconstructing large ill-determined referred to as the ill-posed problem using the iterative method.The efficacy of the tomographic process depends upon the image reconstruction.Images in tomography are vital to control measurand in the process industry.The utmost mathematical problems encounter in tomography are systems of large linear equations.The measured value leads to a large ill-posed matrix system with more than one possible solution.The digitalization of measurand turns the problem into an inverse problem.Various reconstruction algorithms(i.e.,iterative techniques and transform-based methods)solve these inverse problems.There are a variety of ways to gather information in tomography.Electrical capacitance tomography(ECT),Ultrasound,electrical resistance tomography(ERT),and X-ray are prominent techniques and discussed in this research work.ECT measures the change of capacitance at the boundaries of the measuring area.Ultrasound tomography works based on acoustics.ERT measures process variables by reconstruction conductivity distribution from the electrical boundary changes.The data gathered using ECT,ultrasound,and ERT were used to reconstruct the image.In image reconstruction,pixel values can scatter diversely(e.g.,speckling,diffraction,and diffusion).Speckle is a kind of scattering that leads towards blur.Speckled signal values swing from high to low in the concerned pixel.Speckle is not a random error.This error removed by further processing of the image using a suitable deconvolution technique.The digitalization of the problem leads towards linear equations of an ill-posed matrix—Krylov operator such as steepest descent helpful tool to handle such situations.Krylov solvers for linear systems have sophisticated and straightforward formulae for the residual norm.Three Krylov solvers Conjugate gradient for least square(CGLS),minimal residual norm steepest descent(MRNSD),and least square QR factorization(LSQR)are the variations of the steep descent.The steep descent is one of the fundamental iterative techniques used exclusively to solve large sparse square matrices.However,CGLS,MRNSD,and LSQR,the variations of steep descent also solve least-square problems.The methods discussed in this work are CGLS,MRNSD,and LSQR.These three techniques variation of steepest descent,hence the iterative algorithm in nature.Like many other iterative algorithms,these practices suffer from semi-convergence.During the course of deblurring and despeckle images using;CGLS,MRNSD,and LSQR.The ill-posed problems that arise during tomography can be squared or non-squared.In this work,Range restricted GMRES(RRGMRES)algothrim is provided to solve square problems.CGLS,MRNSD,and LSQR are proposed in this work to solve non-square problems.RRGMRES technique is widely applicable to reconstruct two-dimensional tomographic images for the square problem.Reconstruction is achieved after producing the right-hand side of the basic tomographic equation Ux=Y that can be done using any famous tomographic experimental arrangements(X-ray,ECT,ERT ultrasound,etc.).The problem discussed in this research is with the error-contaminated right-hand side.The numerical solution of this matrix is a little bit tricky as the matrix is large and ill-conditioned.For non-square problems,the few steepest descent Krylov solvers,such as CGLS,CRLS,LSMR,etc.,are present but have some issues.The only focus on this work is on CGLS,MRNSD,and LSQR steepest descent Krylov solvers.An adequate stopping criterion to handle semi-convergence for these algorithms presented in this work.Furthermore,this work provides a comparison of CGLS,MRNSD,and LSQR These algorithms are mathematically equivalent,but LSQR is robust and challenging to apply.A large sparse linear least square problem solved by LSQR is Krylov space solver,in fact,based on Lanczos's bidiagonalization.This work applies said variations of the steep descent on a tomographic test problem and compares the three algorithms based on accuracy using MATLAB.The iteration method is implemented using MATLAB to compute the inverse problem.This work focuses on the deblurring of images reconstructed from the received data in industrial tomography along with an effective way to tackle semi-convergence.