Some Non-convex Modeling,Algorithms In Image Processing And Their Applications

Signal and image processing are important research fields and have many applica-tions in science and technology,such as,medicine,remote sensing,security inspection and communication.This thesis studied three kinds of problems in signal and image processing,consisting of non-convex modeling,algorithms design,convergence analysis and applications.The first kind is on l0 regularization problem.Many signals and images can be represented by sparse coefficients in some transform domains.One can use the 0-norm regularization,namely the number of nonzero elements,for sparsity promotion,and this type of problem is called l0 regularization problem.PIHT(proximal iterative hard thresholding)which has global convergence is a popular algorithm for solving it.For further understanding the characteristics of PIHT,under different assumptions,we analyzed PIHT's o(1/k)convergence rate and local linear convergence rate.Then based on PIHT algorithm,we proposed two algorithms:the extrapolated PIHT algo-rithm using function value for modification(EPIHT-FM)and the extrapolated PIHT algorithm using gradient for modification(EPIHT-FM),and proved their global convergence.Finally,we showed their numerical performance and advantages by experiments on compressive sensing,CT image reconstruction and logistic regression classification problems.The second kind is on quantitative photoacoustic tomography(QPAT).It is an emerging medical imaging method,which combines the high resolution of optical imag-ing and the high precision of ultrasound imaging,and has gradually evolved into various clinical and biomedical applications.Based on the background of QPAT problem and the effective regularization technique in image processing,a non-convex non-smooth regularization model was established.We proved that the gradient of the smooth part is Lipschitz continuous at the continuous level,and then proved the convergence of the forward-backward splitting algorithm for solving this non-convex problem.Finally,we discussed the numerical performance of our model and method.The third kind is on dynamic parallel magnetic resonance image(dynamic pMRI).It is a non-invasive indirect imaging technique that has been widely adopted in many medical applications.The key problem is reducing data acquisition time by sampling less data while maintaining high image quality.To overcome this challenge,this thesis proposed a new model for recovering the missing data in auto-calibration region using two kinds of low rank structures in MRI data.It is a non-convex and non-smooth model and can be solved by PALM algorithm.This thesis proved PALM's convergence for solving this problem by verifying the assumption conditions of PALM,especially the Kurdyka-Lojasiewicz property.Finally we showed the advantages compared to some existing methods and robustness of the method by experiments.