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Iterative Image Restoration Algorithms
Published in Vijay K. Madisetti, The Digital Signal Processing Handbook, 2017
The determination of the value of the regularization parameter is a critical issue in regularized restoration. A number of approaches for determining its value are presented in [13]. If only one of the parameters ε or E in Equations 34.40 and 34.41 is known, a constrained least-squares formulation can be followed [9,15]. With it, the size of one of the ellipsoids is minimized, subject to the constraint that the solution belongs to the surface of the other ellipsoid (the one defined by the known parameter). Following the Lagrangian approach, which transforms the constrained optimization problem into an unconstrained one, the following functional is minimized
Intraventricular vector flow mapping 3-D by triplane Doppler echocardiography
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2020
F. Vixege, P. Y. Courand, P. Blanc-Benon, S. Mendez, F. Nicoud, D. Vray, D. Garcia
The objective of 3D-iVFM is to recover the three velocity components in the intraventricular cavity from Doppler velocities alone (i.e. from radial components only) on a limited set of planes (three planes). To provide the 3D-iVFM, we used the echocardiographic triplane mode available on clinical GE scanners (Figure 1). This means that we used Doppler data from three planes separated by an azimuthal angle of 60° in the apical long-axis view, without moving the probe. The Doppler data prior to scan-conversion were retrieved using the EchoPAC software. Volumetric three-component intraventricular blood flows were estimated from the three planes of the color Doppler. The problem was written as a constrained least squares problem, which was solved by the Lagrange multiplier method. We used hemodynamic properties to constrain the problem such as mass conservation for an incompressible fluid (null-divergence) and free-slip conditions on the wall. The mathematical equation of this problem is: with subject to:
A new data preprocessing method for 3D reconstruction of pavement
Published in International Journal of Pavement Engineering, 2021
Yong Xiao, Ya Wei, Chuang Yan, Yalin Liu, Linbing Wang
When using the adaptive estimation method, for a specific point such as Pi, its L nearest neighbours are first identified. Then, a local tangent plane L(Pi) is defined using the L points. The normal vector of the plane L(Pi) is considered as the normal vector of the point Pi. The normal vector of L(Pi) can be expressed as Equation (4).where is the normal vector of the plane L(Pi), is the iteration times, is the fitting residuals after t times, is a Gaussian weight function related to the fitting residuals, , is a normal deviation function , is a distance Gaussian weight function of each neighbours, , are bandwidths of fitting residuals, normal deviation and distance. After the determination of the bandwidths, the weight functions are fixed in each iteration step. Therefore, Equation (4) can be converted to a constrained least squares problem. The covariance can be expressed as M:where , the feature vector corresponding to the minimum eigenvalue of M is the normal vector of point Pi.