-Norm Regularization in Volumetric Imaging of Cardiac Current Sources

Abstract

Advances in computer vision have substantially improved our ability to analyze the structure and mechanics of the heart. In comparison, our ability to observe and analyze cardiac electrical activities is much limited. The progress to computationally reconstruct cardiac current sources from noninvasive voltage data sensed on the body surface has been hindered by the ill-posedness and the lack of a unique solution of the reconstruction problem. Common L2- and L1-norm regularizations tend to produce a solution that is either too diffused or too scattered to reflect the complex spatial structure of current source distribution in the heart. In this work, we propose a general regularization with Lp-norm (1 < p < 2) constraint to bridge the gap and balance between an overly smeared and overly focal solution in cardiac source reconstruction. In a set of phantom experiments, we demonstrate the superiority of the proposed Lp-norm method over its L1 and L2 counterparts in imaging cardiac current sources with increasing extents. Through computer-simulated and real-data experiments, we further demonstrate the feasibility of the proposed method in imaging the complex structure of excitation wavefront, as well as current sources distributed along the postinfarction scar border. This ability to preserve the spatial structure of source distribution is important for revealing the potential disruption to the normal heart excitation.

Paired cycle‐GAN‐based image correction for quantitative cone‐beam computed tomography

“…Furthermore, we found that l 1 ‐norm regularization leads to tissue misclassification, which may be because l 1 ‐norm regularization is a linear programming problem. l p ‐norm regularization, however, can be designed to provide a quadratic cone constraint, which may give the Adam gradient descent method a more stable optimization space . The second component of the loss function is the gradient magnitude distance (GMD).…”

“…l pnorm regularization, however, can be designed to provide a quadratic cone constraint, which may give the Adam gradient descent method a more stable optimization space. 31 The second component of the loss function is the gradient magnitude distance (GMD). Between any two images, the GMD is defined as:…”

Private Incremental Regression

“…If C denotes the convex set centered with radius one with respect to · k,L 1,2 -norm then the Gaussian width of C is O( k log(d/k)) [47]. • Lp-balls (1 < p < 2) are another popular choice of constraint space [40]. The regression problem in this case is defined as:θ…”

A projection‐based algorithm for ℓ2‐ℓp Tikhonov regularization

“…As an attempt to overcome the inherent technical difficulties in the ℓ 1 ‐norm approach, several authors have suggested to relax the ℓ 2 ‐ ℓ 1 regularization and to consider regularization using the penalty term Ω( x ) = ‖ x ‖ p , p ∈ (0,1), (or ), as seen in other studies. () However, the choice Ω( x ) = ‖ x ‖ p (or ) with p ∈ (0,1] or p = 2, may not ensure that all desirable properties of the regularized solution, eg, sparsity and smoothness, are captured . Taking into account these difficulties, another possibility is to consider p ∈ (1,2), ie, to deal with the ℓ 2 ‐ ℓ p Tikhonov regularized problem: which, by selecting p ≈1, may be regarded as an alternative to the ℓ 1 ‐norm minimization problem that attempts to balance the effects of ℓ 1 ‐norm and the ℓ 2 ‐norm .…”