On the Laplacian operation with applications in magnetic resonance electrical impedance imaging

On the edge detection of an image by numerical differentiations for gray function

“…The problem (4) to (7) for finding the approximation to f 1 (s,x 2 ) is ill-posed. When the Tikhonov-type regularization schemes are applied, the penalty term should be chosen carefully to catch the discontinuities of the solution for our edge detection problem.…”

“…When the Tikhonov-type regularization schemes are applied, the penalty term should be chosen carefully to catch the discontinuities of the solution for our edge detection problem. For any fixed x 2 ∈ [c, d], the stable solution to the integral Equation 4 with noisy data satisfying (7) is defined as the minimizer of the cost functional…”

“…Based on this lemma, we can prove the unique determination of the regularizing parameter from (10), with the analogous arguments used in Jin et al 19 Theorem 3.2. For fixed x 2 , and t 1 > 1, assume that the noisy data y (·,x 2 ) satisfy (7) and || || 2 ≥ 4t 2 for some appropriate t 2 > t 1 . Denote by , , 1 (·, x 2 ) the minimizer of (9).…”

“…() There are already several works applying the numerical differentiation models in applied areas, see previous works. ()…”

“…[1][2][3][4] There are already several works applying the numerical differentiation models in applied areas, see previous works. [5][6][7][8][9] The general schemes for dealing with ill-posed problems are regularization, with the mathematical issues of defining the regularizing solution and then establishing its convergence rate under some suitable topology. One of the well-known schemes is to consider the regularizing solution as the minimizer of some cost functional with penalty term.…”