A weighted dual porous medium equation applied to image restoration (Q2857943)

For image restoration modeling, the authors study the following reaction-diffusion equation of dual porous medium type NEWLINE\[NEWLINE u_t = g(|\nabla\,G_{\sigma}\star u_0|)|\Delta u|^{m-1}\Delta u + k(u_0 - u),\qquad (x,t)\in Q_T = \Omega\times (0,T) NEWLINE\]NEWLINE coupled with the homogeneous Dirichlet boundary condition and the initial value condition NEWLINE\[NEWLINE u(x,t) = 0, \quad (x,t)\in\partial\Omega\times [0,T], \qquad u(x,0) = u_0(x),\quad x\in\Omega. NEWLINE\]NEWLINE Here \(g(s) = 1/\sqrt{1 + s^2}\) and NEWLINE\[NEWLINE |\nabla\,G_{\sigma}\star u_0| = \left[\sum_{i=1}^N\left(\frac{\partial G_{\sigma}}{\partial x_i}u_0\right)^2\right]^{1/2} NEWLINE\]NEWLINE where \(G_{\sigma}(x)\) is the Gaussian kernel: NEWLINE\[NEWLINE G_{\sigma}(x) = \frac{1}{(4\pi\sigma)^{N/2}}\exp\left(-\frac{|x|^2}{4\sigma}\right). NEWLINE\]NEWLINE First the authors establish the well-posedness for solutions of the above-mentioned initial-boundary value problem based on nonlinear semigroup theory in Banach spaces. Then they demonstrate some numerical results using three test images, a real nebula image and two artificial noised images. These experiments depend on four parameters: the scale of diffusion, the coefficient \(k\) of fidelity term, the smoothing degree \(m\) and the Gaussian convolution degree \(\sigma\). The first set of images illustrate the performance of the proposed approach on real images. The next tests demonstrate the effectiveness of this model for protecting second-order edge in restoring noised image of a piecewise linear image.