This paper introduces a variational approach for filling-in missing data in digital images. The method involves jointly interpolating gray levels and gradient/isophote directions to smoothly extend isophote lines into the holes of missing data. The interpolation is computed by solving a variational problem using gradient descent, leading to a set of coupled second-order partial differential equations for the gray levels and gradient orientations. The approach is inspired by the Gestaltist's principle of good continuation and can handle holes with arbitrary topology. Applications include restoring old photographs and removing superimposed text. The paper also discusses related work, such as image denoising, film restoration, texture synthesis, and occlusion removal, and provides theoretical results on the proposed variational approach. Numerical experiments demonstrate the effectiveness of the method in various scenarios, including images with complex structures and discontinuities.This paper introduces a variational approach for filling-in missing data in digital images. The method involves jointly interpolating gray levels and gradient/isophote directions to smoothly extend isophote lines into the holes of missing data. The interpolation is computed by solving a variational problem using gradient descent, leading to a set of coupled second-order partial differential equations for the gray levels and gradient orientations. The approach is inspired by the Gestaltist's principle of good continuation and can handle holes with arbitrary topology. Applications include restoring old photographs and removing superimposed text. The paper also discusses related work, such as image denoising, film restoration, texture synthesis, and occlusion removal, and provides theoretical results on the proposed variational approach. Numerical experiments demonstrate the effectiveness of the method in various scenarios, including images with complex structures and discontinuities.