This paper introduces a variational approach for filling-in missing data in digital images. The method involves jointly interpolating image gray-levels and gradient/isophote directions, smoothly extending isophote lines into the missing regions. The approach is based on solving a variational problem using gradient descent, leading to a set of coupled second-order partial differential equations for gray-levels and gradient orientations. The process is interpreted as an application of the Gestalt principle of good continuation. The algorithm can handle holes of any topology and is applicable to various image restoration tasks, such as restoring old photographs and removing superimposed text. The paper presents theoretical results on the variational approach and its gradient descent flow, along with numerical experiments demonstrating the effectiveness of the method. The algorithm requires user interaction to mark regions to be filled-in and uses a band around the hole to ensure continuity of the vector field of gradient directions. The method is validated through various examples, including the restoration of images with holes and the removal of text. The paper also discusses related approaches, such as texture synthesis and disocclusion, and highlights the importance of propagating both gradient directions and gray-levels in the region surrounding the hole. The work is supported by theoretical results and numerical experiments, demonstrating the effectiveness of the proposed method in filling-in missing data in digital images.This paper introduces a variational approach for filling-in missing data in digital images. The method involves jointly interpolating image gray-levels and gradient/isophote directions, smoothly extending isophote lines into the missing regions. The approach is based on solving a variational problem using gradient descent, leading to a set of coupled second-order partial differential equations for gray-levels and gradient orientations. The process is interpreted as an application of the Gestalt principle of good continuation. The algorithm can handle holes of any topology and is applicable to various image restoration tasks, such as restoring old photographs and removing superimposed text. The paper presents theoretical results on the variational approach and its gradient descent flow, along with numerical experiments demonstrating the effectiveness of the method. The algorithm requires user interaction to mark regions to be filled-in and uses a band around the hole to ensure continuity of the vector field of gradient directions. The method is validated through various examples, including the restoration of images with holes and the removal of text. The paper also discusses related approaches, such as texture synthesis and disocclusion, and highlights the importance of propagating both gradient directions and gray-levels in the region surrounding the hole. The work is supported by theoretical results and numerical experiments, demonstrating the effectiveness of the proposed method in filling-in missing data in digital images.