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Restoring Profanity: Applying Mathematics to Digital Image Restoration

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Inpainting, or image interpolation, is a process used to reconstruct missing parts of images. Artists have long used manual inpainting to restore damaged paintings. Today, mathematicians apply partial differential equations (PDEs) to automate image interpolation. The PDEs operate in much the same way that trained restorers do – they propagate information from the structure around a hole into the hole to fill it in.

While artists can work directly on a painting, a PDE requires a mathematical representation of the subject matter, such as a digital image. A digital image is essentially a 2D matrix of integers, with each integer representing the color or grayscale value of an individual pixel. Holes in the image are represented by unknown values in the matrix. PDE -inpainting fills in these missing values based on the values of nearby pixels.

One of Carola’s first explorations of the use of PDEs for inpainting focused on the restoration of 14th-century frescoes. She has found that high-order, nonlinear PDEs enable a wide range of other applications for inpainting, in fields as diverse as medicine, astronomy, and cartography.

Carola is a lecturer in Applied and Computational Mathematics at the Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge, and Fellow of Jesus College, Cambridge. She is head of the Cambridge Image Analysis group.

A joint event with Lucy Cavendish College.

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