Mathematical model for reconstructing a damaged bitmap

The paper describes an algorithm for restoring a damaged image, based on the use of maximum and minimum Lipschitz function defined in a flat area. Namely, we will assume that the image is given by the function = 𝑓(𝑥, 𝑦), where = 0,...,𝑀, = 0,...,𝑁, and its value is a brightness level of point (𝑥, 𝑦), which varies in the range of = 0,...,𝑈. We consider the current window of the size (2𝑛 + 1) × (2𝑛 + 1) with center at the point (𝑥, 𝑦), where = 1, 2,.... As the output luminance of the point corresponding to the center of the window, take the value (𝑥, 𝑦, 𝑧) = min{𝑓(𝑖, 𝑗)+ √︁(𝑥 - 𝑖)2 + (𝑦 - 𝑗)2 + 𝑧2 : |𝑥-𝑖| ≤ 𝑛, |𝑦-𝑗| ≤ 𝑛}, where = 𝑛,...,𝑀 - 𝑛, = 𝑛,...,𝑁 - 𝑛. To suppress the local minima we can use the dual function that looks like this (𝑥, 𝑦, 𝑧) = max{𝑓(𝑖, 𝑗)- √︁(𝑥 - 𝑖)2 + (𝑦 - 𝑗)2 + 𝑧2 : |𝑥-𝑖| ≤ 𝑛, |𝑦-𝑗| ≤ 𝑛}. Next, it is necessary to define for each current point (𝑥, 𝑦) which of these functions must be applied. To do this, we proceed as follows. In one pass through all the points (𝑥, 𝑦) are determined by the image of a local maximum and local minimum points. Repeated passage of this information is taken into account for the determination of the function used. Response 𝐻(𝑥, 𝑦, 𝑧) of our filter is calculated according to the rule 𝐻(𝑥, 𝑦, 𝑧) =⎧⎨⎩ 𝐹,𝑛(𝑥, 𝑦, 𝑧), if (𝑥, 𝑦) is point of local maximum, 𝐺,𝑛(𝑥, 𝑦, 𝑧), if (𝑥, 𝑦) is point of local minimum, 𝑓(𝑥, 𝑦), otherwise. We show examples of operation of this algorithm for images with varying degrees of damage. We consider images having 20 % - 75 % of the defects. Presented algorithm quite well restores the image with different types of lesions: how random nature with a uniform distribution over the entire image (impulse noise), and concentrated in certain areas.