What is the Linear Shading Correction Algorithm accurately?

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I want to right a code in C# without using pre-written libraries for removing shadows on the edges of the image and there is a algorithm for this(Linear Shading Algorithm) but I can not understand some part of the algorithm.

According to algoritm:There is an original image and there are two plates as thin and thick(r1,r2).There are ideal gray colors for plate 1 and plate 2.

Formula is:

enter image description here

Formula for a(i,j) and b(i,j): enter image description here

I found this method in the article that was written for X-ray devices but I could not understand what is the term "plate used for " or what is ideal gray color?Can anybody explain the algorithm for me.If I can understand that terms I am gonna try to write a code for that.

EDIT:I found this algorithm from this book:Computer Vision for X-ray Testing:(Authors:Domingo Merry,Christian Pieringer)