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Linear Interpolation meaning

I am going through a paper in computer vision, and I come through this line :

the L values, or the luminance values, for
these pixels are then linearly and horizontally interpolated between
the pixels on the (one pixel wide) brightest column in region B, and
the pixels in regions A and C.

So, what does linear and horizontal interpolation actually mean wrt image processing?

Paper : http://140.118.9.222/publications/journal/haircut.pdf