What transformations have been applied to this image to yield four derivatives?
This to me looks exactly like the two-dimensional Haar wavelet transform. All four quadrants are a part of the transform, even though different operations have been performed on them. Upper left quadrant is a result of low-pass filtering with a 2x2 two-dimensional rect function (Haar wavelet) and decimation be a factor of 2, upper right is the output of a horizontal high-pass filter, lower left is the output of a vertical high-pass filter, and lower right is the output of a diagonal high-pass filter.