Adaptive Piecewise Constant Approximation (APCA) with wavelets/DWT
Their wavelet coefficients $$W_c = [4.5, 0.5, 1, -1, 1, 1, 0, -1]$$ or $$W_c = [4.5, 0.5, 1/\sqrt{2},-1/\sqrt{2},1/2,1/2,0,-1/2]$$
look perfectly fine to me. In the first case, they use $a = (x+y)/2$ and  $d = x-a$ repeatedly. What might differ is the overall scaling, to account for energy preservation. In Matlab for instance, I get the same result as their by dividing the $3$-level coefficients by $\sqrt{8} = \sqrt{2}^3$:

 wavedec([7, 5, 5, 3, 3, 3, 4, 6],3,'haar')/sqrt(8)


Their algorithm is quite well detailed in the paper, so it should be implementable quite easily. If you want additional help for Matlab like-code (which can be reinterpreted for R or Python), check the tutorial A practical Time-Series Tutorial with MATLAB, starting page 66.