Programming the IDWT for image processing
here is one version of the Haar analysis/synthesis filter bank pair. Oddly, the synthesis filters, up to a scale factor, are mirrors of  the analysis filters.

Other options are:


$\pm 1/\sqrt{2}$ instead or $1/2$ and $1$ (orthogonal version);
reverse: $[1 \;  \pm 1]$ at analysis and $[\pm 1/2 \; 1/2]$ at the synthesis.