Decomposing an Image into High Frequency and Low Frequency Parts Using Bilateral Filter
Since the Bilateral Filter is Spatial Invariant and Non Linear (The output is linear combination of the data, yet the weights depends on the data in Non Linear fashion) it can't be formulated as a Spatially Invariant Filter / Convolution.  

So the Low Pass / High Pass concept in it classic form doesn't hold here.

Yet you can still employ the Image Pyramid concept here with your operator being the Bilateral filter and everything will work perfectly (Namely perfect reconstruction).

It is actually very effective.