Other time-frequency-plane tiling than STFT, DWT, ConstantQ-Transform: multiresolution STFT? It is known that

a) the STFT gives a rectangular tiling of the time-frequency plane

b) the Wavelet transform gives a non-linear tiling (better frequency resolution for low-frequencies, and better time-domain resolution for higher-frequencies)


c) Constant-Q transform (such as NonStationaryGaborTransform) have a logarithmic scale for frequency bins (instead of linear with STFT) and have a time-frequency tiling like this (y-axis logarithmic):



Question:
Is there an transform like this:

i.e. like a normal STFT, but for which the FFT-size would change for successive time frames.
Example: if there is a transient, the FFT-size is small (512) to keep good time-domain resolution, then a few time-frames later the signal is rather stationary, so the FFT-size is higher (8192) to have good frequency resolution.
It's like an adaptative / multiresolution STFT.
Last but not least: of course it's always possible to perform FFT with different window-length on successive time-frames, so the forward transform I'm describing is probably easy to do.
What I'm asking here is a transform which is invertible, i.e. in which we can compute an inverse transform to recover the initial signal.
Does this exist in Matlab or Python?
Your time-frequency grid is Mondrian-shaped: essentially, rectangles supported on dyadic splits of the time or the frequency axes. Hence, you can easily start for any reversible time-frequency tiling, and further slit any of its rectangle onto another invertible tiling. Some call that hierarchical, or nested time-scale/time-frequency decompositions. As long as each is invertible, the combination remains revertible. 

Furthermore, dyadic and power-of-two uniform decompositions can be turned into the other, only by grouping coefficients. This was used for instance in A DCT-based embedded image coder, 1996: an eight (or $2^3$) multiband filter-bank DCT was turned into a 3-level dyadic structure.

So, combining dyadic wavelets and $2^K$ windowed filter-bank, you can get invertible schemes as wished.

One issue though: how does one choose how to nest those different embedded decompositions? Seen as wavelet or local cosine, lapped-transform packets, entropy concepts could be used, but the diversity can be huge.