Meaning of the Transpose of Convolution
The transpose (or more generally, the Hermitian or conjugate transpose) of a filter is simply the matched filter. This is found by time reversing the kernel and taking the conjugate of all the values.

Chapter 5 of Jon F. Claerbout - Earth Soundings Analysis: Processing versus Inversion has a great discussion of the topic of Adjoint operators (or transpose, or back projection, or matched filtering). It's in the context of Seismic imaging, but it really gets to the heart of what you're actually trying to do (in general) with these techniques.