The Remapping technology of the Optimizer is based on the ability to calculate the acoustic field that is produced by a set of loudspeakers. This calculation is possible thanks to the Fourier-Bessel decomposition of the acoustic field into a certain number of coefficients that correspond to the spherical harmonics. Just as the Fourier decomposition is commonly used to analyze a signal in the frequency domain, the Fourier-Bessel decomposition can be used to analyze an acoustic field in the space domain, by decomposing into a sum of elementary radiation patterns that are referred to as spherical harmonics in mathematics.

The function that provides the resulting acoustic field from the input signals is called a "radiation matrix". In a pseudo math notation: Input Signal * Radiation Matrix = Acoustic Field

Real radiation matrix:
the Optimizer first computes the radiation matrix of the real system, in other words the radiation matrix that corresponds to the measured loudspeaker positions. This is possible because the Optimizer knows the exact positions of the loudspeakers in 3D.

This radiation matrix for the real system allows to calculate the actual acoustic field that is produced by the measured loudspeaker placement.

Ideal radiation matrix:
on the other hand, the Optimizer can calculate the radiation matrix for the reference placement, because the loudspeaker positions of the reference placement are, by definition, clearly defined.

This radiation matrix for the reference system allows to calculate the ideal acoustic field that would be produced if the loudspeakers were positioned correctly, according to the reference placement.

Remapping matrix:
the last stage is to find out the additional processing that should be applied to the input signal in order to obtain the ideal acoustic field from the measured loudspeaker system. This is done by inverting the Real Radiation Matrix:

Remapping Matrix = Radiation Matrix of the ideal system * (radiation matrix of the real system)-1

Conclusion:
This Remapping Matrix is computed once (after the loudspeaker positions have been measured) and applied in real time to the input signals to compute the output signals that should be sent to each loudspeaker in order to obtain the reference acoustic field.

Note: in the case where the number of inputs is different from the number of outputs, one could describe this remapping technology as a universal downmixing/upmixing algorithm for 3D audio reproduction.

See Trinnov's AES Convention Paper 6375 for a detailed explanation of louspeaker remapping.