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We present an algorithm for sound analysis and re-synthesis with local automatic adaptation of time-frequency resolution. The reconstruction formula we propose is highly efficient, and gives a good approximation of the original signal from analyses with different time-varying resolutions within complementary frequency bands: this is a typical case where perfect reconstruction cannot in general be achieved with fast algorithms, which provides an error to be minimized. We provide a theoretical upper bound for the reconstruction error of our method, and an example of automatic adaptive analysis and re- synthesis of a music sound.