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We consider the problem of efficient image transmission over noisy time-varying channels subject to a low transmission energy constraint (and fixed bandwidth/delay constraints). We examine the limits of desirability of a highly compressed representation using a joint source-channel coding (JSCC) framework. Specifically, invoking as a platform a state-of-the-art wavelet image coder, we demonstrate how the resulting highly compressed digital stream, appropriately protected against channel noise, is not always the best solution. We show how a hybrid scheme based on simple partitioning of the wavelet image representation into "compressed" and "uncompressed" components can lead to significantly improved performance (of the order of 3 dB in PSNR for Rayleigh channels) over popular JSCC schemes which are based on compressed, entropy-coded, and appropriately unequal error-protected (UEP) source representations.