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In 3D video, texture and supplementary depth data are coded to enable the interpolation of a required number of synthesized views for multi-view displays in the range of the original camera views. The coding of the depth data can be improved by analyzing the distortion of synthesized video views instead of the depth map distortion. Therefore, this paper introduces a new distortion metric for 3D video coding, which relates changes in the depth map directly to changes of the overall synthesized view distortion. It is shown how the new metric can be integrated into the rate-distortion optimization (RDO) process of an encoder, that is based on high-efficiency video coding technology. An evaluation of the modified encoder is conducted using different view synthesis algorithms and shows about 50% rate savings for the depth data or 0.6 dB PSNR gains for the synthesized view.