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This paper presents a novel depth-image coding algorithm that concentrates on the special characteristics of depth images: smooth regions delineated by sharp edges. The algorithm models these smooth regions using piecewise-linear functions and sharp edges by a straight line. To define the area of support for each modeling function, we employ a quadtree decomposition that divides the image into blocks of variable size, each block being approximated by one modeling function containing one or two surfaces. The subdivision of the quadtree and the selection of the type of modeling function is optimized such that a global rate-distortion trade-off is realized. Additionally, we present a predictive coding scheme that improves the coding performance of the quadtree decomposition by exploiting the correlation between each block of the quadtree. Experimental results show that the described technique improves the resulting quality of compressed depth images by 1.5-4 dB when compared to a JPEG-2000 encoder.