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We propose an intra prediction solution to block-based image compression. In order to adapt to local image features during intra prediction, we consider the distinct image singularities within the model of piece-wise smooth functions. With such singularities, i.e., edges in this paper, intra prediction can be performed by solving Laplace equations. Moreover, since edges exhibit spatial correlations, we design a rate-distortion optimized method for edge extraction and edge coding. Our edge-oriented intra prediction thus consists of the prediction of smooth regions as well as the prediction of edges. We compare our intra prediction with that in H.264 and achieve superior performance. Our intra prediction can also be integrated into a block-based image coding scheme, which is comparable to JPEG2000 in terms of objective quality. An important advantage of our intra prediction is the improvement in visual quality at low bit-rate due to the preservation of edges.