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We present a scalable stereo matching algorithm based on a Locally Adaptive Polygon Approximation (LAPA) technique. For accurate local stereo matching, pixel-wise adaptive polygon-based support windows are constructed to approximate spatially varying image structures. Central to building these pixel-wise polygons is a fast algorithm that adaptively decides a set of directional scales, utilizing intensity and spatial information. Thanks to the locally adaptive support window, the proposed method achieves high stereo reconstruction quality both in depth-discontinuity regions and homogenous regions. Moreover, our LAPA-based method offers flexible scalability in terms of quality-complexity trade-off. As a specific instantiation favoring high-quality stereo estimation, our 8-direction stereo method outperforms most of the other local stereo methods and even some global optimization techniques. Another low-complexity alternative is also presented, achieving a significant speedup of up to a factor 20 with graceful accuracy degradation. Within a unified LAPA framework, our stereo method hence facilitates more flexibility in conciliating different algorithm design needs with processing performance issues.