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We present a novel postprocessing utility called adaptive geometry image (AGIM) for global parameterization techniques that can embed a 3D surface onto a rectangular domain. This utility first converts a single rectangular parameterization into many different tessellations of square geometry images (GIMs) and then efficiently packs these GIMs into an image called AGIM. Therefore, undersampled regions of the input parameterization can be up-sampled accordingly until the local reconstruction error bound is met. The connectivity of AGIM can be quickly computed and dynamically changed at rendering time. AGIM does not have T-vertices, and therefore, no crack is generated between two neighboring GIMs at different tessellations. Experimental results show that AGIM can achieve significant PSNR gain over the input parameterization, AGIM retains the advantages of the original GIM and reduces the reconstruction error present in the original GIM technique. The AGIM is also suitable for global parameterization techniques based on quadrilateral complexes. Using the approximate sampling rates, the PolyCube-based quadrilateral complexes with AGIM can outperform state-of-the-art multichart GIM technique in terms of PSNR.