We present a novel 3D reassembly method for fragmented, thin objects with unkwon geometry. Unlike past methods, we do not make any restrictive assumptions about the overall shape of the object, or its painted texture. Our key observation is that regardless of the object's shape, matching fragments will have similar geometry and photometry along and across their adjoing regions. We beging by encoding the scale variabily of each fragment's boundary contour in a multichannel, 2çD image representation. Using this multichannel boundary contour representation, we identify matching sub-contours via 2D partial image registration. We then align the fragments by minimizing the distance between their adjoining regions while simultaneously ensuring geometric continuity across them. The configuration of the fragments as they are incrementally matched and aligned form a grapf structure that we use to improve subsequent matches. By detecting cycles in this graph, we identify subsets of fragments with interdependent alignments. We then minimize the error within the subsets to achieve a globally optimal aligment. We leverage user feedback to cull the otherwise exponential search space; after each new match is found and aligned, it is presented to a user for confirmation or rejection. Using ceramic pottery as the driving example, we demostrate the accuracy and efficiency for our method on six real-world datasets.