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Symmetry Restoration by Stretching



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CCCG 2009 Vancouver BC August 17 19 2009 Symmetry Restoration by Stretching Misha Kazhdan Nina Amenta Shengyin Gu David F Wiley Bernd Hamann Abstract We consider restoring the bilateral symmetry of an object which has been deformed by compression This problem arises in paleontology where symmetric bones are compressed in the process of fossilization Our input is a user selected set P of point pairs on the deformed object which are assumed to be mirror images in some undeformed set AP with some added noise We carefully formulate the problem and give a closed form solution 1 Introduction Much of what we know about evolution comes from the study of fossils From the shapes of the bones of extinct animals we form hypotheses about how they moved what they ate how they are related to each other and so on Yet these shapes are usually deformed by the geological processes which occur during fossilization for example the skull in Figure 1 For some fossils for example skulls and vertebrae we can assume that the original shape was roughly bilaterally symmetric We can use this assumption to reverse the deformation or at least limit the family of possible reconstructions This process is sometimes called retrodeformation Usually the input for retrodeformation is a set of point pairs chosen by the paleontologist on the deformed specimen We assume the point pairs are stored in a 3 2n matrix P with the assumption that point p2i was the mirror image of p2i 1 on the original object before deformation The point pairs are chosen using the expert s understanding of the biological shape Developing automatic methods for finding point pairs or other useful descriptions of the input data is a different also well studied research question see below Under the assumption that the object was compressed we assume that the inverse deformation should be what we call a single axis stretch A single axis stretch is produced by choosing a direction vector and scaling only in that direction it is



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