• Bulletin of the Korean Chemical Society
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• v.35 no.3
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• pp.805-810
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• 2014

The shape invariant potentials are proved to be exactly solvable, i.e. the wave functions and energies of a particle moving under the influence of the shape invariant potentials can be algebraically determined without any approximations. It is well known that the SWKB quantization is exact for all shape invariant potentials though the SWKB quantization itself is approximate. This mystery has not been mathematically resolved yet and may not be solved in a concrete fashion even in the future. Therefore, in the present work, to understand (not prove) the mystery an attempt of deriving exactly solvable potentials directly from the SWKB quantization has been made. And it turns out that all the derived potentials are shape invariant. It implicitly explains why the SWKB quantization is exact for all known shape invariant potentials. Though any new potential has not been found in this study, this brute-force derivation of potentials helps one understand the characteristics of shape invariant potentials.

• Bulletin of the Korean Chemical Society
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• v.33 no.3
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• pp.818-824
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• 2012

When the Langer modification is applied to Coulomb potential, the standard WKB quantization yields an exact energy spectrum for the potential. This Langer modification has been known to be related to the centrifugal term appearing in Coulomb potential. But we find that a similar modification exists for all translationally shape invariant potentials without referring to the centrifugal term. The characteristic shape of the potentials accounts for the generalized version of Langer modification that makes the WKB quantization valid for all translationally shape invariant potentials.

• The Journal of Korea Robotics Society
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• v.4 no.3
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• pp.185-191
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• 2009

This paper presents a new shape-based algorithm using affine category shape model for object category recognition and model learning. Affine category shape model is a graph of interconnected nodes whose geometric interactions are modeled using pairwise potentials. In its learning phase, it can efficiently handle large pose variations of objects in training images by estimating 2-D homography transformation between the model and the training images. Since the pairwise potentials are defined on only relative geometric relationship betweenfeatures, the proposed matching algorithm is translation and in-plane rotation invariant and robust to affine transformation. We apply spectral matching algorithm to find feature correspondences, which are then used as initial correspondences for RANSAC algorithm. The 2-D homography transformation and the inlier correspondences which are consistent with this estimate can be efficiently estimated through RANSAC, and new correspondences also can be detected by using the estimated 2-D homography transformation. Experimental results on object category database show that the proposed algorithm is robust to pose variation of objects and provides good recognition performance.