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Modified Rodrigues parameters (MRPs) are triplets in R 3 bijectively and rationally mapped to quaternions through stereographic projection. We present here a compelling case for MRPs as a minimal degree-of-freedom parameterization of orientation through novel solutions to prominent problems in the fields of 3D vision and computer graphics. In our primary contribution, we show that the derivatives of a unit quaternion in terms of its MRPs are simple polynomial expressions of its scalar and vector part. Furthermore, we show that updates to unit quaternions from perturbations in parameter space can be computed without explicitly invoking the parameters in the computations. Based on the former, we introduce a novel approach for designing orientation splines by configuring their backprojections in 3D space. Finally, in the general topic of nonlinear optimization for geometric vision, we run performance analyses and provide comparisons on the convergence behavior of MRP parameterizations on the tasks of absolute orientation, exterior orientation and large-scale bundle adjustment of public datasets.

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This paper describes a novel method for representing different facial expressions based on the shape space vector (SSV) of the statistical shape model (SSM) built from 3D facial data. The method relies only on the 3D shape, with texture information not being used in any part of the algorithm, that makes it inherently invariant to changes in the background, illumination, and to some extent viewing angle variations. To evaluate the proposed method, two comprehensive 3D facial data sets have been used for the testing. The experimental results show that the SSV not only controls the shape variations but also captures the expressive characteristic of the faces and can be used as a significant feature for facial expression recognition. Finally the paper suggests improvements of the SSV discriminatory characteristics by using 3D facial sequences rather than 3D stills.