This paper presents a new method for the relaxation of multiview registration error. The multiview registration problem is represented using a graph. Each node and each edge in the graph represents a 3-D data set and a pairwise registration, respectively. Assuming that all the pairwise registration processes have converged to fine results, this paper shows that the multiview registration problem can be converted into a quadratic programming problem of Lie algebra parameters. The constraints are obtained from every cycle of the graph to eliminate the accumulation errors of global registration. A linear solution is proposed to distribute the accumulation error to proper positions in the graph, as specified by the quadratic model. Since the proposed method does not involve the original 3-D data, it has low time and space complexity. Additionally, the proposed method can be embedded into a trust-region algorithm and, thus, can correctly handle the nonlinear effects of large accumulation errors, while preserving the global convergence property to the first-order critical point. Experimental results confirm both the efficiency and the accuracy of the proposed method.

A fuzzy multiple objective programming approach to data envelopment analysis (DEA) of imprecise data is proposed in this paper. The problems involving a mixture of imprecise and exact data for all decision making units (DMUs) could be resolved and the discriminating power of imprecise DEA (IDEA) is enhanced. Although Cooper et al. have developed IDEA to overcome the issues of imprecise data, the discriminating power is not satisfactory since too many efficient DMUs are derived. Chiang and Tzeng's approach using fuzzy multiple objective programming techniques is adopted to enhance the discriminating power of IDEA. The same data set of Cooper et al. is employed to illustrate the merit of our approach.

In this paper, the Cramér-Rao lower bound (CRLB) of image registration error using an isotropic fiducial mark is derived. The derived CRLB is a function of the intensity profile of the fiducial mark. Following the development of the CRLB, a new method for designing an isotropic fiducial mark, suitable for digital image registration, is presented. A parameterization method of the fiducial intensity profile is introduced which guarantees no aliasing effect when the fiducial mark is digitized with the proper sampling rate. A method for computing the fiducial intensity profile, based on minimization of the CRLB registration error, and subject to certain practical constraints, is developed. For imaging systems with a significant low-pass effect, it is proposed to pre-emphasize the high frequency components of the fiducial mark by converting the designed gray-scale fiducial marks into binary fiducial marks. Experimental results show that the designed fiducial mark can provide very accurate registration results and that the registration accuracy is independent of its location.