Using Maple to Study the Partial Differential Problems

This paper uses the mathematical software Maple for the auxiliary tool to study the partial differential problem of two types of multivariable functions. We can obtain the infinite series forms of any order partial derivatives of these two types of multivariable functions by using differentiation term by term theorem, and hence greatly reduce the difficulty of calculating their higher order partial derivative values. On the other hand, we propose two examples of multivariable functions to evaluate their any order partial derivatives, and some of their higher order partial derivative values practically. At the same time, we employ Maple to calculate the approximations of these higher order partial derivative values and their infinite series forms for verifying our answers.

A New Model of Bundle Adjustment Used in Camera Calibration

The calibration of the internal and external parameters of each camera is a crucial step for free-form surface measurement based on stereo vision. The nonlinear optimization method of bundle adjustment is usually used in the camera calibration. The model of bundle adjustment can be derived from the nonlinear collinearity equations which contains the nonlinear distortions of camera lens. As the internal parameters of camera, the distortion parameters are implied in the projective transformation matrix, and difficult to be explicitly expressed by an equation. But the establishment of the model of bundle adjustment needs to calculate the first-order partial derivative of each parameter. At present, there is no effective method to decompose the implicit parameters which are taken as a whole in the expansion of first-order partial derivative. The existing bundle adjustment models are not comprehensive and easy to cause ill-conditioned matrix problems. In this paper, we discuss the methods of the nonlinear camera lens distortion correction, establish a new lens distortion expression, and propose a new model of bundle adjustment based on the new distortion expression. Experiments results indicate that the new bundle adjustment model can obtain high camera calibration accuracy, and avoid the ill-conditioned matrix problems.