The twin support vector machine improves the classification performance of the support vector machine by solving two small quadratic programming problems. However, this method has the following defects: (1) For the twin support vector machine and some of its variants, the constructed models use a hinge loss function, which is sensitive to noise and unstable in resampling. (2) The models need to be converted from the original space to the dual space, and their time complexity is high. To further enhance the performance of the twin support vector machine, the pinball loss function is introduced into the twin bounded support vector machine, and the problem of the pinball loss function not being differentiable at zero is solved by constructing a smooth approximation function. Based on this, a smooth twin bounded support vector machine model with pinball loss is obtained. The model is solved iteratively in the original space using the Newton-Armijo method. A smooth twin bounded support vector machine algorithm with pinball loss is proposed, and theoretically the convergence of the iterative algorithm is proven. In the experiments, the proposed algorithm is validated on the UCI datasets and the artificial datasets. Furthermore, the performance of the presented algorithm is compared with those of other representative algorithms, thereby demonstrating the effectiveness of the proposed algorithm.

The twin support vector machine improves the classification performance of the support vector machine by solving two smaller quadratic programming problems. However, for the twin support vector machine and some of its variants, the constructed models are usually transformed from the original space into the dual space to obtain the solutions. Meanwhile, the hinge loss function used in above models is sensitive to noise and unstable in resampling. In order to further improve the performance of the twin support vector machine, the pinball loss function is introduced into the twin bounded support vector machine directly, and the non-differentiable problem of the pinball loss function at zero is solved by constructing a smooth approximation function. Based on this, a smooth twin bounded support vector machine model with pinball loss is obtained. The model is solved iteratively in the original space by using the Newton-Armijo method, then a smooth twin bounded support vector machine with pinball loss algorithm is proposed. In the experiments, the proposed twin support vector machine is validated on the UCI datasets, which shows the effectiveness of the proposed algorithm.

To solve the missing data problem that is caused by reasons, such as occlusion, frame reconstruction by a two-level strategy in multiple images was considered. The method first performed a projective reconstruction combining singular value decomposition (SVD) and subspace method with missing data, which estimated projective shape, projection matrices, projective depths and missing data iteratively. Then it converted the projective solution to a Euclidean one with the unknown focal length and the constant principal point by enforcing constraints. Using the constraints and the fact that scale measurement matrix can recover numberless projection matrices and point matrices, the set equations of the transformation matrix from the projective reconstruction to Euclidean reconstruction were obtained. Experimental results using real images are provided to illustrate the performance of the proposed method.