For fairing requirements of the T-Bézier curve, the T-Bézier curve was smoothed by using the energy method. A control point of the T-Bézier curve was modified by using the energy method to make the T-Bézier curve smooth, while it was shown how the interference factor α influenced the smoothness of the T-Bézier curve. It was obtained a method that a fairing T-Bézier curve would be obtained by moving a control point: the α could be determined before the new control point would be found out, the new T-Bézier curve was produced by these new control points. The whole curve would be smoothed: firstly, the interference factors {α_i}_i=1ⁿ were determined; secondly, the equation system whose coefficient matrix was a real symmetric matrix tridiagonal was solved; thirdly, the new control points {P_i}_i=0ⁿ were obtained; finally, the new T-Bézier curve could be produced. Not only overall fairness of the T-Bézier curve but also C² continuity of data points was achieved. Finally, it was shown that the proposed algorithm is simple, practical and effective by three examples.

The neighboring relationship of sketch patches and photo patches on the manifold cannot always reflect their intrinsic data structure. To resolve this problem, a Locality-Constrained Neighbor Embedding (LCNE) based face sketch-photo synthesis algorithm was proposed. The Neighbor Embedding (NE) based synthesis method was first applied to estimate initial sketches or photos. Then, the weight coefficients were constrained according to the similarity between the estimated sketch patches or photo patches and the training sketch patches or training photo patches. Subsequently, alternative optimization was deployed to determine the weight coefficients, select K candidate image patches and update the target synthesis patch. Finally, the synthesized image was generated by merging all the estimated sketch patches or photo patches. In the contrast experiments, the proposed method outperformed the NE based synthesis method by 0.0503 in terms of Structural SIMilarity (SSIM) index and by 14% in terms of face recognition accuracy. The experimental results illustrate that the proposed method resolves the problem of weak compatibility among neighbor patches in the NE based method and greatly alleviates the noises and deformations in the synthetic image.