The Research Of A Method Of Construction Of Tri-variate Spline S₄²

In reconstructing volumetric data,we had to evaluate the convolution of two Box splines,the computational complexity of which could be very high.To promote efficiency of reconstructing volumetric data in S₄²（R³,△3）,firstly,we induce the explicit piecewise polynomials of locally supported Box spline in S₄²（R³,△3）based on paper [6].Secondly,we get the bases of S₄²（R³,△3）and show its partition of unity.Whether compared with other Box spline with the same C2 reconstruction,or the spline of tensor product,the order of piecewise polynomial remains lowest.This would absolutely reduce the cost of computation and make a more efficient calculation.The main method in evaluation of the seven-direction Box spline in space is decomposition.In the first place,we decompose the seven-direction spline into a three-direction spline and a four-direction spline.The three-direction spline is easy to evaluate.And we can evaluate the four-direction spline by means of integration.So,we convolute them and get S₄²（R³,△3）.In the end,the feasibility and accuracy of the proposed mathematic model and algorithm is demonstrated through the computational experiment.