目的 隐式曲线能够描述复杂的几何形状和拓扑结构，而传统的隐式B样条曲线的控制网格需要大量多余的控制点满足拓扑约束。有些情况下，获取的数据点不仅包含坐标信息，还包含相应的法向约束条件。针对这个问题，提出了一种带法向约束的隐式T样条曲线重建算法。方法 结合曲率自适应地调整采样点的疏密，利用二叉树及其细分过程从散乱数据点集构造2维T网格；基于隐式T样条函数提出了一种有效的曲线拟合模型。通过加入偏移数据点和光滑项消除额外零水平集，同时加入法向项减小曲线的法向误差，并依据最优化原理将问题转化为线性方程组求解得到控制系数，从而实现隐式曲线的重构。在误差较大的区域进行T网格局部细分，提高重建隐式曲线的精度。结果 实验在3个数据集上与两种方法进行比较，实验结果表明，本文算法的法向误差显著减小，法向平均误差由10^-3数量级缩小为10^-4数量级，法向最大误差由10^-2数量级缩小为10^-3数量级。在重构曲线质量上，消除了额外零水平集。与隐式B样条控制网格相比，3个数据集的T网格的控制点数量只有B样条网格的55.88%、39.80%和47.06%。结论 本文算法能在保证数据点精度的前提下，有效降低法向误差，消除了额外的零水平集。与隐式B样条曲线相比，本文方法减少了控制系数的数量，提高了运算速度。

Objective In computer-aided geometric design and computer graphics, fitting point clouds with the smooth curve is a widely studied problem. Measurement data can be taken from real objects using techniques such as laser scanning, structure light source converter, and X-ray tomography. We use these scanned discrete data points for data fitting to perform a general model reconstruction and functional recovery for the original model or product, which is widely used in the field of geometric analysis and image analysis. The data points used in this paper are unstructured scattered data points. Compared with the parametric curves, implicit curves do not need to parameterize scattered data points. Therefore, they are widely studied because of their ability to describe objects with complicated geometry and topology. Because the control points of the conventional implicit B-spline curve need to be arranged regularly in the entire area, a large number of redundant control points are required to satisfy the topological constraints and has some limitations in the local subdivision, which will lead to the phenomenon of control point redundancy. The T-spline effectively solves this problem. Based on the advantages of B-spline curves and surfaces, it admits the structure of T-nodes, and thus, it has many advantages, such as fewer control points and convenient local subdivision. This is the reason why we chose T-spline to perform the implicit curve reconstruction. In some cases, the data points we obtain may not only be scattered coordinate information but also contain some shape constraint conditions, such as the processing of data points with the normal constraints in the field of optical engineering. Therefore, we not only need to constrain the errors of the data points but also have certain requirements for the normal errors. Hence, an implicit T-spline curve reconstruction algorithm with normal constraints is proposed in this paper. Method We first preprocess the data, which adjusts the density of sampling points adaptively by combining with the curvature to remove the redundant data points and add auxiliary points. The step of adding auxiliary points not only avoids the singular solutions but also helps to eliminate the zero level set. Two-dimensional T-meshes are constructed from the scattered point set by using binary tree and subdivision process. Here, we define a maximum number of subdivisions and then count the number of data points in each sub-rectangular block. If the number of data points is greater than the given number of subdivisions, it is subdivided until the number of data points is less than the maximum number of subdivisions and we obtain the initial T-mesh. Then, an effective curve fitting model is proposed based on the implicit T-spline function. Because the number of equations is far more than the number of unknowns, we transform the problem of the implicit T-spline curve reconstruction into a quadratic optimization problem to obtain the objective function. The objective function of our model is divided into three parts: the fitting error term, normal term, and smoothing term. The fitting error term includes the error of data points and auxiliary points. We eliminate the extra zero-level sets by adding offset points and smoothing the term. We also add the normal term to reduce the normal error of the constructed curve. According to the optimization principle, we take the partial derivative of the objective function concerning each of the control coefficients and set it equal to zero. In this case, the original problem is transformed into linear equations. The unknown control coefficients can be obtained by solving the system of linear equations to solve the problem of implicit curve reconstruction. Finally, we insert the control coefficient into the area of the large error to carry out the local subdivision of T-mesh until the precision requirement is reached to improve the accuracy of the reconstruction of the implicit curve. Result The experiment is compared with two existing methods on three datasets, including two concave and convex curves and a complicated hand curve. From the figures in paper, we can see that although the proposed method and the existing method 1 which contructs implicit equations with normal vector constraints, reconstruct the shape of the implicit T-spline curve, some extra zero level sets appear around the curve of method 1, which destroy the quality of the reconstructed curves. The reconstructed results of the proposed method do not have the extra zero level set. The experimental data show that in terms of the error of the data points, the algorithm presented in this paper differs little from the two methods in terms of the average error and the maximum error of data points, which are in the same order of magnitude. However, in the normal error, the proposed algorithm has a significant reduction. In the curves of examples 1 and 2, the proposed algorithm reduces the average error of the normal direction from the order of 10^-3 to the order of 10^-4 and the maximum error of the normal direction from the order of 10^-2 to the order of 10^-3. Meanwhile, in the curve of example 3, the proposed algorithm can still significantly reduce the normal error while method 2 which uses least square fitting method by adding auxiliarg points, has the worst normal constraint. In terms of the quality of the reconstructed curve, the extra zero level set is eliminated by the proposed method while the obvious zero level set exists in method 1 and the reconstruction effect is poor. Meanwhile, compared with the implicit B-spline control grid, the number of control points in the T-mesh of the three data sets was only 55.88%, 39.80%, and 47.06% of that in the B-spline grid. Conclusion Experimental results indicate that the proposed algorithm effectively reduces the normal errors under the premise of ensuring the accuracy of data points. The proposed algorithm also successfully eliminates extra zero-level sets and improves the quality of the reconstructed curve. Compared with the implicit B-spline curves, the proposed method reduces the number of control coefficients and improves the operation speed. Hence, the proposed method successfully solves the problem of implicit T-spline curve reconstruction with normal constraints.