3D curve reconstruction is indispensable in computer vision. The traditional 3D reconstruction considerably relies on point-based correspondences. However

point-based reconstruction ignores the structural information between sample points and may reduce reconstruction accuracy. To avoid this problem

high-level geometric primitives are needed to reconstruct 3D scenes

typically as a 3D curve. This study proposes a vision-based method under the L-infinity norm of curvatures for the 3D reconstruction of nonparametric space curves from a pair of images. An initial cost function for minimization was defined as a weighted sum of reprojection errors and curvatures of the reconstructed curve. Reaching the global minimum is difficult

specifically if the initial cost function is not significantly close to the actual result

because of nonconvexity. In addition

the weight coefficients that control the relative significance between the reprojection errors and curvature were not readily. To address this issue

L-infinity norm replaced with L norm of the curvatures was used as an inequality constraint. The curvatures of all discrete points along the curve should be less than the maximal curvature

which can be estimated from the projection curves of the two images. The reconstructed curve

given a small

was smooth under the inequality constraint. A method utilizing a generalized Lagrange multiplier was used to solve this nonlinear optimization problem with inequality constraint. An experiment using synthetic data showed that the reconstructed curve was significantly similar to the real results and that its reconstruction error was less than 1/7 from the point-based method. These findings illustrate that the proposed method is evidently superior to the point-based method. The proposed method also has higher accuracy than a previously reported method [12]. The influences of the 3D reconstruction errors with respect to in a certain range were not large

thereby validating the robustness of the proposed method. In the noise tests

reconstruction errors linearly increased with the noise. Strong noise probably caused a large stereo ambiguity along the curve and increased uncertainty of the space curve. The intersection line of two crossed cylinders was reconstructed using the proposed method. The reconstructed curve was projected in the XY plane

wherein the projection of the actual intersection line was a circle. The radius of one cylinder can be estimated after fitting the projection curve of the reconstructed curve as a circle. The relative deviation between the estimated and real radii was 2.98%. This value indicates that the proposed method has high accuracy for general industrial applications. The proposed method was used to reconstruct 3D skeletons of a fruit tree trunk; the resultant 3D skeletons were visually acceptable. The proposed method can be used to reconstruct 3D space curves for various applications. However

the stereo ambiguity and uncertainty of camera parameters may reduce the accuracy of the method because only two images were used. The number of projection curves may be increased to solve this problem.