Least-Squares Estimation of Tangent Space for a Triangular Mesh

This paper presents a novel method for computing an orthonormal tangent space at every vertex of a triangular mesh provided with texture coordinates, which is essential in normal mapping. Our method employs least-squares estimation of rotation between two sets of corresponding points: one set from the positions of the 1-ring neighbors of a vertex in texture space and the other set from their corresponding positions in world space. This optimal rotation represents a local space transformation that defines the tangent space at the vertex. Consequently, our method ensures that the resulting tangent space is inherently orthonormal, eliminating the need for intricate operations such as averaging and orthonormalization, which are required in the previous methods. In addition, we introduce texture space alignment of triangles to address vertices on seams, which are inevitable in texture mapping to a closed surface mesh. Experimental results demonstrate the effectiveness of the proposed methods in various experiments.