# Least-Squares Estimation of Tangent Space for a Triangular Mesh
## 4 Experimental Results
Figure 7c,d shows the differences of \(\mathbf{T}_{l}\) and \(\mathbf{T}_{m}\) compared to \(\mathbf{T}_{s}\). The differences are measured by \(\|\mathbf{T}_{l}-\mathbf{T}_{s}\|_{\mathrm{F}}\) and \(\|\mathbf{T}_{m}-\mathbf{T}_{s}\|_{\mathrm{F}}\). We can observe that the differences are also large where the orthonormality errors are large. Figure 8 shows the close-up view of the horn with the red, green, and blue axes corresponding to the tangent, bitangent, and normal vectors. The bright, thick axes represent the orthonormal tangent spaces obtained with our method, while the dark, thin ones represent those with either the linear method or [PERSON]'s method. The three axes at the tip of the right horn has a large error and are far from orthogonal. The experiments were conducted on a MacBook Pro laptop equipped with an Apple M2 Max CPU, utilizing only a single core and thread to measure CPU time for computation. Reading an ASCII file containing a cow model with 2904 vertices, 8706 edges, 5804 faces, 3195 texture coordinates, and 292 seam edges took approximately 8.10 ms. Texture space alignment required approximately 0.32 ms, and tangent space computation with our method took approximately 1.01 ms. Tangent space computation using both the linear method and [PERSON]'s method each took approximately 0.34 ms. Our method demanded more computation than the previous methods due to the heavier workload of the polar decomposition of the covariance matrix employed in our method, compared to the matrix inversion used in the previous method. The next experiment is for the sphere model, where the tangent spaces of adjacent triangles and vertices are expected to be very similar due to the inherent geometry of a sphere. Figure 9 shows the result of this experiment. Although the orthonormality errors were smaller compared to those in the cow model, they remained non-negligible. The first column displays a close-up view, featuring the bright, thick axes representing the tangent spaces obtained with our method, alongside the dark, thin axes obtained with either the linear method or [PERSON]'s method. [PERSON]'s method exhibits lower orthonormality errors than the linear method. For a triangular mesh without vertex-normal vectors, our shape-matching method can generate normal vectors in addition to tangent and bitangent vectors. In the case of a smooth triangular mesh, the generated normal vectors are satisfactory, as illustrated in the sphere model shown in Figure 10a. The normal vectors provided with the mesh were discarded, and new ones were computed using both our shape-matching method and the linear method. The differences \(\|\mathbf{n}_{s}-\mathbf{n}_{l}\|\) are overlaid onto the surface. However, normal vectors for sharp features are less satisfactory, as demonstrated in Figure 10b. It is important to note that the normal vector depends solely on the underlying geometry and is not influenced by the texture map. Consequently, we recommend using the average of face normals, as was done in the linear method, to compute the normal vectors. Our method can then be applied to obtain orthonormal tangent spaces for the constrained normal vectors. Figure 8: Close-up view of the right horn of a cow model. The bright, thick RGB axes represent the tangent spaces obtained with our method, while the dark, thin ones represent those with either (**a**) the linear method or (**b**) [PERSON]’s method.