Figure 9 shows the relationship between the surface normal of the surface point and the phase angle obtained from viewpoints. In Fig. 9, represents the azimuth angle of the surface point observed by the camera , and represents the vector orthogonal to the reflection plane under the coordinate system of the camera . Because is orthogonal to the reflection plane, we obtain Eq. (12) using the phase angle or azimuth angle : Display Formula
(12)The rotation matrix represents the transformation from the world coordinate system to the local coordinate system of the camera indicated by . The transformation from the local coordinate system of the camera to the world coordinate system is the transpose of . Because the transformed vector becomes orthogonal to the surface normal , Eq. (13) holds. Display Formula
(13)If we concatenate Eq. (13) for cameras, we obtain Eq. (14): Display Formula
(14)The surface normal , which satisfies Eq. (14) in the least-squares sense, can be estimated using SVD. The matrix can be decomposed by SVD as follows: Display Formula
(15)Here, is a orthogonal matrix, is a diagonal matrix with non-negative values, and is a orthogonal matrix. The diagonal item of the matrix is the singular value of the matrix and the singular vector corresponding to is . Owing to the relationship between the surface normal and the reflection planes, the rank of the matrix is at most 2; thus, one of the three singular values becomes 0. The proof that the rank of the matrix is at most 2 is presented in the 1. The surface normal can be represented as Eq. (16),40 which can be calculated from the singular vector that has the smallest singular value, namely, the third row of in Eq. (15). Display Formula
(16)In the general case, is an arbitrary scalar coefficient; however, since the surface normal and the singular vectors are normalized vectors, would be either or . Whether must be positive or negative can be easily determined to ensure that the surface normal will face toward the camera. The surface normal estimated by Eq. (16) is the optimal value that minimizes the squared error of Eq. (14) formulated by equations. The input data must be obtained from two or more viewpoints since the rank of the matrix is 2. If we obtain the input data from more viewpoints, the influence of input noise will decrease.