Lighting estimation of a convex Lambertian object using weighted spherical harmonic frames

Abstract

An explicit lighting estimation from a single image of Lambertian objects is influenced by two factors: data incompletion and noise contamination. Measurement of lighting consistency purely using the orthogonal spherical harmonic basis cannot achieve an accurate estimation. We present a novel signal-processing framework to represent the lighting field. We construct a weighted spherical harmonic frame with geometric symmetry on the sphere ${S^2}$. Weighted spherical harmonic frames are defined over the generating rotation matrices about symmetry axes of finite symmetry subgroups of $SO(3)$, and the generating functions are weighted spherical harmonic basis functions. Compared with the orthogonal spherical harmonic basis, the redundant weighted spherical harmonic frames not only describe the multidirectional lighting distribution intuitively, but also resist the noise theoretically. Subsequently, we analyze the relationship of the irradiance to the incoming radiance in terms of weighted spherical harmonic frames and reconstruct the lighting function filtered by the Lambertian BRDF (bidirectional reflectance distribution function). The experiments show that the frame coefficients of weighted spherical harmonic frames can better characterize the complex lighting environments finely and robustly.

Appendices

Appendix A

Euler angles for the operations of the generating matrices are listed in the following. They are corresponding to the rotation matrices between the original z-axis and the symmetry axes of the rotational symmetry groups of the Platonic solids . In Table 2, ${\theta _T} = \mathrm{{1}}\mathrm{{.9106}}$; in Table 3, ${\theta _I} = \mathrm{{1}}\mathrm{{.1071}}$.

Table 2 Euler angles of Tetrahedron

Table 3 Euler angles of Icosahedron

Appendix B

As the noise-resistant property is determined by spherical harmonic frame bounds for the subspaces, the relevant derivation is in terms of spherical harmonic frame. Utilizing the properties of eigenvalue, the explicit expression about eigenvalue can be written as

$$\begin{aligned} {\lambda _{l,i}}&= \left\langle {{{\lambda _{l,i}}{Y_{l,i}}}} \mathrel {\left| {} \right. } {{{Y_{l,i}}}} \right\rangle \nonumber \\&= \left\langle {{S{Y_{l,i}}}} \mathrel {\left| {} \right. } {{{Y_{l,i}}}} \right\rangle \nonumber \\&= \left\langle {{\sum \limits _{l = 0}^L {\sum \limits _{s = 1}^{\left| s \right|} {\sum \limits _{m = d(s)}^{} {\left\langle {{{Y_{l,i}}}} \mathrel {\left| {} \right. } {{Y_{l,m}^s}} \right\rangle } } } Y_{l,m}^s}} \mathrel {\left| {} \right. } {{{Y_{l,i}}}} \right\rangle \nonumber \\&= \sum \limits _{s = 1}^{\left| s \right|} {\sum \limits _{m = d(s)}^{} {{{\left\langle {{{Y_{l,i}}}} \mathrel {\left| {} \right. } {{Y_{l,m}^{{s}}}} \right\rangle }^2}} } \end{aligned}$$

(40)

So ${\lambda _{l,i}} = \sum _{s = 1}^{\left| s \right|} {\sum _{m = d(s)}^{} {{{\left\langle {{{Y_{l,i}}}} \mathrel {\left| {} \right. } {{Y_{l,m}^s}} \right\rangle }^2}} }$, $d(s) \in [ - l,l]$ for the total frame, $d(s)$ is a fixed index in $[ - l,l]$ for the partial frame. Considering the structure of our spherical harmonic frames, at least, there exists one group of original spherical harmonic basis function. Then, ${\lambda _{l,i}} \ge 1$.

For a total spherical harmonic frame, as all the rotated copies which are generated by the $s$-th generating rotation matrix constitute a frame, the eigenvalue satisfies,

$$\begin{aligned} {\lambda _{l,i}}&= \sum \limits _{s = 1}^{\left| s \right|} {\sum \limits _{m = d(s)}^{} {{{\left\langle {{{Y_{l,i}}}} \mathrel {\left| {} \right. } {{Y_{l,m}^s}} \right\rangle }^2}} } \nonumber \\&= \sum \limits _{s = 1}^{\left| s \right|} {\sum \limits _{k = - l}^l {{{\left\langle {{{Y_{l,i}}}} \mathrel {\left| {} \right. } {{{M^s}Y_{l,k}^{}}} \right\rangle }^2}} } = \left| s \right| \end{aligned}$$

(41)

Thus, the spherical harmonic frame Class I is a tight frame. Frame bound equals the number of generating rotation matrices.

Appendix C

The mean squared error (MSE) of an original coefficients with respect to the estimated coefficients are listed below. Each value reveals the variation of one lighting field represented by the listed method. The lighting coefficients are computed by seven methods, $C3, C5$, T, Tii, ICO, ICOii, and basis. For each lighting condition, the estimated coefficients are computed from 1,000 noise samples ( To clarify the reasoning, we only consider one class of noise). The class of Gaussian noise is $15\,\%$ normal plus $30\,\%$ intensity added on the sphere data. To convey concisely, five environment maps are considered, which are shown in the second row of Fig. 5. The MSE values of one row are arranged in sequence according to five environment maps, from left to right. As the noise-resistant property, the MSE values of frame coefficients are less than the ones of basis coefficients, which obey the inequality (28). For tight frames, the frame bounds of $C3, C5$, T, and ICO are 3, 5, 4, and 6, respectively. For Tii, the low frame bound $A$ is 2.5529. For ICOii, the low frame bound $A$ is 4.1810. Compared with T, the suppressing ability of Tii is weaker. The same conclusion applies for ICO and ICOii (Table 4).

Table 4 The MSE of an original coefficients with respect to the estimated coefficients