Analysis of Planar Light Fields From Homogeneous Convex Curved Surfaces Under Distant Illumination

Ravi Ramamoorthi Computer Graphics Laboratory ,
Pat Hanrahan Stanford University


We consider the flatland or 2D properties of the light field generated when a homogeneous convex curved surface reflects a distant illumination field. Besides being of considerable theoretical interest, this problem has applications in computer vision and graphics---for instance, in determining lighting and bidirectional reflectance distribution functions (BRDFs), in rendering environment maps, and in image-based rendering. We demonstrate that the integral for the reflected light transforms to a simple product of coefficients in Fourier space. Thus, the operation of rendering can be viewed in simple signal processing terms as a filtering operation that convolves the incident illumination with the BRDF. This analysis leads to a number of interesting observations for computer graphics, computer vision, and visual perception.


In this paper, we restrict ourselves to the planar case, assuming all illumination and measurements occur in the same plane. We analyse the resulting light field generated by a convex curved surface. For our analysis, we construct the Fourier transform of both the input lighting and the brdf. We demonstrate that the lighting integral transforms to a simple product of coefficients in Fourier space. Thus, the operation of rendering can be viewed in simple signal processing terms. This Fourier analysis of the structure of the light field leads to a number of interesting observations.

For instance, in general, the brdf can be completely determined in theory, given the input lighting and the complete output radiance function. This can be viewed as a simple problem of filter estimation, estimating the filter (brdf) given the output (radiance) and the signal (lighting). The problem is unsolvable when terms in the Fourier expansion of the lighting are zero, corresponding to the signal having no amplitude for certain modes of the brdf filter. Furthermore, this explains the use of a point light source for image-based brdf measurement---the fourier transform of a delta function is constant and nonzero everywhere. We can also think of this as determining the brdf filter by considering its impulse-response. In practical terms, this theoretical result may have applications in efficiently recovering brdfs under uncontrolled illumination conditions, by considering the Fourier transforms of the brdf and lighting.

A similar result holds for the lighting, viz. we can recover the lighting given the brdf and the output radiance. Similarly as above, this becomes impossible when terms in the Fourier transform of the brdf are zero. The theory allows us to easily derive a closed-form formula for the action of distant illumination on a Lambertian object, an important special case. We will show that inverse lighting from a Lambertian object is in general ill-conditioned, allowing only the lowest-order modes of the lighting to be recovered. This implies that an inverse-lighting approach such as that proposed by Marschner can only recover a very rough notion of the lighting. On the other hand, this result may allow us to efficiently prefilter and render environment maps in computer graphics, since only a very low frequency representation of the environment need be used.

Our work may have applications in visual perception. We demonstrate that lighting effects can only induce low-frequency variation in the intensity of a Lambertian surface. Therefore, all high-frequency variation is because of texture, and this explains why we can find the texture on surfaces independently of lighting effects. It also indicates that we will find it difficult to distinguish lighting effects from a slowly varying texture.

Finally, we demonstrate that if reciprocity is taken into account, we can, in general, factor the light field into a product of brdf and lighting terms, without knowing either. Besides obvious applications in computer vision and graphics, this accords with our perceptual ability to ascertain the brdf (or degree of shininess) of a surface, independently of the lighting conditions.

Although our contribution is theoretical and restricted to the planar case, the results can be fairly easily generalized to 3D, and we believe the theory has applications in computer vision and graphics, and can lead to practical algorithms for many important applications.

The figures on the right summarize some of the main results from the paper.

Relevant Links

Figure 1. A diagram showing local and global angles and all the important vectors.

Figure 2. The main result of the paper: a Fourier convolution formula to describe the reflected light field in signal-processing terms as obtained by filtering the illumination using the BRDF.

Figure 3. Plot of Fourier coefficients for Phong BRDFs having different exponents. The Fourier coefficients have a gaussian form with the width depending on the exponent.

Figure 4. Our analytic formula for factoring the reflected light field. This allows us to separate the BRDF and lighting up to global scale, provided reciprocity of the BRDF can be taken into account.

Ravi Ramamoorthi
Last modified: Thu Apr 26 05:27:34 PDT 2001 >