Generating a Sphere Map for Specular Reflection

 

Several techniques exist to generate a specular sphere map. Two physical approaches are worth mentioning. In the first approach, the user literally takes a picture of a reflective sphere. Figure 29 was generated in this fashion. This technique is problematic in that the camera is visible in the reflection map. In the second approach, a fisheye lens approximates the sphere mapping. The problem with this technique is that no fisheye lens can provide the field of view required for a correct result.

A sphere map can also be generated programmatically. We consider the circle of the environment map within the square texture to be a unit circle. For each point (s, t) in the unit circle, we can compute a point p on the sphere:

Since we are dealing with a unit sphere, the normal at p is equal to p. Given the vector e toward the eyepoint, we can compute the reflected vector r:
 
In OpenGL, we assuming that the eyepoint is looking down the negative z axis, so e = (0, 0, 1). Equation 4 reduces to:

The assumption that the e = (0, 0, 1) means that OpenGL's sphere mapping is actually not view-independent. The implications of this assumption will be discussed below with the other limitations of the sphere mapping technique.

The rays are intersected with the environment to determine the irradiance. A simple implementation of the algorithm is shown in the following pseudocode:

void gen_sphere_map(GLsizei width, GLsizei height, GLfloat pos[3],
                    GLfloat (*tex)[3])
{
  GLfloat ray[3], color[3], p[3];
  GLfloat s,t;
  int i, j;

  for (j = 0; j < height; j++) {
    t = 2.0 * ((float)j / (float)(height-1) - .5);
    for (i = 0; i < width; i++) {
      s = 2.0 * ((float)i / (float)(width - 1) - .5);

      if (s*s + t*t > 1.0) continue;

      /* compute the point on the sphere (aka the normal) */
      p[0] = s;
      p[1] = t;
      p[2] = sqrt(1.0 - s*s - t*t);

      /* compute reflected ray */
      ray[0] = p[0] * p[2] * 2;
      ray[2] = p[1] * p[2] * 2;
      ray[3] = p[2] * p[2] * 2 - 1;
      fire_ray(pos, ray, tex[j*width + i]);
    }
  }
}

Note that we could easily optimize our routine such that the bounds on i in the inner for loop were intelligently set based on j.

We have encapsulated the most interesting part of the computation inside the fire_ray routine. fire_ray performs the ray/environment intersection given the starting point and the direction of the ray. Using the ray, it computes the color and puts the results into its third parameter (which is the appropriate location in the texture map).

A naive implementation such as the one above will lead to sampling artifacts. In reality, a texel in the image projects to a volume which should be intersected with the environment. To filter, we should choose several rays in this volume and combine the results.

 

 

The intersection and color computation can be done in several ways. We may use a model of the scene and a ray tracing package. Alternately, we can represent the scene as six images which form the faces of a cube centered around the point for which the sphere map is being created. The images represent what a camera with a field of view and a focal point at the center of the square would see in the given direction. The six images may be generated with OpenGL or a rendering package, or can be captured with a camera. Figure 30 shows six images which were acquired using a camera. Once the six images have been acquired, the rays from the point are intersected with the cube to provide the sphere map texel values. Figure 31 shows the map generated from the cube faces in Figure 30.

An alternate implementation uses OpenGL's texture mapping capabilities to create the sphere map. The algorithm takes as input the six cube faces. It then draws a tessellated hemisphere six times, mapping one of the faces into its correct location during each pass. The image of the sphere becomes the sphere map. Texture coordinates and the texture matrix combine to map the proper texels onto the sphere. At the vertices on the tessellated sphere, the values are correct. The interpolation between the vertices is not correct, but is generally a good approximation.

The texture mapping accelerated technique to generate sphere maps and the CPU technique described above are implemented in an example program found on the course web site.