I implemented the Ikeuchi and Horn shape-from-shading algorithm for the final project in one of my computer vision courses at Berkeley. This is an iterative method for computing surface normals from the image of a shaded 3D surface. The algorithm assumes the surface reflectance properties, the scene lighting, and the viewpoint are all known. The algorithm finds a solution by minimizing an energy function based on a brightness constraint and a smoothness constraint. The brightness constraint ensures that the resulting surface orientations give rise to image luminance values which closely match the original input image. A reflectance map based on the known lighting and surface properties relates surface orientation to image luminance. The smoothness constraint ensures that the surface orientations of neighboring points will vary smoothly across the surface. In order to guarantee that the algorithm will converge to a solution, the known orientations at the occluding boundary serve as a final constraint on the system. My implementation of this algorithm takes grayscale images as input and produces a set of surface normals as output. I then used a simple least-squares method to approximate a 3D surface which fits the resulting normals.

Ikeuchi and Horn represent surface orientations using a stereographic projection of the Gaussian sphere. Every surface orientation visible to a viewer can be represented on a Gaussian hemisphere. The 2D projection of this hemisphere allows us to refer to surface orientation using the (f,g) coordinates of this orientation space. The following image illustrates this projection (adapted from Ikeuchi and Horn 1981).

The Ikeuchi and Horn algorithm assumes the surface reflectance properties, scene lighting, and viewpoint are all known. From this information, we construct a reflectance map which relates images intensites to surface orientation. For a given surface orientation (f,g), this reflectance map reveals what the corresponding brightness value should be in the image. Here is an example reflectance map for a lambertian sphere:

Although the reflectance map relates image intensities to surface orientations, it does not allow us to uniquely recover the surface orientation for any arbitrary point in the image. Most brightness values in the image will be consistent with a set of surface orientations, any of which could give rise to the brightness value in question. Here is an example reflectance map for a lambertian surface illuminated by a light source directed from (0, 0, 1). The highlighted contour (in white) represents the surface orientations consistent with an image intensity I(x,y) = 0.6.

Although this constrains the solution significantly, additional constraints are required before a final solution can be found.

The image irradiance constraint (also known as the brightness constraint) refers to the requirement that the corresponding luminance value R(f, g) for any estimated surface orientation (f, g) at position (i, j) in the image should match the luminance value I(i, j) as closely as possible. This constraint can be formally written as follows:

The smoothness constraint minimizes total variation in surface orientation. Given that an arbitrary image intensity value I(x, y) is consistent with a set of possible surface orientations as described by the reflectance function R(f, g), the smoothness constraint guarantees that the surface orientations of the solution will smoothly vary across surface points. This constraint can be formally written as follows:

The algorithm iteratively updates the surface normals until the sum of the error terms is minimized. In order to guarantee the algorithm converges to a solution, the known orientation of the surface at the occluding boundary is used to further constrain the solution.

The following figures depict some example results using this algorithm.



References

Ikeuchi K and Horn BKP. Numerical shape from shading and occluding boundaries. Artificial Intelligence. 17, Aug. 1981, 141-184.