Photometric Stereo: Equations (PS3)

| Introduction | Advantages | Equations | Gradient & albedo | PS3 vs PS4 |

We use three photometric images to recover the local surface orientation and albedo information for a Lambertian surface where the shadow effect is absent. The geometry of the camera / lighting setup can be seen here. Note that, during image capture, we keep the slant angle constant (50°). Furthermore, we apply the assumptions, that is that the surface is parallel to the image plane of the camera and approximately flat; the light source and camera are located far away from the test sample; and the illumination direction and viewing direction are uniform at each point on the surface.

We transfer the Lambertian surface model into matrix format. Therefore the intensity of pixel in a image can be expressed as follow

where

• i₁(x, y) is image intensity at the point (x, y);
• is a surface normal unit vector to the surface s(x, y) at the point (x, y), and and are surface partial derivatives measured in the x and y directions, respectively;
• L₁ = [ l_x1, l_y1, l_z1 ]^T is a unit illumination vector, which is pointing from the surface towards to the light source; and
• r is surface albedo at the given point (x, y).

Now we consider three light sources with illumination vectors L₁, L₂ and L₃. The equation can be rewritten in matrix form

where

• I = [ i₁, i₂, i₃ ]^T is image intensity vector;
• L = [ L₁, L₂, L₃ ]^T is photometric illumination matrix which incorporates the light intensity for each light source.

Provided that all of three illumination vectors L₁, L₂ and L₃ are not lying in the same plane (non-coplanar), then the photometic illumination matrix L is non-singular and its inverse matrix, L^-1 exists and , where M = [ m₁, m₂, m₃ ]^T

1. For each given point (x, y) on the surface, the image intensity vector I is firstly formed by capturing three images under different illumination directions L₁, L₂ and L₃.
2. The vector M = [ m₁, m₂, m₃ ]^T is obtained by the production of I and L^-1.
3. The surface gradient components can be calculated via
   and .
4. Finally, the surface albedo is recovered by finding the length of vector M.
   

Those computation is straightforward in this case, and a unique result is assured.

| Introduction | Advantages | Equations | Gradient & albedo | PS3 vs PS4 |