Photometric stereo

Under Woodham's original assumptions — Lambertian reflectance, known point-like distant light sources, and uniform albedo — the problem can be solved by inverting the linear equation ${\displaystyle I=L\cdot n}$, where ${\displaystyle I}$ is a (known) vector of ${\displaystyle m}$ observed intensities, ${\displaystyle n}$ is the (unknown) surface normal, and ${\displaystyle L}$ is a (known) ${\displaystyle 3\times m}$ matrix of normalized light directions.

This model can easily be extended to surfaces with non-uniform albedo, while keeping the problem linear.[4] Taking an albedo reflectivity of ${\displaystyle k}$, the formula for the reflected light intensity becomes:

${\displaystyle I=k(L\cdot n)}$

If ${\displaystyle L}$ is square (there are exactly 3 lights) and non-singular, it can be inverted, giving:

${\displaystyle L^{-1}I=kn}$

Since the normal vector is known to have length 1, ${\displaystyle k}$ must be the length of the vector ${\displaystyle kn}$, and ${\displaystyle n}$ is the normalised direction of that vector. If ${\displaystyle L}$ is not square (there are more than 3 lights), a generalisation of the inverse can be obtained using the Moore-Penrose pseudoinverse,[5] by simply multiplying both sides with ${\displaystyle L^{T}}$ giving:

${\displaystyle L^{T}I=L^{T}k(L\cdot n)}$

${\displaystyle (L^{T}L)^{-1}L^{T}I=kn}$

After which the normal vector and albedo can be solved as described above.

The classical photometric stereo problem concerns itself only with Lambertian surfaces, with perfectly diffuse reflection. This is unrealistic for many types of materials, especially metals, glass and smooth plastics, and will lead to aberrations in the resulting normal vectors.

Many methods have been developed to lift this assumption. In this section, a few of these are listed.

• Photometry
• Stereo vision
• 3D scanner