Recover Albedo from Normal

Photometric stereo uses shading variation under known lights to recover both surface normals and albedo (intrinsic reflectance) of a surface. In the Lambertian model, each pixel’s intensity is modeled as the product of albedo and the cosine of the angle between the light direction and the surface normal. This separates geometry (normals) from material (albedo): the normal controls how much light a point receives; the albedo scales how much it reflects.

For a single light, the Lambertian equation is $I = \rho\,(\mathbf{n}\cdot\mathbf{l})$ where $I$ is the observed intensity, $\rho$ the scalar albedo, $\mathbf{n}$ a unit normal, and $\mathbf{l}$ a unit light direction. If $\mathbf{n}$ and $\mathbf{l}$ are known and the surface is front-lit ($\mathbf{n}\cdot$\mathbf{l}$> 0$), this becomes a simple scalar equation in $\rho$. With multiple light sources, you get several independent measurements of the same $\rho$, which allows you to reduce noise by averaging or doing a least-squares fit.

2. Algorithm / Approach

General pattern for recovering albedo from known normal(s) and intensities:

• Use the Lambertian model to express each observation as: $I_k = \rho\,(\mathbf{n}\cdot\mathbf{l}_k)$
• For each valid light (front-lit), solve for an estimate of $\rho$: $\rho_k = \frac{I_k}{\mathbf{n}\cdot\mathbf{l}_k}$
• Combine multiple estimates $\{\rho_k\}$ into a single robust albedo, typically by:
• simple average, or
• weighted average (e.g., weights based on $\mathbf{n}\cdot$\mathbf{l}$_k$ or confidence).

Backlit or grazing-angle observations (dot product $\le 0$ or very small) are discarded since they provide unreliable or undefined albedo estimates.

3. Step-by-Step Strategy

Assume for each pixel you are given:

• observed intensities: I[k] for lights k = 0 … K-1
• unit surface normal: n (3D vector)
• unit light directions: L[k] (3D vectors)

Per pixel:

1. Initialize

• sum_rho = 0
• count = 0 (or sum_weight = 0 if using weighted averaging)

2. Loop over lights

• Compute dot product: d = dot(n, L[k])
• If d <= 0:
• skip this light (backlit or edge-on).
• Else:
• Compute per-light albedo estimate: rho_k = I[k] / d
• Option A (simple average):
• sum_rho += rho_k
• count += 1
• Option B (weighted average, e.g. weight = d):
• w = d
• sum_rho += w * rho_k
• sum_weight += w