Bidirectional Reflectance Distribution Function (BRDF)

The Bidirectional Reflectance Distribution Function (BRDF) is a key concept in photometric image formation. It defines how light is reflected at an opaque surface, describing how incoming light from one direction is reflected into another direction. This function is crucial for modeling surface appearance in computer vision, computer graphics, and realistic rendering.

1. Definition and Mathematical Formulation

The BRDF is a function of four angles: the incident light direction (θ_i, φ_i) and the reflected direction (θ_r, φ_r), and it may also vary with the light’s wavelength (λ).

Mathematically:
fr(θi, φi, θr, φr; λ)

Alternatively, using unit vectors: fr(v̂i, v̂r, n̂; λ)
where:

• v̂_i is the incident light direction
• v̂_r is the viewing/reflection direction
• n̂ is the surface normal

2. Computing Reflected Radiance

The reflected radiance L_r in the direction v̂_r is computed by integrating incoming light L_i from all directions over the hemisphere above the surface:

Lr(v̂r) = ∫ Li(v̂i) × fr(v̂i, v̂r, n̂) × max(0, cos(θi)) dω

For point light sources, this becomes a sum: Lr(v̂r) = Σ Li × fr × max(0, cos(θi))

3. Properties of BRDF

a) Reciprocity

BRDF satisfies the symmetry rule: switching incoming and outgoing directions doesn’t change the result.
fr(v̂i, v̂r) = fr(v̂r, v̂i)

b) Energy Conservation

The total reflected energy must not exceed the incoming energy. This ensures realistic behavior.

c) Isotropy

For isotropic materials, the BRDF is independent of surface rotation and depends only on relative angles.

4. Common Reflection Models

Diffuse Reflection (Lambertian)

Light is reflected equally in all directions. The BRDF is constant: fd = constant

Specular Reflection (Phong Model)

Specular highlights depend on how close the viewer direction is to the perfect reflection direction: fs(θs) = ks × cosk(θs)
Where k_s is the specular coefficient and k is the shininess exponent.

Microfacet Model (Torrance-Sparrow)

Reflectance is modeled based on microscopic surface structure: fs(θs) = ks × exp(−c² × θs²)

5. Applications in Computer Vision

• Shape from shading: Estimating surface geometry based on brightness.
• Material recognition: Identifying surface type (metal, plastic, etc.).
• Inverse rendering: Recovering lighting and material properties from images.
• Realistic rendering: Generating photorealistic images and AR effects.