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Radiometry Image Formation: Key Concepts & Definitions

The document discusses radiometry in image formation, focusing on how light interacts with surfaces in computer vision. Key concepts include radiometric quantities like radiance and irradiance, the BRDF, and the Phong reflection model, along with their respective formulas and derivations. A solved example demonstrates the calculation of radiosity for a Lambertian surface illuminated by a point light source, yielding B(x) = 31.42 W/m².

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47 views3 pages

Radiometry Image Formation: Key Concepts & Definitions

The document discusses radiometry in image formation, focusing on how light interacts with surfaces in computer vision. Key concepts include radiometric quantities like radiance and irradiance, the BRDF, and the Phong reflection model, along with their respective formulas and derivations. A solved example demonstrates the calculation of radiosity for a Lambertian surface illuminated by a point light source, yielding B(x) = 31.42 W/m².

Uploaded by

MI A1
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Radiometry Image Formation

Explain in detail and extract all the formulas (if any) with derivation and a solved example.

The PDF you uploaded, titled “Image Formation: Radiometry”, appears to be a lecture on radiometry
in the context of computer vision and image processing, specifically discussing how light interacts
with surfaces and how this affects image formation. Below is a detailed explanation of the concepts
covered, along with extracted formulas, their derivations, and a solved example at the end.

🔍 Key Concepts & Definitions


1. Radiometric Quantities

These are physical quantities used to measure the amount of light energy:

Radiance L(x, θ, ϕ):


Radiant power per unit area per unit solid angle in a given direction.
Units: W/m2 /sr

L(x, θ, ϕ)

Irradiance E :
Total power arriving at a surface from all directions in a hemisphere.

E(x) = ∫ L(x, θ, ϕ) cos θ dω​

2. Foreshortening

When a surface is tilted relative to the incoming light, the effective area reduces as cos θ , leading to
foreshortening.

3. BRDF (Bidirectional Reflectance Distribution Function)

Defines how light is reflected at an opaque surface:

Lo (x, θo , ϕo ) = ∫ ρbd (x, θo , ϕo , θi , ϕi )Li (x, θi , ϕi ) cos θi dω


​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​

For discrete point sources:

Lo = ∑ ρbd (...)Li cos θi


​ ​ ​ ​ ​

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4. Radiosity B(x)

Total power leaving a point on a surface per unit area, independent of direction:

B(x) = ∫ L(x, θ, ϕ) cos θ dω ​

If radiance is constant w.r.t direction (Lambertian surface):

B(x) = L(x) ∫ cos θ dω ​

Ω
2π π/2
B(x) = L(x) ∫ ​
∫ ​
cos θ sin θ dθdϕ = L(x)π
0 0

5. Directional Hemispheric Reflectance (DHR)

Reflectance of a surface from a specific direction, integrating over outgoing hemisphere:

∫Ω Lo (x, θo , ϕo ) cos θo dω
ρdh (θi , ϕi ) =
​ ​ ​ ​ ​

∫Ω Li (x, θi , ϕi ) cos θi dω
​ ​ ​ ​

​ ​ ​ ​ ​

6. Lambertian Surfaces

For Lambertian surfaces:

Radiance is independent of viewing direction.

BRDF is constant:
ρd
ρbd =
​

​ ​

π
Where ρd is albedo, the fraction of incoming light reflected diffusely.
​

7. Phong Reflection Model (Specular + Diffuse)

Radiance includes a specular component, often modeled as:

L = Ldiff use + Lspecular ​ ​

Lspecular ∝ cosn (α) ​

Where α is the angle between reflection and view direction, and n controls sharpness.

✅ Solved Example
Q: A Lambertian surface is illuminated by a point light source, and its radiance is constant at L(x) =
2
10 W/m /sr. Compute the radiosity B(x).
Solution:

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Given:

L(x) = 10
For Lambertian surface:
2
B(x) = πL(x) = π × 10 = 31.42 W/m

Let me know if you want this turned into a reference sheet, or if you'd like another example with
specular reflection or BRDF derivation!

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