[CV] 5-1. Radiometry and Reflectance

Week 5-1. Radiometry and Reflectance

1. Light and Scene

light and shadow
refraction 굴절
reflection
interrelection: reciprocal reflection 상호 반사(엘베 거울)
scattering
more complex appearances

2. Radiometric Concepts

Solid Angle
projected area of a surface patch onto a unit sphere from a point
it is subtended by a point and a surface patch
unit sphere 위에 있는 surface patch와 점의 사이 부분 (projected area), 그림 참고
$\displaystyle dw = \frac{dA'}{R^2}= \frac{dAcos\theta_i}{R^2}$
subtend: 범위를 정하다
steradian: 입체 호도법
foreshorten: 단축하다

Radiant Intensity of Source: $\displaystyle J = \frac{d\phi_i}{dw}(=\frac{power}{solid angle})(watts/steradian)$
light radiant flux emitted per unit solid angle
radiant flux(power): radiant energy emitted per unit time
solid angle 기준 power
Surface Irradiance 방사율: $displaystyle E = \frac{d\phi_i}{dA}=\frac{power}{surface area}(watts/m^2)$
light radiant flux(power) incident 광속 입사 (received) per unit surface area, not depend on light이 온 곳
surface area기준 power
Surface Radiance
Flux emitted per unit foreshortened 단축된 area per unit solid angle
• Surface can radiate into whole hemisphere 반구
• L depends on reflectance properties of surface
• L depends on direction $\theta_r$
power / (foreshorted area x solid angle)
반구 다 방사 가능, 반사율에 의존, $\theta_r$에 의존

$\displaystyle L = \frac{d^2\phi_r}{(dAcos\theta_r)dw}$

• Radiometric Concepts

3. BRDF (Bi-Directional Reflectance Distributin Function)

Reflectance: Ratio of outgoing energy to incoming energy at a single point
BRDF(Bi-Directional Reflectance Distributin Function) of a surface:
Given an illumination and a viewing direction, measure of how bright the surface appears
BRDF f=L/E = radiance/irradiance = (power/forshorted area x solid angle) / (power/surface area)

Important Properties of BRDFs

• Conservation of Energy
  $\int_{hemisphere}f(\theta_i,\varphi;\theta_r,\varphi_r)dw_r \leq 1$

• Helmholtz Reciprocity (follows from 2nd Law of Thermodynamics 열역학)
  source, viewing directions 바뀌어도 됨
  $f(\theta_i,\varphi_i;\theta_r,\varphi_r) = f(\theta_r,\varphi_r;\theta_i,\varphi_i)$

• Common Assumption: Isotropy (Rotational Symmetry) 등방성
  surface is rotated about the normal해도 안바뀜
  변수 3개로 해도 됨 $f(\theta_i,\theta_r,\varphi_i-\varphi_r)$

4. Mechanisms of Reflection

• Body Reflection:
  diffuse reflection, matte appearance, Non-Homogeneous Medium, Clay, paper, etc.
• Surface Reflection 정반사:
  Specular Reflection, Glossy Appearance, Highlights, Dominant for Metals

Image Intensity = Body Reflection + Surface Reflection

Diffuse Reflection and Lambertian BRDF
Lambertian surface: equally bright from any view direction, independent of outgoing direction $\vec{v}$
Lambertian BRDF: $\displaystyle f(\theta_i,\varphi_i;\theta_r,\varphi_r) = \frac{\rho_d}{\pi}$
$\rho_d$ albedo- 달, 행성이 반사하는 태양 광선의 비율

BRDF = L/E, L = BRDF * E
Lambert's Cosine Law: reflected energy는 방향, surface norma 사이 각도의 cosine에 비례

Specular Reflection and Mirror BRDF
Mirror BRDF:
모든 입사각은 한 방향으로 반사 only when $\vec{v}=\vec{r}$
Mirror BRDF (double-delta function):
$f(\theta_i,\varphi_i;\theta_r,\varphi_r) =\rho_s \delta(\theta_i - \theta_v)\delta(\varphi_i + \pi - \varphi_v)$
Surface Radiance: $L = I\centerdot \rho_s \delta(\theta_i - \theta_v)\delta(\varphi_i + \pi - \varphi_v)$

Combining Specular and Diffuse: Dichromatic Reflection
Observed Image Color = a x Body Color + b x Specular Reflection Color