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==== Radiance ====
**Radiance** is defined as the [[yanding:成像基础知识:光学:辐射度学与光度学:辐射通量|radiant flux]] emitted by a radiation source in a given direction per unit projected area and per unit [[yanding:成像基础知识:光学:辐射度学与光度学:立体角|solid angle]], denoted as \(L\).\\
**Unit:** \(\mathrm{W \cdot sr^{-1} \cdot m^{-2}}\) (watts per steradian per square meter).\\
{{ :yanding:成像基础知识:光学:辐射度学与光度学:辐射亮度3.png?350 |}}
**Mathematical Expression:**\\
$$L = \frac{dI}{dA \cos\theta} = \frac{d^2 \Phi}{dA \cos\theta \, d\Omega}$$
where: $L$ is the radiance; $I$ is the radiant intensity; $\Phi$ is the radiant flux; $A$ is the differential area of the radiation source; $\theta$ is the angle between the observation direction and the normal to the differential area; $\Omega$ is the differential solid angle in the observation direction.


**Relationship Between Radiance and Imaging**\\
An extended radiation source can be considered as consisting of many differential area elements, and the radiation may vary in different directions.\\
Radiance $L$ is used to describe the intensity of radiation in a specific direction, defined as the radiant flux per unit area per unit solid angle. In the imaging process, what the human eye or a camera receives is precisely the radiance of the scene in various directions; therefore, an image can be regarded as a sampling of the radiance distribution.\\
Thermal imaging provides an intuitive example, as shown in Figure 1.\\
|  {{ :yanding:成像基础知识:光学:辐射度学与光度学:微波炉内腔.png?300 |}}  |
^  Figure 1: Thermal image of the interior of a microwave oven  ^
(Figure 1: https://commons.wikimedia.org/wiki/File:Opened_oven_seen_with_thermal_camera.jpg)\\
The different colors in the image correspond to the spatial distribution differences of the target's radiance in the infrared band, with brighter areas indicating higher radiance.\\

**Relationship Between Radiance and Irradiance for a Lambertian Surface**\\
For an ideal Lambertian surface with a reflectance of $\rho$:
  * Relationship between incident irradiance and surface radiant exitance:
  $$ M = \rho E $$

  * Relationship between radiant exitance and radiance:
  $$ M = \pi L $$

  * Combining these yields the relationship between radiance and incident irradiance:
$$ L = \frac{\rho E}{\pi} $$

**Physical Significance**: The radiance $L$ of a Lambertian surface is directly proportional to the incident irradiance $E$ and the surface reflectance $\rho$. This relationship serves as a fundamental model in computational imaging and lighting design, used to derive the radiance of a target surface from the irradiance of the light source.