Radiance is defined as the radiant flux emitted by a radiation source in a given direction per unit projected area and per unit solid angle, denoted as \(L\).
Unit: \(\mathrm{W \cdot sr^{-1} \cdot m^{-2}}\) (watts per steradian per square meter).
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.

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.