Definition
Radiance(symbol: $L$) is the radiant flux emitted, reflected, transmitted, or received by a surface, per unit solid angle per unit projected area in a specified direction.As the fundamental property of extended sources, it characterizes the radiant intensity per unit projected area of a surface element along a given line of sight.
Figure 1:Schematic Diagram of Radiance
Source: https://en.wikipedia.org/wiki/Luminance#/media/File:Etendue.svg

Unit:
The SI unit of radiance is \(\mathrm{W \cdot sr^{-1} \cdot m^{-2}}\) (watts per steradian per square meter).

Mathematical Expression:
Radiance $L$ is mathematically defined by the following formula:

$$L = \frac{dI}{dA \cos\theta} = \frac{d^2 \Phi}{dA \cos\theta \, d\Omega}$$ 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 source can be regarded as a collection of infinitesimal surface elements, whose emitted radiation may vary with direction.
Radiance is used to describe the strength of radiation in a given direction; it is defined as the radiant flux per unit area per unit solid angle. In the imaging process, the radiation received by the human eye or a camera corresponds to the radiance of the scene along each viewing direction. Consequently, an image can be interpreted as a sampling of the radiance distribution.
Thermal imaging is a typical application of radiance, as shown in Figure 1. Differences in color (or brightness) in the image correspond to variations in the spatial distribution of radiance in the infrared band, with brighter regions indicating higher radiance.

Figure 1: Thermal image of a microwave oven cavity.
Source: https://commons.wikimedia.org/wiki/File:Opened_oven_seen_with_thermal_camera.jpg

Relationship Between Radiance and Irradiance for a Lambertian Surface
For an ideal Lambertian surface with reflectance $\rho$:

• Relation between irradiance and radiant exitance:

$$ M = \rho E $$

• Relation between radiant exitance and radiance:

$$ M = \pi L $$

• Combining the above relations gives:

$$ L = \frac{\rho E}{\pi} $$

Physical interpretation
For a Lambertian surface, the radiance $L$ is proportional to the incident irradiance $E$ and to the surface reflectance $\rho$. This relation provides a simple model for estimating surface radiance from illumination conditions and is widely used in imaging and lighting analysis.