Tag Archives: etendue

Exposure, imaging system, irradiance, luminance, radiance, Radiometry, reflectance, Uncategorized

Angles and the Camera Equation

February 26, 2023 Jack Leave a comment

Imagine a bucolic scene on a clear sunny day at the equator, sand warmed by the tropical sun with a typical irradiance ( $E$ ) of about 1000 watts per square meter. As discussed earlier we could express this quantity as illuminance in lumens per square meter ( $lx$ ) – or as a certain number of photons per second ( $\Phi$ ) over an area of interest ( $\mathcal{A}$ ).

(1) $\begin{equation*} E = \frac{\Phi}{\mathcal{A}} \; (W, lm, photons/s) / m^2 \end{equation*}$

How many $photons/s$ per unit area can we expect on the camera’s image plane (irradiance $E_i$ )?

Figure 1. Irradiation transfer from scene to sensor.

In answering this question we will discover the Camera Equation as a function of opening angles – and set the stage for the next article on lens pupils. By the way, all quantities in this article depend on wavelength, which will be assumed in the formulas to make them more readable.

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Color, Exposure, Photometry, Radiometry, Sensors

Connecting Photographic Raw Data to Tristimulus Color Science

December 9, 2021 Jack Leave a comment

Absolute Raw Data

In the previous article we determined that the three $r_{_L}g_{_L}b_{_L}$ values recorded in the raw data in the center of the image plane in units of Data Numbers per pixel – by a digital camera and lens as a function of absolute spectral radiance $L(\lambda)$ at the lens – can be estimated as follows:

(1) $\begin{equation*} r_{_L}g_{_L}b_{_L} =\frac{\pi p^2 t}{4N^2} \int\limits_{380}^{780}L(\lambda) \odot SSF_{rgb}(\lambda) d\lambda \end{equation*}$

with subscript $_L$ indicating absolute-referred units and $SSF_{rgb}$ the three system Spectral Sensitivity Functions. In this series of articles $\odot$ is wavelength by wavelength multiplication (what happens to the spectrum of light as it progresses through the imaging system) and the integral just means the area under each of the three resulting curves (integration is what the pixels do during exposure). Together they represent an inner or dot product. All variables in front of the integral were previously described and can be considered constant for a given photographic setup. Continue reading Connecting Photographic Raw Data to Tristimulus Color Science →

Color, Exposure, Photometry, Radiometry, Sensors

The Physical Units of Raw Data

November 13, 2021 Jack Leave a comment

In the previous article we (I) learned that the Spectral Sensitivity Functions of a given digital camera and lens are the result of the interaction of light from the scene with all of the spectrally varied components that make up the imaging system: mainly the lens, ultraviolet/infrared hot mirror, Color Filter Array and other filters, finally the photoelectric layer of the sensor, which is normally silicon in consumer kit.

Figure 1. The journey of light from source to sensor. Cone Ω will play a starring role in the narrative that follows.

In this one we will put the process on a more formal theoretical footing, setting the stage for the next few on the role of white balance.

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Exposure, Radiometry

Photons Emitted by Light Source

September 16, 2014 Jack 6 Comments

How many photons are emitted by a light source? To answer this question we need to evaluate the following simple formula at every wavelength in the spectral range of interest and add the values up:

(1) $\begin{equation*} \frac{\text{Power of Light in }W/m^2}{\text{Energy of Average Photon in }J/photon} \end{equation*}$

The Power of Light emitted in $W/m^2$ is called Spectral Exitance, with the symbol $M_e(\lambda)$ when referred to units of energy. The energy of one photon at a given wavelength is

(2) $\begin{equation*} e_{ph}(\lambda) = \frac{hc}{\lambda}\text{ joules/photon} \end{equation*}$

with $\lambda$ the wavelength of light in meters and $h$ and $c$ Planck’s constant and the speed of light in the chosen medium respectively. Since Watts are joules per second the units of (1) are therefore $photons/m^2/s$ . Writing it more formally:

(3) $\begin{equation*} M_{ph} = \int\limits_{\lambda_1}^{\lambda_2} \frac{M_e(\lambda)\cdot \lambda \cdot d\lambda}{hc} \text{ $\frac{photons}{m^2\cdot s}$} \end{equation*}$

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