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Angles and the Camera Equation

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 and position in the Field of View, which will be assumed in the formulas to make them readable, see Appendix I for a more formally correct version of Equation (1).

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A Simple Model for Sharpness in Digital Cameras – I

The next few posts will describe a linear spatial resolution model that can help a photographer better understand the main variables involved in evaluating the ‘sharpness’ of photographic equipment and related captures.   I will show numerically that the combined spectral frequency response (MTF) of a perfect AAless monochrome digital camera and lens in two dimensions can be described as the magnitude of the normalized product of the Fourier Transform (FT) of the lens Point Spread Function by the FT of the pixel footprint (aperture), convolved with the FT of a rectangular grid of Dirac delta functions centered at each  pixel:

    \[ MTF_{2D} = \left|\widehat{ PSF_{lens} }\cdot \widehat{PIX_{ap} }\right|_{pu}\ast\ast\: \delta\widehat{\delta_{pitch}} \]

With a few simplifying assumptions we will see that the effect of the lens and sensor on the spatial resolution of the continuous image on the sensing plane can be broken down into these simple components.  The overall ‘sharpness’ of the captured digital image can then be estimated by combining the ‘sharpness’ of each of them.

The stage will be set in this first installment with a little background and perfect components.  Later additional detail will be provided to take into account pixel aperture and Anti-Aliasing filters.  Then we will look at simple aberrations.  Next we will learn how to measure MTF curves for our equipment, and look at numerical methods to model PSFs and MTFs from the wavefront at the aperture. Continue reading A Simple Model for Sharpness in Digital Cameras – I

Linearity in the Frequency Domain

For the purposes of ‘sharpness’ spatial resolution measurement in photography  cameras can be considered shift-invariant, linear systems when capturing scene detail of random size and direction such as one often finds in landscapes.

Shift invariant means that the imaging system should respond exactly the same way no matter where light from the scene falls on the sensing medium .  We know that in a strict sense this is not true because for instance pixels tend to have squarish active areas so their response cannot be isotropic by definition.  However when using the slanted edge method of linear spatial resolution measurement  we can effectively make it shift invariant by careful preparation of the testing setup.  For example the edges should be slanted no more than this and no less than that. Continue reading Linearity in the Frequency Domain

Information Transfer – The ISO Invariant Case

We know that the best Information Quality possible collected from the scene by a digital camera is available right at the output of the sensor and it will only be degraded from there.  This article will discuss what happens to this information as it is transferred through the imaging system and stored in the raw data.  It will use the simple language outlined in the last post to explain how and why the strategy for Capturing the best Information or Image Quality (IQ) possible from the scene in the raw data involves only two simple steps:

1) Maximizing the collected Signal given artistic and technical constraints; and
2) Choosing what part of the Signal to store in the raw data and what part to leave behind.

The second step is only necessary  if your camera is incapable of storing the entire Signal at once (that is it is not ISO invariant) and will be discussed in a future article.  In this post we will assume an ISOless imaging system.

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How to Measure the SNR Performance of Your Digital Camera

Determining the Signal to Noise Ratio (SNR) curves of your digital camera at various ISOs and extracting from them the underlying IQ metrics of its sensor can help answer a number of questions useful to photography.  For instance whether/when to raise ISO;  what its dynamic range is;  how noisy its output could be in various conditions; or how well it is likely to perform compared to other Digital Still Cameras.  As it turns out obtaining the relative data is a little  time consuming but not that hard.  All you need is your camera, a suitable target, a neutral density filter, dcraw or libraw or similar software to access the linear raw data – and a spreadsheet.

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