The pinhole model

The first very simple way to acquire images is with the pinhole model. Here, part of the light coming from the object passes through a small aperture $O$ and is projected onto the focal plane. We do this so that each point of the object is represented by a ray. Otherwise, the image would not be formed.

The distance $f$ is called the focal length. Let’s suppose we have the following model to describe the pinhole:

Since the aperture $O$ is finite, the rays produce blur. This effect can be modelled by

$$ g_z \ast S $$

where $S$ would be the “real” image and $g_z$ is the indicator function of a disk of radius $\frac{(z + f) D}{z}$.

Note
An indicator function on a set $E$ is zero anywhere except in $E$, where it is always $1$.

Using lenses

For a sharper image, we can add a lens. It behaves as shown in the following image.

To remember:

  1. Horizontal rays converge at the focal point;
  2. Rays passing through the origin $O$ do not change;
  3. Rays passing through a focal point exit horizontally.

The focal distance $f$ can be calculated using Descartes’ Relation.

$$ \frac{1}{d} + \frac{1}{d^\prime} = \frac{1}{f} $$

Depth of field

Depth of field is the distance separating the sharp object closest to the camera from the farthest sharp object.

$$ \Delta \approx \frac{2 \varepsilon p^2}{D f} $$

where $p$ is the object distance.

Info
The aperture number $N$ is defined by $N = \frac{f}{D}$. It usually appears in geometric progression (at $q = \sqrt{2}$) and is represented by $f / \cdot$.
  • e.g. $f / 4$ means $N = 4$.

When $N$ increases (at constant $f$):

  1. $\Delta$ increases;
  2. Diffraction defects increase;
  3. Vignetting defects (darker image near bords) increase;

Sampling

Now the aim is to transform the photons in the image into something electrical that can be digitized.

$$ \left( f \colon \mathbb{R}^2 \mapsto \mathbb{R} \right) \rightarrow (\{f(k)\}_{k \in \Omega}). $$

Photon counting can be formalized mathematically by convolution with $g_{text{capt}}$, which is the sensor’s indicator function.

$$ I_{i, j} = (I \ast g_{\text{capt}})(i, j) $$

The following image shows an example of a sensor grid.

The sampling of a function $f$ at a point $\mathbf{x}$ is modeled by

$$ f \cdot \delta_{\mathbf{x}} $$

So, to model image acquisition, we’ll use Dirac’s comb, $\Pi_\Gamma$, which is defined shortly.

$$ \Pi_\Gamma = \sum_{\gamma \in \Gamma} \delta_\gamma $$

where $\Gamma$ is the set of positions where the detectors are placed (e.g. $\Gamma \subset \mathbb{N}^2$).

We can also place the detectors in a hexagonal array. This can sometimes be advantageous.