# Object Space Resolution

This is Section 2.2 of the Imaging Resource Guide

In order to determine the absolute minimum resolvable spot that can be seen on the object, the ratio of the field of view to the sensor size needs to be calculated. This is also known as the Primary Magnification (PMAG) of the system.

(1)$$\text{PMAG} = \frac{\text{Sensor Size} \left[ \text{mm} \right]}{\text{Field of View} \left[ \text{mm} \right]}$$

The ratio associated with system PMAG allows for the scaling of the imaging space resolution which tells us the resolution of the object.

(2)$$\text{Object Space Resolution} \left[ \tfrac{\text{lp}}{\text{mm}} \right] = \text{Image Space Resolution} \left[ \tfrac{\text{lp}}{\text{mm}} \right] \times \, \text{PMAG}$$

Generally when developing an application, a system’s resolution requirement is not given in lp/mm, but rather in microns (μm) or fractions of an inch. There are two ways to make this conversion:

(3)$$\text{Object Space Resolution} \left[ \large{\unicode[Cambria Math]{x03BC}} \normalsize{\text{m}} \right] = \frac{1000 \tfrac{\large{\unicode[Cambria Math]{x03BC}} \normalsize{\text{m}}}{\text{mm}}}{2 \,\times \, \text{Object Space Resolution} \left[ \tfrac{\text{lp}}{\text{mm}} \right]}$$
(4)$$\text{Object Space Resolution} \left[ \large{\unicode[Cambria Math]{x03BC}} \normalsize{\text{m}}\right] = \frac{\text{Pixel Size} \left[\large{\unicode[Cambria Math]{x03BC}} \normalsize{\text{m}} \right]}{\text{PMAG}_{\text{System}}}$$

While one can quickly jump to the limiting resolution on the object by using the last formula, it is very helpful to determine the imaging space resolution and PMAG to simplify lens selection. It is also important to keep in mind that there are many additional factors involved, and this limitation is often much lower than what can be easily calculated using the equations.

## Resolution and Magnification Calculation Examples using a Sony ICX625 sensor

Known Parameters:
Pixel Size = 3.45μm x 3.45μm
Number of Pixels (H x V) = 2448 x 2050
Desired FOV (Horizontal) = 100mm

Limiting Sensor Resolution:

\begin{align} \text{Image Space Resolution} \left[ \tfrac{\text{lp}}{\text{mm}} \right] & = \frac{1000 \tfrac{ \large{\unicode[Cambria Math]{x03BC}} \normalsize{\text{m}}}{\text{mm}}}{2 \, \times \, \text{Pixel Size} \left[ \large{\unicode[Cambria Math]{x03BC}} \normalsize{\text{m}} \right]} \\ \text{Image Space Resolution} \left[ \tfrac{\text{lp}}{\text{mm}} \right] & = \frac{1000 \tfrac{\text{lp}}{\text{mm}}}{2 \, \times \, 3.45 \left[ \large{\unicode[Cambria Math]{x03BC}} \normalsize{\text{m}} \right]} \approx \boldsymbol{145} \large{\unicode[arial]{x03BC}}  \textbf{m} \end{align}

Sensor Dimensions:

\begin{align} \text{Horizontal Sensor Dimension} \left[ \text{mm} \right] & = \frac{\left( 3.45 \large{\unicode[Cambria Math]{x03BC}} \normalsize{\text{m}} \right) \left( 2448 \right)}{1000 \tfrac{\large{\unicode[Cambria Math]{x03BC}} \normalsize{\text{m}}}{\text{mm}}} & = \boldsymbol{8.45} \large{\unicode[arial]{x03BC}}  \textbf{m} \\ \text{Vertical Sensor Dimension} \left[ \text{mm} \right] & = \frac{\left( 3.45 \large{\unicode[Cambria Math]{x03BC}} \normalsize{\text{m}} \right) \left( 2050 \right)}{1000 \tfrac{\large{\unicode[Cambria Math]{x03BC}} \normalsize{\text{m}}}{\text{mm}}} & = \boldsymbol{7.07}  \large{\unicode[arial]{x03BC}} \textbf{m} \end{align}

PMAG:

$$\text{PMAG} = \frac{8.45 \text{mm}}{100 \text{mm}} = \boldsymbol{0.0845} \textbf{X}$$

Resolution:

$$\text{Object Space Resolution} = 145 \tfrac{\text{lp}}{\text{mm}} \times 0.0845 = 12.25 \tfrac{\text{lp}}{\text{mm}} \approx \boldsymbol{41 \large{\unicode[arial]{x03BC}} } \textbf{m}$$
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