How to calculate the resolving power of a lens

In a camera system, the image sensor receives incident light (photons) – either focused through a lens or any other optics. Hence, lens selection plays a major role in determining image quality, FoV (Field of View), DoF (Depth of Field), etc.

In today’s blog, let’s look at how to determine the resolving power of a lens, which is one of the most critical parameters to consider while choosing a lens for your application.

What is resolving power of a lens, and why is it important?

The resolving power of a lens is its ability to distinguish between two lines or points in an object or scene.

Let’s say you have to decode an object like a small barcode and that too at a longer distance. You would have to differentiate the bars by allowing a certain amount of space between them. This minimum distance depends on the resolution of the camera. However, when you use a sensor with the desired resolution, it is also important to choose a lens that can help capture the level of detail expected from that resolution. This is where knowing the resolving capacity of the lens becomes important.

For instance, if you are using See3CAM_CU135 – a 13 MP high-resolution USB camera from e-con Systems™ – to read barcodes, you need to make sure that the lens you pick can meet the maximum desired resolution.

 

Theoretical calculation of the resolving power of a lens

The resolving power of a lens is measured in line pair per millimeter or lp/mm. It is a measurement of spatial resolution used to calculate how small a detail in an image can be resolved by a lens. The unit expresses the number of line pairs you can fit within one millimeter.

A line pair is a pair of black and white lines next to each other with the same width and orientation. The ability to differentiate two bars as separate entities in a specific resolution would be based on the contrast level. It means that calculating resolution in terms of lp/mm is extremely useful when comparing lenses. This can act as one of the criteria while choosing the best-fit lens for a given sensor and application.

Resolving power is calculated as object space resolution. And object space resolution is derived from what is called the image space resolution. We will now look at both the terms and learn now they are calculated.

 

Image space resolution

Image Space Resolution is the resolution in the image plane in consideration of the sensor pixel size. Generally, two pixels or one line pair is the highest frequency which can be resolved by a sensor – using the Nyquist frequency. Therefore, image space resolution is theoretically calculated as:

Image space resolution in lp/mm = 1000 / (2 * pixel size in μm)

Image space resolution is inversely proportional to the sensor’s pixel size. This means the smaller the pixel size, higher will be the image space resolution value.

 

Object space resolution

As mentioned above, Object Space Resolution denotes the resolving power of a lens. It defines the size elements of the object that can be resolved. It is calculated as:

Object space resolution in lp/mm = Image space resolution * PMAG.

Where PMAG = Sensor size / Field of view (FoV).

 

Calculating the resolving power of a given lens practically

Calculating the resolving power value of a camera lens practically involves taking into consideration the real aspect ratio of the camera.

For instance, if you want to calculate the resolving power of See3CAM_CU135 – 4K USB camera – with the default product lens, you should capture the image of the resolution chart with the desired aspect ratio at the prescribed working distance.

To give you an example, if the aspect ratio considered for See3CAM_CU135 – 13MP USB camera – is 4:3, the image of the resolution chart has to be captured at that aspect ratio to finally arrive at the resolving power. Below is a sample image of the resolution chart taken at an aspect ratio of 4:3.

 

To understand how the theoretical and practical methods of calculating resolving power differ, we will look at how these are done for e-con Systems’ See3CAM_CU135 in the next section.

 

Calculating the resolving power of See3CAM_CU135 – a 4K USB camera

The resolving power of a lens can be calculated manually as well as in automated manner. The manual method is called the human eye perception method, and the automated technique is called the IMA Test method. We will look at both these in detail in this section.

 

The human eye perception method

To calculate the resolving power or object space resolution value using the human eye perception method, you need to first find the LW/PH (line width per picture height) value. To do this, you must observe the line pair highlighted in red (horizontal or vertical) in the resolution chart given in the previous section (Figure 1).

As shown in the chart, you can count the number until the line pair degrades. It is where you can’t distinguish the black and white lines (due to merge, grey colors will appear). Typically, the values of this line will be mentioned in the resolution chart (100 x per picture height).

The reason why this is called the human eye perception method is that the number of lines counted may differ depending on the ability of the observer to distinguish between two consecutive lines.

Let us now see, how the calculation is done. Following are the data points with respect to the sensor:

 

• Pixel Size: 1.1 μm
• Active area Height in pixels: 4208
• Horizontal FOV: 661.8 mm

 

Sensor Height	= Sensor size * active area height
	= (1.1μm * 4208)/1000
	= 4.6288mm.

 

Theoretical calculation of See3CAM_CU135’s resolving power

Here is how the theoretical calculation goes.

 

Image space resolution	= 1000 / (2 * pixel size)
	= 1000/ (2 * 1.1)
	= 454.54 lp/ mm

 

PMAG	= Sensor Height / HFOV
	= 4.6288 / 661.8
	= 0.007.

 

Object Space resolution	= Image Space Resolution * PMAG
	= 454.54 * 0.007
	= 3.18 lp/mm

 

Practical calculation of See3CAM_CU135’s resolving power

Here’s the subjective resolution observed from ISO resolution chart, considering that you can resolve the line at 23.

 

Image space resolution in line width per picture height	= 2300
Image space resolution in line pair per picture height	= Image space resolution in line width per  picture height / 2
	= 1150

 

Image space resolution in lp/mm	= Image space res olution in line pair per picture height / sensor height
	= 1150 / 4.6288
	= 248 lp/mm

 

Object space resolution	= Image space resolution * PMAG
	= 248 * 0.007
	= 1.73 lp/mm

Given below is the comparison of image space resolution and object space resolution values obtained using the theoretical and practical methods.

 

	Theoretical	Practical
Image space resolution	454.54 lp/ mm	248 lp/mm
Object space resolution	3.18 lp/mm	1.73 lp/mm

 

IMA Test Method:

In the IMA test method – which is the automated method – the LW/PH value is calculated by considering the MTF30 value. This is an objective measurement of sharpness, which is better than the subjective analysis. The data in the IMA chart is in the units LW/PH, and it can be converted to lp/mm as shown in the above calculation.

Given below is the IMA chart for See3CAM_130 – 13MP autofocus USB camera from e-con Systems.

 

Using the above data, you can validate how much lp/mm a particular lens would resolve at a specific working distance. Based on this, you can select the right lens for your application. It is pertinent to note that a change in FoV will affect the magnification factor, which will end up affecting the practical image space resolution.

 

Get it right the first time with e-con Systems

If you are looking for help in selecting a camera solution with the best-fit lens for your application – no matter the industry, please write to us at camerasolutions@e-consystems.com. You can also visit our Camera Selector to get a full view of e-con Systems’ camera portfolio.