Quote of the Day

I went through the whole gamut of organization, which was a vain effort so to say, because in the end, I ended with electronics, which have no life at all. But I don't think it vain because to understand life, one must understand electrons too.

— Albert Szent-Györgyi, Nobel Prize winner in Medicine. His research focused on the role of electrons in biological processes.

## Introduction

I use a Samsung S5 phone for my daily smart phone work. Recently, I have even started to use its camera/video system for some rough measurement work (examples here and here). This work has made me a bit curious about the how the camera subsystem was designed. In this post, I will document a very rough experiment that I performed over lunch today in which I measured the S5 camera's Field of View (FOV). I will also compute the field of view using some information that Samsung provides. The agreement between the results seem reasonable considering my crude approach.

I am always surprised at the quality of the images that I can get from this phone, however, there are serious optical compromises required to put a camera into this phone's tiny form factor. While I prefer to use my DSLR, I rarely have it with me when I need to take a quick photo – most of my photos are of whiteboard discussions. In these situations, any photo is better than trying to copy the board quickly.

## Background

### Definitions

As usual, I will use the Wikipedia as my source for the definitions of terms.

- Field of View (symbol
*FOV*) - In photography,
*FOV*describes the angular extent of a given scene that is imaged by a camera. It is used interchangeably with the term Angle of View (*AOV*). - Crop Factor (symbol
*CF*) - In digital photography, a crop factor is related to the ratio of the dimensions of a camera's imaging area compared to a reference format; most often, this term is applied to digital cameras, relative to 35 mm film format as a reference.
- In the case of digital cameras, the imaging device would be a digital sensor. The crop factor is also commonly referred to as the Focal Length Multiplier ("FLM") since multiplying a len's focal length by the crop factor or FLM gives the focal length of a lens that would yield the same field of view if used on the reference format.
- Focal Length
- The focal length of an optical system is a measure of how strongly the system converges or diverges light. For an optical system in air, it is the distance over which initially collimated rays are brought to a focus. I will not be using the phone camera's focal length because Samsung has not published that number. Instead, I will use the equivalent 35 mm focal length, which they do publish.
- Equivalent Focal Length (Symbol
*f*) - The focal length of a phone camera is usually stated in terms of an equivalent 35 mm focal length. The equivalent focal length is the focal length in a 35 mm SLR that a would provide the same FOV as the camera's lens/sensor system.

### S5 Technical Characteristics

In Table 1, I gathered the following S5 technical information from this web page. I will use this information to compute the S5's FOV (horizontal and vertical) based on the lens and image sensor dimensions.

Parameter | Value |
---|---|

Effective Focal Length | 31 mm |

Image Sensor Horizontal Dimension | 5.08 mm |

Image Sensor Vertical Dimension | 3.81 mm |

Note that the image sensor dimensions have a ratio (horizontal/vertical) of 16/9 ≈ 1.78. This is the aspect ratio seen with HDTV and is different than the 3/2 ratio used in standard 35 mm film format.

### 35 mm Film Reference

Focal length is a characteristic of the camera's lens. The S5's focal length of 31 mm is always described as "effective", which means that the lens has a focal length provides an image similar to that of a 31 mm lens on a 35 mm analog camera. As a point of reference, 35 mm analog cameras often had a 50 mm focal length lens because this gave the camera an FOV similar to the human eye.

I find the use of effective focal length confusing – a focal length of 31 mm ≈ 1.25 inches is much larger than the S5 is thick. I was unable to find a drawing of the optical path for an S5, but you can get a feel for the complexity present in a cell phone camera by looking at a typical reference design I found while searching on the web (Figure 3). This lens assembly contains one glass and two plastic lenses.

35 mm film (Figure 4) has a much larger image gathering area than a phone camera's image sensor. For example, the horizontal dimension of 35 mm film is 36 mm, but the corresponding dimension of the S5's image sensor is 5.08 mm. This means that the S5 will only capture the central portion of the image (Figure 5). The overall aspect ratio of 35 mm film is 3/2 (i.e. 36 mm/ 24 mm). Modern aspect ratios (e.g. HDTV, DSLRs) are moving to 4/3 or even larger – my Sony α55 DSLR is configurable for either 4/3 and 16/9.

Unfortunately, Figure 5 does not show an image sensor the same size as the S5, which is 1/5" wide (horizontal). This means that the S5 sensor area would comfortably fit within the 1/3" box. This also means that the image captured by the S5 is only a tiny portion of the image that a 35 mm analog camera would capture.

Many people choose to view the cropping that occurs as a magnification because the image does look magnified. You can view what is happening any way you want. But you need to remember that all of your sensor's pixels are in that small box in the center of the image – the rest of the image area is unused.

This becomes important in DSLR cameras when you use a lens designed for an 35 mm analog system with an image sensor with a smaller capture area than 35 mm film.

### FOV Formula

The Wikipedia give Equation 1 as the formula for the field of view. The discussion there is excellent, and I will elaborate no further here on this formula.

Eq. 1 |

where

- FOV is the field of view along a given direction (e.g. horizontal or vertical)
*d*is the sensor length along a specific direction (e.g. horizontal or vertical)*f*is the focal length of the lens.

## Analysis

I will process the same data using two different methods and see how the results vary.

### Theoretical FOV Calculation

Figure 6 shows my application of Equation 1 to compute the horizontal and vertical *FOVs* of the Samsung F5.

### My Measurement of the S5's FOV

#### Angle Measurements

It is lunch time and I took a couple of minutes (literally) to take photographs from known distance to a letter-sized sheet of paper at known distances that I taped to the wall of my cube. I then used a generic graphic program (PicPick) to measure the dimensions (in pixels) of the letter-sized sheet of paper as seen in the photographs. Figure 7 shows two photograph examples.

Figure 7(a): Sheet of Letter-Sized Paper at 24 inches. | Figure 7(b): Sheet of Letter Sized Paper at 48 inches. |

Note that my measurements might not be very accurate because my camera hold at the various distances was only approximate and I was not holding the camera's line-of-sight perfectly perpendicular to the paper.

My raw measurements are documented in the Mathcad screenshots included below.

#### Average of FOV Calculations

This approach is related to that used in this blog post. Equation 2 shows a modified version of Equation 1 that is a bit more useful for this type of testing.

Eq. 2 |

where

*d*is the image size measured in pixels._{ImagePixel}*d*is the object size (e.g. sheet of paper) in pixels._{ObjectPixel}*d*is the object size in standard linear units (e.g. inches, mm)._{Object}

Figure 8 shows how to derive Equation 2.

Figure 9 shows my application of Equation 2 to multiple data inputs and averaging the results.

#### Least Squares to FOV Curve

A related approach is to assume that the object range is long enough that we can ignore triangles and use spherical measurements (i.e. *FOV = d _{Object}/r_{Object}*). This approach is very similar that used in this blog post.

Equation 4 shows the how the ratio between the paper and image size varies with *1/r*_{Object}.

Eq. 3 |

where

*k*is the ratio of the object size to the image size.

The data of Figure 10 will plot linearly versus 1/*r*, which allows us to generate a least-square linear fit – I am specifically interested in the slope m. The slope of the line, . Since we know the size of the sheet of paper, we can compute the *FOV*. The results obtained are similar to the previous two methods, at least considering how crudely I took the measurements.

## Conclusion

All three methods produced good agreement for the vertical FOV (37°). There was a slight discrepancy between the horizontal FOV values (60° versus 67°). I do not consider this discrepancy significant considering how crudely I took the measurement, especially in the horizontal dimension (i.e. my camera position was not exactly perpendicular to the sheet of paper).

I am amazed at how well these cell phone camera's perform, particularly when you consider how inexpensive they must be. I plan on buying an Arduino shield camera (example) with cameras to do some experimenting on my own.

Interesting stuff, but not entirely coorect. Film / dslrs have an aspect ratio of 3/2 (30mm*24mm) NOT 4/3 which is the format of old TVs and computer screens.

Thank you for bringing this to my attention. I am an old video engineer and have the 4/3 ratio stuck in my brain – even when staring at a 36 mm/ 24 mm image on 35 mm film. I have corrected the post.

Again, thank you.

mark

This is the correct attitude to a correction.

Bravo.

I'm trying to figure out how to calculate the field of view and distance from a jpg of a panoramic landscape, with an image size of 18960px x 1328px, taken using the panoramic mode on the Samsung Galaxy S5's camera. I'm struggling a little with the maths. How do you figure FOV and distance when the picture is of a landscape? Many thanks for your help.