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Relationships  between

 

Crop  factor,  field  of  view,  photographic  reach,

 

image size,  pixel density,  and  pixel  size

 

 

These notes are a supplement to  this article  and are designed to give you further information about the methods used to compare the cameras listed in  this index. You should read these notes in conjunction with the detailed comparison of a "theoretical" 24 megapixel APS-C camera with a "theoretical" 36 megapixel full frame camera as demonstrated  here.

 

These notes deal  only  with arithmetical analysis, because the assessment of the  quality  of cropped images and the  quality  of cameras, are separate issues

 

 

Relevant specifications of the cameras

 

To illustrate the relationships between crop factor, field of view, photographic reach, image size, pixel density, and pixel size, we shall compare the relevant specifications of two “theoretical” cameras which have the following specifications:

 

24 megapixel (mp) APS-C camera: Image dimensions: 6000 pixels x 4000 pixels; sensor size: 23.4mm x 15.6mm

 

36 mp full frame (FF) camera: Image dimensions: 7353 pixels x 4902 pixels; sensor size: 36.0mm x 24.0mm

 

 

The objectives of this camera comparison are to:

 

1.    Calculate the crop factor;

 

2.    Establish the field of view of a 24mp APS-C camera when it has a 300mm lens attached;

 

3.    Establish the field of view of a 36mp FF camera when it has a 300mm lens attached;

 

4.    Compare the photographic reach of the two cameras based on the same fields of view;

 

5.    Compare the photographic reach of the two cameras based on the same image widths;

 

6.    Compare the pixel density of the two cameras;

 

7.    Compare the pixel pitch of the two cameras; and

 

8.    Discuss the following claim: "A 24mp APS-C camera provides 54% more reach than a 36mp FF camera."

 

 

 

Objective 1: Calculate the crop factor

 

The crop factor represents the relationship between the sensor width of the FF camera and the sensor width of the APS-C camera. Therefore, in this example, the crop factor is approx. 1.54x (36mm / 23.4mm). Click  here  to see a full discussion about the meaning and calculation of the crop factor.

 

 

Objective 2: Establish the field of view of a 24mp APS-C camera when it has a 300mm lens attached  

 

The field of view of the above 24mp APS-C camera when it has a 300mm lens attached represents 300mm x the crop factor of 1.54, which gives a field of view of 462mm (300mm x 36mm /23.4mm).

 

 

Objective 3: Establish the field of view of a 36mp FF camera when it has a 300mm lens attached

 

The field of view of a 36mp FF camera with a 300mm lens attached is  300mm.

 

 

Objective 4: Compare the photographic reach of the two cameras based on the same fields of view

 

It is important to first read  this article, which deals with the definition of "photographic reach".

 

In this example, the sensor width of APS-C is 23.4mm and the sensor width of FF is 36.0mm. Therefore, to obtain the same fields of view for images from both cameras, the uncropped image width of FF (7353 pixels) is multiplied by 23.4 and dividided by 36.0.

 

This results in the images having the same fields of view but different image widths as shown below:

 

Uncropped image width of APS-C = 6000 pixels

 

Cropped image width of FF

to same field of view as APS-C      = 4780 pixels  (7353 x 23.4 / 36.0)

 

Relationship: The photographic reach of APS-C is approximately 25.5% greater than that of FF (6000 pixels / 4780 pixels).

 

Note: With the same field of view, APS-C has an image width of 6000 pixels compared with 4780 pixels for FF. For further information about the mathematical "reach" formulas used on this page, click  here  to go to an article titled  Advantages and disadvantages of cropping images to gain extra "reach".

 

 

Objective 5: Compare the photographic reach of the two cameras based on the same image widths

 

As a check on the “photographic reach advantage” obtained using Objective 4, the image width of the FF image (7353 pixels) can be cropped to the same image width as the APS-C image (6000 pixels). It is then necessary to compare the field of view (in millimetres) of the APS-C image with the  changed field of view  (in millimetres) of the FF image when it is cropped to the same image width as the APS-C image.

 

This results in the images having the same image width but different fields of view, as follows:

 

Field of view of APS-C =  focal length of lens  x  sensor width of FF  /  sensor width of APS-C  = 462mm  (300mm x 36.0mm / 23.4mm) OR 300mm x the crop factor of 1.54 gives 462mm.

 

Changed field of view of a FF image when it is cropped to the same image width as an APS-C image

 

= uncropped image width of FF  / cropped image width of FF  x  focal length of lens  = 368mm  (7353 / 6000 x 300mm)

 

Relationship: The photographic reach of APS-C is 25.5% greater than that of FF (462mm / 368mm). This agrees with the photographic reach of 25.5% that was obtained when applying Objective 4.

 

For further information about the mathematical "reach" formulas used on this page, click  here  to go to an article titled  Advantages and disadvantages of cropping images to gain extra "reach".

 

 

Objective 6: Compare the pixel density of the two cameras

 

Approximate pixel density  (in pixels per linear centimetre)

 

Pixel density in pixels per linear centimetre = image width in pixels  divided by  width of sensor in centimetres

 

APS-C   =    2564.1   (6000 / 2.34)

FF          =    2042.5   (7353 / 3.60)

 

Pixel Density Advantage:  APS-C is 25.5% greater than FF (2564.1 / 2042.5)

 

Note: The pixel density advantage of APS-C is exactly equal to the reach advantage of 25.5% that was calculated under Objectives 4 and 5.

 

 

Objective 7: Compare the pixel pitch of the two cameras

 

Refer to the reservations  here  about calculating the "true" width and area of an individual pixel.

 

Pixel pitch in microns  = width of sensor in millimetres  divided  by  image width in pixels  multiplied by  1000

 

APS-C   =   3.900    (23.4 / 6000  x 1000)

FF          =   4.896    (36.0 / 7353 x 1000)

 

Relationship: The pixel pitch of FF is 25.5% greater than that of APS-C (4.896 / 3.900).

 

Note: The relationship between the pixel pitch of APS-C and FF is exactly equal to the reach advantage of 25.5% calculated under Objectives 4, 5, and 6.

 

 

Objective 8: Discuss the following claim: "A 24mp APS-C camera provides 54% more reach than a 36mp FF camera"

 

It's true that an APS-C camera has a crop factor of about 1.54x if it has the specifications set out above under "Relevant Specifications of the Cameras". Therefore, a 300mm lens on this APS-C camera provides a field of view of 462mm as shown in "Objective 5"  (300mm x the crop factor of 1.54x). This can be a considerable advantage to photographers who, for example, photograph wildlife and who want to crop images to place greater emphasis on a distant subject.

 

It's also true that a 300mm lens on a full frame camera provides a field of view of 300mm. However, it's incorrect in this example to say that,  because of  the 1.54x crop factor, the APS-C camera provides  54%  more reach than a 36mp FF camera (that is, 462mm vs 300mm when a 300mm lens is attached to both cameras).

 

Before an accurate reach comparison can be made, the FF image needs to be cropped so that it provides the same field of view as the APS-C image. The correct comparison is shown in  "Objective 4"  where the FF image is cropped to the same field of view as the APS-C image. The uncropped image width of APS-C (6000 pixels) then gives 25.5% greater reach than the cropped FF image (4780 pixels) when both images provide the same field of view.

 

The only time when the reach advantage of an APS-C camera is equal to the crop factor is when the image widths of both the APS-C camera and the FF camera are the same. This is the case, for example, when a 24mp APS-C camera is compared with a 24mp FF camera, as demonstrated  here. With this example, the reach advantage of the 24mp APS-C camera is about 54%, which is the same as the crop factor of 1.54x.

 

However, the reach advantage of an APS-C camera is always equal to the pixel density advantage of that camera. If the pixel density of an APS-C camera and a FF camera are identical, there is no reach advantage. Therefore, the best method of determining the reach advantage is to first compare the pixel densities of the two cameras under review.

 

When determining the reach advantage (or telephoto advantage), it is important to check that the answers obtained under Objectives 4, 5, 6, and 7 are the same.

 

 

SUMMARY

 

*    When a full sized image from FF is cropped so that it has the same  field of view  as a full sized image from APS-C, if the image width (expressed in pixels) of APS-C is greater than the image width of the cropped image from FF (expressed in pixels), this is because the pixel density of APS-C is greater than that of FF. This assumes that the same lens and focal length are used on both FF and APS-C.

 

*    When a full sized image from FF is cropped so that it has the same  image width  as a full sized image from APS-C, if the field of view (expressed in millimetres) of APS-C is greater than the field of view of the cropped image from FF, this is because the pixel density of APS-C is greater than that of FF. This assumes that the same lens and focal length are used on both FF and APS-C.

 

*    The pixel density advantage in favour of APS-C is exactly the same as the photographic reach advantage calculated under Objectives 4, 5, and 7 above.

 

*    The relationship between the pixel pitch of APS-C and FF is exactly the same as the photographic reach advantage calculated under Objectives 4, 5 and 6 above.

 

*    In most cases, the crop factor does not provide an accurate measure of the reach advantage of an APS-C camera when it is compared with a FF camera. However, the reach advantage of an APS-C camera is always equal to the pixel density advantage of that camera. If the pixel density of an APS-C camera and a FF camera are identical, there is no reach advantage. Therefore, it is good practice to first compare the pixel density of the two cameras under review when you want to determine the reach advantage.

 

 

Click  here  to see an index of camera comparisons showing the mathematical relationships between crop factor, field of view, photographic reach, image size, pixel density and pixel size.

 

Click  here  to go to an article titled "Advantages and disadvantages of cropping images instead of using lenses with longer focal lengths".

 

Click  here  to go to the full explanatory article about the crop factor and “telephoto advantage” of an APS-C camera.

 

Note that Appendix 2 of the above article includes the following sections:

 

Calculation of pixel pitch

 

Calculation of the area of one pixel

 

Why is pixel size important - increasing pixel count increases noise?  (Click  here  to read the full article)

 

Pixel level vs image level in digital photography (also refers to the Studio Test Scene  published by Digital Photography Review)

 

Case study: High ISO low light images of the Sony A77 compared with those of the Sony A55

 

Downscaling and upscaling images for comparative purposes  (can apples be compared with oranges?)

 

Practical examples of the application of the "pixel density advantage" template

 

Click  here  to see a comparison of two “theoretical” cameras, which permits the reconciliations between the percentages shown for pixel density, and pixel area, to be exactly equal.

 

Click  here  to go to the home page of Rob’s Photography New Zealand.

 

The notes on this page are designed to be read in conjunction with the full explanatory article that you can see  here. Note that the analysis on this page does not include a discussion of the various complex issues that can arise in practice when estimating pixel density and the the pixel pitch or area of individual pixels. It is recommended that you study a detailed technical article if you would like to become familiar with these issues. For example, you may find this  DPR forum discussion  about pixel density and pixel size to be helpful. Therefore, the calculations set out below are presented for the purpose of calculating only a very approximate measurement of pixel density, pixel pitch, and the area of one pixel, which can be used for comparing the approximate mathematical relationships between image size and the pixel density and pixel size of different cameras.

 

 

 

 

 

 

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