﻿ Crop Factor Full frame cameras vs APS-C cameras. An analysis of the crop factor and "telephoto advantage" for images taken with an APS-C camera.
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"Full  Frame"  Cameras  vs  "APS-C"  Cameras

Analysis  of  the  Crop  Factor  and

“Telephoto  Advantage”  of  an  APS-C  Camera

It is often stated that an APS-C camera has an advantage over a full frame camera (FF) when telephoto lenses are used. The reasons why this advantage can occur, and the extent to which it can it occur, are discussed in this article. Note that the abbreviation "APS-C", means "Advanced Photo System type-C" (as explained in "Wikipedia").

This is an important topic, because if you are mainly interested, for example, in wildlife or sports photography, you will probably be using telephoto lenses to capture many of your images. In these circumstances, you may find some difficulty in deciding whether to purchase an APS-C camera or a FF camera. Therefore, when making this decision, it will be helpful if you compare the features of several APS-C and FF cameras, taking particular note, for example, of the sensor size and resolution of each camera, and the density at which the pixels are packed on to the sensor.

The purpose of this article is to explore the reasons why an APS-C camera (referred to as APS) may have an advantage over a full frame camera (referred to as FF) when telephoto lenses are used.

This article demonstrates mathematically that, when a full sized image from FF is cropped so that it produces the same field of view as a full sized image from APS, if the image size of APS is greater than the size of the cropped image from FF, this is because the pixel density of APS is greater than that of FF.

If you are already familiar with the concepts illustrated in this article, click  here  if you wish to skip directly to comparisons made for selected cameras. These comparisons show the approximate mathematical relationships between pixel density, pixel size, and image size (for both linear and area relationships).

Click  here  to go directly to Appendix 3 of this article. This appendix summarises the exact mathematical relationships between image size, pixel density, and pixel size, for the two "theoretical" cameras described below (for both linear and area relationships). The information in Appendix 3 can be used as a template for calculating the pixel density and pixel size for any camera.

It is stated in "Wikipedia - The Free Encyclopedia", that:

"In digital photography, the crop factor, format factor or focal length multiplier of an image sensor format is 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 most commonly used definition of crop factor is the ratio of a 35 mm frame's diagonal (43.3 mm) to the diagonal of the image sensor in question; that is, CF=diag35mm / diagsensor. Given the same 3:2 aspect ratio as 35mm's 36 mm × 24 mm area, this is equivalent to the ratio of heights or ratio of widths; the ratio of sensor areas is the square of the crop factor."

To illustrate the principles explained in this Wikipedia article, we shall assume that the sensor size of FF is 36mm x 24mm, and for APS it is 24mm x 16mm. Therefore, both cameras have a 3:2 aspect ratio, and FF's sensor is 50% larger than APS's sensor. There are several ways of calculating a crop factor for the two "theoretical" cameras above:

Based on measurement of long side  (width) of sensors:  =  1.5x  (36mm / 24mm)

This is calculated using the following formula:

crop factor  =  width of sensor of full frame camera       divided by      width of sensor of second camera

Based on measurement of short side of sensors (height):  =  1.5x  (24mm / 16mm)

Based on diagonal measurement of sensors: = 1.5x  (43.2666mm / 28.8444mm)

Based on area measurement of sensors: = square root of 2.25  =  1.5x  (864mm / 384mm = 2.25mm)

Note:  The crop factors calculated above are all 1.5x because the aspect ratio of FF is 3:2 (36mm : 24mm), and the aspect ratio of APS is also 3:2 (24mm : 16mm). However, if the sensor of APS did not have a 3:2 aspect ratio (such as a 4:3 aspect ratio), the above crop factor calculations would  not  be equal.

We shall now provide a practical application of the above 1.5x crop factor. We shall assume that a lens with a focal length range of 70mm – 300mm, is attached to FF. The images captured by FF with this lens will show the “full view” that the lens is capable of providing, and will provide a genuine 70mm field of view. In addition, when zoomed in to 300mm, the images captured will show a genuine 300mm field of view, just as they would if this lens was attached to a 35mm film camera.

However, when this same lens is attached to APS, the photographer might say that this 70mm - 300mm lens now "acts like" its focal length has been multiplied by 1.5, meaning that it has the same field of view that is provided by a 105mm - 450mm lens on FF. But, in fact, the actual focal length of a photographic lens is fixed by its optical construction,  and does not change  when it is attached to APS.

Therefore, when this 70mm - 300mm lens is attached to APS, the smaller sensor of APS  crops  the image the lens produces so that, in effect, the image is magnified 1.5 times. This 1.5x magnification is referred to as the “crop factor”, and in the above example, because FF's sensor is 50% larger than APS's sensor, the crop factor is 1.5x. The Wikipedia article referred to above explains that:

"The crop factor is sometimes referred to as "magnification factor", "focal length factor" or "focal length multiplier". This usage reflects the observation that lenses of a given focal length seem to produce greater magnification on crop-factor cameras than they do on full-frame cameras. This is an advantage in, for example, bird photography, where photographers often strive to get the maximum "reach". A camera with a smaller sensor can be preferable to using a teleconverter, because the latter affects the f-number of the lens, and can therefore degrade the performance of the autofocus."

Note that, with regard to capturing wider angle images, APS is at a considerable disadvantage in comparison with FF, because it can’t take full advantage of the 70mm lens. With a 70mm lens, the widest focal length that can be captured by APS is 105mm.

Note: The formula for calculating the crop factor is also discussed in this article:

The meaning of the term "telephoto advantage"

It is often claimed that, when compared with FF, the 1.5x crop factor of APS (in the above example) gives it a significant "telephoto advantage", “crop factor advantage”, "lens reach advantage", or “magnification advantage” over FF when telephoto lenses are used. This is because, for example, the photographer can use a 300mm lens on APS, and obtain the same field of view that is provided by a 450mm lens on FF. This can be an advantage to APS owners, because a 300mm lens may be smaller, lighter, and less expensive than a 450mm lens.

Note: The advantages and disadvantages of cropping images to gain greater magnification, instead of using  lenses with longer focal lengths, are discussed in  this article.

The purpose of this example is to demonstrate that, if a full sized image from FF is cropped so that it produces the same field of view (FOV) as a full sized image from APS, provided that the pixel densities of both cameras are the same, the full sized image from APS will be the same size as the cropped image from FF.

For this example, assume that the sensor sizes and image dimensions of two "theoretical" cameras are as follows:

The first camera is referred to as “FF”, and it has a “full frame” sensor which measures 36mm x 24mm. Note that this is also the size of a 35mm film frame. FF produces a maximum image size of 6000 pixels x 4000 pixels, and is therefore a 24 megapixel camera (6000 pixels x 4000 pixels gives an overall image area of 24 megapixels).

The second camera is referred to as "APS", and it is an APS-C camera which has a sensor that measures 24mm x 16mm. APS produces a maximum image size of 4000 pixels x 2667 pixels, and is therefore a 10.67 megapixel camera.

Images "A" and "B" below, show approximately to scale, examples of the full sized images that are produced by FF and APS.

Image A (below): Full Sized Image Captured by Camera FF

Captured with a 300mm lens providing a FOV of 300mm

Image Size 6000 pixels x 4000 pixels Image B (below): Full Sized Image Captured by Camera APS

Captured with a 300mm lens providing a FOV of 450mm

(300mm x crop factor of 1.5 = 450mm)

Image Size 4000 pixels x 2667  pixels When you compare Image "A" with Image "B", you could conclude that APS has a "telephoto advantage" of 50% over FF because, when the same 300mm lens is used on both cameras, FF provides a field of view of 300mm, while APS provides a field of view of 450mm.

Cropping the image of FF to the same field of view as captured by APS

If the owners of FF want to obtain an image that has an identical field of view to that captured by APS, they can use the same 300mm lens that was used on APS, and then  crop out  the parts of FF’s picture that are not in APS’s picture. The resulting image size, after cropping the image from FF to the same field of view  as captured by APS, is 4000 pixels x 2667 pixels, or approximately 10.7 megapixels, as shown below in Image “C”.

Image C (below):  Image Captured by Camera FF (300mm) Cropped to a FOV of 450mm

Represents Image A (FOV 300mm) cropped to the same FOV (450mm) as Image B captured by Camera APS

Image Size 4000 pixels x 2667 pixels In these circumstances, the width of an uncropped APS image (Image B) is 4000 pixels, and the width of a cropped FF image (Image C), with the same field of view of APS, is also 4000 pixels.

Note: The calculation of the image width of the cropped image of FF is as follows:

Cropped image width of FF to the same field of view as APS = Full image width of FF  multiplied by  sensor width of APS  divided by sensor width of FF

=  6000 pixels  x  24mm  /  36mm  =  4000 pixels

This 33% reduction in the width of FF’s image from 6000 pixels to 4000 pixels, is because, in this example, the sensor width of APS (24mm) is 33% less than that of FF (36mm).

Therefore, in this example, by cropping the image from FF, to the same field of view as the image from APS, the original 50% “telephoto advantage” or “lens reach advantage” of APS has been eliminated. However, to achieve this result, the image size of FF has been reduced from its original dimensions of 6000 pixels x 4000 pixels (24 megapixels) to 4000 pixels x 2667 pixels (10.67 megapixels).

Note:  If the owners of FF wish to obtain a telephoto image that has the full dimensions of 6000 pixels x 4000 pixels (rather than the cropped dimensions of 4000 pixels x 2667 pixels) then, with the above example, it would be necessary to use a 450mm lens on FF in order to obtain the same field of view that is provided by a 300mm lens on APS.

Pixel density calculations for Practical Example "A"

The term  pixel density  means the density at which the pixels are packed on to a camera's sensor. For example, under the pixels per linear centimetre method, the pixel density of a camera may be calculated as follows:

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

For Practical Example "A", the pixel densities (in pixels per linear centimetre) of both FF and APS are 1666.67, calculated as follows:

Pixel density of FF in pixels per linear centimetre  =  image width in pixels (6000)  divided by  width of sensor in centimetres (3.6) =  1666.67

Pixel density of APS in pixels per linear centimetre  =  image width in pixels (4000)  divided by  width of sensor in centimetres (2.4) =  1666.67

Therefore, for Practical Example "A", because the pixel densities of both FF and APS are equal, there is no  pixel density advantage  to either camera.

In this example, we shall alter the specifications for APS so that it now produces a maximum image width of 4800 pixels, instead of the previous maximum of 4000 pixels. The specifications for FF remain unaltered.

The purpose of this example is to demonstrate that, when an image from FF is cropped so that it produces the same field of view as a full sized image from APS, the full sized image width of APS (4800 pixels) is 20% greater than the cropped image width of FF (4000 pixels). This 20% “gain in image width” has arisen because the pixel density of APS (in pixels per linear centimetre) is 20% greater than that of FF.

In this example, the specifications for FF are the same as shown in Practical Example "A", that is, FF has a "full frame" sensor which measures 36mm x 24mm. FF produces a maximum image size of 6000 pixels x 4000 pixels, and is therefore a 24 megapixel camera.

APS has a sensor that measures 24mm x 16mm. APS produces a maximum image size of 4800 pixels x 3200 pixels, and is therefore a 15.36 megapixel camera.

Images "A" and "B" below, show approximately to scale, examples of the full sized images that are produced by FF and APS.

Image A (below): Full Sized Image Captured by Camera FF

Captured with a 300mm lens providing a FOV of 300mm

Image Size 6000 pixels x 4000 pixels Image B (below): Full Sized Image Captured by Camera APS

Captured with a 300mm lens providing a FOV of 450mm

(300mm x crop factor of 1.5 = 450mm)

Image Size 4800 pixels x 3200  pixels Cropping the image of FF to the same field of view as captured by APS

If the owners of FF want to obtain an image that has an identical field of view to that captured by APS, they can use the same 300mm lens that was used on APS, and then  crop out  the parts of FF’s picture that are not in APS’s picture. The resulting image size, after cropping the image from FF to the same field of view as captured by APS, is 4000 pixels x 2667 pixels, or approximately 10.7 megapixels, as shown below in Image “C”.

Image C (below):  Image Captured by Camera FF (300mm) Cropped to a FOV of 450mm

Represents Image A (FOV 300mm) cropped to the same FOV (450mm) as Image B captured by Camera APS

Image Size 4000 pixels x 2667 pixels You can see from the above images that, the full sized image width from APS in Image B (4800 pixels) is now 20% wider than the image width from FF in Image C, that has been cropped to the same field of view as captured by APS (4000 pixels).

This 20% “gain in image width” in favour of APS, has arisen because the pixel density (in pixels per linear centimetre) of APS, is 20% greater than that of FF, as shown in the calculations below.

Pixel density calculations for Practical Example "B"

For Practical Example "B", the pixel density (in pixels per linear centimetre) of FF is 1666.67 and for APS it is 2000, as calculated below:

Pixel density of FF in pixels per linear centimetre  =  image width in pixels (6000)  divided by  width of sensor in centimetres (3.6) =  1666.67

Pixel density of APS in pixels per linear centimetre  =  image width in pixels (4800)  divided by  width of sensor in centimetres (2.4) =  2000

Therefore, for Practical Example "B", there is a  pixel density advantage  in favour of APS of 20%  (2000 /  1666.67).

To summarise, in the above example, when the same 300mm lens is used on both APS and FF, the width of an uncropped APS image (4800 pixels), is 20% larger than the width of a cropped FF image with the same field of view as APS (4000 pixels). However, as demonstrated above, this 20% "gain in image width"  has arisen because, in this example, the pixel density of APS is also 20% greater than that of FF.

Note: In this section of the article, the pixel densities of the cameras have been calculated using the pixels per linear centimetre method, but in Appendix 1  to this article, we show how an area measure, known as the megapixels per square centimetre method, can also be used to calculate pixel density.

Appendix 2  to this article provides calculations of the estimated width and area of one pixel, and shows the relationship of these calculations to image size and pixel density.

Appendix 3  to this article provides a summary of the approximate mathematical relationships between image size, pixel density, and pixel size, of the two "theoretical" cameras dealt with in this article. The information in Appendix 3 can be used as a template for calculating the pixel density and pixel size of any camera.

Gain in comparable widths of print sizes as a result of the 20% "gain in image width"

In terms of print sizes, if the uncropped image of APS (of 4800 pixels width) is printed at 200 pixels per inch (ppi), the width of the print is 24 inches.

In contrast, if the cropped image of FF (of 4000 pixels width) is printed at 200 ppi, the width of the print is 20 inches.

So, the net effect of the “gain in image width” of APS, is to produce a print at 200 ppi, that is 4 inches wider (or 20% wider) than that produced with the same field of view from FF's cropped image.

Practical Example "C" :  Cropping an image from FF to the same image width as an image from APS

As an alternative to cropping an image from FF to the same  field of view  as an image from APS, we shall crop an image from FF to the same  image width  as an image from APS. We shall then compare the changed field of view of FF with that of APS. We shall assume that a 300mm lens is on both FF and APS.

Note: We are dealing in this article  only  with the arithmetical aspects of this analysis, because the assessment of the  quality  of cropped images and the  quality  of the cameras, are separate issues.

In this example, the specifications for FF and APS are the same as shown in Practical Example "B", that is:

FF has a "full frame" sensor which measures 36mm x 24mm. FF produces a maximum image size of 6000 pixels x 4000 pixels, and is therefore a 24 megapixel camera.

APS has a sensor that measures 24mm x 16mm. APS produces a maximum image size of 4800 pixels x 3200 pixels, and is therefore a 15.36 megapixel camera.

Therefore, when a 300mm lens is on APS, the field of view is 450mm (300mm x crop factor of 1.5).

When a 300mm lens is on FF, the field of view is 300mm.

However, when an image from FF with a width of 6000 pixels is  cropped to the same image width  as a full sized image from APS (4800 pixels), the field of view of the original image from FF is changed from 300mm to 375mm (6000 / 4800 x 300mm). The 20% difference between the fields of view of 375mm and 450mm arises because, in this example, the pixel density of APS (in pixels per linear centimetre) is 20% greater than that of FF.

Click  here  to see a practical pictorial illustration of the principles dealt with above.

Click  here  to see an article that deals with the principles underlying the above calculations. This article also deals with the advantages and disadvantages of cropping images to gain extra reach.

*  When the same lens and the same focal length are used to take a photograph with a camera that has, for example, a "full frame" 36mm x 24mm sensor (FF), and with an APS-C camera that has a 24mm x 16mm sensor (APS), the resulting images produce different fields of view. This is because APS crops the image that is provided by the lens, so that in comparison with the image produced by FF, in effect, the image from APS is magnified 1.5 times. This 1.5x "magnification" is referred to as the "crop factor". It is also referred to as the "magnification factor", "focal length factor", or "focal length multiplier". However, note that the actual focal length of a photographic lens is fixed by its optical construction, and does not change when it is attached to APS.

*  It is often claimed that, when compared with FF, the 1.5x crop factor of APS (in the above example) gives APS a significant "telephoto advantage" or "lens reach advantage" over FF when telephoto lenses are used.

*  Practical Example "A" of this article demonstrates that, if a full sized image from FF is cropped so that it produces the  same field of view  as a full sized image from APS,  provided that  the pixel densities of both cameras are the same, the full sized image from APS will be the  same size  as the cropped image from FF. Therefore, in this example, there is no "telephoto advantage" or "pixel density advantage" in favour of APS.

*  Practical Example "B" of this article demonstrates that, when an image from FF is cropped so that it produces the  same field of view  as a full sized image from APS, the full sized image width of APS (4800 pixels) is 20% greater than the cropped image width of FF (4000 pixels). This 20% "gain in image width" has arisen because, in this example, the pixel density of APS (in pixels per linear centimetre) is 20% greater than that of FF.

*  Practical Example "C" of this article demonstrates that, if a 300mm lens is on both FF and APS, when an image from FF is cropped so that it produces the  same image width  as a full sized image from APS, the field of view of  an APS image is 450mm, and the changed field of view of a FF image is 375mm. The 20% difference between the fields of view of 375mm and 450mm arises because, in this example, the pixel density of APS (in pixels per linear centimetre) is 20% greater than that of FF.

Note: The above conclusions are based on  linear  measurements, and deal only with the arithmetical aspects of the analysis of the pixel density / telephoto advantage of an APS-C camera. Therefore, the assessment of the  quality  of cropped images and the  quality  of the cameras, are separate issues that are not dealt with in this article.

Appendix 1  to this article shows the image size and pixel density calculations based on  area  measurements.

Appendix 2  to this article provides calculations of the estimated width and area of one pixel, and shows the relationship of these calculations to image size and pixel density.

The following supplementary notes are designed to give you further information about how to compare the cameras listed in the above index:

Appendix 3  to this article provides a brief summary of the approximate mathematical relationships between image size, pixel density, and pixel size. The information in Appendix 3 can be used as a template for calculating the pixel density and pixel size of any camera. Rob's  Photography  New  Zealand