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Appendix  3

Analysis  of  the  "pixel  density  advantage"

of  two  "theoretical  cameras"

Template  for  calculating  the

approximate  mathematical  relationships  between

Pixel  density,  pixel  size,  and  image  size

(for  both  linear  and  area  relationships)

Note:  This template should be studied in conjunction with the full explanatory article that you can see  here.  This article gives further details about the calculations, and it uses the same specifications and "theoretical cameras" that are demonstrated below. In particular, note the reservations expressed  here  about calculating the "exact" width and area of one pixel.

Note that this template is not designed to provide information about the quality of images or the quality of the cameras, because these are separate issues. In addition, this template is designed only to compare cameras that have a 3:2 aspect ratio.

In addition, there is further information in  these supplementary notes  about the relationships between the crop factor, field of view, photographic reach, image size, pixel density, and pixel size that will help you to interpret the calculations set out below.

Specifications  of  Two  "Theoretical"  Cameras

APS: Image dimensions: 4800 pixels x 3200 pixels  (15.36 megapixels);  sensor size: 24mm x 16mm

FF:   Image dimensions: 6000 pixels x 4000 pixels  (24.00 megapixels);  sensor size: 36mm x 24mm

Crop  Factor

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.

Linear  Relationships

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 = 2000       (4800 / 2.4)

FF =    1666.67  (6000 / 3.6)

Pixel Density Advantage: APS is 20% greater than FF

Note: With this example, if the full frame camera (FF) had the same pixel density as the APS-C camera (APS), it would have 34.56 megapixels, and image dimensions of 7200 pixels x 4800 pixels.

Pixel pitch (in microns)

Refer to the reservations expressed in  Appendix 2  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 = 5   (24 / 4800 x 1000)

FF =    6   (36 / 6000 x 1000)

Relationship: FF is 20% greater than APS

Crop an image from FF to the same  field of view  as an image from APS

Gain in image width (in pixels) as a result of the above 20% pixel density advantage

Uncropped image width of APS   = 4800 pixels

Cropped image width of FF

to same field of view as APS        = 4000 pixels  (6000 x 24 / 36)

Relationship: APS is 20% greater than FF

Note: The cropped image width of FF to the same  field of view  as an image from APS, has been calculated using this formula:

Uncropped width of an image from FF  x  sensor width of APS  /  sensor width of FF     (6000 x 24 / 36 = 4000)

Crop an image from FF to the same  field of view  as an image from APS

Gain in comparable widths of print sizes as a result of the above 20% pixel density advantage

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 (4800 / 200).

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

Relationship: The net effect of the 20% pixel density advantage 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 the cropped image of FF.

Crop an image from FF to the same  field of view  as an image from APS, and compare the changed field of view of FF with that of APS: Assume that a 300mm lens is on both cameras and that the field of view of an uncropped FF image is 300mm

Field of view of APS = focal length of lens  x  sensor width of FF / sensor width of APS = 450mm  (300mm x 36.0mm / 24.0mm)

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

= uncropped image width of FF  /  cropped image width of FF  x  focal length of lens     =  450mm  (6000 / 4000 x  300mm)

Relationship: The fields of view of APS and FF are the same, that is, 450mm.

Note: The image width of a FF image, when it is cropped to the same field of view as an APS image, is 4000 pixels (6000 x 24 / 36).  Click  here  to go to an article titled "Advantages and disadvantages of cropping images instead of using lenses with longer focal lengths". This article gives further details in support of the formulas used above.

Crop an image from FF to the same  image width  as an image from APS, and compare the changed field of view of FF with that of APS: Assume that a 300mm lens is on both cameras

Field of view of APS =  focal length of lens  x  sensor width of FF  /  sensor width of APS  = 450mm  (300mm x 36.0mm / 24.0mm)

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

= uncropped image width of FF  / cropped image width of FF  x  focal length of lens  = 375mm  (6000 / 4800 x 300mm)

Relationship: APS is 20% greater than FF.

Note: Click  here  to see a forum discussion titled: "How do you calculate the reach advantage? Sony A900 vs Nikon D3S" Digital Photography Review, Sony SLR Talk Forum, April 2010.  Click  here  to go to an article titled "Advantages and disadvantages of cropping images instead of using lenses with longer focal lengths". This article gives further details in support of the formulas used above.

Area  Relationships

Pixel density (in megapixels per square centimetre)

Pixel density in megapixels per square centimetre = number of megapixels on the sensor  divided by  sensor area in square centimetres

APS = 4    (15.36 / 3.84)  or  pixel density in pixels per linear cm. squared / 1,000,000   (2000 x 2000 / 1,000,000)

FF = 2.77778     (24 / 8.64)      or  pixel density in pixels per linear cm. squared / 1,000,000    (1666.67 x 1666.67 / 1,000,000)

Relationship: APS is 44% greater than FF

Pixel area (approximate area of one pixel in square microns)

Refer to the reservations expressed in  Appendix 2  about calculating the "true" width and area of an individual pixel.

Area of one pixel in square microns = area of sensor in square microns  divided by  the number of pixels on the sensor

APS = 25    (384,000,000 / 15,360,000) or pixel pitch squared   (5 x 5)

FF =    36   (864,000,000 / 24,000,000) or pixel pitch squared    (6 x 6)

Relationship: FF is 44% greater than APS

Crop an image from FF to the same  field of view  as an image from APS

Gain in image area  (based on image sizes in megapixels)

Uncropped image area of APS = 15.36 megapixels     (4800 pixels x 3200 pixels)

Cropped image area of FF

to same field of view as APS    = 10.6667 megapixels  (4000 pixels x 2666.67 pixels)

Relationship: APS is 44% greater than FF

Note: The above arithmetical reconciliations work exactly, only when the number of megapixels on the sensor is exactly the same as the image width in pixels, multiplied by the image height in pixels. In addition, the image width divided by the image height, must give exactly the same answer as the sensor width divided by the sensor height. In this example, exact reconciliations can be made because the above conditions have been met.

However, in practice, when comparing two cameras, the above arithmetical reconciliations will not work exactly because of roundings in the quoted specifications and also because of the way the effective number of pixels of the cameras is calculated. Therefore, it is important to realise that, any practical application of this template will provide only approximate answers and relationships. In addition, the template is not designed to provide information about the quality of images or the quality of the cameras, because these are separate issues.

Click  here  to see an index of further camera comparisons showing the mathematical relationships between image size, pixel size, pixel density, and reach etc.

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

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

Appendix 2  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:

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 see examples of the outstanding resolution of the full frame Sony A900.

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

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