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Crop factor explained

What effect do the sensor size have on images.
by Gisle Hannemyr
Published: 2005-05-16.

Introduction

The negative of a 135-format film (aka. 35 mm film) has an image area that is 36 mm by 24 mm. Most digital cameras have sensors that are smaller than this (figure 1). The amount you need to multiply the diagonal of the digital sensor with to match the diagonal of 35 mm film (aka. 135-format film) is usually called the digital sensor's “crop factor”1).

There exists digital cameras where the sensor is roughly the same size as the 135-format film negative, for example the Canon EOS 5D II, the Nikon D700, and the Sony Alpha A-900. We say that these cameras have a crop factor of 1.0x, and refer to them as cameras with FX-format or Full-format (FF) sensors.

The sensors in most DSLR bodies from Nikon, Fujifilm, Pentax, Sigma and Sony, MILC bodies from Sony and Samsung, and the Fujifilm X100 compact camera, have a crop factor of 1.5x. We refer to them as cameras with DX-format sensors. Cameras built according to the Four-thirds standard, uncluding µFour-thirds (Olympus, Panasonic and some Leicas) have a crop factor of 2.0x. The sensors in the Nikon 1 MILC bodies have a crop factor of 2.7x. We refer to them as cameras with CX-format sensors. The Pentax Q MILC is the system camera with the smallest sensor, with a crop factor of 5.6x.

Most digital consumer compacts have small sensors. For instance, the 1/1.7" type CCD used in a number of premium digital compact cameras has a crop factor of 4.6x.

Sensor sizes.
Figure 1: The relative sizes of different sensor formats, from FX to 1/2.3".

Canon users three different sensor sizes in its DSLR lineup. In addition to the top of the line models with FX-format sensors, Canon also use sensors that have a crop factor of 1.3x and 1.6x.

Sigma uses a DX-format sensor with 1.5x crop in the SD1, but sensors with a 1.7x crop factor in the SD9, SD10, SD14 and SD15 DSLRs and DP1 and DP compact cameras.

The crop factor is a physical property of the image sensor. Some people become confused by the crop because they have observed that there exists DSLR lenses that designed to be used on bodies with cropped sensors (e.g. the lenses from Nikon carrying the “DX” designation, or the lenses from Canon carrying the “EF-S” designation). However, the only significance of such designations is that the lens is designed with a reduced image circle that makes it unsuitable for use on a camera with an FX-format sensor. The focal length of such lenses are the same as the focal length of lenses designed for the FX-format. In other words, the Nikkor 35 mm DX (designed for a DX-format sensor) and the Nikkor 35 mm AF-D (designed for an FX-format sensor) will cover exactly the same field of view when placed on a camera with a DX-sized sensor such as the Nikon D3200. In both cases, the field of view will be 44°. This is what most people will call a “normal” field of view.

Below I discuss the effects of the crop factor in terms of field of view, perspective, depth of field, diffraction, f-number, hand holding, and finally in terms of macro ratio.

Field of View

The field of view (FOV) of a lens is the angular cone extending from the focal plane of the camera into space. The FOV depends on two things, the diagonal of the sensor (D) and the focal length (f) of the lens. The formula below shows how to compute the diagonal FOV (the answer will be in degrees or radians, depending upon what unit the atan function puts out).

FOV = 2 x atan (D / 2 / f)
(D = diagonal of the sensor, f = focal length of the lens)

The table below show the results of computing the diagonal FOV for three different lenses on three different sensor sizes. The first lens is a 135 mm lens used on a camera with an FX-format sensor (i.e. a crop factor of 1.0x), the second lens is a 90 mm lens used on a camera with an DX-format sensor (i.e. a crop factor of 1.5x), and the third lens is a 28 mm lens used on a camera with a 1/1.8" type CCD sensor (i.e. a crop factor of 4.8x).

Sensor Crop f. D (mm) f (mm) FOV (rad.) FOV (deg.)
FX/135-format 1.0x 43.3 135 0.318 18.2
DX-format 1.5x 28.4 90 0.313 17.9
1/1.8" type CCD 4.8x 8.9 28 0.316 18.1

The table shows that all three lenses in this case will produce about the same field of view: 18°. Lenses with a more narrow FOV of view than 34° is generally regarded as telephoto lenses. I.e.:given a small enough sensor, a 28 mm lens is a telephoto lens.

Some photographers, however, have worked for a long time with 135-format film and are used to thinking about FOV in terms of focal lengths of lenses attached to 35 mm film cameras instead of degrees. For them, the crop factor is a convenient way of computing the 35 mm film “equivalent” or “effective” focal length from the real focal length. Those photographers recommend that you multiply the real focal length with the crop factor for whatever type of camera you are using to get the 35 mm film “equivalent” or “effective” focal length.

Example: Given that you use a camera with sensor that is DX-format and a 90 mm lens, the FX/35 mm film “equivalent” focal length of this combination is:

90 mm x 1.5 = 135 mm.

Note, however, that the real focal length of the lens does not change. The crop factor multiplication is just a trick to help a photographer that is used to working with 35 mm film to visualise the FOV when using a certain lens on a digital camera with an unfamiliar sensor size.

I do not think it is a good idea to talk about “equivalent” or “effective” focal lengths. Too often, introducing these terms into a discussion only adds to the confusion, so I shall say more about them. Use of these terms should in my opinion be discouraged.

An alternative way of looking at this is to start with the common names used for different classes of lenses that provides the photographer with a different perspective (i.e. super-wide, wide, normal, short tele, tele and super-tele), and how these classes of lenses maps onto specific focal lengths depending upon sensor size. This is done in the two tables below.

The table shows the focal length (in millimetres) for for a number of common DSLR formats. The rows labelled “FOV” shows the field of view in degrees for the focal lengths listed. The top FOV row shows diagonal field of view, and the bottom FOV shows horizontal field of view for sensors with a 3:2 aspect ratio (i.e. all except Four-thirds, which has a 4:3 aspect ratio). You can use the table to figure out the focal lengths to look for if you want a specific class of lens, given the sensor size of the camera you want to use the lens on.

FormatCrop f.Super-wideWide NormalShort teleTeleSuper-tele
FOV (diag.): 114°84°75°63°57°47°37°34°18°16°6.2°5.0°3.1°
FX: 1.0x14.024 28 35 40 50 65 70 135150400500800
Canon (1.3x):1.3x10.818 22 27 31 38 50 54 104115308385615
DX: 1.5x9.3 16 19 23 27 33 43 47 90 100267333533
Canon (1.6x):1.6x8.8 15 18 22 25 31 41 44 84 94 250313500
Sigma: 1.7x8.2 14162124293841 79 88235294471
Four-thirds: 2.0x7.0 12141820253335 68 75200250400
CX: 2.7x5.1 9101315182426 50 55147183267
FOV (hor.): 104°74°65°54°48°40°31°29°15°14°5.2°4.1°2.6°

When using the table, you will need to match focal length to the nearest focal length that is actually available. E.g. if you want a “normal” fixed focal length lens, matching the popular 50 mm on a camera with an FX-format sensor on a camera with a DX-format sensor, the table tells you that an exact match is 33 mm. However, a 33 mm lens does not exist. In this case, you should pick a 35 mm or a 30 mm.

Sensor sizes.
Figure 2: Frames shows the FOV of 14 mm lens on FX, DX, Four-thirds and CX.

Let's say that you want to buy a super-wide class lens. You can see from the table that for a camera with an FX-format sensor, you need to find a lens where the focal length is 24 mm or shorter. If your camera uses a DX-format sensor, you need to use a lens that is 16 mm or shorter for the same super-wide effect.

This means that a lens such as the Nikkor 14-24 mm will be clearly classified as super-wide for use on a camera with a FX-format sensor, but only marginally so for a camera with a DX-format sensor. If we use this Nikkor with an adaptor on a camera with a Four-thirds sensor, it will be classified as wide-to-normal (not super-wide).

We can also see from the table that for a camera with a DX-format sensor, you would need a lens with a zoom range 9.3 mm to 16 mm to match the FOV of the Nikkor 14-24 mm when used on a FX-format sensor. Unfortunately, such a lens does not exist for the DX-format, so one must use something like the Sigma 10-20 mm instead. This lens, however, does not go as wide as 114° on the wide end. On a camera with a DX-format sensor, f=10 mm gives a diagonal FOV=110°.

While there exists lenses with zoom ranges to goes from wide to telephoto (e.g. the Canon EF 28-300mm f/3.5-5.6 L IS USM and the Nikon 18-200mm f/3.5-5.6G IF-ED AF-S DX VR), the compromises that is inherent in such designs make many photographers build a collection of quality zooms that covers the range from super-wide (FOV ≈114°) to telephoto (FOV ≈12°) spread over three lenses. The table below shows the typical zoom lenses in such a collection. The table also shows the diagonal FOV at both ends of the range.

FormatCrop f.FOVSuper-wide zoomStandard zoom Tele zoomFOV
FX: 1.0x114°14-24 mm24-70 mm70-200 mm12.4°
DX: 1.5x110°10-20 mm17-55 mm50-135 mm12.0°
Canon (1.6x):1.6x107°10-22 mm17-55 mm50-135 mm11.4°
Four-thirds: 2.0x114°7-14 mm 14-35 mm35-100 mm12.4°
CX: 2.7x114°5.1-9 mm 9-26 mm26-73 mm12.4°

Currently, users of cameras with FX-format sensors have access to original zooms with a maximum aperture of f/2.8 in the FOV-range from super-wide) to telephoto. For users of smaller sensors, third party lenses must be used to cover at least part of this range if one want wide or fast zooms.

Perspective

Some people believe that “perspective” in photography is determined by the focal length of the lens used. They will argue that cer­tain focal lengths have a certain, inherent perspective or “look” that will be present irrespective of the size of the sensor. In other words, they will argue that a 135 mm lens will always give the “compressed” look one gets from using a telephoto lens, and a 28 mm lens will always give the “expanded” look of a wide-angle lens.

This is wrong. There is nothing special about the focal length. Perspective in a photograph is deter­mined by one thing: the position of the photographer relative to the scene. (To be precise, perspective is determined by the distance from the front nodal point to the scene).

The reason people believe that a specific focal length effect a certain look is because the narrow FOV of a telephoto lens forces the photographer to take up a position far away from the scene, and the large FOV of a wide-angle lens allows the photographer to move closer. It is these differences in position, and not the focal length as such, that effect the difference in perspective.

It follows that as long as you stay in the same position, you get the same perspective no matter what focal length lens you put on the camera.

Below is a demonstration of this. It shows two photographs of the same scene. The first is taken with a 135 mm lens, the second with a 28 mm lens. The reason they show the same perspective is because the 28 mm lens is used on a camera with a much smaller sensor. The second photograph can be considered to be a 7.18 mm by 5.32 mm crop from the middle of a 36 mm by 24 mm film frame.

135 mm.
Figure 3: The photograph above is taken with a 135 mm lens on 135-format film (35 mm) film camera and scanned.

28 mm.
Figure 4: The photograph above is taken with a 28 mm lens with a digital compact camera with a very small sensor (4.8x crop factor).

Depth of Field

Selective use of focus, and in particular the ability to control what parts of the image are sharp, and what parts are not, is important for creative photography. However, “unsharpness” in photography is really controlled by two different properties. The first is the quality of the unsharp regions of the photograph. This property is called bokeh after the Japanese word for “confused”. Bokeh will become the subject of another (forthcoming) article. The second is the size of the unsharp region. This property is called depth of field (usually abbreviated to DOF). DOF shall be the topic of this section.

DOF Definition

According to Leslie Stroebel et al, Basic Photographic Materials and Processes, Second Edition, depth of field (DOF) is defined as the range of object distances within which objects are imaged with “acceptable” sharpness (p. 151).

Of course, only one single plane will be in perfect focus. But because humans do not have perfect eyesight, and because lenses and sensors do not have infinite resolution, there is a region surrounding the point in perfect focus that most viewers will perceives as sharp. This perceived region is the DOF.

An old rule of thumb says the region of acceptable sharpness extends from 1/3 in front of perfect focus plane to 2/3 behind it. However, this rule of thumb is only strictly true at 1/3 of the hyperfocal distance. In the macro region, the perfect focus plane is in the middle of the region of acceptable sharpness, and when the lens focused at the hyperfocal distance; then region of acceptable sharpness extends from 1/2 the focusing distance to infinity.

It is important to understand that DOF is not an objective, intrinsic or inherent property of a film negative or a digital image file. DOF is mainly determined by magnification. A postcard sized and poster sized print of the same negative or file will show different DOF if both are viewed from the same distance. The smaller print will appear to have a greater DOF. DOF is a property of an photographic image that depends upon, and varies with, the context of presentation, including such things as the physical size of the printed image, the distance of the observer relative to the printed image, and even the visual acuity of the observer. The DOF numbers one can find in manufacturers' DOF-tables or engraved upon lenses are not universal and ubiquitous. They are just guidelines for what the DOF will be like, based upon assumptions about the how the DOF in an average print will be perceived when it is viewed by an average observer from an average distance.

Shallow DOF

Portrait photographers usually want to have the main subject tack sharp, set against a creamy, smooth background. To achieve this effect, you want the DOF to be as shallow as your equipment let you.

As we shall see below, DOF gets less and less shallow for a certain FOV when the size of the camera's image sensor or film gets smaller. This explains why it is so difficult to capture a really shallow DOF with a compact camera.

But before we move on to determining how sensor size impacts upon DOF, here is just some rules of thumb to help you achieve the blurriest backgrounds that your camera is capable of.

  1. Set longest focal length that you have. If you're using a compact camera, zoom it out to the tele end.
  2. Use widest aperture (smallest f-number) available at that focal length.
  3. Get as close as possible to your subject.
  4. Arrange for subject to be as far away from the background as possible.
  5. Select a background that is as discrete as possible.

These rules of thumb work for all cameras, compacts as well as DSLRs. But please bear in mind that compact cameras with small image sensors are not very suitable for shallow DOF. With this type of camera, you may not get the creamy background you want, even if you follow all the rules above.

Hyperfocal distance

Landscape photographers, however, tend to want everything from some near object to infinity to appear sharp. To achieve this, you must set focus to the so-called hyperfocal distance (HFD).

The hyperfocal distance is defined as the shortest focus distance you can set where DOF reaches infinity. When this distance is set on the lens' distance scale, the maximum depth of field is obtained. Everything from half the HFD up to infinity is rendered sharp.

Back in the days of film, lenses used to have markings that indicated DOF at various apertures for 135-format film. The picture below is of such a lens (a Nikkor 85 mm f/1.8 pre-AI). Notice that some of the apertures are colour coded, and that surrounding the focus mark there are symmetrical markings in the same colours that indicates DOF at that aperture.

HFD.
Figure 5: Setting the hyperfocal distance using the lens' DOF markings.

In the illustration, the lens is set to its smallest aperture (f/22), which is blue. To preset the hyperfocal distance for 135-format film, based on the markings on the lens, we put the infinity symbol next to the left blue DOF-mark. We can then see that the HFD in this case becomes 10.6 meters, and that DOF in this cases reaches from HFD/2, or 5.3 meters, to infinity. Notice that the right blue DOF mark is next to 5.3 meters.

However most modern lenses comes without such markings, and if you have lenses with DOF-markings on them, you may need to adapt them. The markings on old lenses are for 135-format film. Many DSLRs use a sensor that is smaller than this. If you're using a camera with a DX or APS-C digital sensor, you can still use the old scale by using the markings for one stop less (e.g. use the lens' markings for f/16 if you are shooting with a DX/APS-C sensor at f/22).

DOF calculators

If you need to figure out the hyperfocal distance or the DOF for any particular combination of focal length, aperture, sensor size, and subject distance, and you don't have a lens with DOF markings, or you're not sure whether the DOF markings on the lens apply to the DSLR you use, you can use a DOF calculator.

Several websites offer DOF-calculators, both for use directly on the web, and for use with PDAs and SmartPhones. Below are links to those I know about. The colums tagget “µm” indicate what CoC they use for different sensor formats, and the column tagged “z” lists the embedded z-number. I find it helpful to know these parameters. If you do not understand what these are, you may want to read our technical note on depth of field in Gaussian optics, or you may just want to ignore them.

Links to different DOF calculators and what key values they use.
 1/1.8" 4/3  1.6x    DX    1.3x FX/FF
ResourcezCoC in µm
Canon Europe 1442  19 2430
Don Fleming (DOFMaster) 144261519202430
Paul van Walree 144261519202430
Warren Yong (f/Calc) 1730 1316162025

The calculators will give slightly different results, based upon what values the designers have picked as the z-number. The most popular value for z-numbers seems to be 1442. This article in Wikipedia argues that the value 1730 is apocryphal and propagated by mistake.

However, since DOF is a perceived value, there is no single right answer. The value of 1442 for z-number was originally derived from experiments to determine the resolving power of human vision. This particular value is found to match a person with average eye-sight viewing a photograpic print with of typical size (about 16.7 cm x 25 cm, 30 cm diagonal) from a typical viewing distance (about 30 cm). If the print is larger, the viewing distance shorter, or if the person looking at it is more eagle-eyed than the norm, we need to use a larger z-number to calculate the perceived DOF.

It is important to realise that the DOF markings on the lens or DOF calculators only apply if the end result is a paper print of typical size viewed from a typical distance by a typical person. If you pixel-peep (i.e. look at the pixels that make up a digital photograph at 100 % – or larger – on a computer screen), you will not see the DOF predicted by the markings on your lens or by any of the DOF calculators available. Instead, you'll find that regions of your photo that according to the DOF markings on the lens or the DOF calculator should be sharp, are blurry. Again: DOF is a perceived property. By pixel-peeping you are perceiving the image at a higher magnification than the typical viewing situation. The assumptions embedded in the DOF markings on the lens or DOF calculator is not valid in such an atypical viewing situation.

DOF and sensor sizes

Many photographers have noticed that it is very difficult to achieve a shallow DOF with compact digitals cameras, and a assume that a small sensor is “has” a deep DOF. But the real reason for this is that the focal length of the lenses used on these cameras are extremely short.

Part of the confusion stems from manufacturers' tendency to obfuscate the focal lengths of compact cameras. For instance, the Canon PowerShot G5 is usually advertised as having a 4x zoom lens with a range from 35 mm to 140 mm. This is not so. The Canon PowerShot G5 has a FOV that is equivalent to this range (on a camera that uses 135-format film). Its actual focal lengths, however, only goes from 7.2 mm to 28.8 mm. It is these very short focal lengths, along with the lens' aperture, that determines the camera's DOF.

To visualise just how much deeper DOF becomes when focal lengths are shortened to the keep FOV across formats identical, I have created the following table. The first five colums has information about the sensor and camera, while the last columns show the shift in DOF for the digital sensor formats listed, compared to 135-format film.

Shift in DOF between FX-format and other sensor sizes, same FOV.
Type CropØ (mm)WxH (mm) ExamplesShift (EV)
1/1.8"4.8x8.9 7.2x5.3 Canon G54.5
CX 2.7x15.9 13.2x8.8Nikon V12.9
4/3 2.0x21.8 17.4x13.1Olympus E-12.0
? 1.7x24.9 20.7x13.8Sigma SD141.6
? 1.6x27.0 22.5x15.0Canon EOS 60D1.3
DX 1.5x28.4 23.7x15.6Nikon D70001.2
APS-C 1.4x30.1 25.1x16.7Nikon Pronea 6i, Canon EOS IX1.0
APS-P 1.4x31.7 30.2x9.5 Nikon Pronea 6i, Canon EOS IX0.9
? 1.3x34.3 28.7x18.7Canon EOS 1D Mk III0.7
APS-H 1.3x34.5 30.2x16.7Nikon Pronea 6i, Canon EOS IX0.7
FX/FF 1.0x43.3 36.0x24.0Canon EOS 5D, Nikon D7000.0

For instance, if you shoot with a Canon EOS 60D, and you want the same DOF as you used to have when you used film, you need to open up the lens 1.3 stops.

The shift in DOF can also be computed from the crop factor. To get the same DOF on two cameras with different size sensors, we must open up the aperture on the camera with the smaller sensor to the f-number we get when we divide the f-number on the camera with large sensor with the crop factor quotients. Example: If we use f=90 mm, f/4.0 on the camera with the FX/FF sensor, we need to open up to 4.0/1.6 = f/2.5 and use f=90 mm/1.6=56 mm lens to get an identical FOV and DOF if we use a body with a sensor that have crop factor equal to 1.6x.

When you examine the table above, you should see why it is so difficult to get a shal­low DOF on compact digicams with small sensors. The shift on a camera with a 1/1.8" type sensor is 4.5 EV. This means that a Canon Powershot G5, at its maximum aperture f/2.0 will have the same DOF as a camera with an FX-format sensor have at f/9.6, given the same FOV.

What this means is that there is no way to duplicate both the FOV and the very shallow DOF that you get when you use a 85 mm f/1.4 portrait lens at full bore on a camera with a FX-format sensor, on a camera with a DX-format sensor. In theory, you should get the same effect by using a 57 mm f/0.9 lens – but no such lens exist.

The converse, however, is not true. As R.N. Clark explains in his article The Depth-of-Field Myth and Digital Cameras, the very deep DOF of small-sensor cameras can also be obtained by using a camera with a larger sensor, just by upping the ISO and stop­ping down the lens. Because of the better noise characteristics of large sensors, and because small sensor cameras are more diffraction limited than large sensor cameras, we find that when making the images from two cameras with different sized sensors, and arranging for them being identical in terms of resolution, FOV, exposure time, and signal-to-noise ratio, we find that the DOF is also identical. Example: To reproduce the deep DOF of a Canon Powershot G5 (1/1.8" type sensor) stopped down to f/8.0 (it's minimum aperture), we must stop down the aperture of a DSLR with a FX-format sensor a further 4.5 stops. To maintain the same shutter speed, we must also increase the ISO on the DSLR by 4.5 stops. I.e.: if we use ISO 100 and f/8.0 with the small sensor, we must use ISO 2300 and f/38 on the camera with the FX-format sensor. This will result in both images having identical DOF, and very similar image quality. If one does the math, it nicely works out that one get the same amount of diffraction, and the same amount of noise, in both images.

For more about DOF, see our technical note on depth of field in Gaussian optics.

Diffraction

Diffraction limits the resolving power of an optical system at small apertures. Why this happens is beyond the scope on this article. For an explanation, see the tutorial on diffraction in Sean McHugh's Cambridge in Colour. [Note: The diffraction calculator at the end of the tutorial has a bug that will give you the wrong results for cameras with more than 3 Mpx if you tick the checkbox marked “Set circle of confusion* based on pixels?”. Ticking this box will make the CoC a function of the pixel pitch for all sensors. While this is the correct thing to do for a camera with less than 3 Mpx, it will (with the presets used by McHugh), give bogus results for cameras with roughly more than 3 megapixels.]

When uniform light passes through an aperture, it is diffracted and creates a pattern of light and dark regions known as an Airy disc. The angular resolution of the system, which is important in astronomy when one is photographing point light sources, is limited by the Rayleigh criterion, which is met when two Airy discs from two point light sources are so close together that they overlap.

In digital photography, we say that an optical system is diffraction limited when the size of the Airy disc is larger than half the CoC, or larger than about 3/4 of the pixel pitch2) (whichever is largest). The CoC depends on the diagonal of the sensor (D). The size of the Airy disc (AD-size) depends upon the wavelength of light (λ), and the aperture (N) of the optical system, and a constant (1.22) derived from a calculation of the position of the first dark ring surrounding the central Airy disc of the diffraction pattern. The formulas to compute the AD-size (derived from the articles linked to above) and CoC, are:

AD-size = 1.22 x λ x N
CoC = D / z-constant
(λ = wavelength of light, D = diagonal of the sensor)

Setting CoC equal to twice the AD-size and solving with respect to aperture (N) gives us a function to compute the aperture where the systems becomes diffraction limited for different sensor sizes (assuming that about 3/4 of the sensor's pixel pitch is smaller than this).

N = CoC / (2 x 1.22 x λ)

Note that the sensor diagonal (D) used to compute CoC and wavelength of light should be expressed in the same units (e.g. nm).

In the table below, I've computed the aperture/f-number (N) where this happens for a number of different sensor sizes, for light at wavelengths 400 nm (blue), 550 nm (green) and 700 nm (red). Since the eye's resolution is mainly determined by green, the numbers for the column with the heading “λ=550nm” is the most relevant. The z-constant used to compute the CoC is set equal to 1500.

Aperture where the system becomes diffraction limited
FormatCrop f.D (mm)λ=400nmλ=550nmλ=700nmExample
1/2.7"6.6x6.6f/4.5f/3.3f/2.6Nikon 775
1/1.8"4.8x8.9f/6.1f/4.4f/3.5Canon G5
1/1.7"4.6x9.5f/6.5f/4.7f/3.7Canon G12
CX2.7x15.9f/10.8f/7.9f/6.2Nikon V1
4/32.0x21.8f/14.9f/10.8f/8.5Olympus E-1
?1.7x24.9f/17.0f/12.4f/9.7Sigma SD14
?1.6x27.0f/18.5f/13.4f/10.6Canon EOS 40D
DX1.5x28.4f/19.4f/14.1f/11.1Nikon D3200
APS-C1.4x30.6f/20.9f/15.2f/12.0Nikon Pronea 6i
?1.3x34.5f/23.5f/17.1f/13.5Canon EOS 1D
FX1.0x43.3f/29.6f/21.5f/16.9Nikon D800

Looking at this table, it becomes clear why the smallest aperture available on the Canon Powershot G5 is f/8. In the green channel (550 nm), the G5 becomes diffraction limited at f/4.4, so even at f/5.6, the image is degraded by diffraction.

F-number

As noted above in the section discussing field of view (FOV) the real focal length of a lens is a fundamental property of its design and does not change if the size of sensor, and therefore the FOV, changes.

The f-number is a numerical designation for a lens' aperture. The f-number is derived from the lens' real focal length. It expresses the diameter of the entrance pupil of the lens as a fraction of the real focal length of the lens. The “f” in “f-number” is actually shorthand for focal length. For example, f/4.0 represents a entrance pupil diameter that is one-quarter of the lens' focal length. If the lens has f=100 mm, f/4.0 designates an entrance pupil diameter equal to 100/4=25 mm.

It follows that the size of the sensor does not affect the f-number. In other words, a lens that is f/4.0 on a camera with an FX-format sensor will also be f/4.0 on a camera with an sensor that is a different size.

Unlike a tele-converter, which enlarges the image an therefore spreads the light over a larger area (resulting in a light-loss in stops equivalent to the enlargement factor of the tele-converter), no light-loss happens when you put a specific lens on a body with a cropped sensor. My Nikon D700 (FX) and my Nikon D80 (DX) picks the same shutter speed when I mount the same Nikkor 50 mm lens on them set to f/1.4, and spot-meter a standard Kodak 18 % grey card in aperture priority mode.

Hand Holding

The so-called “focal length reciprocal rule” is an old rule of thumb that is known by almost every photographer using 135-format film. The “rule” says that to avoid blur from camera shake, you should use a shutter speed equal to the reciprocal of the focal length (or faster). For example, if you are shooting with a 100 mm lens, for hand held shots you should use a shutter speed of at least 1/100th of a second.

For digital, to get identical results with different sensor sizes, you need to take the crop factor into account. In other words, if you believe the focal length reciprocal rule is a good guideline, you must multiply the focal length with the camera's crop factor if you shoot with a camera that has a sensor smaller than FX-format. I.e. an updated version of the old rule of thumb looks like this:

min. shutter speed = 1/(focal length x crop factor)

Example: Let's say we are using a 100 mm lens on a camera with a DX-format sensor. Our slowest stutter speed for hand held shots is 1/(100 x 1.5) = 1/150th second.

The reason you need to figure in the crop factor is that an image projected onto a physical DX-format sensor need to be magnified more than an image projected onto a FX-format sensor for any given print size. As part of the process, camera shake blur is also magnified, and the increased magnification must be compensated for by using a faster shutter speed.

It should be pointed out that the focal length reciprocal rule for hand-holding is not a hard, scientific fact. It is just a rule-of-thumb that some people claim works reasonable well if the photographer has average steady hands, and the resulting average sized print is viewed by a person with average eyesight from a normal viewing distance (e.g. a print sized up to 17 x 25 cm viewed from a distance of around 30 cm). If your hands are less steady, and/or you use a crop from the total frame, and/or you print larger than 25 cm, you may need to use a higher shutter speed than indicated by the reciprocal rule to get a sharp print – and vice versa.

Not everyone agree with the rule. The rule indicates that with a normal lens, 1/50 second is safe for hand holding. Ansel Adams wrote:

Tests I conducted some years ago, photographing leafless trees against the sky, indicated that, using a normal lens with a hand-held camera, the slowest shutter speed that ensured maximum sharpness was 1/250 second. I found that even with firm body support image sharpness was noticeably degraded at 1/125 second, a speed that many photographers consider safe for hand-holding a camera with normal lens.
— Ansel Adams, The Camera, p. 116.

Adams' observation are confirmed by a test conducted in 2008 by Erwin Puts. He photographed a Siemens star with three top class bodies (Nikon, Leica and Olympus) from a distance of two meters. Each body was fitted with very good lens with a FOV equal to around 100 mm on a body with a FX-format sensor. According to the “focal length reciprocal rule”, 1/100 should be a safe shutter speed for hand holding. However, Puts found that hand-held shots at much shorter shutter speeds did not have the same sharpness as shots taken on a tripod. He concluded:

Overall one might say that a shutter speed below 1/200 does not do justice to the potential image quality in all three cases.
— Edwin Puts, Vibration Reduction compared with handholding.

Another of Puts' findings that is worth mentioning is that VR actually works:

The Nikon offers in-lens vibration reduction and at 1/100 the results are very close to what you get on tripod!! Only the very fine details get lost in the process, but a contrast of 20% at 1200 lp/ph is an excellent performance. This result holds for shutter speeds as low as 1/15. One may say that VR at 1/15 is much better than non-VR at 1/180. For number aficionados this implies an improvement of 3.5 to 4 stops.
ibid.

Without VR, I think the old “focal length reciprocal rule” is of dubious value. Without any form of support, the minimum shutter speed to use for hand held shots should be:

min. shutter speed = 1/(focal length x crop factor x 2)

But the again, as Magnum photographer Henri Cartier-Bresson once said: “Sharpness is a bourgeois concept.”.

Macro Ratio

In macro photography, the term macro ratio (or magnification ratio) denotes the ratio between real life dimensions and how those dimensions are projected on to the sensor.

A macro ratio of 1:1 means that the projection of an object on to the sensor will have the same dimensions as the actual object (i.e. life size). A macro ratio of 1:2 means half life size, and a macro ratio of 3:1 will magnify objects to three times life size, and so on.

The macro ratio for a specific lens is constant and not a function of the sensor's physical size. A 1:1 macro lens on a 135-format film camera or a DSLR with a FX-format sensor will still be a 1:1 macro lens when you put it on a DSLR with a smaller sensor.

All a macro ratio of 1:1 tells you, is that 1 millimetre in “real life” will measure 1 millimetre when projected on to the camera's sensor. This does not means that the acquisition will be the same if you use a DSLR with a FX-format sensor and one with a DX-format sized sensor. If you, for example, were to photograph a ruler at 1:1, you would capture 36 mm of it along the long side if your camera has a FX-format sensor, but only 24 mm of it if you use a camera with a DX-format sensor.

In 35 mm film photography, a lens is not regarded as a “real” macro lens unless its macro ratio is at least 1:13). The image of a ruler photo­graphed at 1:1 on 135-format film will show a 36 mm long section of it across the long side of the negative frame. How­ever, if one used a Four-thirds camera (2.0x crop factor) to photograph the same ruler, a macro factor of 1:2 would be sufficient to capture 36 mm of the ruler across the long side. This has lead to some people insisting that on the Four-thirds system, a macro ratio equal to 1:2 is “equivalent” to 1:1 on a film camera or a digital camera with a FX-format sensor, and that it is proper to use the term “macro” to designate Four-thirds lenses with a macro ratio equal to 1:2, or more.

IMHO, the notion of an “equivalent” something (e.g. macro factor, focal length, etc.) when discussing the effects of using sensors of different sizes should be avoided, Whatever pedagogical qualities such as an approach may have, there is also a lot of evidence that this approach leads to a lot of confusion among those mistaking equivalence for actuality. It is much better to use the actual value – and to educate oneself and others to understand what this entails.


Endnotes

1) A brief note on terminology. Some people dislike the term “crop factor” and insist on using another term, such as “lens factor” or “focal length conversion factor”. I think those terms are misleading because they suggest that the lens' focal lens somehow is “converted” to some other focal length. It is not. The only thing that happens is that the parts of the image circle that is outside the sensor is cropped, and that makes “crop factor” the most appropriate term.

2) Strictly speaking, for a Bayer camera the true value lies somewhere between half the pixel pitch and half the diagonal of one RGGB block, which is equal to the pixel pitch. The exact value depends on the quality of Bayer demosaicing algorithm used, but 3/4 of the pixel pitch may be a reasonable approximation.

3) Except by marketing people. Marketing people attach the word “macro” to any lens that has a minimum focusing distance marginally shorter than the norm.

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3 responses:

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I've read your papers on different photographic themes. They are very instructive and well written.

Thank you!

One of the most detailed, comprehensive and clear explanations I've found on the web, on the pros and cons of different sensor formats in photography. I only regret not having found it earlier!

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Without any prior knowledge every one can understand most confusing terms linked to cameras, very clear explanation.

Thank you

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