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A primer on sensor sizes
What effect to the sensor size have on images
Table of Contents
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. FX size, 135 format or full frame) is usually called the digital sensor's “crop factor”.
(A brief note on terminology. Some people dislike the term “crop factor” and insist on using another term, such as “lens factor” or “lens 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, and - as I shall show - 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.)
The sensors in the DSLR bodies from Nikon (except the D3), Fujifilm, Pentax and Sony have a crop factor of 1.5x, Canon's consumer DSLR bodies use sensors that have a crop factor of 1.6x. Sigma's DSLR bodies have a crop factor of 1.7x. Cameras built according to the Four-Thirds standard (Olympus, Panasonic and some Leicas) have a crop factor of 2.0x. Digital consumer compacts have sensors with an even larger crop factor. For instance, the Sony 1/1.8" type CCD used in a number of popular digital compacts has a crop factor of 4.8x.
There exists digital cameras where the sensor is exactly the same size as the 135-format film negative, for example the Kodak DCS Pro SLR/n, the Canon EOS 5D and the Nikon D3. We say that these cameras have a crop factor of 1.0x, and refer to them as cameras with FX size sensors.
Figure 1: The relative sizes of different sensor formats, from
the FX/135 film format to the 1/1.8" type CCD format used in many compact
cameras.
Below I discuss the effects of the crop factor in terms of field of view, perspective, depth of field, diffraction, f-stop, 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 in radians and degrees.
FOV (radians) = 2 x atan (D / 2 / f)
FOV (degrees) = 2 x atan (D / 2 / f) x 57.3
(D = diagonal of the sensor, f = focal length of the lens)
The table below show the results of computing the diagonal FOV for two different lenses on two different sensors. The first lens is a 135 mm lens used with 35 mm film (i.e. a crop factor of 1.0x), and the second 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 degrees. Lenses with a more narrow FOV of view than 45 degrees is generally regarded as telephoto lenses. I.e.:given a small enough sensor, a 28 mm lens is a telephoto lens.
Most photographers, however, 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” focal length from the real focal length: Just multiply the real focal length with the crop factor to get the 35 mm film “equivalent” focal length.
Example: Given the 1/1.8" type CCD sensor discussed above and a 28 mm lens, the 35 mm film “equivalent” focal length of this combination is:
28 mm x 4.8 = 134 mm.
Note, however, that the real focal length of the lens does not change. The crop factor multiplicator is just a convenient device to help the photographer visualize the FOV when using a certain lens on a digital camera with an unfamiliar sensor size. Other characteristics of focal length, such as depth of field, remains tied to real focal length.
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 normally associated with telephoto lenses, and a 28 mm lens will always give the “expanded” look normally associated with wide-angle lenses.
This is wrong. Perspective in a photograph is determined by one thing: the position of the photographer relative to the scene. (To be precise, perspective is dermined by the distance from the front nodal point to the scene).
Focal length doesn't enter into it. The reason people believe that a specific focal length effect a certain look is because the narrow FOV of telephoto lenses forces the photographer to take up a position far away from the scene, and the large FOV of wide-angle lenses 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 postion, 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.
Figure 2: The photograph above is taken with a 135 mm lens on
135-format film (35 mm) film camera and scanned.
Figure 3: 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
According to Leslie Stroebel et al, Basic Photographic Materials and Processes, 2nd ed., Focal Press, 2000, depth of field (DOF) is defined as the range of object distances within which objects are imaged with acceptable sharpness.
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 deem to be in focus. At distances shorter than the hyperfocal distance (explained below), this region is intersected by the perfect focus plane: 1/3 of the region lies in front of perfect focus plane and 2/3 lies behind.
A lot of the confusion surrounding the term DOF in photographic circles stems from a failure to understand what DOF is, and is not.
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, with a deeper DOF in the smaller print. 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 measure for DOF most people are interested in when moving between equipment with different sensor sizes, is the shift in DOF that happens at identical FOV. As explained in the FOV-section, to maintain FOV we need to change the focal length by the amout indicated by the crop factor. For instance, a f=90 mm lens gives us a FOV of 27 degrees on a camera with a FX-sized sensor. To get the same 27 degrees FOV on a camera with a DX-sized sensor (crop factor 1.5x), we must use a lens with 90 mm/1.5x = 60 mm focal length. The shift in DOF is also determined by the crop factor. To get the same DOF on both cameras, we need to open up the aperture on the camera with the smaller sensor to the f-stop we get when we divide the f-stop on the camera with the FX-sized sensor with the crop factor. Example: If we use f=90 mm, f/4.0 on the camera with the FX-sized sensor, we need to open up to 4.0/1.5 = f/2.7 on the f=60 mm lens fitted to the body with the DX-sized sensor. to get an identical DOF. This constitutes a shift of 1.2 EV. The table below shows the aperture shift needed to maintain DOF, comparing a number of popular sensor sizes to FX.
| Type | Crop | Examples | Shift (EV) |
|---|---|---|---|
| 1/1.8" | 4.8x | Canon G5 | 4.5 |
| 2/3" | 3.9x | Canon Pro 1 | 4.0 |
| 4/3 | 2.0x | Olympus E-1 | 2.0 |
| ? | 1.7x | Sigma SD14 | 1.6 |
| ? | 1.6x | Canon EOS 40D | 1.3 |
| DX | 1.5x | Nikon D300 | 1.2 |
| APS-C | 1.4x | Nikon Pronea 6i, Canon EOS IX | 1.0 |
| ? | 1.3x | Canon EOS 1Dn Mk II, Kodak DCS 760 | 0.7 |
| 135/FX | 1.0x | Canon EOS 5D, Nikon D3, Nikon F6 | 0.0 |
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 Powerhot G5, at its maximum aperture f/2.0 will have the same DOF as a camera with an FX-sized sensor have at f/9.7, given the same FOV.
What this means is that there is no way to duplicate the “look” created by 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-sized sensor, on a camera with a smaller 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-sized 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-sized 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.
If we want to compute numbers telling us what the DOF actually is, and related things such as the hyperfocal distance, there exists a number of different models that let us do this, given certain variables, such as lens aperture, focal length, camera-to-subject distance and magnification. They all have one thing in common: They recognize that DOF is a subjective property depending upon context, and they are only valid within their assumed context.
The model I shall use to compute DOF-values is a simple gaussian model of an optical system, taken from Allen R. Greenleaf: Photographic Optics, MacMillan, New York, 1950, pp. 25-27. In the formula shown below, f is the focal length, d is the subject distance, CoC is the circle of confusion, and N is the f-stop.
As can be easily seen, this is a hyperbolic function that converges towards zero in the macro region. To be precise, it is zero when the subject distance (d) is equal to the focal length (f). It converges towards infinity as the distance (d) increases. The value of the distance d when the function becomes infinty is known as the hyperfocal distance (H), and can be computed as follows:
H = f2/(CoC x N)
The circle of confusion is an estimate of the largest spot in the image that an human observer of average eyesight will consider to be a point, rather than a disk. It can be computed by the following formula:
CoC = D / zeiss-factor
The term D in this formula is the diagonal of the film or sensor (i.e. the film negative frame or the digital sensor) used to capture the image. The so-called zeiss-factor is not an arbitrary number, but is closely linked to the resolving power of the human visual system. A often quoted value for the zeiss-factor is 1730, but 1500 is probably more appropriate according to this Wikipedia article.
Assumming a zeiss-factor of 1500, we find that CoC for FX/135-format film (43.3 mm diagonal) is 29 µm, that the CoC for a DX-sized digital sensor (28.4 mm diagonal) is 19 µm, and that the CoC for a 1/1.8" type digital sensor (8.9 mm diagonal) it is 6 µm. If we make a print where the diagonal measures 30 cm, the CoC, given a zeiss-factor of 1500, will measure 200 µm on the magnified print. This is 1/5th of a millimeter and corresponds nicely to the fact that in good light a person of average eyesight viewing a print from a distance of 25 cm should be capable of resolving 5 lines per millimeter.
This simple model of an optical system assumes that singular points are infinitely small, that all lenses are perfect and symmetrical, that diffraction and airy disks do not exist, and that cross-talk, sensor pixel pitch, bayer interpolation and film grain does not interfere with resolution.
While by no means totally accurate, the simple gaussian model will suffice for the purposes of this essay, which is to predict how sensors of different physical sizes will impact upon the DOF of a photograph. For digital sensors, it should be noted that this model assumes that the CoC is larger than the sampling limit according to the Nyquist theorem (i.e. larger than twice the photosite pitch) . This contraint is, however, satisfied for all modern digital cameras with a pixel count of 6 Mpx or more.
Also note that at smaller apertures, resolution is limited by diffraction and not by the CoC. All the computations below are done for aperture equal to f/2, which is not diffraction limited, even for the smallest sensors (10 mm diagonal) plotted. However, a sensor of that size will become diffraction limited already at f/4 in a 10 Mpx camera.
All the computations used to generate the two figures below is based upon the gaussian model of the optical system presented above. I have used a zeiss-factor of 1730 to compute the circle of confusion. This may be too strict, and I plan to redo this with the more moderate figure of 1500.
Below is two figures demonstrating how depth of field (DOF) changes with various sensor diagonals and focal lengths of a digital camera if aperture is kept constant.
Figure 4 shows how DOF changes with we keep the field of view (FOV) constant - i.e. the focal length is at all times kept is equal the sensor diagonal. (This translates into a so-called “normal” lens). The figure clearly shows that given the same field of view, the wider focal length that must be used with a small sensor size (f=10 mm) will result in a deeper DOF than a lens that must be used with a large diagonal sensor (f=45 mm).
Figure 4: The Y-axis shows of depth of field (DOF) for a
“normal” lens for various sensor dimensions from 10 mm
to 45 mm.
If we, instead, keep the focal length constant but vary the sensor size, a larger sensor also results in a larger DOF, given that we keep all other factors (aperture, position with respect to the subject, print size and viewing distance) unchanged.
Figure 5 is a 3D plot showing how the DOF changes when both the focal length and the sensor diagonal vary. As in the previous figure, the vertical Y-axis shows the DOF and the X-axis shows the focal length. The sensor diagonal is now plotted indpendent of focal length, along the Z-axis. Note that if we keep the focal length constant (i.e moving up and down in the Z-dimension), DOF actually decreases with smaller sensor diagonals. The reason digital compacts has large DOFs is not because a smaller sensor gives a larger DOF, but because the short focal lengths that these cameras are equipped with gives you a large DOF.
Figure 5: The Y-axis shows the depth of field (DOF) for
various sensor dimensions from 10 mm to 45 mm (Z-axis) and
for focal lengths from 10 mm to 55 mm (X-axis).
In both figures, DOF is computed in millimeters for an aperture of f/2 and focus set to 1 meter.
I've also put up a small DOF Test Page that demonstrates the effect of sensor size or film size on depth of field.
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 = Twice Pixel Size?”. 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 pitch1) (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 / zeiss-factor
(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 differaction limited for different sensor sizes (assuming that about 3/4 of the sensor's pixel pitch is smaller than this).
N = (D / zeiss-factor) / (2 x 1.22 x λ)
Note that the diagonal (D) and wavelength of light should be expressed in the same units (e.g. nm).
In the table below, I've computed the aperture/f-stop (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 zeiss-factor is set equal to 1500.
| Format | Crop f. | D (mm) | λ=400nm | λ=550nm | λ=700nm | Example |
|---|---|---|---|---|---|---|
| 1/2.7" | 6.6x | 6.6 | f/4.5 | f/3.3 | f/2.6 | Nikon 775 |
| 1/1.8" | 4.8x | 8.9 | f/6.1 | f/4.4 | f/3.5 | Canon G5 |
| 1/1.7" | 4.6x | 9.5 | f/6.5 | f/4.7 | f/3.7 | Canon G9 |
| 2/3" | 3.9x | 11.0 | f/7.5 | f/5.5 | f/4.3 | Canon Pro 1 |
| 4/3 | 2.0x | 21.8 | f/14.9 | f/10.8 | f/8.5 | Olympus E-1 |
| ? | 1.7x | 24.9 | f/17.0 | f/12.4 | f/9.7 | Sigma SD14 |
| ? | 1.6x | 27.0 | f/18.5 | f/13.4 | f/10.6 | Canon EOS 40D |
| DX | 1.5x | 28.4 | f/19.4 | f/14.1 | f/11.1 | Nikon D300 |
| APS-C | 1.4x | 30.6 | f/20.9 | f/15.2 | f/12.0 | Nikon Pronea 6i |
| ? | 1.3x | 34.5 | f/23.5 | f/17.1 | f/13.5 | Canon EOS 1D |
| FX | 1.0x | 43.3 | f/29.6 | f/21.5 | f/16.9 | Nikon D3 |
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-stop
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-stop number is a numerical desigation for a lens' aperture. The f-stop number is derived from the lens' real focal length. It expresses the diameter of the diaphragm aperture as a fraction of the real focal length of the lens. The “f” in “f-stop” is actually shorthand for focal length. For example, f/4.0 represents a diaphragm aperture diameter that is one-quarter of the focal length. If the lens has f=100 mm, f/4.0 designates an aperture equal to 100/4=25 mm.
It follows that the size of the sensor does not affect the f-stop number. In other words, a lens that is f/4.0 on a camera with an FX-size sensor will also be f/4.0 on a camera with an sensor that is a different size.
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, this rule of thumb only apply if you use a camera with an FX-sized sensor. For other digital formats, the rule need to take the crop factor into account. In other words, you need to multiply the focal length with the camera's crop factor, and this becomes the minimum shutter speed to use for hand held shots, i.e.:
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-sized 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 the image from the DX-sized sensor need to be magnified more than the image from a FX-sized sensor for any given print size. Camera shake blur is also magnified, and the increased magnification must be compensated for by using a faster shutter speed.
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-sized 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 millimeter in “real life” will measure 1 millimeter 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-sized sensor and one with a DX 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-sized sensor, but only 24 mm of it if you use a camera with a DX-sized sensor.
In 35 mm film photography, a lens is not regarded as a “real” macro lens unless its macro ratio is at least 1:12). 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 suffiscient 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-sized 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.
1) 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.
2) Except by marketing people. Marketing people use the word “macro” to describe any lens that has a minimum focusing distance less than 50 cm.
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