The Holding Tank

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Depth of Field, Image Quality and Sensor Size

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Sensor Size and Angle of View

Most current (2005) DSLRs other than a few top end professional versions such as the Canon 1Ds series have sensors smaller than a normal 35mm frame and are APS-C sized compared to Full Frame 35mm.

The effect of this is to crop the frame causing the angle of view for a given lens focal length to be reduced.

The cameras are often described as having a lens focal length multiplier of some factor such as 1.6X in the case of the Canon 20D.

This is very confusing to many novices, often encountered misconceptions are:

1. The focal length of the lens changes: No it doesn't the crop changes.

2. The perspective changes: Not of the same angle of view it doesn't. For example a 50mm lens on a 20D will have the same angle of view and perspective as  an 80mm lens on a 35mm film camera or a full frame digital camera such as the 1Ds.

The camera angle of view (AoV) is defined by the maximum image size Li and the lens focal length F and the magnification M with the following equation:

AoV=2.arctan(Li/F/(1+M)) - Eq 1

In fact the rule of multiplying the lens focal length by the crop factor is useful over the range of focal lengths of interest to connect the AoV of a smaller sensor to that of full frame 35mm. Note that as magnification increases from M=0 at infinity focus to M=1 (life size the lens AoV decreases by approximately 2.

It is interesting to look at internal focus lenses. These focus by moving the rear principle point with the byproduct that the focal length is shortend, this can be significant for macro lenses.

For example the focal length of the EF 100mm f2.8 Macro lens can be estimated to vary from 93mm at infinity to 64mm at 1:1 life size. For Li=22mm (APS-C width) this gives an AoV of 26.5 to 19.5 degrees. If the focal length remained fixed at 93mm and magnification was achieved only by overall linear extension as assumed in equation 1 above then the AoV would be 13.5 degree. Thus internal focus lenses mitigate vibration sensitivity slightly compared to OLE lenses, in this case by around 0.5 stop.

So lets define the terms:

1. Instead of focal length multiplier lets call it crop factor. So a 1.6X crop factor sensor has approximately half the area of a 35mm full frame camera.

2. When I say full frame lets understand this means the standard 35mm film frame and not have any semantic whinging like "...to me full frame is an 8X10 land camera".

However, one thing that is different is Depth of Field.

Hand Hold Shutter Speeds

For 35mm photography the 1/f rule of thumb is often used to indicate a minimum shutter speed for reasonably sharp photos. Like DOF and circle of confusion size this is print size dependent, so it is advisable to use a stop or two faster than the minimum if you possibly can, or a lot more for longer telephotos.

For digital cameras with a smaller format sensor the increased magnification for printing requires that the crop factor be taken into account. So for a APS-C camera the rule is modified to be 1/f/crop_factor, or 1/(f*1.6). So for a 100mm lens on APS-C the minimum safe shutter speed will be 1/160.

The are a couple of other things which also effect the safe minumum speed.

Increased magnification

If magnification is increased by extentension to life size the lens angle of view is halved. As a result it is ncessery to double the shutter speed used for safe handholding. This seems applies however you get the magnification, ie by internal focus macro lenses or close-up dioptre filters. So a 100mm macro lens focused to give life size reproduction on and APS-C camera will require1/320 as a minimum shutter speed. (See first section)

The actual increase is proportional to 1+M, where M is the magnification so we arive at the below plot:

Very wide angle lenses.

As we say earlier in this section the angle of view of a lens focused at infinity is 2.atan(r/f). Assuming that the r is the distance from the sensor centre to the sensor corner and f is the lens focal length, and without loss of generality neglecting issues of asymmetric focal lengths in SLR lenses.

So if c is the desired size of the circle of confusion as a sharpness criteria we note the angle subtended at the centre of the sensor by the CoC is a1=atan(c/f) and the angle subtended by a CoC in the corner is a2=atan(r/f)-atan((r-c)/f)

So we can introduce the shake sensitivity ratio for the corner compared to centre of the image as Shake_Ratio=a1/a2.

Plotting this factor Shake_Ratio factor gives the below graph assuming c=0.02mm for APS-C and c=0.035mm for 35mm.

We can see that the factor converges asymptotically to one for lenses with angles of view above 85mm on full frame 35mm.

For focal lengths below 25mm on 35mm the factor starts to become significant and becomes about a stop (factor of 2) for ultra wide lenses.

However the above is the peak difference in shake sensitivity for the corners. The increase in sensitivity is dependent on the offset from optical centre as shown in the next two plots below.

Because the area covered by pixels at the lower shake sensitivity make up more of the picture than those in the corner we can look at the distribution of shake sensitivity in terms of the number of pixels as shown in the next two plots.

So we can see that the RMS (root mean square) shake factor corresponds to about 68% of the pixels

So we can plot the RMS (~68% of pixels are less sensitive than this) and 90% (of pixels are less sensitive than this) shake factors against focal length rather than looking just at the peak or worst case shake factor.

One of these two plots gives a better compromise if the photographer is seriously exposure challenged with a wide or ultra wide lens.

So we can see that for ultra wide lenses a shutter speed increase in about a stop is sufficient for 90% of the image area and for wide angle an increase of about 1/2 stop is sufficient for 90% of the image area. For 100% of the area values of 1.5 stop and 1.0 stop above the format specific 1/f rule should be used ideally.

On top of the minimum guide values another stop is highly recommended.

For 90% of pixels this gives the following recommend minimum shutter speeds.

And repeating the shutter speed plot in tabular form.

 35mm focal length mm 14 16 22 28 35 50 85 35mm shutter speed 1/ 34 33 34 38 43 55 88 APS-C focal length mm 10 12 17 22 28 35 50 APS-C shutter speed 1/ 35 35 38 44 51 61 84

Depth of Field in General Photography

Some lenses have depth of field markings allowing you to estimate the range of distances that will appear reasonably sharp before and after the point of focus.

This is normally important for landscape subjects where the photographer wants to estimate if he can keep an interesting foreground detail and the distant mountains in good focus simultaneously requiring a small lens f-stop.

Another area where DoF is important is in portrait and wildlife photography for example, where the photographer wants the subject in focus but distracting background and foreground detail blurred. This requires a large f-stop.

The equations in Eq 2 for the Near and Far point of the DoF depend on the quantity of hyperfocal distance. This is the distance you set your lens to for a given f-stop so that infinity is just within the DoF.

- Eq 2

Here H is the hyperfocal distance, F is the lens focal length, f is the f-stop used to take the picture, S is the distance of sharp focus and c is the diameter of the circle of confusion.

So c is an arbitrary sharpness factor to be selected based on the users required sharpness. The value used to calculate any given lens engraved DoF scale depend upon the manufacture.

Calculating the hyperfocal distance for a range of angle of views (AoV), expressed on the x-axis as the equivalent 35mm focal length, for full frame 35mm and 1.6X crop factor camera formats at f/8 gives Figure 2-1.

Figure 2-1

The green trace shows the hyperfocal distance for a full frame camera with c=0.03mm, a typical 35mm format sharpness factor.

The red trace shows the hyperfocal distance for the same AoV on a 1.6X crop factor, again with c=0.03mm.

The shorter hyperfocal distance indicates that there is considerably more DoF for the smaller format camera at a given focal length.

However, the size of the circle of confusion should be divided by the crop factor to produce the same level of sharpness, this gives the blue trace with less DoF, this still has more DoF the full frame trace for the same AoV and f-stop.

This has two important conclusions for practical photography:

1. If using the DoF scale on lenses designed to work with full frame 35mm cameras evaluate the DoF using a stop wider f-stop than the one you are using to take the shot for 1.6X crop facror camera.

2. DoF control (limiting) is more of a problem for cameras with sensors smaller than full frame, so wide aperture lenses are essential if this is an important factor for you.

3. Conversely the increased DoF for smaller sensors gives a more easily obtained DoF for landscape photography when compared to full frame.

This format dependent DoF ratio is constant at all focal length as shown in Figure 2-2:

Figure 2-2

A set of DOF plots for APS-C cameras showing near and far points for typical Canon lens marked distances and focal lengths is provided here and the f-stops required for various nature subjects is plotted here.

Below is a simple table worked out for distance settings normally marked on Canon lenses. The an EXCEL version with instructions can be downloaded from here

Harold M. Merklinger has published a very interesting article "Depth of Field Revisited" where he argues for an alternative approach to the traditional hyperfocal focusing method. Some discussion analysis of this is looked at here.

The below plots show how the circle of confusion diameter increases away from the plane of ideal focus and beyond the APS-C DOF and 35mm DOF lines.  This is plotted against focus distance and magnification.

DoF with Macro Photography

DoF becomes very limited for close-up and macro photography.

The relevant equations are given in Eq-3 below:

There are two equations, the first for a reversed lens and the second for a lens mounted normally. (In the days of manual focus cameras it was common to reverse lenses for application at magnifications of 1 or more for quality reasons.)

Here M is the image magnification, P is the lens pupilary magnification, f is the f-stop used to take the picture and c is the diameter of the circle of confusion.

Typically for the 35mm format P is approximately 1 for a 50mm lens, and P>1 for wide angle and P<1 for telephoto. Note that P will change for a zoom dependent upon the zoom focal length setting, and may change slightly for modern prime lens with internal focusing dependent upon the focus distance.

The image magnification M is defined as the ratio of the image size on the film or sensor to the physical subject or object size.

The lens pupilary magnification P is defined as the exit (rear) pupil size divided by the entry (front) pupil size with lens in normal or forward mount.

If you need to you can estimate P by squinting with a ruler close to the front and back lens elements to measure the apparent diameter of the aperture.

The equations are valid for c.f/focal_length << M and so are usable for moderate closeup work also. The forward DOF equations is derived here.

Considering only the normally mounted lens case for P=1 as an example and setting the size of the circle of confusion c to 0.03mm for full frame and 0.03mm/crop factor for smaller sensors. Similarly the smaller sensor magnification will change so that magnification required for full frame 35mm M35 will only need to be M35/crop_factor to fill the smaller sensor with the subject.

Note that the exposure correction factor will be (M/P+1)2 and ((M.P+1)/P)2 for normal and reverse lenses respectively.

We then can calculate the DoF at f/16 for full frame and 1.6X crop factor sensors as in Figure 3-1:

Figure 3-1

Comparing the DoF for full frame to the 1.6X crop factor sensor gives Figure 3-2:

Figure 3-2

We can see that the smaller sensor provides improved DoF for a given image size as a percentage of the format frame. However this advantage reduces asymptotically as magnification increases.

Note that macro DOF depends on magnification and pupilary magnification for a set f-stop. If however the light loss is figured into the equation so the set f-stop is adjusted for various magnifications and pulpilary magnifications so the effective f-stop is constant the DOF is then only dependent on magnification, This is illustrated in the below plots showing the APS-C DOF for a f-stop set of f11 and and effective f-stop or f11 respectively.

The tradeoff of shooting photographs at lower magnification and cropping to improve depth of field is investigate here.

Diffraction in Macro Photography

Diffraction normally limits lens sharpness at infinity from about f16 to f22 on APS-C and f22 to f32 on 35mm depending exactly on the lens characteristics. However this is at infinity. For macro work the amount of diffraction scales linearly with M as the angle of diffraction remains the same for the same physical aperture as shown below.

The angle of diffraction is inversely proportional to the size of the diffracting feature, so making the simplifying assumption that sin(a)~= a for small a the diffraction effects become significant about two f-stops earlier for every doubling in magnification because f-stops change diameter by the factor square root 2 every stop.

This works out to being the same as the effective aperture f-stop*(M/P+1) for P=1.

 Magnification Diffraction Limit APS-C Approx Set f-stop Diffraction Limit 35mm Approx Set f-stop 0 (infinity focus d=f) f22 f32 1 (d=2.f) f11 f16 2 (d=3.f) f7 f11 3 (d=4.f) f5.6 f8 4 (d=5.f) f4.5 f6.5 5 (d=6.f) f4 f5.6

However there remains a tradeoff between peak sharpness and depth of field in macro work, a good example of this has been provided by the well known macro photographer LordV.

Degree of Out of Focus Blur

Blur at Constant Distance

We can solve eq-2 in terms of the circle of confusion (CoC) to get a sense of the amount of blur in out  of focus areas. Plotting the size of the circle of confusion normalized (CoCN)  relative to the size of the CoC required for acceptable focus, in these cases CoC acceptable is 0.019mm which is OK for an APS-C sized sensor like the Canon EOS 20D.

So in the following plots, if the CoCN is 1.0 or less the object is in focus. For good blur we probably want CoCN > 10. In all cases the set focal length is 3M.

Figure 4-1a

Figure 4-1b

Figure 4-1a and 4-1b show how not only the depth of field but the gradient against distance of blur and indeed the maximum level of blur change with focal length.

Repeating the same exercise for a standard lens of 35mm for APS-C format at various f-stops.

Figure 4-2a

Figure 4-2b

Figure 4-2a and 4-2b show the effect on not only depth of field but also a significant change to the maximum blur for distant objects.

We can use the method of calculating the size of the blurred objects CoCN by comparing the performance obtained using two equivalent portrait focal lengths for 35mm and APS-C format, 85mm and 50mm respectively. The typically available apertures for these lenses are f/1.8 and f/1.4 respectively. This gives rise to the CoCN plots of Figure 4-3a and 4-3b, these are both relative to the appropriate acceptable CoC sizes of 0.03 and 0.019mm respectively.

This shows that the maximum blur available to an APS-C sensor like to EOS 20D’s is about 75% of that compared to a similarly framed shot on 35mm although the depth of field is not vastly different.

Perusing the format differences further in Figure 4-4a and 4-4b to establish the f-stop for equivalent performance.

This shows that as far as available out of depth of field blur is concerned, a 50mm f/1.4 lens on the EOS 20D is equivalent to a  85mm f/2.4 lens on 35mm. This is a 1.6 stop penalty for APS-C.

This has important implications for the creative control of the image. It is often said that painting is an additive process and photography is a subtractive process. A major tool for subtraction of elements from an composition is control over depth of field and blur. This control is maximized with large aperture lenses. This tool is harder to use on APS-C format than 35mm.

Blur at Constant Magnification

Instead looking at the degree of blur for a constant magnification gives a rather different view.

The equation for near and far depth of field can be written as:

Sn=vn-v=[(P-1)·(v-F)·c·F+P*F2·v/f] / [P·c·(v-F)+P·F2/f]

Sf=vf-v=[(P-1)·(v-F)·c·F-P*F2·v/f] / [P·c·(v-F)-P·F2/f]

Where v, vn and vf are the distances of the planes of sharp focus and near and far DOF limits for CoC size c and f-stop number f using a lens of focal length F and pupilary magnification P.

Noting that the Gaussian lens equation applies with the distances from the lens principle planes for subject and image v and b respectively and the definition of magnification M:

1/F=1/v+1/b and M=b/v

So, v=f·(1+1/M)

Modern lenses with internal focus will vary the value of F and P as the focus (change magnification), however in this analysis the simplifying assumption is F and P are fixed. Typical values for P for given focal lengths are measured from various EF and EF-S lenses.

The CoC size is plotted below for various focal lengths at certain magnifications and f-stop numbers. For APS-C and 35mm typical depth of field values for c are 0.019 or 0.03mm, so values of 0.2 to 0.3 represent substantial blur for subject isolation.

So we see from these plots that a shorter focal length tends to provide more far depth of field and less near depth of field than a longer lens for the same magnification.

However, at significant close-up and macro magnifications the telephoto lens tends to provide more blur (less depth of field) than the wider angle lens for a given set f-stop.

Re-ploting for life size for a Set f/5.6 and an effective f/5.6 produces less difference in the degree of blur for the same light transmission.

This last plot suggests that when exposure and diffraction is considered there is minimal difference in the degree of blur isolation possible at different focal lengths at macro magnifications.

At magnifications above about 0.5 the effect of focal length begins to be significant again even when effective f-stop is taken into account.

So we can conclude that in terms of background isolation through blur there is minimal advantage in using a longer focal length above about 0.5X magnification.

At lower magnifications typical of the magnification used for larger subjects such as flowers there is a small advantage even when the effective aperture is compared.

Last Updated 05/06/2008

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