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Home , Hyperfocal distance

Stacking

Associated key issues: - Depth-of-Field - Image Quality

Overview 1. Reproduction of slides (pages 2-11) 2.Two-page summary (pages 19-20)

Detailed notes

1. Depth-of Field (pages 12-15) 2. Large Prints (page 14) 3. Focus stacking (page 16) 4. Resolution & Diffraction Limit (pages 17-18)

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Depth-of-Field

The basic concept of Depth-of-Field

In the image below the camera is focused at infinity on the most distant part of the scene. While the image is strictly focused only at the plane at infinity, to the human eye much of the scene in the mid-ground is perceived to be in focus. As we come to the foreground the image is seen to become out-of-focus and the cross-over line at a distance H from the camera is the demarcation of our perception from out-of-focus to the in-focus part of the image. The region from H to infinity is known as the Depth-of-Field (DoF) of the image. With the camera focus set at infinity the distance H is known as the Hyperfocal Distance and is a crucial factor in defining DoF. With the camera focussed at infinity the DoF range is H - ∞. However, if we re-focus the camera to the distance given by H, the image still remains in focus at infinity but the lower limit of DoF now extends to H/2 as shown later in Figure 1. Thus, in practice, we need only focus from close-up to the hyperfocal distance at H (0 < x ≥ 1). Focussing at H gives the maximum possible DoF range of H/2 - ∞. As the plane of focus reduces from the hyperfocal distance the DoF range reduces considerably to be only a few mm either side of the focal plane at small distances.

The Hyperfocal Distance

The formula for H is well-known and given by

H = 1/C x f2 / f-stop

It is clear that H increases by the square of the lens and decreases linearly with increasing f-stop. The constant C (known as the circle-of-confusion) is a measure of the maximum resolution we can perceive. For detail smaller than C we perceive to be in focus and it is only for objects larger than C than we can begin to perceive out-of-focus effects. The value for C is a crucial number and depends upon three main factors:

1. the sensor or film size (given by the “Crop Factor”); 2. the image size and viewing distance; 3. human vision capabilities. We will deal with each of these factors in turn. Page 13 1. The sensor or film size The sensor or film size of the camera is conveniently related by the relative diagonal dimension used to define lens requirements. This is commonly expressed as the Crop Factor (CF) given by

Crop Factor = 43.3mm /(diagonal of sensor or film in mm)

Where 43.3 mm is the diagonal length of a 35mm or full frame sensor. Thus a full frame sensor has a Crop Factor of 1 and, for example, an APS-C sensor (15.6mm x 23.5mm) has a Crop Factor of 1.5 and a 20cm x 25cm (8 x 10 inch) has a Crop factor of 0.14. The Crop Factor for other sensor or film sizes are readily calculated.

2. The image size and viewing distance The image size and from how far way we view it affects our perception of the Depth-of-Field of the image. From far away much of the image may appear to be in focus until we take a closer look to find areas which are clearly out-of-focus giving us a perceived reduction in Depth-of-Field. For Depth-of-Field calculations to have any use we need to standardise an acceptable viewing distance, Vdist. It is generally accepted that the best value for Vdist is that given by around 1.5x to 2x the length of the diagonal of the image. Taking the lower number, this can be calculated from the Crop Factor as Vdist ≈ 43.3mm x 1.5/CF or simply estimated directly.

3. Human Vision The resolution of human vision for standard 20/20 eyesight (about 75% of the adult population including the use of glasses and contact ) is illustrated in the following diagram.

Objects smaller than D in diameter cannot be resolved which limits our ability to focus on fine detail. For objects where the diameter is larger than D we can begin to detect any out-of-focus detail in the image. The value for C For best performance we need to match the value of D in the above figure to the constant C (Circle-of- Confusion) at the viewing distance, Vdist, taken here as given by ~1.5x the image diagonal. At full frame where CF = 1, Vdist ≈ 70mm and we calculate D for 20/20 vision (0.017 degree in radians is 0.00029 so D = 0.00029 x Vdist) as 0.02mm giving, with D = C, 1/C in the Hyperfocal equation as 50. For different format sizes the value for C must be scaled to the diagonal dimension (the viewing distance, Vdist) of the format so for all format sizes we have 1/C = 50 x CF. Resolution on a print is described in terms of points-per-inch (ppi) or line-pairs per millimetre (lp/mm). For line-pairs of 50% contrast (MTF = 0.5) the relationship between the diameter C and the print resolution in line- pairs per millimetre is lp/mm = 0.72/C. With C = 0.1mm, 5x the magnification of the value just estimated for a full frame sensor, we get 7lp/mm or 360ppi printer resolution. Note this is the limit of Epson printers and beyond the limit of resolution of other commercial inkjet printers such as from HP or Canon (300ppi). The implication is that for prints smaller than 5x full frame (12cm x 18cm) or 22cm diagonal from an Epson printer and 26cm diagonal from an HP or Canon printer, the resolution of 20/20 vision will not be meet. Prints larger than 12cm x 18cm present no problem as they demand decreasingly smaller values of ppi resolution as D increases in value. Specifically, minimum ppi is given by ~118/(Vdist in metre value).

Page 14

The Hyperfocal Distance equation With the value for 1/C just calculated, we now have the Hyperfocal equation for a viewing distance equal to around 1.5x the diagonal of the print or projected image for 20/20 vision given as

H = 50 x CF x f 2 / f-stop

This Hyperfocal distance equation applies to all images printed or projected. For a given Crop Value, focal length and f-stop this value for H does not apply if the viewing distance (Vdist ) is much different than ~1.5 the diagonal of the printed or projected image.

Values for C and the Crop Factor are given in the table below for commonly-used together with the hyperfocal distance calculated at f 8 (rounded to a convenient number) for a standard lens. As seen, the value for H decreases rapidly with the smaller formats.

Format C CF Standard Lens H @ f 8

20cm x 25cm 0.143mm 0.14 360mm 113m

Digital 0.031mm 0.65 75mm 23m Full Frame 0.02mm 1 50mm 16m APS-C 0.013mm 1.5 35mm 12m Micro Four Thirds 0.01mm 2 25mm 8m

High-End compact 0.004mm 4.8 10mm 2.8m

The traditional value for the Hyperfocal distance was established in the 1930s when the printing and projection of images was a lot softer than is the case now. It is based in viewing a 20cm x 25cm print at 30cm with an assumed value for C of 0.25mm. This is applicable to someone with 20/60 vision. Clearly the traditional value is inadequate and needs to be revised. Despite the urgings of many to do so there is great inertia to change and the traditional definition largely remains as seen, for example, when using Hyperfocal distance calculators on the internet. Large Prints

As discussed above, at a comfortable viewing distance equal to around 1.5x the diagonal, Vdist, of the image, we need a print ppi resolution of 118/(Vdist numerical value in metres) for normal 20/20 vision to completely resolve the image. At a maximum resolution of 360ppi this translates to a minimum print size of 12cm x 18cm when the viewing distance is ~1.5x the diagonal dimension of the image. The total count is around 4.5MB and this is all that is required for larger prints given the proportionately reduced resolution required for larger prints. The implication is that any sized print from 12cm x 18cm upwards can be printed from a 4.5MB camera!

For „pixel peepers‟ viewing a 24cm x 36cm print at a distance of 33cm appropriate to a 12cm x 18cm print we would need a 24MB camera for full viewing resolution. In addition, the perceived Depth-of-Field will be halved for the pixel peeper. It is difficult to imagine a pixel peeper would view an even larger print of 48cm x 72cm at this same distance (and further halving the Depth-of-Field), but if so, our inveterate peeper would need a 96MB camera for full satisfaction! Page 15 Depth-of-Field Overview

The above figure is a complete summary of the Depth-of-Field. The horizontal axis is the focus position of the camera from 0 to H and the vertical axis is the distance from the camera through the subject. At a focus position S (as shown) the Depth-of-Field around the subject at S is the distance DoF shown in the figure within the in-focus region. Out-of-focus regions are shown in blue.

At a focus distance of H the greatest Depth-of-Field is achieved with DoF ranging from H/2 to infinity. As the focus to the subject distance decreases the range of DoF is seen to decrease quite dramatically until for close-in subjects it becomes very small indeed. When S is less than 0.2 x H the Depth-of-Field extends either side of the focal plane by an equal amount of [S x (S - f ) /H] giving a total Depth-of-Field

DoF = 2 x S x (S - f )/H mm

On page 19 we provide a full annotation of this figure so the Depth-of-Field can be calculated accurately. Page 16 Focus Stacking Close-up subjects The main application of focus stacking will be for close-up subjects where the Depth-of-Field is small and unable to focus the entire subject of interest with a single image. Figure 2 below illustrates an example of a close-up of a subject where four images with the same Depth-of-Field but taken at different focal lengths are necessary to encompass the entire subject. We place the first focal plane, S1 in the figure, in front of the subject. To have a continuous Depth-of-Field for subsequent focal planes we set S2 = S1 + DoF, S3 = S2 + DoF, S4 = S3 + DoF where DoF is given from the previous page as 2 S1 (S1 –f) /H.

In taking the several images the camera must be on a tripod and the only change between images is the different focus settings S1 to S4 using manual focus.1 You will need to check with a target beforehand to determine how much rotation of the focusing ring is required to alter the focus position between planes. Alternatively, you can make smaller (guessed) changes of the focusing ring with each image to try and ensure all parts of the subject have been captured in focus. While this is likely to result in more images than strictly necessary it should not be a subsequent processing problem provided there are not more than about 20 images. Above all, avoid camera movement between takes and for this reason it is best to minimise the number of images required. To reduce the number of images choose the f-stop at the diffraction limit fdiff of the lens (see next section) as this will provide the largest value for DoF and thereby reduce the number of images to be taken. You may be tempted to increase the f-stop to even higher values but this will result in increasingly un- sharp images and should be avoided.

Once the images have been taken, stack the images in Photoshop or Photoshop Elements. In the latter case you will need to purchase „elements plus‟ (download for under $20 from htpps://elementsplus.net). In the case of Photoshop go to Edit > Auto Align followed by Edit > Auto Blend. The latter process selects the in-focus parts of the subject from each image in the stack. Once processed (and this can take some time) flatten the image and it becomes a standard image for any further work.

H

Figure 2:

Distant subjects In taking distance subjects, such as landscapes, setting the focus at the hyperfocal distance, H, will give you a Depth-of-Field of H/2 – ∞ which will be sufficient in most cases. If a greater Depth-of-Field is required, then the same principles apply as for close-up photography but is unlikely you will need more than 2-3 images. Be aware, however, that with the larger change in focus compared to close-up subjects, there will be some perspective issues with small changes in Field-of-View which will require some judicious blending of the images.

1. An alternative to re-focusing is to mount the camera on a rack and physically move the camera in the axial direction. However, this will provide more likelihood of minor changes in perspective than re-focusing and best avoided. Page 17 Resolution and Diffraction Limit

In a system the overall resolution depends on both the lens and the sensor.

Lens Resolution The resolution achieved by a perfect lens decreased with f-stop at a rate of 720/f-stop lp/mm or 36,000/f-stop ppi.1 The reduction in resolution with higher f-stops is significant as seen from the table.

Resolution of a perfect (diffraction-limited) lens with f-stop

f-stop f 1 f 2 f 4 f 8 f 16 f 32 f 64 Resolution 36,000 ppi 18,000 ppi 9,000 ppi 4,500 ppi 2,250 ppi 1,125 ppi 563 ppi

Sensor Resolution The resolution of the sensor has a fixed value depending on the size and megapixel count of the sensor.2 At the f-stop where the lens resolution is equal to the sensor resolution is the diffraction limit fdiff as illustrated in the figure. For f-stops smaller than fdiff the sensor is the limiting factor on resolution while for f-stops above fdiff the lens becomes the limiting factor. Page 18

Overall Camera Resolution To properly assess a camera system it is necessary to consider the lens and sensor (camera body) in combination. One of the most usual resources in a realistic assessment of overall camera performance is provided by DxO Labs (.com). DxO have a vast range of data on lens and camera combinations. For resolution they have simplified the procedure with their concept of Perceptual Megapixels, P-Mpix, a megapixel number3 to replace the MP value for the sensor. P-Mpix is always equal or smaller than the sensor megapixel count, MP, and provides a simple method of accounting for the realistic resolution performance of a given camera system. (Reduction in resolution from the ideal case given by MP can be substantial and values for P- Mpix of around half of MP or less are very common.)

Taking the value for P-Mpix for your lens/camera body combination given by DxO, the following table enables you to choose the appropriate value for fdiff 4 within the given range of P-Mpix values in the table. It is best to avoid f-stops above fdiff unless you especially want a large Depth-of-Field and at the same time producing relatively smaller prints, as for every increase in f-stop above fdiff the resolution is reduced by √2 with a consequent reduction in resolution and maximum print size.

Table of P-Mpix values to determine the diffraction limit f -stop for common formats

Common Formats P-Mpix values (https://www.dxomark.com) P-Mpix range for Medium Format 10 - 20 21 - 39 40 - 100 - P-Mpix range for Full Frame 6 - 12 13 - 23 24 - 54 - P-Mpix range for APS-C 3 - 6 7 - 11 12 - 24 -

P-Mpix range for Micro Four Thirds 1 - 2 3 - 5 6 - 13 14 - 20

Diffraction Limited f -stop fdiff = 16 fdiff = 11 fdiff = 8 fdiff = 5.6

Notes

1 These figures are for a 50% contrast level (MTF=0.5).The resolution of an actual lens can vary markedly from the perfect lens at small f-stops depending on the quality of the lens with imperfections in manufacture. For larger f-stops greater than about f 11 a well-manufactured lens comes close to the resolution of a perfect lens. 2 The maximum resolution of a sensor in ppi is given by 850 x (Crop Factor) x sqrt (sensor megapixel value, MP). 3 The single P-Mpix value for a given by DxO is the best result achieved over the range of and for a zoom it is averaged over the best results over the entire range of focal lengths. For a more accurate result, especially in the case of a , you can dig down into the measurements provided by DxO and find the individual P-Mpix value for both a given aperture and focal length.

4 The diffraction limit fdiff = 42 /[(Crop factor) x sqrt (P-Mpix)]. For an ideal system P-Mpix = MP

Page 19 Depth-of-Field Summary

Chart Title DoF/H 2.5

2

DoF upper limit

1.5 [ x/(1 - x) ]

1

Plane of

focus DoF/H

0.5

DoF lower limit

[ x/(1 + x) ] 0

0 0.2 0.4 0.6 0.8 1 1.2 x = S/HS/H

Close-up{ Distant subjects Series1 Series2 Series3 subjects

Figure 1: Depth-of-Field (DoF/H) with distance to plane of focus (S/H)

For all projected images or prints where the viewing distance is 1.5x the diagonal, the Hyperfocal distance, H, for 20/20 vision is given by

H = 50 x (Crop Factor) x f 2 /f-stop; mm

[Note: this is almost three times the traditional value established in the 1930s and is now inadequate, being limited to the equivalent of 20/60 vision.]

The practical application of Depth-of-Field curves For a given distance (S/H) to the plane of focus the DoF either side of the focal plane can be determined from Fig. 1 (approximated assuming S >> f ).

The particular case of close-up subjects when x is less than 0.2 and S may be similar in size to f, the Depth-of-Field extends either side of the focal plane by an equal amount of [S x (S - f ) /H] giving a total Depth-of-Field

DoF = 2 x S x (S - f )/H mm

For greater Depth-of-Field, H should be as small as possible. Page 20 Camera Image Quality (IQ) Key elements 1. Dynamic Range

High performance Typical reduction in dynamic range with ISO sensor @ 100 ISO 400 ISO 1600 ISO 6400 ISO ≥ 13 stops By 1-2 stops By 2-4 stops By 4-5 stops 2. Diffraction Limit (see dxomark.com for P-Mpix values) Format P-Mpix values and associated Diffraction Limit Full Frame 6 - 12 13 - 23 24 - 54 - APS-C 3 - 6 7 - 11 12 - 24 - Micro 4/3 1 - 2 3 - 5 6 - 13 14 - 20

Diffraction f = 16 f = 11 f = 8 f = 5.6 limit diff diff diff diff

Format Crop factor

Focus stacking Summary Full Frame 1 APS-C 1.5 Setting up the camera Micro 4/3 2

1. Set at 100 ISO and f-stop set at fdiff for your lens/camera combination (macro or standard lens) 2. Turn off image stabilisation and mount on a tripod 2 3. Calculate the hyperfocal distance H = 50 x (Crop factor) x f / fdiff mm 4. For given distance to subject, S, calculate DoF1 = 2 x S x (S - f) / H mm. Similarly, for the distance S+X, where X is the subject extent, calculate DoF2 = 2 x (S+X) x (S+X-f) / H mm. 5. From the DoF values just determined, and with the aid of a test target, establish how far to rotate the focusing ring* between each shot at each end of the subject. 6. Beginning at S+X take a series of images, if possible, by remote shutter release. 7. Re-focus (gently!) between shots rotating the focus ring by the amount determined in 5.

* for a given lens this is required to be determined only once for any objects of similar distances and extent.

Post processing

In Photoshop – stack and highlight all the images in the stack. Got to Edit > Auto Align following by Edit > Auto Blend. Flatten. Done!

In Elements – you will need to add ‘elements plus’ (see http://elementsplus.net) downloaded for less than $20. For a demonstration see: https://www.youtube.com/watch?v=NgoUF1DA_T4