General: Merklinger's Approach to Estimating Depth of Field (Extended Version)

On this page, I would like to investigate Harold M. Merklinger's approach to estimating depth-of-field in landscape photography, which can be seen as an alternative to using the standard depth of field approach. A special case of Merklinger's approach, namely setting the focus to infinity can also be regarded as an alternative to setting the focus to the hyperfocal distance in order to achieve sharpness from a certain distance up to infinity.

Note: On page Merklinger's Approach to Estimating Depth of Field (Short Version), I cover this topic without calculations and more concise. The page you are currently viewing is for the people who need the proofs and more details...

Introduction

A Sony RX100 M1 user from Russia made me aware of Harold M. Merklinger's approach to estimating depth-of-field in landscape photography. Admittedly, I had never heard of this approach before. He also pointed me to Kevin Boone's article Hyperfocal distances and Merklinger's method in landscape photography. In this article, Boone, among others, compares the practice of setting the focus to the hyperfocal distance in order to have everything sharp from a certain distance (which is, in fact, half the hyperfocal distance) up to infinity with setting the distance to infinity. This is actually a long-standing practice for photographers, but it can also be regarded as a special case of Merklinger's approach. Merklinger himself discusses the relationship between the two methods as well, and provides some numbers and information about the "infinity case". Thus, even if we would prefer not to subsume this case under the label "Merklinger approach", he at least, contributed useful information about it.

For those, who just want to use Merklinger's approach, Boone's article provides already the most important things that you need to know, although I do not find the paper easy to understand in every aspect... Nevertheless, you can find tables that help you employ the method (including added minimum distances = hyperfocal distances, see here), as well as a discussion about the traditional approach to depth of field and hyperfocal distance versus Merklinger's approach, including a discussion of the simple approach to just set aperture to f/8 and distance to infinity.

Merklinger also published a shorter article Depth of Field Revisited about his approach with less mathematics, but easier to understand for me.

Merklinger's Approach

Merklinger presents his approach in the paper The INs and OUTs of Focus - An Alternative Way to Estimate Depth-of-Field and Sharpness in the Photographic Image and calls it an object field approach (see side note) to depth of field - in contrast to the conventional image field approach based on the circle of confusion (CoC). His counterpart to the CoC in the object field is the disk of confusion (DoC). Merklinger explains:

... instead of asking what would make all my images look acceptably sharp, I might ask what objects which I see before me will be recorded in this particular image. What will be too small or too out-of-focus to be outlined distinctly? What objects will be resolved? What surface textures will be apparent in the final image? These questions are fundamentally different from that of asking what will achieve a uniform standard of image resolution. And as might be expected, we will not get the same answers or the same advice from our calculations.
... we are concentrating our attention on characteristics of the scene to be recorded - the object field (see side note) - as opposed to the characteristics of the final image. There is a very real distinction to be made here. When we concentrated on the image alone, we did not take into account what was being photographed. We had decided in advance on an across-the-board image quality standard.

Circle of Confusion (CoC)

Merklinger's "across-the-board image quality standard" mentioned above is represented by the circle of confusion (or CoC), which plays an important role in depth of field and hyperfocal distance calculations. For the following discussion, it is important to note that the circle of confusion lies on the film - that is, in the image field. The CoC is based on assumptions about how much detail the human eye can resolve when people view a photo at arm's length (about 25 cm), resulting in the CoC convention of 1/30 mm for 35 mm film (for different film/sensor formats adopted values are in use). For more information about the CoC, see the glossary.

Disk of Confusion (DoC)

The disk of confusion (DoC; diameter d) is, according to Merklinger, an exact analog of the circle of confusion (CoC; diameter c) to describe depth of field. The disk lies, however, in the object field, that is, in the scene to be photographed. Merklinger explains it as follows:

Let's think of it another way. Suppose we have in our camera, located on the film, a very tiny but bright source of light: a very tiny "star". That star will project its light through our camera lens (acting now as a projector). Wherever that starlight falls on a flat surface we will see a disk of light. The size of that disk of light will depend upon where the surface is relative to where the lens is and where it is focused. If the surface happens to be right where the lens is focused, we will see only a tiny bright point of light. A little ways in front of or behind where the lens is focused, we would see a small disk of light. ... We'll call this disk the disk of confusion.

Calculating the Diameter of the Disk of Confusion (but Actually, We Want to Know the f-Number...)

The Merklinger paper confused my a bit, because the arguments go somehow "upside down". In practice, you decide on the size (or diameter) of the objects you want to resolve (the disk of confusion), and all you need is an f-number for the focal length that you want to use. The theory, however, delivers a formula that gives you the size of the disk of confusion, depending on distances and the aperture opening, which again depends on the focal length and the f-number. Thus, you need to rearrange Merklinger's formula to get what you need...

For arriving at the diameter of the disk of confusion Sx (or Sy), Merklinger asks the question (with additions by me):

"What object at distance X (between lens and point of focus D; or distance Y between point of focus D and infinity) would be imaged on the film plane as a circle of diameter c (circle of confusion) if the lens were stopped down so as to image the object sharply?"

I will not present Merklinger's mathematical derivation of the result here, but would rather like to refer you to his paper. He seems to do a lot of algebra at the end of his derivation*, whereas I found that the intercept theorem leads to the same solution in a simpler way. Anyway, here is Merklinger's result:

Sx = (D - X) / D * d, Sy = (Y - D) / D * d
With d = f / N we get:
Sx = (D - X) / D * f / N, Sy = (Y - D) / D * f / N
What we really need in practice is:
N = (D - X) / D * f / Sx, N = (Y - D) / D * f / Sy

Addition: If you express the object distance from the point of focus in fractions or multiples x or y of the focus distance D, you get even simpler formulae and "rules of thumb"*:

Y > D: Y - D = y * D, y < 1 => Sy = (Y - D) / D * d = y * D / D * d = y * d
X < D: D - X = x * D, x > 1 => Sx = (D - X) / D * d = D / x / D * d = d / x

Legend: Sx = size of disk of confusion (object between lens and point of focus D); Sy = size of disk of confusion (object between point of focus D and infinity); D = distance of point of focus; d = f / N = diameter of working aperture; f = focal length; N = f-number (f-stop)

*) Merklinger writes in another paper: "The rule for determining what objects are resolved is ultra-simple. We focus our lens exactly at some distance, D, from our lens. An object one-tenth of the way back from D, towards our camera (that is, at a distance of 0.9 D), will be resolved if it is at least one-tenth as big as the opening in our lens diaphragm. If the object is one-quarter of the way from the point of exact focus to the camera lens, it will have to be one-quarter as big as our lens to be resolved. And so on. The same rule will hold on the far side of that point of exact focus also. An object twice as far from the lens as the point of exact focus will have to be as large as the lens aperture to be resolved."

*) I was able to reproduce Merklinger's calculations, but still do not understand, how he arrived at the starting formula Sx = c * (X / B) for the diameter of the disk of confusion (he does not explain this; B = distance between lens and film plane, X = distance between lens and object x). Just using the interception theorem lead me directly to the above formulae - without any assumptions about the relationship between c and d and any use of the lens formula...

Merklinger summarizes his result as follows:

The only things that matter are the working diameter of the lens d (size of the lens aperture as seen from the front of the lens = f/N; f = focal length, N = f-number), the distance D at which the lens is focused, and where the other significant objects are in relation to our lens (X, Y) and the point or plane of exact focus (D - X, Y - D).
X is the distance from the lens to the place where we wish to estimate the diameter of the disk of confusion. Sx is the diameter of the disk of confusion at distance X.

Thus, whenever you specify a DoC criterion for the size of objects to be resolved (for example, 4 mm for blades of grass), you do it for distance X (not for D, there Sx = 0...)! You specify such a criterion also for a distance Y beyond the point of focus D. In the case that both criteria are the same (as we have for the CoC), we get:

Sx = Sy = S => Diff = D - X = Y - D => S = Diff / D * d
N = Diff / D * f / S (S = DoC, Diff = difference between distance of objects and distance of point of focus)
If we express the difference Diff in multiples n or fractions 1/n of D, we get: S = n * D / D * d = n * d (n > 1) or : S = D / n / D * d = d / n (n < 1)

That is, if the criteria are the same, the far point (or plane) Y is at the same distance behind the point of focus D as the near point (or plane) X is before D.

A special, but important case is that the lens is focused at infinity. I deal with it further down.

Some Characteristics of the Disk of Confusion

According to Merklinger, "the disk of confusion is about the size of the smallest object which will be recorded distinctly in our image. Smaller objects will be smeared together; larger objects will be outlined clearly - though the edges may be a bit soft."

The size of the disk of confusion is easily estimated for two specific distances:

At half the distance from the camera to the point of exact focus, the disk is half the working diameter of our lens.
At twice the distance to the point of focus, the disk is equal to the lens diameter.

The latter two statements can be easily derived from Merklinger's formulae for the disk of confusion:

Sx = (D - X) / D * d and X = D/2 => Sx = (D - D/2) / D * d = d / 2
Sy = (Y - D) / D * d and Y = 2*D => Sy = (2*D - D) / D * D = d

The Relationship between the Diameter of the Disk of Confusion and the Circle of Confusion - And a Surprise for Me...

Merklinger starts his calculations of the size of the disk of confusion from a relationship between it and the size of the circle of confusion:

Sx = c * X / B (Sx = diameter of disk of confusion, c = diameter of disk of confusion, X = distance of object x, B = distance between lens and film plane)

In the course of the calculations, however, he eliminates the CoC from the formula by substituting it with the DoC. Therefore, the formulae tell us nothing about the CoC and its relation to the DoC.

Since the relationship formula above contains an unknown distance B (distance between lens and film plane*), the CoC cannot be calculated directly. I therefore asked myself whether it would be possible to arrive at a relationship formula that contains only well-known parameters. I found that B can be calculated from the lens formula (1 / f = 1 / D + 1 / B) and got: B = f * D / (D - f). Substituting B with the right-hand expression lead to the following formulae (no guarantee for its correctness, the final form of the formulae depends an taste...):

Sx = c * X / B = c * X * (D - f) / ( f * D) = c * X / f - c * X / D
c = Sx * B / X = Sx * f * D / (X * (D - f)) = Sx * f / (X * (1 - f/D))
X = Sx * B / c = Sx * f * D / ((D - f) * c) = Sx * f / ((1 - f / D) * c)

For the case that D = infinity (i.e., focus on infinity), we get the following simplified formulae:

Sx = c * X / f; with Sx = d, we get d = c * X / f
c = Sx * f / X; with Sx = d, we get c = d * f / X = f * f / (X * N)
X = Sx * f / c; with Sx = d, we get X = f * d / c = f * f / (c * N)

Thus, for the infinity versions of the formulae, the object distance X is more or less "normalized" for multiples of the focal length. At an object distance of the focal length, the diameters of the circle of confusion and the disk of confusion are identical.

Using the Relationship for Calculating the Minimum Distance - And a Surprise...

I assumed that this formula might also be used to calculate the minimum distance, based on focal length of the lens, a criterion for the diameter of the CoC (image field) and one for the diameter of the DoC (object field). This would be fairly easy for the case that distance is set to infinity because of the simpler formula. To verify this, I created an example, and since it seemed to work, I also calculated a complete distance table in Excel for the Sony RX100 M1 - only to find out that it was nearly identical to the table of hyperfocal distances for this camera. Only then, I discovered that indeed the hyperfocal distance is just the minimum distance based on the CoC criterion, when distance is set to infinity (see here for a derivation). So here we know at least, both of them. At least, this makes plausible that my formulae above are correct... I deal with the "infinity" case further down.

Side Note: Object Field Methods (From Wikipedia)

Traditional depth-of-field formulas and tables assume equal circles of confusion for near and far objects. Some authors, such as Merklinger (1992), have suggested that distant objects often need to be much sharper to be clearly recognizable, whereas closer objects, being larger on the film, do not need to be so sharp. The loss of detail in distant objects may be particularly noticeable with extreme enlargements. Achieving this additional sharpness in distant objects usually requires focusing beyond the hyperfocal distance, sometimes almost at infinity. For example, if photographing a cityscape with a traffic bollard in the foreground, this approach, termed the object field method by Merklinger, would recommend focusing very close to infinity, and stopping down to make the bollard sharp enough. With this approach, foreground objects cannot always be made perfectly sharp, but the loss of sharpness in near objects may be acceptable if recognizability of distant objects is paramount.

Other authors (Adams 1980) have taken the opposite position, maintaining that slight unsharpness in foreground objects is usually more disturbing than slight unsharpness in distant parts of a scene.

Moritz von Rohr also used an object field method, but unlike Merklinger, he used the conventional criterion of a maximum circle of confusion diameter in the image plane, leading to unequal front and rear depths of field.
(From Wikipedia: en.wikipedia.org/wiki/Depth_of_field )

Special Case of Merklinger's Approach: Setting Distance to Infinity

The method of setting distance to infinity is a long-standing photographic practice, but can also be regarded as a "special case" of Merklinger's approach (see also Merklinger p. 29ff).

Determining the Disk of Confusion or the f-Number

Setting distance to infinity, simplifies Merklinger's formulae and thus, makes calculations of the disk of confusion easier. For the infinity condition, you can simply set the object distance X (or Y) to zero (in comparison to D = infinity) and get the following result (in bold):

X => 0, S = D / D * d = 1 * d = 1 * f / N = d => S = d, S = f / N (D = focus distance = infinity; S = diameter of disk of confusion, f = focal length of lens, N = working f-number)
What we need in practice, is: N = f / S

Thus, under this condition, the disk of confusion S assumes, regardless of the distance to the object, the constant size d of the diameter of the working aperture, which can be calculated from the focal length f and the working f-number N for any given lens. This can be condensed into a simple rule of thumb (from Michael Reichmann at The Luminous Landscape):

Set the lens to infinity, then divide the focal length of the lens by the working aperture (f-number). This will give you, in millimeters, the subject resolution limit from the far distance to right in front of the camera.

For example, if you are using a 50 mm lens focused at infinity, and the aperture is f/8, then 50 divided by 8 is about 6 millimeters, meaning that objects of 6 mm size and more will be identifiable within the resolution capabilities of the lens and the sensor. 6 mm objects may be a bit soft, but are still identifiable (from The Luminous Landscape and Merklinger, adapted).

Merklinger makes the following comments on this condition:

When a lens is focused at infinity, the disk of confusion will be of constant diameter, regardless of the distance to the object.
The diameter of the disk of confusion, S, will be equal to d, the working diameter of the lens, at all object distances.

Both statements are direct consequences of the formula S = d presented above.

In practice, this may mean that you have to use the formula "the other way round" because you started from specifying the disk of confusion. According to Merklinger, all you need in this special case is to calculate the f-number N for a given focal length f and size of the disk of confusion d (which is your criterion of what is to be resolved in the image), that is, N = f / d. You can calculate this easily in your head.

What About the Near Limit?

What is still missing here, is a criterion for the near limit, albeit Merklinger probably would state that the disk of confusion is the near limit, according to his approach. But since his methods does not tell anything about the CoC at closer distances, you might get a bit nervous about this... I can assure you that you are not completely "left in the dark". As I derived elsewhere, the hyperfocal distance (HFD) is just the near limit if you adopt the conventional CoC as your "sharpness "criterion. I also show below (and Merklinger also mentions this) that at half the HFD, the CoC is twice the "acceptable" or "standard" size.

Tables for Determining the f-Number

Instead of doing the DoC math in your head, you can also calculate tables of d-values. I published such tables for the disk of confusion for my cameras (except for the Ricoh GXR) on this site - see the sections for the individual cameras. For my tables, however, I did it the "original" way and used fixed f-numbers, as given by cameras, and the most important focal lengths of my cameras to calculate tables of d-values. When you use these tables, you first have to decide on a value of d, based on the scene to be photographed and a focal length f . Then search the table for a suitable d-value and extract the corresponding f-number for that focal length. Often, you will not find the exact d-value in the table and have to select one that comes close. Boone proposes to choose the f-number conservatively in this case, that is, to select the next larger f-number to be "on the safe side".

Here is a step-by-step procedure for using the tables (there are two variants of it, one for cameras/lenses without a DOF scale and one for lenses with a DOF scale - they differ in the last step 4):

(1) Choose a diameter for the disk of confusion based on the size of the features in the images to be resolved; this is also the diameter of the working aperture.
(2) Select from the d-value table an appropriate f-number for the focal length of the lens that you will use*. Round in the direction of a larger f-number to be "on the safe side"**.
(3) Take care that the f-number is not at or above the diffraction limit aperture of your camera (details).
(4a) For cameras and lenses without DOF scales (in my case: Ricoh GR, Leica X Vario, my wife's Sony RM100 M1):
- Consult a table of hyperfocal distances for the respective camera, look for the focal length and f-number to be used and determine the hyperfocal distance. This can be used as "allowable" minimum distance based on the traditional CoC criterion.
(4b) For lenses with a DOF scale (on my Leica M (Typ 240)):
- Read the "allowable" minimum distance from the "near marker" of the DOF scale of the lens for the f-number to be used (thus, the "allowable" minimum distance is based on the traditional CoC criterion).

*) For my tables, I used fixed f-numbers, as given by the cameras, and the most important focal lengths of my cameras to calculate tables of d-values. When you use these tables, you first have to decide on a value of d based on the scene to be photographed and a focal length. Then search the table for a suitable d-value and extract the corresponding f-number for that focal length. Often, you will not find the exact d-value in the table and have to decide for one that comes close. Choose the f-number conservatively in this case, that is, select the next larger f-number to be "on the safe side".
**) Choose the f-number conservatively, that is, select the next larger f-number to be "on the safe side".

Table for the General Method (Limited Focus Distance) versus the Special Case "Distance Set to Infinity"

As a "specialist for tables" I try to capture the essence of Merklinger's approach for a limited focus distance D versus the special case focus at infinity" in a small table:

	Focus at D, Object at X, Y			Focus at Infinity
Distance	CoC	DoC	DoC Formulae	CoC	DoC	DoC Formula
Near Limit*	n.a.*	n.a.	Y > D: Sy = (Y - D) / D * d Sy = n * d, with: Y - D = n * D X < D: Sx = (D - X) / D * d Sx = d / n, with: D - X = D / n	n.a.**	d	S = d
D / n	n.a.	d / n => d
D / 2	n.a.	d / 2
D (focus)	0	0
*2 D**	n.a.	d
*11D**	n.a.	10 * d
*n D**	n.a.	n * d => inf
Far Limit*	n.a.*	n.a.
Infinity	n.a.	inf		0
Further useful formulae >			d = f / N	H (approx.) = f * f / (N * c) = f * d / c

*) Not dealt with by Merklinger's method
*) To be calculated from DOF formulae; **) for X = H (HFD), the CoC assumes the "acceptable" value

Or in "compressed" format:

	Focus at D, Object at X, Y			Focus at Infinity
Distance	CoC	DoC	DoC Formulae	CoC	DoC	DoC Formula
D / n	n.a.	d / n => d	Y > D: Sy = (Y - D) / D * d Sy = n * d, with: Y - D = n * D X < D: Sx = (D - X) / D * d Sx = d / n, with: D - X = D / n	n.a.*	d	S = d
D (focus)	0	0
*n D**	n.a.	n * d => inf
Infinity	n.a.	inf		0
Further useful formulae >			d = f / N	H (approx.) = f * f / (N * c) = f * d / c

*) For X = H (HFD), the CoC assumes the "acceptable" value

Legend: Sx = size of disk of confusion (object between lens and point of focus D); Sy = diameter of disk of confusion (object between point of focus D and infinity); D = distance of point of focus; X, Y = distance of object (X<D, Y>D); n = abs (focusing distance - object distance)/(focusing distance); H = hyperfocal distance (H); d = f/N = diameter of working aperture; f = focal length; N = f-number (f-stop); c = diameter of circle of confusion

Everything Sharp from "Here" to Infinity...

Often, we want everything sharp from "here" to infinity. There are two alternative methods for dealing with this requirement: Either ...

Set the distance to the hyperfocal distance. This provides sharpness based on the traditional CoC criterion at the near and far limits. Or ...
Set the distance to infinity. Here you can at least, be sure that everything is perfectly sharp at infinity.
It also turns out that the hyperfocal distance can act as the near limit based on the conventional CoC criterion (easily to see on lenses with a DOF scale).

In the following, I attempt to compare both methods.

Hyperfocal Distance versus Merklinger's Approach

Above, I mentioned that Merklinger's simplified approach is an alternative to setting distance to the hyperfocal distance. Merklinger (p. 31ff) therefore discusses "what we gain and what we lose when we focus at infinity instead of the tried-and-true hyperfocal distance" and makes the following comments on this:

At the inner limit of the conventional depth-of-field the disk of confusion is half the diameter of the lens opening (because the distance to the inner limit of the depth-of-field is one-half the hyperfocal distance).
Thus at the inner limit of depth of field the most we lose by focusing at infinity is a factor of two in resolution of the subject.
On the other hand, for subjects beyond the hyperfocal distance, the story may be quite different. At a subject distance of twice the hyperfocal distance, the disk of confusion is equal in size to the lens diameter. At this distance either method gives the same result.
At three times the hyperfocal distance, the disk of confusion is twice the lens diameter. At four times the hyperfocal distance, it is three times the lens diameter and so on. At ten times the hyperfocal distance, the disk of confusion is nine times the lens diameter. => Thus, the DoC grows without limit until infinity.

To verify these statements for focusing at the hyperfocal distance, I used the Merklinger formulae and set X (smaller than D) or Y (larger than D) to the hyperfocal distance H, fractions of it, or multiples of it:

If D = H and X = H/2, which is the DOF near limit (we focus hyperfocal and ask for objects at the near limit), we get Sx = (D - X)/ D * d => Sx = (H - H/2)/ H * d = d/2, that is, half the size of the disk of confusion (see statement 1 above).
If D = H and Y = 2*H (we focus hyperfocal and ask for twice the HFD), we get Sx = (Y - D)/ D * d => Sx = (2*H - H)/ H * d = d, that is, the size of the disk of confusion (see statement 2 above).
If D = H and Y = 3*H (we focus hyperfocal and ask for three times the HFD), we get Sy = (Y - D)/ D * d => Sy = (3*H - H)/ H * d = 2*d, that is, double the size of the disk of confusion (see statement 3 above). And so on...

When we, instead, focus at infinity, we get Sx = d at any distance, which is half the resolution at HFD/2, the same at double the HFD, and better for any other multiple of the HFD compared with focusing at the hyperfocal distance.

Merklinger concludes "Thus, if we are using a good lens, good film, and careful technique, we potentially have a lot to lose in the resolution of distant subjects by focusing the lens at the hyperfocal distance. In practice, by focusing instead at infinity, we will lose a factor of two in subject resolution at the near limit of depth-of-field but gain about a factor of six in the resolution of distant subjects! It's often worth the trade."

Another question would be, which diameter the CoC will have at half the HFD (X = H/2) when we focus at infinity (at HFD, that is, for X = H, it is the "nominal" value). I start from the formula for c from above:

c' = Sx * f / X = d * f / X = f * f / (N * X) (c' = actual CoC; focus on infinity, Sx=d, d = f/N; N = f-number, f = focal length)

X can be expressed in fractions or multiples of H (X = H/x for fractions):

c' = x * f * f / (N * H) (focus on infinity, N = f-number, f = focal length; H can be taken from HFD tables)

Now, we should be able to replace H using its simplified definition H = f * f / (N * c):

c' = x * f * f / (N * f * f / (N * c)) = x * c

It looks as if the CoC is simply a fraction or multiple in size at HFD/x when focused at infinity - in contrast to focusing at the HFD (x = 1), where it is just "normal" size. I guess I have to verify this...

Note that Kevin Boone also includes an intensive discussion of hyperfocal distance versus Merklinger's approach in his article Hyperfocal distances and Merklinger's method in landscape photography. He also shows an example of how the "far limit" is negatively affected by setting the focus point too close when using the hyperfocal distance (due to settings errors or insufficient CoC). The far limit may move dramatically closer in such a case, which might ruin a shot. Boone therefore cautions the readers:

"When using the hyperfocal method, it's advisable to calculate the hyperfocal point for a particular f-stop, and then stop down one position, as otherwise small errors in the focus position can have dramatic effects on the far point. If you can't do that, it's better to set the focus distance too long rather than too short as, although this will affect the near point, the effect on the near point is far less pronounced."

Test Photos

I prepared some "quick-and-dirty" test photos with some of our cameras, where I compare the following conditions: (1) focus set at infinity and (2) focus set at the hyperfocal distance. I also used several different apertures for each camera. I used all cameras at the same equivalent focal length of 28 mm. In hindsight, I realized that only the focal length is relevant. Thus, I could as well have taken the photos with one camera at different focal lengths.

Here are links to the respective pages (located in the sections of the cameras):

While the photos could be of better quality, they more or less confirm the above statements.

What is so Special about the Hyperfocal Distance?

Many examples in Merklinger's paper deal with the hyperfocal distance (HFD), and I asked myself, "What is so special about a distance that probably 95% of all photographers do not even know of?" First, I thought, that perhaps the hyperfocal distance is special in that certain relations are valid for its fractions and multiples so that is especially easy to deal with. But then I found that this is valid for any distance that you focus on. Thus, if you express the object distance X or Y in fractions or multiples x or y of the focus distance D, you get:

Y > D: Y = y * D, y > 1 => Sy = (Y/D - 1) * d = (y - 1) * d
X < D: X = x * D, x < 1 => Sx = (1 - X/D) * d = (1- x) * d

In these formulae only the ratio of the focus and the object distance is relevant, but not that the fact that the focus distance is hyperfocal. Above, I listed already Merklinger's statement on this:

The size of the disk of confusion is easily estimated. At half the distance from the camera to the point of exact focus, the disk is half the working diameter of our lens. At twice the distance to the point of focus, the disk is equal to the lens diameter.

But over all the fuss with the hyperfocal distance, I forgot this gem...

So, I think the only relevant aspect from a calculation point of view is that the for the HFD, half the HFD, and infinity, we know the diameter of the circle of confusion (CoC), when we focus at HFD. From a practical point of view, it is, of course, also relevant by providing us with a distance setting that delivers a depth of field (DOF) that extends from a near point up to infinity (according to the CoC criterion, that is, with "acceptable sharpness" at the near and far limits).

Tabular Comparison with Hyperfocal Distance

In the following, I attempt to capture the differences between the hyperfocal distance approach and Merklinger's approach in an overview table:

Criterion	Hyperfocal Distance (HFD)	Merklinger		Comment
Criterion	Hyperfocal Distance (HFD)	Special Case: Focus at Infinity	Focus at Object at Finite Distance	Comment
Distance set to...	HFD	Infinity	Distance of most relevant objects in scene (<< infinity)	Typically farther away when using Merklinger's method
Near Limit	HFD/2 (CoC criterion)	HFD (CoC criterion); can be easily read from lenses with a DOF scale The size of the DoC = aperture opening (f/N) is constant across the whole distance range up to infinity	Given by the size of the DoC for the near limit (distance X) specified by the photographer	Infinity: Merklinger's method does not specify a near limit, only a "minimum resolved" object size that is constant up to infinity
Far Limit	Infinity (CoC criterion)	Infinity (optimal sharpness)	Given by the size of the DoC for the far limit (distance Y) specified by the photographer	Infinity: Sharpness is optimal at infinity
Best Sharpness at...	Hyperfocal distance	Infinity	Object that is in focus	Merklinger's method puts the emphasis on objects farther away or at infinity; the HFD is typically fairly close to the photographer
DOF	Between half the HFD and infinity Determined by the choice of aperture, CoC, and focal length of lens	Between HFD and infinity (the near limit is not specified by Merklinger's method, but via the CoC) Determined by the choice of aperture, CoC, and focal length of lens	From X to Y, according to the DoC criteria for the near limit X and the far limit Y	You can specify different DoC criteria for near and far objects and look for a "compromise" f-number
Role of Aperture	Determines DOF (near limit; far limit is infinity)	End result of the calculations	End result of the calculations	The f-number is the "final result" of Merklinger's method
Choice of Aperture	On the basis of HFD tables or calculations (or DOF scales)	On the basis of the DoC tables or individual calculations (which is easy)	On the basis of individual DoC calculations ("infinity" tables can also be used)
Formulae	HFD formula (or approximation)	Special case DoC formulae with object distance X or Y = infinity (these formulae are much simpler than the regular ones and can be used for calculating DoC tables, from which the f-number can be chosen)	"Regular" DoC formulae	For focus at infinity (D = infinity) the formulae can be simplified
Formulae for DoC	---	S = d	Sx = (D - X) / D * d Sy = (Y - D) / D * d With d = f / N you get: Sx = (D - X) / D * f / N Sy = (Y - D) / D * f / N	Note that Sy and Sy differ from D in the general case (for focus at infinity, S and d are the same)

Legend: Sx = size of disk of confusion (object at distance X between lens and point of focus D); Sy = diameter of disk of confusion (object at distance Y between point of focus D and infinity); D = distance of point of focus; d = f/N = diameter of working aperture; f = focal length; N = f-number (f-stop); c = diameter of circle of confusion

Summary

The following table summarizes the results from this section:

	Focus at H (HFD), Object at X, Y			Focus at Infinity
Distance	CoC	DoC	DoC Formulae	CoC	DoC	DoC Formula
H / 2	c (near DOF limit)	d / 2	Y > H: Sy = (Y - H) / H * d X < H: Sx = (H - X) / H * d	2 * c	d	S = d
H	0 (focus)	0		c (near DOF limit)
*2 H**	< c	d		c / 2
Infinity	c (far DOF limit)	inf		0 (focus)
Further useful formulae >			d = f / N	H (approx.) = f * f / (N * c) = f * d / c

Legend: Sx = size of disk of confusion (object between lens and point of focus H); Sy = diameter of disk of confusion (object between point of focus H and infinity); H = hyperfocal distance (HFD); X, Y = distance of object (X<H, Y>H); n = multiple or fraction of ratio between focusing distance and object distance; d = f/N = diameter of working aperture; f = focal length; N = f-number (f-stop); c = diameter of circle of confusion

Merklinger's "Rules of Thumb"

Merklinger published a number of "rules of thumb" (chapter 10, p. 69ff), which are direct consequences of the formulae that he derives in his paper. In the following, I cite some of them (I shortened rules, if this seemed appropriate), namely the ones that I find myself most important for my work:

1. If we want an object to be resolved, we make sure that the disk of confusion is smaller than the object.
2. If we want an object to be blurred out, we make sure that the disk of confusion is larger than the object.
3. Between the point of exact focus and the camera, the disk of confusion can never be larger than the working diameter of our lens.
Beyond twice the distance from lens to point of exact focus, the disk of confusion grows larger than the lens opening - without limit!
4. If we want anything at infinity to be critically sharp, focus at infinity.
5. If we want all objects of, say, 5 millimeters diameter to be recorded - at whatever distance - we should use a lens aperture of 5 millimeters or smaller and focus no closer than half the distance to the furthest object.
6. The zone of acceptable delineation of the subject falls equally in front of and behind the point of exact focus (not 1/3, 2/3!).

Comments

Ad 1 and 2) Explanation to be provided; but see my test photos (Sony RX100 M1 (28 mm equiv.), Leica X Vario (28 mm equiv.), Leica M (Typ 240) with M-Rokkor 28 mm)

Ad 3) This is a consequence of Merklinger's formulae for the disk of confusion.

Ad 4) The consequences of this strategy are discussed above.

Ad 5) The focus rule can be derived from Merklinger's formulae, but needs to be explicated here (the DoC value is probably arbitrary in this case)...

Ad 6) This is derived in Merklinger's paper, and I list it here, because this is useful information in my view.

Why is Merklinger's Approach not Used More Widely?

A Russian RX100 M1 user who had initially pointed me to Merklinger's approach also asked me: "What is the PRINCIPAL PRACTICAL usage (help) of this theory and why very few people use it in practice?" Before I haven't put the approach into practice, I cannot answer the first question. I also cannot give definitive answers to the second question, but at least, I can offer a few guesses here...

One reason for the lack of acceptance might be that the method is "simply" too complex for real-world use, particularly, if you want to take photos "spontaneously", as my friends once characterized their style of photographing. Merklinger only needs a few sketches and does most of the calculations in his head. For me, it looks at the moment as if you need a computer at hand to do the calculations (actually, no issues in the times of smartphones...).

And, of course, you must also be willing to think about the structure of the scene that you want to photograph, particularly about the size of the details that you want to capture (resolve). You may also have to decide, whether you set distance simply to infinity or to another value, which would require more calculations.

And last, but not least, the theory makes calculations easier if you think in fractions and multiples of the hyperfocal distance. First of all, I fear that only few photographers know of this concepts, my friends do not.

The other issue that I see is that while we are used to estimate and think in distances, the HFD is more difficult to deal with, because it is different for each f-number. The larger, the f-number, the smaller the HFD and the larger the DOF. That is, what most advanced photographers understand. But a larger f-number also decreases the disk of confusion if you focus closer than infinity. Now you have two effects together: the objects can get closer and the disk of confusion gets smaller and thus, smaller objects can be resolved. Now the question is, "Is the better resolution at larger f-numbers just the result of being able to get closer to the object, or is there actually a 'practical' increase?" When I looked at my test photos, this question came up, and I still have to find an answer to it. Maybe, it is a simple as that: The larger f-number allows you to get closer to an object (at the same "conventional" resolution) and thus, you can resolve more details. In this case, no "Merklinger theory" would be needed...

A Practical Example

The same RX100 M1 user asked me what Merklinger's approach would deliver for the following example (or better, which would be the optimal aperture value for this): The camera should resolve blades of grass, for which a diameter of the disk of confusion of 5 mm (0.5 cm) is assumed. The question is:

Which aperture value (f-number) delivers the optimum resolution for this case for my camera (Sony RX100 M1) - or, to extend this question, for some sample cameras (or sensor formats)*?

Actually, this is not the "correct Merklinger question", since the point of focus always delivers optimal sharpness. Merklinger's theory is based on geometric optics and does not say anything about this. In a short paper on his method, Merklinger writes: "True, diffraction effects won't let me resolve 11 mm at 5000 ft. ...." (the resolving power of the lens and the sensor may also limit what can be resolved). Thus, this is another story and not the point here. The point here is that we have to make decisions about where we want to resolve 5 mm, either in the foreground, the background, or both (which may make it hard to find a compromise...). These are our "Merklinger" near and far limits, corresponding the the CoC-based near and far limits, but defined in the object space.

Let's say that we want to resolve objects of 5 mm diameter at X = half the distance D at which we focus (that is, in the foreground).

I distinguish the following two cases:

Focusing at a finite distance, which will be "without loss of generality" the hyperfocal distance H, see below,
and
Focusing at infinity, which is the "boundary condition": Up to an object distance of 2 X, it is worse than focusing at a finite distance, beyond that it is better.

*) I calculated the f-numbers for several cameras at different focal lengths, but for the calculations, only the focal length is relevant, not the sensor size or the camera type. This was only done to make it easier for the reader to find useful equivalent focal lengths for the respective cameras.

Here are the basics for my calculations:

For the infinity condition we get: S = d with d = f / N => N = f / d (d = 5 mm => N = f / 5). That is, our criterion S is equal to the lens opening d and, using the formula d = f / N for a given value of d, we can calculate the f-number N (N = f / d = f / 5).
For any finite distance X, we have to consider not only the criterion S, but also the distance D we want to focus at and the distance X at which our resolving criterion should be satisfied. This is typically the near limit. The far limit Y may have a different criterion or the same as the near limit. If both are different, a compromise between the two resulting f-numbers has to be found.
Without loss of generality, I set the focusing distance to the hyperfocal distance H.

If we decide that our near limit X will be at half the focusing distance D (and the far limit X extends half the focusing distance D beyond it, meaning that DOF = D), we get

Sx = d * (D - X) / D = d * (D - D/2) / D = d / 2

With d = f / N, we get:

Sx = d / 2 = f / (2 * N) => N = f / (2 * Sx) => Sx = 5 mm => N = f (mm) / 10 (mm)

When we think in fractions and multiples of the focus distance, the resulting d-values are the same for the same fraction or multiple of the focus distance, regardless of its actual value (this can be derived from the formulae above). Therefore, I decides - without loss of generality - that the focus distance D be the hyperfocal distance H, because focusing at this distance is a competitive approach to Merklinger's. This also allows me, to present some numbers not only for the DoC but also for the CoC...

Thus, in the results table below I use the hyperfocal distance H as point of focus (D) and consider the near limit X = H / 2 (or D / 2 in the general case). The basics for my calculations can be found here. On this page, I just present the results, that is, the f-numbers for a DoC of 5 mm (note that the table does not include or need a "real" distance D because the near limit is half the focus distance):

Criterion: Sx = 5 mm (DoC)			Focus at H (HFD)						Focus at Infinity
Camera*	Focal Length (mm)		f-Number for Sx at X = H/2		Focus	Near Limit		CoC at H/2 (mm)	f-Number (f/d)		Near Limit (CoC)	Resolving Limit (0 - inf)	CoC at H/2 (mm)
Camera*	Real	Equ.	Calcul.	Nearest*	H (m)	H/2 (m)	DoC at H/2 (mm)**	CoC at H/2 (mm)	Calcul.	Nearest*	H (m)	DoC (mm)	CoC at H/2 (mm)
Sony RX100 M1 1" Sensor Diffr. Limit: f/8	10.4	28	1.04	1	--	--	5	0.011	2.08	2	4.93	5	0.022
	13.0	35	1.3	1.5	--	--	5	0.011	2.60	2.8	5.44	5	0.022
	18.5	50	1.85	1.8	17.48	8.74	5	0.011	3.70	4	7.80	5	0.022
	25.9	70	2.59	2.8	21.59	10.80	5	0.011	5.18	5.1-5.6	10.81	5	0.022
	37.1	100	3.71	4	31.32	15.66	5	0.011	7.42	7.1-8	15.68	5	0.022
Leica X Vario APS-C Sensor Diffr. Limit: f/16	18	28	1.8	1.8	--	--	5	0.019	3.6	3.5-4	4.28	5	0.038
	23	35	2.3	2.5	--	--	5	0.02	4.6	4.5-5.1	5.27	5	0.04
	33	50	3.3	3.5	4.80	2.40	5	0.02	6.6	6.3-7.1	7.67	5	0.04
	46	70	4.6	4.5	5.91	2.96	5	0.02	9.2	9-10	10.54	5	0.04
Leica M Full-frame Sensor Diffr. Limit: f/22	15	15	1.5	1.4-1.5	5.32	2.66	5	0.03	3.0	3.2	2.38	5	0.06
	21	21	2.1	2.0-2.4	5.85	2.93	5	0.03	4.2	4.5	3.30	5	0.06
	25	25	2.5	2.4-2.8	7.39	3.70	5	0.03	5.0	5.1 (5.04)	4.16	5	0.06
	28	28	2.8	2.8	9.27	4.64	5	0.03	5.6	5.6 (5.66)	4.65	5	0.06
	35	35	3.5	3.5	11.49	5.75	5	0.03	7.0	7.1-8	5.76 (5.14)	5	0.06
	50	50	5.0	5.1	16.59	8.30	5	0.03	10.0	10-11	8.32 (7.42)	5	0.06
	75	75	7.5	7.1-8	23.51	11.76	5	0.03	15.0	16	11.79	5	0.06
	90	90	9.0	9.0	30.16	15.08	5	0.03	18.0	18	15.12	5	0.06
	135	135	13.5	14-16	38.10	19.05	5	0.03	27.0	28	21.44	5	0.06

*) f-number set to nearest "infinity" value. The f-numbers should not be taken too seriously, in part they are "nominal", in part "exact" (sometimes, the exact ones are given in parentheses).
**) At H/2, the DoC is d/2, at H, the DoC is 0, at 2*H, the DoC is d again, and then it grows without limit.
*) For the calculations, only the real focal length is relevant, not the sensor size or the camera type.

Discussion

Focus at Infinity

For focusing at infinity, the answer is straightforward: The aperture values are valid for any distance. If you increase the f-number further, the disk of confusion gets smaller, and you can resolve even smaller structures. Be sure not to increase the f-number beyond the diffraction limit.

The method also tells us that 5 mm can even be resolved if objects are hundreds of meters away (and more...). But then the angular separation should be so small that the blades are no longer be resolvable. Since the Merklinger method is based on geometric optics, it does not know anything about resolving limits caused by the media (film: grain; sensors: pixel size, Bayer filter, AA filter) and by the characteristics of real lenses - as it does not know anything about diffraction... So please take such results with a "grain of salt."

Focus at the Hyperforcal Distance (a Finite Distance)

For focusing at a finite distance, we get the theoretical result that the diameter of the disk of confusion (DoC) will continually decrease from the lens to the point of focus D, where it is zero, and then increase again, until it grows above all limits. When we specify the near limit in multiples or fractions of the focus distance D, we can generalize our results and need not use "real" distances.

When I calculated the values for the finite distance, I was somewhat shocked at the beginning, because the f-numbers were so small, smaller than some cameras offer. What does this tell us? The answer is that any f-number larger than the calculated one will do (below, the diffraction limit, of course...). One might say, the criterion is too lax, and there is room for making it more strict. If I would, for example, adopt a disk of confusion of half the size, that is, of 2.5 mm in diameter as my criterion, I could just use the f-numbers that I calculated for the infinity condition. This also showed me the way to how the tables for the infinity condition can also be used for finite distances: Just calculate or estimate (X - D) / D (it's even simpler for fractions of D) and multiply the f-number for infinity with it - voila!

And Now the Answers...

Returning to the starting question, a simple answer to a Sony RX100 M1 user might be:

If infinity need not be sharp, set the focus at a distance so that the relevant parts are within sufficient reach (not farther away than twice the focus distance). You can use any focal length and any f-number from the lowest possible (depending on the focal length) up to f/8 (the diffraction limit).
Alternatively, you can make your criterion stricter. A disk of confusion of 2.5 mm would just correspond to the "infinity" data.
If infinity needs just to be "acceptably sharp" (CoC criterion)*, set the focus at the hyperfocal distance. You can read the distance and corresponding f-number from tables of the hyperfocal distance or the DOF scale of your lens (depend on the focal length). Use only f-numbers up to f/8 (the diffraction limit). Note that the near limit (CoC criterion) is half the distance of the "infinity" condition.
If infinity needs to be "optimally sharp" (better than the CoC criterion), set the focus at infinity. You can use any focal length, and an f-number up to f/8 (f/8 is needed for 100 mm equiv.; smaller f-numbers may not be sufficient for some focal lengths - see the table above).

Do not expect grass blades that are farther away to be resolved, because there is more to photography than geometric optics...

*) Note: Elsewhere on this Website you will find a discussion about whether "acceptably sharp" at infinity is sufficient in comparison with focusing on infinity. You will find that a decision on this question depends on your requirements for viewing images on the TV or computer screen and for printing (print size).

Conclusions

At the moment, I can only make some preliminary remarks. Merklinger's approach may be difficult to handle for some people, because, in the general case, it requires calculations or the use of sketches. The special (or simplified) case that distance is set to infinity is easier to handle and can be dealt with using individual calculations or using tables, much like the DOF and HFD tables.

For me is, however, more important that Merklinger's method cautions me of some of the dangers or difficulties (for example, on cameras without a distance scale) that are implied by using the HDF. The biggest danger seems to be for me that you set the distance (meant as HFD...) too short (which is easy with some lens's distance scales) so that objects at large distances (or infinity) get too fuzzy. Boone demonstrates in his article that it is easy to err "on the wrong side" (= near limit) and thus, arrive at unsatisfactory results. Erring on the far limit has much less impact, particularly for distant objects, where nothing changes. Only close objects may get a little bit fuzzier.

Merklinger's paper and Boone's article also show that the strategy to set distance at infinity and "waste" some DOF in the foreground is a "safe" strategy for landscape shots where foreground objects do not play a major role. In the end, I have used this strategy all over my photographic life, but always felt guilty of "not using the optimum", that is, the hyperfocal distance.

As both authors point out, with the hyperfocal distance approach, objects at infinity are just "acceptably" sharp. But what "acceptably" means, can vary from situation to situation. Better monitors and larger prints may require a smaller diameter of the CoC than is used in the camera (or on the distance scale of the lens) as standard. Under these conditions, your photos may look too fuzzy at infinity. Setting distance to infinity will ensure that you always get the optimum at large distances, whatever output device is being used.

References

Merklinger

Kevin Boone: Hyperfocal distances and Merklinger's method in landscape photography (www.kevinboone.net/hyperfocal.html)
Harold M. Merklinger: The INs and OUTs of Focus - An Alternative Way to Estimate Depth-of-Field and Sharpness in the Photographic Image (Internet edition; www.trenholm.org/hmmerk/TIAOOFe.pdf)
Harold M. Merklinger: Depth of Field Revisited (www.trenholm.org/hmmerk/DOFR.html)
Lester Wareham: Depth of Field, Image Quality and Sensor Size (www.zen20934.zen.co.uk/photography/dof/dof_Merklinger.htm)
Merklinger focusing technique - Fred Miranda Forums (www.fredmiranda.com/forum/topic/844331/0?keyword=merk)
dpreview thread started by Detail Man: How Useful is Merklinger's "Object Field" Focusing ? (www.dpreview.com/forums/thread/3521836)
Michael Reichmann (The Luminous Landscape): Digital Focusing Part Two (luminous-landscape.com/digital-focusing-part-two/) (requires subscription!)

Hyperfocal Distance, DOF

DOFMaster Website
Depth of Field (Wikipedia)
Hyperfocal Distance (Wikipedia)
Profondeur de champ et capteurs numériques (English version: Depth of field and digital sensors)
Profondeur de champ (DOF calculator, French), version 1.6

Diffraction

Bob Aktions: Optimum Aperture - Format size and diffraction (www.bobatkins.com/photography/technical/diffraction.html)
The Airy Disc - An Explanation of What it is and Why You Can’t Avoid it (Oldham Optical UK): www.oldham-optical.co.uk/Airy%20Disk.htm
Lens Diffraction & Photography (Cambridge in Colour, Sean McHugh): www.cambridgeincolour.com/tutorials/diffraction-photography.htm
Digital Camera Diffraction, Part 2 (Cambridge in Colour, Sean McHugh): www.cambridgeincolour.com/tutorials/diffraction-photography-2.htm

ExifTool

ExifTool: www.sno.phy.queensu.ca/~phil/exiftool
Download links are provided on the page; as they include the version number, they would soon break if provided here...
ExifTool - Composite Tags: www.sno.phy.queensu.ca/~phil/exiftool/TagNames/Composite.html