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Understanding physical meaning of sharpness
Is Harold Merklinger's theory correct?
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(Third edition)

In my previous articles [1, 2] I have already explained the basic ideas of depth of field and described all the necessary formulas to calculate it. But according to my bitter experience, people do not like to analyze boring mathematical expressions. They prefer to implement simple recommendations instead. However, as a Russian saying goes, simplicity can be worse than larceny. Excessive simplifications often lead to false understanding.

This article was written to deflate two mistakes that are quite common nowadays:

1. If we change the focal distance of our lens while maintaining the same image magnification and keep the lens set to the same f-number, the zone of acceptable sharpness does not change.
2. If we focus our lens at infinity instead of focusing it at the hyperfocal distance, objects at infinity will be shown considerably sharper in our picture.

Both ideas originated from Harold Merklinger's book “The INs and OUTs of focus” [3]. I accept that this book contains a lot of correct and interesting ideas. But, ironically, the book also gave rise to a number of misleading recommendations.

This article compares Merklinger's approach with the traditional theory and puts his ideas to the test. All the explanations here put an emphasis on graphs and real photographs instead of mathematical formulas. As a result, I hope, this physical text is easier to understand.

And finally, I would like to remind you that the traditional theory is not exact either. It is based on a number of reasonable assumptions, however it still works perfectly most of the time. Any special cases, like micro-photography, for example, are not analyzed here.

Measuring unsharpness
(Traditional approach)

Unfortunately photographers often do not know how to interpret the results of the classical theory. They believe the formulas give them the exact description of the zone of acceptable sharpness. But they often do not understand that in some cases objects became fuzzy very quickly beyond the area of sharpness, while in other cases objects out of focus look only slightly diffused.

Let us explain the classical theory of sharpness, analyzing the degree of fuzziness. In article [2] I showed how to derive the formula that describes fuzziness behind the focusing point. Applying that approach to the general situation, we can obtain the universal mathematical expression:

Do not worry! I put this tiny formula here just for your reference. You will not find any other pieces of math in this text. :-)

c = c' | 1 - d / d ₀ |,

where
c - diameter of the circle of confusion for the point
          that is located at the distance of d₀ from the lens;
d - lens-to-sharp image distance;
c' - diameter of the circle of confusion for the infinitely distant point
          c' = f ²/(dN) = (Mf)/N;
f - focal length;
N - f-number (1,4; 2; 2,8; 4, 5,6; 8; �);
M - magnification (M = f / d).

The two vertical lines |…| stand for the absolute value.

The parameter c can be interpreted as the diameter of the imaginary round photobrush, with which the image on film is created. The smaller this value, the sharper the image.

Now let us analyze the graph that corresponds to this simple formula (see Fig.1).

It is quite naturally that c = 0 at the point of exact focus (d0 = d). Strictly speaking, c is not actually equal to zero at this point due to diffraction. The red dashed curve shows how the actual function c = f(d0) looks like. But this effect can be considered negligibly small for the purposes of this article. Moreover, the differences between the classical theory and Merklinger's approach lie ouside this small area. Beyond the distance of exact focus fuzziness grows. However, this growth is limited by the value of c'.

Fig. 1

In front of the focusing point, the degree of fuzziness grows sharper. At the halfway point (d₀ = d/2) objects are as blurry as at infinity. If the camera-to-object distance is four times shorter than d, the objects are three times unsharper than at infinity.

The graph in Fig. 1 demonstrates that the function that describes fuzziness is substantially non-linear. Because of this, it is not easy to derive simple practical formulas from it. However within a limited vicinity of the focusing point it is not that difficult to develop a good approximation.

Now let us look at the same graph calculated exactly in accordance with our formula (Fig.2).

	One can easily notice that within the green oval (i.e. when 0.75d < d0 < 1.5d) our red curve can be substituted with two straight lines (brown dash lines). In this area, c/c' does not exceed 0.4 (40 per cent).
Fig. 2

Let us call this area the green zone. All approximate theories work perfectly within the boundaries of this area. However, we have to use the initial non-linear model, when our parameters do not comply with the green zone requirement.

Harold Merklinger offered another approach to linearization. His object space, where fuzziness varies linearly, can be obtained from the traditional film plane model with the help of a non-linear transformation. But let us be patient. Merklinger's theory wil be discussed a little bit later.

How to calculate depth of field graphically

The procedure is very simple. First we have to draw a horizontal line that determines the acceptable circle of confusion c₀, i.e. the acceptable size of the elementary dot in our image. Typically, in 35mm photography c₀ = 0.03mm, but you may choose any other suitable value as well. For example, if you want to enlarge your negatives significantly, you may want to choose c₀ = 0.01mm.

Two intersections of such a horizontal line with the fuzziness curve will show us the limits of depth of field (DOF). The exact formulas to calculate DOF can be found in [1].

Now let us consider most important cases. In the table below, the green horizontal line shows the acceptable level of fuzziness c₀, while the dash blue line corresponds to the fuzziness of an infinitely distant point c'. The relatively thick green band under each graph shows the DOF area.

!!! IMPORTANT NOTE: The graphs in the table serve exlusively to demonstrate the general arrangement of curves and lines. All the distances are meaured in units of d, while fuzziness is measured in units of c'. Thus, the equal distance between the focusing point and the origin of coordinates in cases A to D does not mean the distance between the camera and the object is the same in all those cases.

Again, I have to remind you that all of the above is the classical theory. The description of its results is not traditional, but the results themselves are well-known. Now let us analyze Merklinger's approach.

A different approach—the object field

In his book “The INs and OUTs of focus” [3] Harold Merklinger criticized the traditional rules. In his opinion, it is better to describe DOF in terms of the resolution in the object field rather than concentrating on the characteristics of the final image. The idea looks vague, doesn't it? Do not worry! The general idea of Merklinger's approach is quite simple.

The sketch in Fig. 3 is usually drawn by all advocates of the object field theory. The lens “sees” the object with its working diameter of f/N. The lens is focused at the distance d, just where the object is located. The DOF area is determined by the acceptable divergence (two green arrows) of the blue dash lines that represent a cone in the object field. According to Merklinger, the bigger the diameter of the cone at a certain distance, the fewer details can be resolved. And poor resolution means poor sharpness. In a sense, Merklinger's theory is a linear approximation of the traditional theory. With the help of a non-linear transformation, the object field, where fuzziness varies linearly, can be recalculated into the image field model, where fuzziness is described by non-linear functions.

Fig. 3

In this article I can give only a brief explanation of Harold Merklinger's approach. If you want to learn more details, please read the original book.

It is fair to say that some of Merklinger's conclusions are quite reasonable. But at the same time, it is important to remember that there are many differences between his rules and traditional recommendations. Of course, Merklinger's approach is basically correct. However, the problem is that his theory perfectly works when the resolution is discussed, but it is not suitable for the purposes of sharpness.

One may also ask a natural question: Is it possible to compare two so diferent approaches? The answer is: it depends. Of course, the two theories describe different things, i.e. the object field and the film plane. But at the same time they both deal with sharpness. When I compare them, I try to find out which theory better matches the intuitively obvious concept of sharpness.

Thus, a person, who is far from physics and math, typically faces the following contradictions:

1. According to the traditional theory, the degree of fuzziness grows quite sharply in front of the focusing point. Moreover, this growth is non-linear. Merklinger claims the disk-of-confusion in the object space grows linearly.

2. The traditional circle of confusion that operates in the image field is limited by the value c' = f ²/(dN) = (Mf)/N. Merklinger's disk-of-confusion in the object field can be infinitely large.

3. In the object field, the zone of acceptable resolution is absolutely symmetrical. The traditional zone of sharpness (DOF) can be asymmetrical under certain circumstances.

4. If focusing can be described by Fig.3, than we must admit that DOF does not depend on the focal length provided image magnification is maintained constant. The latter condition means that we must increase (or decrease) d and f simultaneously. And if d/f is constant, than the apex angle of the cone in the object space is also constant. According to the traditional approach, beyond the limits of the green zone (see Fig.2) DOF do depend on the focal length, even if magnification is kept constant.

Merklinger wrote that in spite of many differences, both methods can be used in practice. This sounds quite strange, because the recommendations of the two theories critically diverge. Merklinger wanted to clarify the subject matter, but ironically many photographers got perplexed. So the question is, which method works best? Which theory treats sharpness in a reasonable way?

There is only one way to resolve this dispute. I mean experiments.

Sharpness and the focal length

Does sharpness depend on the focal length at a constant magnification? To answer this question, I had to take several photographs with lenses of different focal lengths. During the experiment, I changed d in proportion to f to maintain magnification constant (f/d = const).

A good friend of mine, Mr. Sparkys the Toucan, was selected to participate in my experiments because he had always been my most patient model. You can see this guy in the photograph that I took in the light of my flash unit (Fig. 4). This image serves to show you the actual appearance of the bird. The experimental photos were taken without flash; therefore the illumination was constant for all the shots. The toucan was seated on a stool in front of a curtain. The distance between the curtain and the front edge of the stool was 70 cm (2�3�). This distance was fixed during the experiments. The shots were taken at f/4.5. The lenses were focused on the curtain.

Fig. 4

Before discussing the results of the experiments, let us recall what the two theories predict for our conditions.

According to Merklinger�s theory, the degree of unsharpness must be the same for all focal lengths. Or, at least, resolution of details must be the same.

The traditional approach treats the situation as follows. If we start with a lens with a relatively short focal length, our parameters will be beyond the limits of the green zone (Fig. 2). In that case, the front DOF will be smaller than the rear DOF. When focal length increases, the front area of sharpness will grow, while the rear area of sharpness will shrink. This process will continue, until the front DOF almost equals the rear DOF. Then the parameters will be within the green zone, and DOF will stop reacting to changes in focal length.
Thus, to detect the differences between the two theories, we have to start shooting outside the green zone.

Now let us return to the results of my experiments.

Fig 5.   f = 35 mm	Fig. 6.   f = 70 mm	Fig. 7.   f = 140 mm

When producing the photos shown in Fig. 5, Fig. 6 and Fig. 7, I equally enlarged all the negatives. The constant magnification means the size of the plate, which was attached to the curtain, must be the same in all photographs. The size of the toucan should vary, of course.

The results are self-explanatory. In the first photo the toucan�s eyes are vague, while in the last photo they are almost sharp. The stool in Fig. 7 is undoubtedly sharper than its peer in Fig. 6. Moreover, it is easy to notice that an increase in focal length results in better resolution of details. The letters on the toucan�s wings perfectly illustrate this idea.

Advocates of Merklinger�s theory may not be happy with such illustrations because the image of the toucan varies in size. To make things clear, I also prepared another version of those pictures (Fig. 8, Fig. 9 and Fig. 10). In this case, the enlargement of the negatives differs from one image to another, and the size of the toucan is maintained constant.

Fig. 8.   f = 35 mm	Fig. 9.   f = 70 mm	Fig. 10.   f = 140 mm

Again, it can be clearly seen that both sharpness and resolution do depend on the focal length when magnification is kept constant. The experiment showed that, when we look at pictures, the traditional theory outperforms Merklinger�s rules. The latter approach is only a special case and should be used with care. For example, you must never use Merklinger�s recommendations for the objects that are located closer to the camera than half the hyperfocal distance.

Now it is time to check Merklinger�s recommendations for the area behind the point of exact focus.

Focusing at infinity vs. focusing at the hyperfocal distance

Harold Merklinger was very unhappy with the traditional recommendation to focus the lens at the hyperfocal distance to obtain the largest DOF. No wonder. His disk-of-confusion in the object field unlimitedly grows behind the object in focus (see the diverging blue dash lines in Fig. 3). Due to this fact, resolution diminishes as the distance from the camera increases. Merklinger claims this fact is critical for the pictures of distant objects.

So what are his recommendations? He suggested a simple way out. If the lens is focused at infinity, the two blue dash lines run in parallel (Fig. 11). This means, the object field is scanned by the tube with a constant diameter of f/N, and resolution is maintained constant. If we accept such arguments, than we have to compare the size of each our object with the working diameter of our lens to assess resolution and sharpness.

Fig. 11

But a number of questions still remain unanswered. Does it make any sense? Is the traditional recommendation to focus the lens at the hyperfocal distance incorrect? If so, how many new details will we see, if we focus the lens at infinity? Will the distant objects look significantly sharper?

First of all, let us recall what the traditional theory recommends.

The graph in Fig. 12 shows the degree of unsharpness as a function of the distance from the camera. The red curve corresponds to the case when a lens in focused at the hyperfocal distance (h), while the blue curve represents the lens focused at infinity. As we can see from this graph, if any important objects are closer to the camera than two hyperfocal distances (2h), we should focus our lens at the hyperfocal distance. If all our objects are located behind this point, the focus should be set at infinity.

Fig. 12 (c0 = c')

The traditional recommendation to focus the lens at the hyperfocal distance assumes that infinitely distant points will be rendered sharp enough, because fuzziness of remote objects cannot exceed c' (Fig.12).

Again, a few experiments can help us to understand the subject matter correctly.

First of all, I would like to demonstrate you that the divergence of the blue dash lines in Fig. 3 is not that critical. Moreover, it is quite natural, and it does not lead to unsharpness.

The easiest way to demonstrate circles of confusion is to take several photographs on a street of a night town (Fig. 13). By the way, Merklinger wrote that his method outperforms the traditional theory when there are many similar objects at different magnifications (Chapter 9 in his book). Exactly this case is shown in Fig. 13. So this scene can tell us a lot about both methods. Of course, Fig. 13 exclusively serves to show you the cityscape I selected for my experiments. The experimental photographs will follow.

Fig. 13

Now let us see what happens if the lens in focused on a close object.

Fig. 14. f = 50 mm; N = 4; d = 1.5 m	Fig. 15. Enlarged fragment of Fig. 14

The photo in Fig. 14 corresponds to the case when the lens was focused at 1.5 m. Under such conditions, the lamps were transformed into circles. The enlargement of the fragment in the yellow frame is shown in Fig. 15. The traditional theory proved to be true again. The green spot represent the green traffic light that was 100 m away from the camera. It is absolutely the same size as the neighboring faint circles from the lamps that were over 300 m away.

	The photograph in Fig. 16 was taken at f/2. Without enlargement, it can be clearly seen that circles of confusion do not increase for the distant sources of light. The imaginary photobrush is the same for objects 10 m away and for objects 500 m away. Because of this, the divergence of blue dash lines in Fig.3 is not that dangerous.
Fig. 16. f = 50 mm; N = 2; d = 1.5 m

But our judgments must be fair to Merklinger. He described the situation in terms of resolution, while the traditional approach deals with sharpness of contours. The question is — Is it correct to substitute sharpness with resolution?

It must be admitted that the larger the distance from the camera, the more details are captured by a circle of confusion of a constant diameter. In this sense, Merklinger is right. Resolution for distant objects does decrease. But it is absolutely natural. It is more than likely that a person with perfect eyesight will not recognize a very familiar face from the distance of 100 m. Nobody would anticipate the opposite outcome. Even if our eyes are focused at infinity, we cannot expect to see the same number of details on remote objects as we see on close objects.

Sharpness of contours is a different thing. When we focus our eyes at infinity, we will not be able to distinguish small details at large distances. But contours of large objects will be sharp and clear. I mean that it is better to describe sharpness in terms of an imaginary photobrush that works in the image field than in terms of resolution in the object space. Let us have a look at Fig. 8 again. Does it matter that we can distinguish the toucan�s eyes in the picture? The eyes remain blurry, and that is the point.

When a painter creates a picture, he may not show a lot of details. The picture will look sharp, if the contours are distinct. And vice versa, if the contours are blurry, the picture is considered to be unsharp, no matter how many details can be distinguished.

If you admit my arguments, then you should also admit that focusing at the hyperfocal distance is quite reasonable.

Now that enough has been said about the two theories and the actual circles of confusion have been demonstrated, you are invited to compare two photographs. To test Merklinger�s recommendations, I decided to take two picture of a cityscape (Fig. 17). My camera was 20 m away from the nearest lamp post. The red advertising board with the inscription BAZAAR on it was approximately 250 m away. I used a 50 mm lens, so the hyperfocal distance was 21 m (c0 = 0.03 mm, f/4). The goal of my experiment was to detect any differences between two shots, one being taken with a lens focused at infinity, and the other being taken with the same lens focused at the hyperfocal distance. It was also interesting to see how much new details can appear in a picture, if the lens is focused at infinity. The cityscape I selected is shown in Fig. 17. Of course, we cannot draw any conclusions on the basis of such a small picture. For our purposes we have to look at enlarged fragments.

Fig. 17

First, let us see how remote objects are rendered (Fig. 18 and Fig. 19) at the two focusing distances. Since the BAZAAR board was 10 times farther from the camera compared to the hyperfocal distance, it could be considered to be infinitely distant.

Fig. 18 The lens was focused at the hyperfocal distance	Fig. 19 The lens was focused at infinity

The advantages of focusing at infinity did not prove to be obvious. When preparing the illustrations, I scanned the negatives @2820 dpi, which gives 3 pixels per the diameter of the circle of confusion. Had I scanned the negatives @4000 dpi, the differences would have been more noticeable. Nevertheless, I believe the gains would not have been considerably evident. If we are not going to enlarge a 2 x 3 mm fragment of a negative, the advantages of focusing at infinity are negligibly small.

Now let us see at the foreground (Fig. 20 and Fig. 21).

Fig. 20 The lens was focused at the hyperfocal distance	Fig. 21 The lens was focused at infinity.

The rear part of the car was approximately 6 m away from the camera. The degree of enlargement is the same for all fragments (Fig. 18 to Fig. 21). It is quite obvious that the image in Fig. 21 is smeared. Our ability to resolve all the details does not save that picture. It remains to be quite blurry. I believe, in this case it is better to focus the lens at the hyperfocal distance. It will give us the largest DOF in our picture.

Focusing at infinity should be preferred to focusing at the hyperfocal distance, only if all of the following conditions are observed:

1. There are no important objects closer than the hyperfocal distance.
2. The negatives are going to be enlarged significantly (larger than 8� x 12�).
3. The camera is installed on a tripod, and a fine-grain film is used.

If at least one of the above conditions is not observed, it is worth focusing the lens at the hyperfocal distance.

Conclusion

The traditional theory works perfectly in most practical cases. Moreover, it is not that difficult at all to be substituted with an alternative that pretends to be its peer.

Harold Merklinger�s approach can also be used in practice. However, it is a very special and limited method. For the purposes of resolution it works perfectly. But it may fail when it deals with sharpness.

Generally speaking, depth of field is determined by magnification, f-number, and the focal length. Only if the focusing distance is significantly smaller than the hyperfocal distance, depth of field does not depend on the focal length at a constant degree of magnification.

The largest depth of field can be obtained if we focus the lens at the hyperfocal distance. It should be kept in mind that when we focus lenses at infinity, we always make close objects fuzzier.

And, of course, a photographer's experience also does matter.
The theory is dry, experience is vivid. :-)))

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Links

1. Igor Yefremov. Depth of field. A combination of unknown and obvious facts. (The article is published on this web site.)
2. Igor Yefremov. Bluring the background. A scientific approach. (The article is also published on this web site.)
3. H. Merklinger. The INs and OUTs of Focus. Internet Edition
             (The book can be downloaded from: http://www.trenholm.org/hmmerk/download.html).

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