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Jon Grepstad Large Format Photography
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A LIMITED LOOK AT THE MATHEMATICS OF DEPTH OF FIELDMichael K. DavisEmail: zilch0@primenet.com October 11, 1995 Depth of field is defined as the range of distances from a camera that will be acceptably sharp in the finished picture. For the photographer whose creative energies are often in competition with the need to think analytically, this parameter of image quality is probably second only to exposure determination in the attention it demands. This text will describe the variables that affect depth of field, offering in summation, a simple formula and some techniques that can be used to achieve acceptable sharpness from foreground to infinity. This discussion will be limited to cameras that lack perspective control, and will only provide the formulae associated with obtaining a depth of field that includes infinity. We are going to reverse the chronology of image making, beginning with a discussion of the final event - viewing the print. Prints are often viewed at a distance equal to their diagonal. More on viewing distances later, but for the moment, imagine you are looking straight down on a 6 inch by 8 inch print, from above its center. This print has a diagonal of exactly 10 inches. Your eye is placed at the apex of an equilateral triangle that is formed with the print diagonal as the base leg. Each side of this triangle is 10 inches long. We will call this a viewing distance of 10 inches (even though the distance from your eye to the center of the print is a little less than that). Let's have a better look at this print. It's a classic lake and mountain scenic with an out of focus foreground. The foreground objects fell short of the near-distance sharp. The depth of field is bounded by the near-distance sharp and the far-distance sharp. Objects at distances between these two sharps are perceived to be acceptably sharp in our example print, when viewed at a distance of 10 inches. The foreground appears unfocused because thousands of circles of confusion are overlapping each other and they are of an unacceptable diameter. A circle of confusion is formed when the image for a given point on the subject is focused in front of or behind the film plane instead of precisely at the film plane. The circles of confusion for those points in the image that correspond to objects that are precisely focused are extremely small. Points imaged from objects nearer or farther from the camera than those objects residing in the plane of sharpest focus have increasingly larger circles of confusion at the film plane. At the near- and far-distance sharps, the circles of confusion have increased to a size that is unacceptable. The maximum acceptable diameter of these circles of confusion, as viewed in the final print after possible cropping and enlargement, must be subjectively selected before we can proceed to discussions of how to control depth of field. You can decide to what diameter these circles of confusion should be limited and then implement controls to adhere to that decision. A survey of lens manufacturers revealed that their depth of field scales and tables were based on values ranging from 1/70th to 1/200th of an inch for 10 inch diagonal enlargements viewed at 10 inches. The depth of field scales on lenses you own are expanded or compressed by the manufacturer to produce circles of confusion at a diameter of their choosing. A conservative choice for the maximum acceptable circles of confusion, after enlarging the format diagonal to a print diagonal of 10 inches, to be viewed at a distance of 10 inches, is 1/175 inch. Most would agree that at a viewing distance of 10 inches, a print with circles of confusion under 1/175 inch will be found acceptably sharp. This would correspond to a 35mm frame having circles of confusion no greater than 1/1021 inch before enlarging the negative 5.87 times to reach the 10 inch diagonal of the print. 1/1021 inch is about 0.025mm. Since our 6x8 print has a diagonal of 10 inches, we know that exactly 1750 acceptable circles of confusion of 1/175 inch diameter could fit end-to-end along the print's diagonal. Depending on the subject, of course, there could actually be many more than this overlapping each other along the diagonal. If they were end to end, though, no more than 1750 would span the 10 inch length. If this were a contact print, made without enlarging the original negative, we would always have acceptable sharpness as long as the diameters of the circles of confusion in the original negative do not exceed 1/175th of an inch. If we enlarge the negative to dimensions greater than 6x8 and increase our viewing distance proportionately, there would be no perceived loss of sharpness. What happens if we make the same 6x8 print from a smaller negative? Our first formula will provide the answer. For a specified format diagonal, what diameter circles of confusion are acceptable at the film plane, prior to enlargement for viewing at a distance equal to the print's diagonal, such that the perceived sharpness will be equivalent to viewing 1/175 inch circles of confusion at a distance of 10 inches?
-1- Where did the constant 1750 come from? 1750 circles of confusion, of 1/175 inch diameter, could fit end-to-end across a 10 inch print diagonal. This formula shows that as the format diagonal decreases, the acceptable diameter for circles of confusion at the film plane decreases, but for any format, these circles of confusion will not be enlarged beyond the goal diameter of 1/175 inch in a 10 inch diagonal print. Using this formula to determine the acceptable circle of confusion at the film plane, any enlargement made from any format size, when viewed at a distance equal to the print diagonal, will be found acceptably sharp. (It will be perceived to have the same sharpness as a 10 inch diagonal print viewed from 10 inches with circles of confusion not exceeding 1/175 inch.) This calculation can be done for each format you work with, taking into account any cropping you expect to do. For example, the 35mm format has an exposed area of roughly 24x36mm, but if this is regularly cropped to extract 4:5 aspect ratio enlargements, only a 24x30 mm area will be used. Thus the net format diagonal is the square root of the sum of the squares of 24 and 30, which is 38.42mm, not the gross diagonal for 24x36 mm (43.27mm). Using the net diagonal, 38.42mm, converted to inches, 1.51 inches, let's calculate the maximum acceptable circle of confusion at the film plane for a 35mm negative that will be used to make a 4:5 aspect ratio enlargement. Using formula -1-: 1.51 inches / 1750 = 0.000864 inch or 1/1157 inch This maximum acceptable circle of confusion at the film plane is considerably smaller than the 1/175 inch circles we will see after enlargement. The ratio of 1157 to 175 represents how much you have to enlarge the 24x30mm portion of a 35mm negative to expand its 1.51 inch diagonal to the 10 inch print diagonal.
-2- Armed with the figure 0.000864 inch and the focal length of our 35mm camera's lens, we are prepared to calculate the hyperfocal distance. We can calculate a value for any selected aperture, to produce 4:5 aspect ratio enlargements of any size that when viewed from a distance equal to the print diagonals, will be found acceptably sharp. To calculate the hyperfocal distance, use the following formula:
-3- For every lens you use for a given format, you can calculate hyperfocal distances for each aperture. Let's calculate the hyperfocal distance for a 50mm (1.97 inch) lens, using the previously calculated maximum acceptable circle of confusion for a 24x30mm negative, selecting an aperture of f/16:
Using formula -3-: The near-distance sharp is at half the hyperfocal distance. This would be 140 inches (11.7 feet). By focusing at the hyperfocal distance of 23.4 feet, the near-distance sharp will be at 11.7 feet and the far-distance sharp will be infinity. Points on every object at distances from 11.7 feet to infinity will be imaged with the circles of confusion not exceeding 1/175 inch in a 10 inch diagonal print. (If you focus on infinity, the near sharp will be the hyperfocal distance of 23.4 feet.) Once the hyperfocal distance is known, there are several ways to adjust the depth of field.
-4- Similarly, four times the object distance yields sixteen times the depth of field. If you can not achieve the depth of field you need by other methods, consider increasing the object distances to the camera. Later, when you crop the negative to achieve the perspective had before increasing the object distances, the loss in sharpness due to cropping will be linear, not exponential like the gain in depth of field. Grain might become an issue due to the increased magnification of the negative, but the desired depth of field will have been achieved. You could for example, get a net gain of three times the depth of field by increasing the object distance by a factor of three (a 9x increase in depth of field) and later compensate for the change in perspective by cropping by a factor of three (a 3x decrease in depth of field). To implement this concept at the time of exposure, simply move backward until the nearest object that must be sharp has fallen within the calculated near-distance sharp (one-half the calculated hyperfocal distance for the focal length and aperture being used).
-5- For a given format, going from a 100mm lens to a 50mm lens yields four times the depth of field, but if you have to close the distance to achieve the same perspective with the 50mm lens that you would have had with the 100mm lens, you will have exactly undone the gain had by switching lenses. By choosing the 50mm lens and later cropping to achieve the desired perspective, instead of halving the object distance by moving closer, there will still be a net gain in depth of field after enlargement, similar to item -4- above.
-6- If you double the f-number, you double the depth of field. The depth of field at f/16 is twice that at f/8. If the hyperfocal distance for f/8 is 24 feet, it will be 12 feet at f/16. Focusing at the hyperfocal distance in both cases, would give far-distance sharps of infinity and near-distance sharps of 12 feet for f/8 and 6 feet for f/16 (the near-distance sharps being half the hyperfocal distances).
-7- Thus far, we have assumed that all viewing distances would be equal to the print diagonal. This is not an unrealistic assumption, but it is said that the best perspective is achieved at a viewing distance determined using the formula above. A 50mm (1.97 inch) lens on a 35mm camera, generating a 10 inch diagonal print with a 4:5 aspect ratio, having a negative-to-print magnification ratio of 6.62 (10 inch diagonal / 1.51 inch diagonal), should be viewed at a distance of: 1.97 inch * 6.62 = 13.0 inches For the 50mm focal length, a 13 inch viewing distance is greater than the more critical 10 inch viewing distance for which we subjectively decided to use 1/175 inch circles of confusion. For any degree of magnification from the 24x30mm useful area of the 35mm format (for prints with a 4:5 aspect ratio), it is not until focal lengths under 1.5 inches (38mm) are used that the so-called best viewing distance would bring the viewer inside the 10 inch distance for a 10 inch diagonal print. This would jeopardize the apparent sharpness of our 1/175 inch circles of confusion in our 10 inch print and is why the final print size and viewing distances might be considered when selecting the constant used in formula -1-. It should be apparent by now that the constant 1750, is really a variable. Fortunately, for any value selected as the maximum acceptable diameter of circles of confusion, increasing the negative-to-print magnification simultaneously moves the best viewing distance farther away, enhancing the apparent sharpness at the same rate at which it is being lost due to enlargement. Only the focal length component of this equation presents a threat.
-8- You do not have to generate several depth of field tables for each lens to deal with various anticipated viewing distances. Starting with the value 1750 in formula -1-, you can generate a list of the hyperfocal distances for the apertures of a given lens that will give you 1/175 inch circles of confusion in a 10 inch diagonal print. From there, you can simply increase or decrease the calculated hyperfocal distances to handle viewing distances that are not equal to the print diagonal. If you know at the time of the exposure that you intend the final enlargement or projection to have a diagonal of 40 inches and expect it to be viewed from a distance of 40 inches, your calculated hyperfocal distances are fine. If you anticipate 20 inch viewing distances for the 40 inch diagonal, you must double your calculated hyperfocal distance. A near-distance sharp would be doubled, too. With this new information you can alter your f-number, object distance or focal length to achieve the necessary depth of field. The limited scope of this article does not cover other factors that affect sharpness. Diffraction, for example, worsens as apertures are reduced to improve depth of field. So too, camera and subject movement become a greater issue when slow shutter speeds are used with narrow apertures. Other formulae are available that allow you to calculate near- and far-distance sharps when focused at distances other than the hyperfocal distance, including those typical in macro-photography. The books I reference below are highly recommended for additional information.
View Camera Technique Leslie Stroebel
Basic Photographic Materials and Processes
The Camera Ansel Adams
Jon Grepstad gjon@online.no | Home | Depth of field | Camerabuilders |
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