Telephoto Figure of Merit: Explanation and Uses

By Rodger Carter and Rick Matthews


 During the past several years digital cameras have become a major commodity in the photography market.  There are now dozens of manufacturers producing more than one hundred models with a variety of features and capabilities.  Buyers have more choices than ever, so much so that the selection process has become somewhat difficult, if not downright intimidating

 One group of hobbyists that might have a special headache when choosing a digicam are those who are primarily concerned with photography of distant subjects such as wildlife, sporting events and other subjects normally requiring a long lens or big zoom.  In the field of film photography a number of options are available to improve one's chances of obtaining large, high-resolution images of a distant subject.  These include approaching the subject more closely (when possible), increasing the focal length of the lens, and using a finer grain film.  If these are not sufficient, going to a larger format camera and/or a higher quality lens may help accomplish the task.  Similar options exist for the digicam photographer except that higher resolution equates to more pixels and a larger format might mean going to a high-cost professional digicam.

 The most practical solution for digicamers other than reducing the distance between the subject and the camera would be to increase focal length and number of pixels.  That is, to purchase a digicam with a high number of pixels and a long zoom, or one to which a long zoom or lens can be attached.  The problem is somewhat more difficult than with a film camera because the filmers can increase media resolution just by going to a finer grain film.  For digicam owners, increasing resolution means going to an entirely different camera.  Further, with such a plethora of zooms and pixel counts available in today's consumer digicams, narrowing down the choice to those which are most suitable for distant subject photography can be a bewildering task.  Current optical zoom ratios include 3X, 4X, 5X, 6X, 7X, 8X, 10X and 14X.  Cameras with large zoom ratios generally have large maximum focal lengths as well.  Pixels counts vary even more, starting with 640 X 480 pixels for minimally serious photography, and up to 5 mega pixels or more.  Between these figures exist half a dozen or so sensors of intermediate resolution.

 It is not intuitively obvious which digicam or digicams are most suitable for telephoto photography when you have a choice of so many focal lengths and sensor sizes.  More pixels can offset the disadvantage of a smaller zoom and vice versa.  Selecting among cameras A, B, C, D, and E with combinations of 14X zoom and 786K pixels; 10X zoom and 1.92MP; 8X zoom and 1.23MP; 5X zoom and 1.38MP; and 3X with 5MP may send the prospective purchaser to the medicine cabinet, especially when each of those digicams may vary greatly in other useful features and specifications.

 The purpose of the Figure Merit (FM) originated by Rick Matthews of the Wake Forest University Physics Department is to provide an objective, accurate method of determining the relative ability of digicams with varying focal lengths and pixels to provide the greatest number of pixels in the subject area when the camera is used at longer focal lengths.  That is, to help select which particular combination of focal length and pixels will best do the job for those primarily interested in purchasing a digicam for distant-subject photography.  The FM formula combines the two key digicam specifications, maximum focal length and sensor pixels into one, simple to compute, mathematical equation that provides a reliable indicator of a digicam’s ability to reach out and touch someone.  To determine the FM of a particular digicam you merely multiply the number of pixels of the sensor times the square of the zoom’s maximum focal length (35mm equivalency).  That is, FM = P x FL x FL.  For example, The Sony FD91 has 786,432 pixels and a maximum focal length of 518mm.  Multiplying the number of pixels times the focal length squared, we arrive at an FM of 211 billion (786,432 x 518 x 518 = 211,018,579,968).  The 35mm equivalent FL is used rather than the true digicam FL for two reasons:  (1) the 35mm equivalent is more commonly cited by manufacturers than the actual sensor FL, and (2) the 35mm equivalent FL corrects for variations in sensor physical size which might otherwise cause the computed FM to be somewhat inaccurate.

Focal Lengths

 As rays of light enter a camera through its lens they converge to a point of focus some distance behind the lens.  Objects closer to the lens will focus at a point farther behind the lens than objects more distant from the camera.  For other than the most simple of cameras, a means of obtaining a sharp focus for subjects at various distances must be provided; however, as the subject becomes relatively distant from the camera, perhaps fifty feet or more, the entering rays become parallel for all practical purposes (similar to a point at an infinite distance) and the point of sharp focus becomes stationary no matter how much farther away the subject is from the camera.  That stationary distance between the lens and the point of focus is known as the focal length (FL) of that particular lens.  One source defines the focal length as “The distance between the optical center of a lens and the plane where parallel light rays from an object at infinity come to a sharp focus (Bruce:38).”

 How do the FL and number of pixels affect the final image and why are they combined in Rick's equation in the specific manner given above?  Let’s do a short review of focal lengths and pixels.  Those who are familiar with varying focal length lenses or zooms know that exchanging a shorter FL lens with a longer FL lens results in enlargement of the image (Davis and Binau:44).  The actual enlargement depends on the relationship between the FLs of the two lenses.  For example, a 50mm lens (approximately equal to a 35mm frame diagonal) is considered typical for a non-zoom 35mm camera.  Changing to a 100mm lens would cut the angle of the camera’s view in half.  That is, an image only half as wide and half as high as the previous image would now fill the frame.  We would consider the magnification power of the 100mm lens to be 2X (100mm divided by 50mm = 2).  The enlarged portion would be twice as wide and twice and high as previously - four times the sensor or film area that it occupied in the 50mm lens image.

 Another way of describing the above situation would be to say that a portion of the original image which was only ½ half as wide and ½ as high as the total image was enlarged by the 100m lens until it became the entire image.  The area of the portion that was enlarged consisted of ¼, or 25% of the original image (1/2 X 1/2 = 1/4).  This means that every item visible in the enlarged image also occupies four times the area on the film or sensor as it did in the original image.  Since the number of pixels in the digicam sensor does not change when the focal length is changed, it means that every item visible in the enlarged image contains four times as many pixels as it did in the original image.  Because we are primarily concerned with the number of pixels in the chosen subject, the increase in the area of the subject is of primary interest rather than the increase in height or width.  That is why the FM equation depends upon the FL squared rather than the FL alone.

ANGLES OF VIEW                      100mm SUBJECT AREA

 Can the ratio of two FLs squared be used to compare the magnification effect of two different FLs?  Yes!  Examine the figures above.  The magnification is directly proportional to the ratio of the FLs squared.  Compare the area of 50mm FL to the area of 100mm FL.  100mm x100mm = 10,000mm.  50mm x 50mm = 2,500mm.  10,000mm divided by 2,500mm = 4.  The 100mm lens enlarges the smaller area by four times.  Since the enlarged area now fills the frame, it means that it will contain four times as many pixels as previously.  Again, the reason we compare  FLs squared rather than just the FLs themselves is because we are concerned with the changes in area of a subject in the image as shown on the film or sensor, not the increase in height or width.

Number of Pixels

 The effect of the second factor in the FM equation, number of sensor pixels, is even easier to visualize.  If images from two digicams, one with twice as many pixels as the other, are printed at the same size, the print from the camera with twice the number of pixels will contain twice as many pixels as the other (surprise).  It will thus appear to have greater resolution, that is, finer details and smoother edge lines (less jaggies).  Because the number of pixels a sensor has can be equated to image resolution the way larger format film provides better resolution than smaller format film, an increased number of pixels  will provide prints that are improved in sharpness due to the increased number of pixels that appear in any given area of the print.

 How does pixel number relate to telephoto photography?  A trick to mimic a long focal length lens is to crop images so that they only contain the area of interest.  The more pixels a sensor has, the more pixels will remain in the cropped area of the photo.  If one camera has four times as many pixels as another, one fourth of the original image can be enlarged and cropped and still contain the same number of pixels as an uncropped image from the camera with the lesser number of pixels.  That is, when digitally enlarging photos from two different cameras, the number of pixels remaining in the cropped areas is directly related to number of pixels contained in each of the camera sensors.  Since this is a direct one-to-one relationship, the FM equation involves only the number of pixels, P, rather than being squared as is done with the focal lengths.

The FM Equation

 Now we come to the important part.  Because both the FL squared and the number of sensor pixels are each used in determining the number of pixels per square inch in an image, and because each one is independent of the other, the combined effect is determined by multiplying the two factors together.  That is, if the zoom of camera #1 provides 3 times as many pixels in the subject area than does camera #2 (FL #1 squared = 3 times that of FL #2 squared), and there are four times as many pixels in the sensor of camera #1 than in camera #2, then the combined effect will be twelve times as many pixels in an equal-sized subject area of digicam #1 as compared to digicam #2 (3 X 4 = 12).  An example computation is below.

Camera #1:  34-68mm FL; 640 x 480 pixel sensor.  FM = 68 x 68 x 307.2K = 1.42 Billion.

Camera #2:  32-102mm FL; 1280 x 960 pixel sensor.  FM = 102 x 102 x 1.2288MP = 12.78Billion.

 That is, camera #2 would have a 9 to 1 advantage over camera #1 when both are used at their maximum zoom (12.7 billion divided by 1.42 billion = 9).  This means that when an equal image segment from camera #1 is digitally enlarged to equal the same image segment in camera #2, it would only have 1/9th the number of pixels in the image as would be in the image from camera #2 - a big difference in resolution and sharpness

Examples of some sample digicam Figures of Merit are shown in the chart below:

Figures of Merit for Various Digicams of Interest.

Updated FM Chart as of 22 June 03
Provide by digital_ray_of_light

Camera Model                        Pixels          35mm FL       Figure of Merit               Zoom

Kowa TD-1                     2048 x 1536       1350          5,733,089,280,000             3x
Olympus c750                  2288 x 1712         380             565,622,886,400             10x
Olympus c740, c730        2048 x 1536         380             454,243,123,200             10x
Toshiba m700                  2048 x 1536         350             385,351,680,000             10x
Nikon 5700                     2560 x 1920         280             385,351,680,000             10x
Canon pro90 IS               1856 x 1392         370             353,688,268,800             10x IS
HP 850                            2272 x 1712         300             350,069,760,000             8x
Panasonic fz-1                 1600 x 2000         420             338,688,000,000             12x IS
Olympus c720                 1948 x 1488         320             296,819,097,600             10x
Olympus c2100               1600 x 1200         380             277,248,000,000             10x
Fuji s602 interpol             2832 x 2128         210             265,768,473,600             6x
Sony CD1000                 1600 x 1200         370             262,848,000,000             10x
Toshiba m500                  1600 x 1200         350             235,200,000,000             10x
Olympus E100RS            1368 x 1024         380             202,280,140,800             10x IS
Minolta 7, 7i, 7Hi             2560 x 1920         200             196,608,000,000             7x
Minolta 5                         2048 x 1536         250             196,608,000,000             7x
Sony F717, F707            2560 x 1920         190             177,438,720,000             5x
Fuji s602                         2048 x 1536         210             138,726,604,800             6x

Limitations of the FM

 Uses for the FM are strictly limited to comparing the long-distance image capabilities of various digicams which have differing FLs and number of pixels.  Further, the effect of lens quality, digicam internal programs, use or non-use of a tripod, and photographer expertise can also affect image resolution, but are not considered by the FM equation  The FM is merely a mathematical formula for comparing telephoto capabilities of various digicams when all other factors are equal.

Value of Long Lenses/Zooms

 Zooms are useful for obtaining close-ups of many subjects that often cannot be obtained in any other way:

1.  Wildlife photography.  Most wildlife will not let you approach near enough to capture the desired image.

2.  Composition.  A zoom has the advantage of quick composition without taking the time to switch between lenses, time that may mean the subject is long gone.

3.  Candid photography.  Good candid shots require that the subject be kept unaware that they are being photographed.  A long zoom can make you part of the distant background.

4. Architectural/building features.  A zoom takes the place of a ladder.

5.  Distant ships and aircraft.  Big zooms can perform photographic wonders at air shows.  That distant speck at sea can become a frame-filling keeper
6.  Sporting events.  A zoom can provide close-up views of the athletes in action.

7.  Concerts.   Obtaining a good photo of the cast depends primarily on having a good zoom.

8.  Race cars.  A good zoom is the next best thing to being in the pits.

9.  Animals in zoos.  A good zoom wisely used can provide images similar to those taken in the wild.

10.  Macro photography.  A long zoom is a good substitute for a macro lens when the subject is not likely to allow the photographer to approach close enough to use the macro.

11.  Graduations.  Individuals will often be nothing more than unrecognizable dots in a photo unless a zoom or long lens is used.

12.  Weddings.  A zoom is often necessary to obtain the desired photos without intruding upon the ceremony

13.  Aerial photography.  Shots taken from aircraft of subjects on the ground may show little in the way of useful images unless a zoom is utilized.

14.  Surveillance.  As with candid photography, it is critical that the subject not be aware of the camera (Might even save you a punch in the nose!).

15.  Astro-photography.  Having a long zoom to begin with means that astro-photos can be done with a relatively small, easy to handle (and inexpensive) telescope.

16.  Framing/composition.  A zoom allows the photographer to easily try a number of compositions and then select the one most preferred

17.  Perspective.  As with framing, a zoom allows a seamless choice of perspectives from the same shooting position.

18.  Image quality.  Other things being equal, a long zoom will normally provide better image quality than a short zoom with a teleadapter.

19. Time and convenience.  Having a built-in zoom rather than employing add-ons or SLR lenses eliminates the time and effort spent in the field switching back and forth from one lens to another.

20.  Storage.  Add-on lenses or SLR lenses equal to built-in zooms in magnification may require significantly more space and weight in the camera bag.

21.  Expense.  Built-in zooms generally add less to total cost than a camera plus teleadapters, or additional long focal length lenses for an SLR camera.

22.  Macro bellows.  Use of a bellows with a close-up lens requires that the digicam be equipped with a long zoom lens, otherwise the subject will only occupy a small circle in the center of the image.

23.  Image stabilization.  Several digicam models with long zooms also have built-in image stabilization that allows sharp, hand-held photos at maximum magnification—you can leave the tripod home!


 The Figure of Merit equation originated by Rick Matthews provides a quick, easy way to evaluate various digicams based on their ability to use focal lengths and pixels to obtain the best possible results when the photographer is primarily engaged in the capturing of distant images.

Zooms – the next best thing to being there!


Bruce, Helen Finn.  Your Guide to Photography: A Practical Handbook.  USA.  Barnes & Noble, Inc.  1965.

Davis,  F.W. and H.G. Binau.  Basic Photography.  Columbus, Ohio.  The Department of Photography of the Ohio State University.  1956.

NPDI (NPD INTELECT).  Internet., Feb. 29, 2000