Resolution of Digital Cameras:  How Much is Needed and How Much Have You Got?

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Introduction

           We are concerned with pixels and resolution because we want our pictures to look good. Therefore the two basic questions concerning resolution are how much we need, and how well does any particular camera do? Here we are assuming that printer capabilities are not limiting. In the past, when film was used to record photographic images, it was logical that lens resolution be described in terms of distances on the film, e.g. the number of line pairs/mm that could be resolved. With the shift to digital photography, the elimination of film, and the understanding that resolution of an image depends somehow on the number of pixels in the camera’s sensor, “resolution” is now more often described in terms of number of pixels in the image. Only if the optics are sufficiently good, however, is counting the pixels an adequate measure of a camera’s resolution. If a camera lens is of low quality and spreads the image of a point over a number of pixels, then the camera’s resolution will be less than that suggested by the number of pixels in its sensor. Not surprisingly, measures of a camera’s resolution other than lp/mm or pixel counts are gaining in popularity. One the most meaningful measures is the number of line pairs (one black line and one equal width “white line”) that can be resolved per picture width, lp/pw or picture height.  

How Much Resolution Do We Need?

        The amount of detail that is needed in an image depends on one's use of the image. If we are examining prints, then the question becomes determining the limit of resolution of the human eye when examining prints. I tested my resolution under the light conditions under which I display my prints. The test used the target of line pairs described and provided below. At a distance of 13 feet (156 inches) I can just tell that the bottom pattern of the most closely spaced 20 line pairs, which is in total 1.6 inches wide, is indeed a set of vertical black and white lines and not a gray rectangle. If I were viewing a 4" x 6" print from a distance of 18", the number of line pairs that I could just distinguish across the 6"  would be 20 x 6/1.6 x 156/18 = 650. Thus, a camera able to resolve 650 line pairs across a picture width generates as sharp an image as my eyes are capable of discerning. Any greater resolution is unnecessary as long as I don't significantly crop an image before printing.  I find that I observe 8" x10" images from about 36" and 13" x19" images from about 60". These sizes and distances turn out also to require a resolution of about 650 line pairs per image width. As shown below, a good prosumer compact digital requires about four pixels to resolve a line pair This corresponds to about 2500 pixels across an image and 1667 pixels from top to bottom, giving an active sensor size of a little over 4 megapixels. Leaving space for cropping, it seems that 6-7 megapixels is the maximum sensor size one would ever need--if the optics are adequate.

       My eyes under the lighting I used can just resolve 0.6 line pairs/arc-minute, arctan(1.6/20 x 1/156). This is somewhat less than the limit of human resolution that is about 1.7 line pairs/arc-minute. Possibly then, an argument can be made for using cameras with sensors capable of resolving 2000 line pairs across an image, which would require a sensor of about 24 megapixels. Clearly, there is no need for anything larger than 24 megapixels. If when you print the line pair image given below at 8.5" x 11" and you can tell from a distance of 39 feet that the bottom set of line pairs is truely line pairs and not a grey rectangle, then you need 24 megapixels. I don't.

A Direct, but Subjective Measure of Resolution

        A good idea of the quality of a camera's resolution can be obtained merely by looking closely at  how well it images an edge. If the sharp edge of a black rectangle is imaged on the center of a pixel, then its output response should be the average of the white response and the black response, a mid level gray. The pixel immediately to one side of this central pixel should be full white, and the pixel on the other side should be full black. Blur in the optical system will smear black into the white area and white into the gray area and may generate a band several pixels wide over which the transition between white and black occurs. Close examination of such pixel densities requires the use of an image editing program capable of enlarging images to the point that individual pixels are easily discerned. The programs ImageJ, Picture Window Pro, and Adobe Photoshop are all capable of such magnification (See related web sites below.)

       A portion of the greatly enlarged output in jpeg format from my Canon S70 camera of  a white-black edge is shown below and reveals an interesting feature. The image was exposed such that the white areas were not fully white in the exposure, that is, not of intensity 1.0, and the black areas were not fully black, that is, not intensity 0.0. Just to the side of the edge are several pixels that are whiter than those away from the edge. Similarly, on the other side of the edge are pixels that are more black than those away from the edge.

These enhanced contrast areas are a result of digital sharpening (see The Principles Behind Digital Image Sharpening) of the image that has been performed in the camera as it converts the information recorded by the sensor into the final jpeg format file of the image. This sharpening is always performed by the camera when it stores images in the jpeg format. While it generally enhances the picture quality, it does not always do so. For the most refined computer-based image editing, it is best that this in-camera image sharpening be prevented. As we will see below, sharpening affects measurement of the optical resolution of a camera.

        Many cameras allow images to be stored in raw format in addition to jpeg format. The raw format images are not sharpened, and computer programs for the conversion of raw files allow sharpening to be omitted from the conversion of raw to jpeg or tiff files. The illustration below shows another highly magnified portion of an image of a white-black boundary. This image was converted from raw to tiff format without sharpening. The extra-white and extra-black halos from sharpening are absent. The target was slightly tipped with respect to the camera so that the edge would fall in varying positions on the pixels.  At its narrowest, the transition region from white to black encompasses three pixels. At such a point, as illustrated, the edge must have fallen close to the center of the middle pixel in the transition. Clearly some black blurred into the adjacent pixel and some white blurred into the pixel on the other side. Thus, we can see that the camera blurrs significantly beyond half a pixel but not much beyond one and a half pixels.

An Objective Measure of Resolution

Before proceeding, it is helpful to provide an objective measure of resolution. It is the modulation transfer function, MTF, that can be defined for black-white line pairs. Take a cross section of the output from a perfect optical system that is generating an image of the lines on a sensor, and let the sensor consist of infinitely small pixel elements. Pixels recording the image will see light of intensity 0 for the black lines and intensity 1.0 for the white lines. The MTF of an image of the lines is defined as the maximum intensity difference between a pixel imaging a black line and a pixel imaging a white line divided by the intensity difference between white and black areas when blurring is irrelevant (taken from large areas of black and white on the image). In the case of a perfect optical system in which there is no blurring or spreading of images, MTF = (1.0 – 0.0)/ (1.0 – 0.0) = 1.0

Resolution With Finite-sized Pixels

        A highly imperfect optical system could blur the lines and completely mix them. In this case an image of line pairs is recorded as uniform grey, 50%. Images of large black and white objects would still contain regions of intensity 0 and intensity 1. The MTF in this case equals (0.5 – 0.5)/(1.0 – 0.0) = 0.

            The drawings below show that, even with perfect opics, as the number of line pairs increases, the MTF must drop from 100% to 0%. Thus a reasonable measure of the optical quality of a camera is how much more rapidly the measured MTF decreases that it would if the optics had been perfect.

            Suppose the widths of the lines are such that a line pair has a width approximately equal to the width of one pixel. In this case, the number of line pairs across the image, lp approximately equals the pixel width of the image, or  lp ≅ pixels wide. No matter what the exact phasing of line pairs is on the pixels, each pixel images the equivalent of one black line and one white line and its output will be 0.5. Thus, the MTF in this case, even with perfect optics, is 0.

            If the number of pixels across the image is doubled or the number of line pairs across the image is halved, lp ≅ pixels wide/2. Equivalently, this is two pixels per line pair.   In this case sometimes pixels will line up exactly in register with the lines and with perfect optics, the local MTF in this little region will be one. At other places across the image, the pixels will not be exactly in register. (Let us ignore the special and physically unrealistic case of line pairs exactly matching the pixels, pixel for pixel across the entire image.) For every place across the image that they are perfectly in register and give a MTF of 1, there will be a place where they are perfectly out of register and have a MTF of 0 and so on. Averaged across a significant portion of an image, the MTF equals 0.5. If the optics are less than perfect, the areas of black and white will spread somewhat into each other and the minimum intensity by any pixel recording a black line will be greater than 0 and the maximum pixel intensity of a white line will be less than 1.0, and as a result, the MTF will be lower than 0.5.

If the number of pixels across the image is doubled again, lp ≅ pixels wide/4,  that is, four pixels per line pair,

then with perfect optics, no matter what the precise positioning of the lines with respect to the borders of the pixels, one pixel will receive a full black image and the pixel two positions away receives a full white image. Thus, with perfect optics the MTF is 1.0.

            With a little effort one can show that the theoretical best MTF for one line pair per three pixels, lp ≅ pixels wide/3, is 0.75. The precise behavior of MTF as a function of line pairs per pixel in this range is quite interesting, but not important for our questions.

            The end result of our analysis is that one measure of the optical quality of a digital camera can be obtained by comparing the measured MTF when imaging the vertical black and white lines in the region of two to four pixels per line pair, that is  at lp ≅ pixels wide/2  to  lp ≅ pixels wide/4.  In this latter case, a good optical system will have an MTF close to 100%.


Measuring Resolution

            It is relatively easy to make resolution measurements by photographing a target containing groups of 10 or 20 appropriate lines (ranging from 5 to 20 line pairs per inch)  from a distance of 5 to 15 feet, then analyzing with an image editing program like Picture Window Pro, Photoshop, or ImageJ. It should also be possible to display a suitable target on one's monitor and photograph that. From the pixel positions of the edges of a set of lines and the total number of pixels across the image, calculate the total number of line pairs possible across the image. Pick images and sets of lines where lp  is close to pixels wide/4.  Measure the lightness value of the image in white areas and black areas and then expand the image and read out the lightness values across several black and white lines to obtain the MTF by comparison to the white and dark values.

An optical target for use in resolution measurement.

Downloadable version of the above target (0.5 MB).

Resolution Measurements on a Canon S70

          The table below shows the maximum possible MTF as a function of line pairs per pixels wide and the values obtained with a Canon S70 high end consumer camera with images stored as jpg files and with images converted from raw format with no sharpening. At lp ≅  pixels wide/4 the camera does well both in wide angle and telephoto in both the center of the image and in the corners. Surprisingly, however, the resolution falls unexpectedly rapidly at greater resolution. As one would expect, (The Principles Behind Digital Image Sharpening), sharpening can enhance detail, as shown by the fact that at pixels per line pair = 4, the MTF of a sharpened image is 0.8 whereas without sharpening it is 0.4. Sharpening is incapable of generating resolution where there is none, that is, at pixels per line pair =2.

Some related web sites.

http://www.clarkvision.com/index.html.  This site has nice pictures and a very thorough analysis of imaging and sensors.

ImageJ, http://rsb.info.nih.gov/ij/index.html. This wonderful program is free. It is directed towards quantitative analysis of images.

Picture Window and Picture Window Pro, http://www.dl-c.com/. This is an inexpensive but very powerful program is designed for serious photography and serious photographers.

Norman Koren's site,, http://www.dl-c.com/, http://www.normankoren.com/Tutorials/MTF.html,   provides excellent tutorials on many aspects of digital photography, including resolution analysis and MTF as well as the use of Picture Window Pro.

Clark Vision, very nice site, http://www.clarkvision.com/. This is another excellent site with wonderful tutorials and beautiful pictures.

Imatest, http://www.imatest.com/. This commercial site and bulletin board sells and services a program for analysis of lens resolution developed from the ideas that are described on Koren’s site.

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