How do depth of field and the circle of confusion relate to pixel size on the sensor?

Question

I have been reading about depth-of-field and presume to have a sufficient understanding of what it means and how it is related to lens properties, lens aperture, focal length, sensor format size and probably even print dimensions in case the picture is being printed. I do have a question about how/if depth-of-field depends on the dimension of a pixel. Let me elaborate:

Given two sensors with differing pixel sizes: x and 4x, the latter sensor integrates more light per-pixel than the former, however, it might have a lower resolution if the sensor dimensions remain fixed. Theory suggests that the depth of field is determined by those circles-of-confusion which are very close to being an in-focus point, and this perhaps also means that these very small CoCs fall within the integration area of the same pixel. Now when a CoC becomes larger than a pixel (like also in the case of diffraction blur), there is some spreading of intensities among neighboring pixels and this would clearly lead to blur. However, if one was using the 4X sized pixel, even with a slightly larger CoC, the intensity would still integrate into the same pixel area, and would thus perhaps be in-focus? Is this the right assumption? And if so, is there any analysis of pixel-area dependence of depth-of-field? Moreover, depth-of-field is in some sense independent of the integration medium as it clearly occurs in analog film devices as well, so is there a trade-off or a difference in depth-of-field between film and digital cameras?

Please point me to the correct references for this question. Please correct me if there seems to be a fundamental issue with the assumptions made above.

Answer 1

To understand how a camera's pixel pitch may affect Depth of Field (DoF), you must first understand what DoF is as well as what it isn't.

Regardless of the aperture of a lens, there will only be one distance that will be in focus. That is, there will only be one distance at which a point source of light will be focused to a single point on the recording medium. Point sources of light at other distances will be projected on the sensor (or film) plane as a blur circle, or circle of confusion (CoC). If this CoC is sufficiently small enough to be perceived as a point by human vision at a specific display size and distance, it is said to be within the DoF. The limits of DoF change based on aperture, focal length, and focus distance as well as the display size and viewing distance of the image. You can print two copies of the same image file and if one is displayed at twice the size of the other at the same viewing distance by a person with the same visual acuity the smaller print will appear to have more DoF than the larger one (assuming the resolution of the image file itself is not the limiting factor). There is no magical barrier at which everything on one side is in perfect focus and everything outside of that line is blurred. Rather, as the distance from the true point-of-focus increases, so does the size of the blur circle and we gradually begin to perceive that objects are not absolutely sharp.

Your fundamental understanding of how an image is created out of the data from a sensor with a Bayer mask in front of it (the overwhelming majority of digital cameras) is not entirely accurate. There is no direct correlation to a single sensel (pixel well) on a Bayer sensor and a single pixel in an image produced from the data provided by that sensor. The numbers for each color of each pixel in the image produced are interpolated from the data produced by multiple adjacent sensels.

You also assume, incorrectly in most use cases, that a two pixel blur will be detectable at typical viewing sizes and distances. It won't be. A typical circle of confusion for FF cameras is 0.03mm (30µm). A typical 20MP FF sensor has pixels around 6.5µm wide. Even accounting for the 2x2 grid of RGGB masked pixels used to produce 4 RGB pixels in the image produced, the 13µm width of a 2x2 cell is still less than half the width of the 0.03mm CoC needed for an 8x10 print viewed at 10-12 inches by a person with 20/20 vision. Most APS-C cameras have pixels that are slightly larger than 4µm wide. So the recommended CoC for an APS-C sensor of around 0.019mm is still over twice as wide as a 2x2 cells on a typical APS-C sensor.

If the theoretical sensor in your question with pixels 4X larger are sufficiently large enough to limit the perceived resolution of the image, then everything in the image with a CoC smaller than the resolution limits of the sensor will appear to be equally in focus at the expense of also appearing equally pixelated/blurry. This would occur when the CoC needed for a particular display size and viewing distance is less than twice the width of the pixel pitch of the camera's sensor. It would not, however, be a hard limit but instead would the point at which we gradually begin to perceive that the picture is made up of individual pixels that our eyes can discriminate.

Answer 2

I think your reasoning is correct.

The problem comes from people mixing abstract terms with their experiences without actually understanding the terms mean.

circle-of-confusion (CoC) (in German: "Unschärfekreis")

is one of these terms. The literal translation from German means: "circle of non-sharpness" which is IMHO a better description (and the reason why I've included it). So. What is this about?

As a photographer, it's easy to tell if an image is in focus or not: you just look at it and see whether it is in-focus. But your wife (or your children or your customers for that sake) may have a very different opinion about this image.

As a scientist you want hard facts that apply to everybody: A tool that will tell whether an image is in- or out of focus no matter what the circumstances are.

What we need to understand is the following:

CoC is not a value that is somehow calculated or fixed, it is assumed on a case-by-case basis.

As such, the CoC puts a precise number to the opinion of the photographer. For the scientist it is a crutch to get the "fluffy stuff" into a precise number. With this precise number, the scientist can use the optical formulas and calculate depth-of-field for specific situations.

Answer 3

When studying geometric optics we learn that the lens forms a sharp image of an object at a given distance downstream from the lens; objects at different distances will not image as sharp because their images to lens distances will be different. In other words an object at infinity ( ∞ ), comes to a focus nearer the lens than a close by object. Practical observation however reveals that objects before and behind the point focused on often appear sharp. Thus we have a zone fore and aft of the point focused upon that is perceived to be in satisfactory focus. This is the zone of depth-of-field.

Now the lens images each point on the various subjects and renders them as small circles at the focal plane. The term for these is “circles of confusion”. So named because they overlap and intermingle and under the microscope they appear as indistinct circles with scalloped edge. We perceive an image, or portion of an image to be sharp when these circles of confusion are so tiny that are not observed to be circles, that is, they appear as points without disenable shape.

Tiny circles appear as points when viewed from a distance that is about 3000 times their diameter. A 1 inch coin viewed from 3000 inches appear as a point ( 25.4mm viewed from 76 meters). This criteria proves to be far too stringent for viewing photographs. This is due to lens performance and the typical contrast ratio of graphic images. The widely adopted standard is a circle with a diameter of 1/1000 of the viewing distance. This is the stuff of depth-of-field tables and calculation. That works out to 1/100 of an inch ( 0.25mm) viewed from 10 inches (250mm). In other words a circle size of 2/100 inches = ½ mm viewed from 500mm.

Because todays cameras yield minuscule images, we must enlarger to get a useful picture. Typically we view images from a distance about equal to their diagonal measure. Therefore, depth-of-field calculations typically take all this into account by generalizing that the circle size permissible at the focal plane is 1/1000 of the focal lenght. Technical imaging often request more stringent standards. Kodak adapted 1/1750 of the focal length. Again, most tables and charts use 1/1000. Thus for a 50mm lens, the depth of field tables use a circle size of 0.05mm. Consider an 8 x 12 print or displayed image of a full frame. The magnification will be about 10X. The customary viewing distance is about 20 inches. With a circle size of 0.05 at the focal plane, the circle size on the print (display) will be 0.05 x 10 = 0.5 (1/2 mm).

This is the stuff of depth-of-field. The math takes into account the scope of human vision, the contrast of the image, the degree of magnification and the viewing distance. Again the standard most often uses is 1/1000 of the focal length. These values are valid for the size of photosites.

More gobbledygook from Alan Marcus