Within the context of creative photography (which is how all questions here should be understood), Depth of Field (DoF) is defined based upon blur perceptible by the viewer.¹ Any blur circle in an image at a specific display size and distance small enough to be perceived as a point by a viewer with a defined visual acuity is deemed to be within the DoF. Any blur circle large enough that it can be perceived as larger than a point by the same viewer at the same display size and distance is deemed to be outside the DoF.
With that context in mind, a word about what depth-of-field is and is not:
In a way, depth-of-field is an illusion. There is only one plane of focus. Everything in front of or behind the point of focus is out
of focus to one degree or another. What we call DoF is the area
where things look, to our eyes, like they are in focus. This is
based on the ability of the human eye to resolve certain minute
differences at a particular distance. If the slightly out-of-focus
blur is smaller than our eye's capability to resolve the detail then
it appears to be in focus. When you magnify a portion of an image
by making it larger or moving closer to it you allow your eye to see
details that before were too close together to be seen by your eyes as
separate pieces of the image.
Since things are gradually blurrier the further they are from the point of focus, as you gradually magnify the image the perceived depth
of field gets narrower as the near and far points where your eyes can
resolve fine details moves closer to the focus plane.
There is no intrinsic physical property that makes everything within a specific distance 'sharp' and everything outside that distance 'blurry'. There's no magic line inside of which everything is equally sharp and outside of which everything is equally blurry. There is the focus distance that is the 'least blurry' and there is everything else that gets increasingly blurrier the further one moves from the point of focus. Within the context of creative photography (which is how all questions here should be understood), the question to be answered is not, "Is it blurry?" The question to be answered is, "Is it blurry enough that a viewer with a specific visual acuity looking at it at a certain magnification from a certain distance will be able to tell that it is blurry?" Or to put it in the way most people in the photographic field state it, "Is it acceptably sharp?"
In fact, you can take the same exact photo and print it at two different sizes, view them from the same distance, and they will have different depths of field. It is a gradual transition from sharp to blurry. How much we magnify the virtual image cast by the lens onto the sensor as it is recorded by the digital sensor determines the exact size of blur (measured on the sensor) that we can perceive when we view it. There are several variables that affect just how far from the point of focus things start becoming noticeably blurry to our eyes.
The blur becomes noticeable at smaller distances from the point of focus (and thus the perceived DoF is smaller) if we:
- Use a longer focal length/narrower angle of view
- Use a shorter subject distance
- Use a wider aperture
- Use a larger display size
- View the displayed image from a closer distance
- Have better vision
The blur increases more gradually at larger distances from the point of focus (and thus the DoF is perceived to be larger) if we:
- Use a shorter focal length/wider angle of view
- Use a longer subject distance
- Use a narrower aperture
- Use a smaller display size
- View the displayed image from a larger distance
- Have weaker vision
Additionally, if the imaging system is either diffraction limited or resolution limited so that everything smaller than a specific size is equally blurry, this will affect the perceived DoF. When nothing is seen as 'sharp' anything with the same amount of blur is seen as within the DoF.
To properly calculate depth of field all of these factors must be taken into account. Many DoF calculators make (often unspoken) assumptions about some of them. Most DoF calculators, such as DOF Master, assume an 8x10 display size viewed from a distance of 10 inches by a person with 20/20 vision.
Since depth-of-field is dependent upon viewing size and distance as well as the visual acuity of the viewer it is hard for a DoF calculation to indicate depth-of-field if it doesn't know what the display size of the photo will be. The same goes for lenses that may be used with different cameras that have different sensor sizes. The DoF scale for the same lens will be different for an APS-C camera than it would be for a Full Frame camera if we plan to display images from both at the same size and distance.
Assuming the standard circle of confusion used for an image produced with a 36x24mm sensor, displayed at 8x10, and viewed at 10 inches by a person with 20/20 vision will accurately predict perceived DoF for most images is too broad in today's environment. The digital photography revolution has pretty much eliminated any idea of a standard display size and viewing distance. For Depth of Field calculations to be accurate they must be based on all of the variables listed above including the display size and viewing distance as well as the focal length, f-number, and angle of view determined by the sensor size (which directly affects the magnification ratio needed to display an image at a specific size). What is 'acceptably sharp' when viewed scaled to fit entirely onto a monitor will be different than the same image file viewed at 100% (1 image pixel equals one screen pixel). Viewing a 24MP image on a 23" HD (1920x1080) monitor at 100% is like looking at a small part of a 60x40" print!
If you want to account for differing display sizes and distances, you can use the Flexible Depth of Field Calculator from Cambridge in Colour and click show advanced to enter those variables.
Now, let's look at the idea of true equivalency:
There's no such thing.
- When we change the size of the camera's sensor and the focal length of the lens we don't change the wavelength of visible light. The light does not scale at the same proportions as our lenses/cameras do. Specifically with regard to the question, the same f-number on a smartphone with a focal length of 3.3mm that yields the same FoV as a FF camera with 25mm lens is 7.2 times narrower. If we are using f/1.8 the FF lens has an opening 13.9 mm wide. The smartphone has an opening 1.83 mm wide. The size of the light waves trying to squeeze through that 1.83 mm hole are NOT 7.2 times smaller than the light waves passing through that 13.9 mm hole, though. The effects of diffraction will substantially impact the image projected onto the smaller smartphone sensor compared to the image projected onto the FF sensor because a much higher percentage of the light rays going through the smaller aperture of the smartphone will be scattered by interacting with the surface edges of the aperture diaphragm. Light rays travel in waves, not in straight lines as we like to draw them in ray diagrams.
- We can't change the camera position without also changing the perspective which is solely dependent upon the camera position and the relative positions of everything in the field of view. To get the same photo, the optical center of the camera's lens must be in the same location for both cameras. To get an 'equivalent' photo we must use different focal lengths that are in the same proportions as the linear measurements of the sensors.
- When we change the f-number to compensate for the combined changes in magnification and focal length, we also change the exposure. To keep the exposure constant with the same ISO and shutter time, we can't change the f-number at all. If we change the shutter time to compensate for the difference in f-number, any scene that includes objects in motion (or any image taken with the camera in motion) will no longer look the same. Don't forget that an f-number is a dimensionless ratio between the focal length of a lens and the diameter of the entrance pupil. Also keep in mind that any aperture wider than f/13.6 with the 25mm lens for the FF camera will be less affected by diffraction than the 3.3mm lens at f/1.8 in the smartphone. And that is before we even begin to consider the difference in pixel sizes between the two cameras.
Depth of field is all about angles and whether the narrowest angle the viewer can discriminate is larger or smaller than the angle between the smallest discreet details in a photograph as they are viewed.
Unless I'm missing something, you haven't answered my question. Either way, true DOF does exist, and it has a well-defined physical definition, which is very much independent from the observer (I won't go into here). There's no illusion in a smartphone and a full frame DSLR which have the same f# (which supposedly determines your DOF), yet given the same FOV, the two will yield a vastly different DOF.
Within the context of creative photography (which is how all questions here should be understood), the 'well-defined physical definition' would only be applicable if the image is enlarged enough that the viewer can discriminate individual pixels in the image. In the case of current hardware, that would require enlargements and viewing distances that are well beyond what is typical.
At one foot viewing distance a person with 20/15 vision can resolve only about 290 ppi. At 8 inches it is 440 ppi. So the image from a camera such as the Nikon D810 would need to be enlarged to 25x17 inches yet still viewed from only a 12 inch distance before the circle of confusion is reduced to the same size as the camera's pixel pitch. To display that image at full resolution one would need a 30 inch monitor with a resolution of 7360x4912 (that is, a pixel pitch of 290 ppi). Typical 30 inch monitors have resolutions of only around 1920x1080 which is around 75-80 ppi. So even looking at the thing with a magnifying glass we would not be able to discriminate details smaller than about 4 pixels wide (in the original 7360 pixels wide image file before it was scaled down to be displayed at 1920 pixels wide). But who looks at a 30 inch monitor from only one foot away? When we back up to a more typical viewing distance of two to three feet, the circle of confusion grows even larger and the perceived DoF increases.
¹ From the comments:
Depth of field is anything but subjective. It can be quantified using
with points spread functions and Reighley's criterion for optical
I've edited the answer to make clearer that within the context of creative photography, which is what this entire site is about, the perception of blur by the viewer is the defining factor in the definition of DoF. Any definition of DoF that includes differences too small to be perceived by the viewer are moot within the context of creative photography. Even if an optical system and/or recording medium is capable of much higher resolution than the viewer can perceive those details are irrelevant when talking about DoF within the context of creative photography.
Point spread functions and Reighley's criteria are useful for defining the absolute limits within which an optical system can discriminate details. But they must take into account all of the limiting factors in an optical system.
In the context of a human viewing a photograph, the most limiting factor in the entire system is most often the human's ability to perceive the difference between minute amounts of blur and points.