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Let's assume I have a prime lens for full frame, with fixed specs (let's say 85mm f/1.8). Changing the lens is not an option, but I have a choice of a full frame camera and a crop sensor camera. The lens is compatible with both cameras. I want to fill the frame with the subject.

How does background blur and depth of field vary depending on sensor size in this case, when the same lens is used for both sensor sizes and framing is equivalent?

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If the goal is to use the same focal length lens and yield the same cropping FX (full frame) vs. DX (compact digital), the camera to subject distance is decreased when using the FX and elongated when using the DX.

Thus for this lash-up, the DX working from afar, delivers expanded DOF resulting in reduced background blur. Conversely, the FX, working in closer delivers contracted DOF resulting in heightened background blur.

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You should realize that with the Same lens from the Same distance, the smaller sensor (called a cropped sensor) crops the field of view proportionately. So it is Not the same picture, its view is cropped. To see the same full field of view, the cropped sensor has to stand back more, to distance x crop factor.

Depth of Field is greater if a shorter focal length, or a longer focal distance, or a stopped down aperture. And also if a larger sensor, which is accounted for in the CoC size.

If all else is the same (same lens and f/stop and distance. but then the field of view is Not the same), the smaller sensor has a little less Depth of Field than the larger sensor, simply because the smaller sensor suffers greater enlargement to be the same standard viewing size (DOF computes for an 8x10 inch viewing size). This enlargement is magnification of the blur too.

However, in practice, smaller sensors normally use a shorter lens, because the smaller sensor crops the field of view. In order to have the same field of view, small sensors normally use a shorter lens (wide angle), which expands the view of the smaller sensor. For a very small sensor, then a very short lens. This focal length is a larger factor than sensor size, so in practice, we think the small sensor has greater depth of field (due to its normal shorter lens), when in fact, the effect of just the sensor size alone is the opposite. These factors offset each other, but the short lens wins.

That is speaking of the Depth of Field zone around the subject.

But if speaking of the background (with either sensor), background blur is worse (or better if that was the goal) by standing back with a longer lens. This standard trick can easily improve the DOF at the subject while degrading and hiding the background more (if it is back substantially behind the subject)

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The background blur disc size in millimeters on the sensor can be obtained from the slightly simplified equation

b = f^2 / (x_d * N)

where b is the blur disc size in millimeters, f is the focal length, x_d is the subject distance and N is the aperture number. In the equation, it is assumed that background is at infinity (if it's not at infinity, the equation would become slightly different, slightly more complicated).

I can further introduce crop factor into the equation:

b = f/N * (f*crop)/x_d * 1/crop

Now, I want to express b as a fraction of sensor diagonal:

relative_b = b/diagonal = (f/N) * ((f*crop)/x_d) * (1/(crop*diagonal))

Note crop*diagonal is constant. So, the first term, f/N is the lens specification. The second term, ((f*crop)/x_d) is constant for equal framing because f*crop is the 35mm equivalent focal length, and it divided by the subject distance is constant for equal framing. The third term too is constant: if you multiply crop factor by sensor diagonal, you get full frame sensor diagonal which is a constant.

So, the relative background blur as a percentage of sensor size stays constant for a given lens, assuming you shoot wide open and use equal framing.

Now let's take a look at depth of field. The following equation can be derived for it:

DoF = 2 * x_d^2 * N * C / f^2

where DoF is the depth of field, x_d is the subject distance, N is the aperture number, C is the circle of confusion and f is the focal length.

I can further introduce crop into the equation:

DoF = 2 * (x_d/(f*crop))^2 * (N) * (C*crop^2)

Now C * crop is C_FF, the circle of confusion for a full frame sensor, so

DoF = 2 * (x_d/(f*crop))^2 * (N) * (C_FF*crop)

In this equation, 2 is constant, (x_d/(f*crop))^2 is constant given equal framing, N is dependent on the lens and (C_FF*crop) is proportional to crop factor.

So, the crop camera has more depth of field although it has exact same background blur.

If the photographer desires both large background blur and deep enough depth of field, it actually is beneficial to use a crop camera, or alternatively to emulate a crop camera on a full frame camera (walk back, and crop the final image to be smaller). It depends on the lens whether any crop camera benefit can be seen compared to cropping the full frame image: a poor lens can be the limiting factor in resolution, giving crop camera no edge, whereas a good lens can benefit from the higher pixel density of a crop camera.

A full frame camera can always emulate a crop camera albeit with lower megapixel count (and without crop lens compatibility if it's DSLR and not mirrorless), whereas a crop camera cannot emulate a full frame camera.

Summary: with a given lens, crop camera gets more depth of field, but background blur is the same for crop and full frame, if the background is at infinity.

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    I don't understand why you've multiplied by crop factor where you have in the formulas. Can you add explanations to your answer? – xiota Jul 8 at 23:56
  • What does "equal framing" mean in this context? Are you saying that the subject at the plane of focus takes up the same amount of the field of view? (i.e., a head shot will always have the same height/width when the subject is in focus)? – scottbb Jul 11 at 21:19
  • @xiota I believe juhist multiplied by 1 (that is, crop/crop). It's just that the numerator and denominator were split to different places for other motivations. Similar in concept to adding a value and later subtracting it in order to complete the square. Just a math trick. – scottbb Jul 13 at 2:54
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It seems you want to change sensor size, while keeping the same framing with the same lens. (Formulas are from Wikipedia.)

  • Depth of FieldDOF increases by the crop factor because distance to subject (u) is multiplied by crop factor (to maintain framing) and Circle of Confusion (C) is divided by crop factor (to maintain output size). Since distance is squared, only one of the crop factors cancels out. Here is the standard formula for DOF:

    DOF = 2 u2 N C / f2

    N = aperture F-number
    C = circle of confusion
    u = distance to subject
    f = focal length

    All of the advice people give to minimize DOF are in the formula – use larger apertures, use longer focal lengths, and get closer to the subject.

  • Background BlurThe amount of blur decreases because the distance to subject (s) increases (by crop factor) in the denominator. Here is a formula for amount of background blur:

    b = f ms xd / (N (s + xd))

    b = blur
    f = focal length
    N = aperture F-number
    ms = subject magnification (what's this?)
    xd = distance between subject and background
    s = subject distance

    WayneF explains magnification ratio:

    Magnification is just ratio of distance behind lens (focal length) / distance in front of lens... See [Circle of confusion] for a simpler formula, including for infinity.

    Since ms is "subject magnification", it is focal length (f) / subject distance (s). The blur formula can be rewritten: b = f2 xd / (N s (s + xd))

    As the subject-background distance increases, xd/(s + xd) approaches 1. The formula simplifies to: b = f2 / N s

    The blur that results from using different focal lengths to maintain the same framing can be evaluated by comparing f/N (f/s stays constant when framing is kept constant with different focal lengths). However, in this scenario, f and N are kept constant, while s is increased to maintain framing. So blur decreases by a factor of 1/crop-factor.

  • Depending on the lens and camera, you may have the option to (more or less) cancel out the crop factor with a focal reducer. (There is about an 8% difference.) This allows you to maintain (about) the same DOF and blur as a full-frame camera. The trade off is loss of features such as auto focus and image stabilization, as well as the possible addition of some image defects, such as distortion or vignetting, depending on the lens.

Here is a similar question in which sensor and framing are kept the same, but focal length changes: Does amount of background blur change with focal length given equal framing?

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How does background blur and DoF vary with sensor size for a prime lens?

It doesn't. At least not in the sense that the only thing that matters is sensor size.

It varies with the enlargement ratio between the sensor/film format size and the final display size. The exact same image can have different DoF when displayed at different sizes and/or viewed from different distances.

Comparing different sensor sizes is only meaningful if all other assumptions are held constant:

  • Subject size
  • Subject distance
  • Focal length
  • Display size
  • Viewing distance
  • Observer's visual acuity

If one enlarges images from both sensors to the same display size, then the crop body enlargement will be enlarged by 1.5X or 1.6X (depending on the crop body in question) more then the FF camera. This results in all blur in the image also being enlarged by 1.5 to 1.6 times that of the FF image. DoF is defined by how much blur can still be perceived by the viewer as a sharp point.

Let's look at an example using Cambridge in Color's excellent flexible DoF calculator:

With a FF sensor using an 85mm lens at f/2.8, if one shoots at five meters subject distance the DoF for a 10" print viewed at 10 inches (25mm) by a person with 20/20 vision will be 0.13 m or 13 cm, with 6 cm in front of the subject distance and 7 cm behind the subject.

If we back up to 8 meters with a 1.6X crop sensor and use the same 85mm at f/1.8 and display/view the result under the same conditions, the DoF is 0.21 m or 21 cm, with 10 cm in front of and 11 cm behind the subject distance.

The effect on background blur will vary depending on the ratio of the distance to the background from the camera compared to the subject distance.

There will also be an effect on perspective as a result of moving back 3 meters without changing the distance between the subject and background. If the background is five meters behind the subject and ten meters from the camera, that's 200% of the subject distance at five meters. But when the subject distance is eight meters and the background is still five meters behind the subject and thirteen meters from the camera, background is only 163% the subject distance from the camera. For more regarding the effect on perspective (even when using the same focal length), please see this answer to Is there a difference between taking a far shot on a 50mm lens and a close shot on a 35mm lens?

Here's a reminder about Depth of Field in general:

There's only one distance that is in sharpest focus. Everything in front of or behind that distance is blurry. The further we move away from the focus distance, the blurrier things get. The questions become: "How blurry is it? Is that within our acceptable limit? How far from the focus distance do things become unacceptably blurry?"

What we call depth of field (DoF) is the range of distances in front of and behind the point of focus that are acceptably blurry so that things still look like they are in focus.

The amount of depth of field depends on two things: total magnification and aperture. Total magnification includes the following factors: focal length, subject/focus distance, enlargement ratio (which is determined by both sensor size and display size), and viewing distance. The visual acuity of the viewer also contributes to what is acceptably sharp enough to appear in focus instead of blurry.

The distribution of the depth of field in front of and behind the focus distance depends on several factors, primarily focal length and focus distance.

The ratio of any given lens changes as the focus distance is changed. Most lenses approach 1:1 at the minimum focus distance. As the focus distance is increased the rear depth of field increases faster than the front depth of field. There is one focus distance at which the ratio will be 1:2, or one-third in front and two-thirds behind the point of focus.

At short focus distances the ratio approaches 1:1. A true macro lens that can project a virtual image on the sensor or film that is the same size as the object for which it is projecting the image achieves a 1:1 ratio. Even lenses that can not achieve macro focus will demonstrate a ratio very near to 1:1 at their minimum focus distance.

At longer focus distances the rear of the depth of field reaches all the way to infinity and thus the ratio between front and rear DoF approaches 1:∞. The shortest focus distance at which the rear DoF reaches infinity is called the hyperfocal distance. The near depth of field will very closely approach one half the focus distance. That is, the nearest edge of the DoF will be halfway between the camera and the focus distance.

We must also remember that hyperfocal distance, like the concept of depth of field upon which it is based, is really just an illusion, albeit a rather persistent one. Only a single distance will be at sharpest focus. What we call depth of field are the areas on either side of the sharpest focus that are blurred so insignificantly that we still see them as sharp. Please note that the hyperfocal distance will vary based upon a change to any of the factors that affect DoF: focal length, aperture, magnification/display size, viewing distance, etc.

For more about why this is the case, please see:

Why did manufacturers stop including DOF scales on lenses?
Is there a 'rule of thumb' that I can use to estimate depth of field while shooting?
How do you determine the acceptable Circle of Confusion for a particular photo?
Find hyperfocal distance for HD (1920x1080) resolution?
Why I am getting different values for depth of field from calculators vs in-camera DoF preview?
As well as this answer to Simple quick DoF estimate method for prime lens

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    "if all other assumptions are held constant" – including enlargement ratio, DOF stays the same. But OP wants to keep the same subject framing with the same lens (focal length, aperture), which requires changing other parameters (subject distance) that can affect DOF. – xiota Jul 9 at 2:10
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    (This answer looks like it was ported from a different question and needs some tweaks to apply to this question.) – xiota Jul 9 at 2:14
  • @xiota The fact remains that DoF changes with enlargement ratio, even with the exact same image file. – Michael C Jul 9 at 17:09
  • I added a middle section containing material I though was added when I posted the original answer, but apparently didn't post. I'm currently on a summer road trip and do not have access to the site via my normal devices. – Michael C Jul 9 at 17:34

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