The f/ratio myth and camera size

There are often questions about smaller versus larger pixels
and the corresponding camera size. Many of us would like smaller
cameras that did just as good a job as larger ones. Is that
possible? No, at least in terms of signal-to-noise that can
be recorded. Here is why.

There is a common idea in photography that exposure doesn't
change between different size cameras when at the same f/ratio.
For example, the sunny f/16 rule says a good exposure for a daylight
scene is 1/ISO at f/16. Thus for ISO 100 film, you use a 1/100
second exposure on an 8x10 camera at f/16, a 4x5 camera at f/16,
a 35mm camera at f/16, an APS-C digital camera at f/16, down to the
smallest point and shoot camera at f/16 (assuming the small camera
goes to f/16). This leads people to think cameras scale
easily and still give the same image. But there is a fallacy
in this idea, and that is spatial resolution on the subject.
The smaller camera, even at the same f/ratio, has a smaller lens
which collects a smaller number of photons per unit time.
The smaller camera gets the same exposure time because the UNIT
AREA in the focal plane represents a larger angular size on the
subject.

The rate of arrival of photons in the focal plane of a lens
per unit area per unit time is proportional to the square of
the f-ratio. Corollary: if you keep f/ratio constant, and change
focal length then the photons per unit area in the focal plane is
constant but spatial resolution changes.

So how does this apply to making smaller cameras?

The problem is that if you scale a camera down, say 2x, the
aperture drops by 2x, the focal length drops by 2x (to give the
same field of view), the sensor size drops by 2x, and the pixel
size drops by 2x (to give the same spatial resolution
on the subject). It should be obvious by this point
that per unit time the aperture has collected only
1/4 the number of photons. Also, the smaller pixels each
collect 1/4 less photons since their area is
divided by 4 to keep spatial resolution constant.

Another way to look at the problem is aperture collects light, the
focal length spreads out the light, and the pixels are buckets that
collect the light in the focal plane. BUT THE TOTAL NUMBER OF PHOTONS
DELIVERED TO THE FOCAL PLANE IS ONLY DEPENDENT ON APERTURE
(ignoring transmission losses of the optics).

Back to the camera example: scale a camera down by 2x keeping
f/ratio and spatial resolution constant. You lose 4x the
photons entering the lens with the smaller camera, and since you
you must use 2x smaller pixels, the area is 4x less, so you LOSE
ANOTHER 4x photons/pixel. Thus, photons delivered to a pixel for a
given resolution on the subject goes as the 4th power of the
aperture (and camera size)! Decreasing your camera by 2x
means 16x less photons per pixel if you want to maintain field of
view and megapixel count!

This is just what we observe with small cameras: their
smaller sensors have smaller full well capacities, that get filled
for a given exposure time with a smaller number of photons.
That in turn means higher noise because there are fewer
photons.

Check out this web page for more info on this subject:
http://home.earthlink.net/~stanleymm/f_ratio_myth.htm

Roger Clark
Photos, other digital info at: http://www.clarkvision.com

Higher noise under what conditions? If the shot has a sufficient
illumination level
and the ISO setting on the camera is low enough, you should see no more
noise
from most small pixel cameras than from a large pixel camera. However,
this only
applies to scenes with relatively small illumination "spreads." If
there are
deep shadows (essentially, underexposed areas) then they will
demonstrate
noise to some extent. The question is, what pixel size is needed to
keep noise
relatively unnoticeable in shots with wide illumination ranges?
My guess is that no current camera can produce a completely noise free
image when all
areas are taken into account, because none of them have the pixel well
capacity and
dynamic range to handle scenes with wide-ranging illumination levels.
This is especially true when people insist on underexposing areas in
order to "control" highlights in other areas. When they manipulate the
curves of the image to bring out
detail in the underexposed areas, those areas display noise, lack of
contrast, depressed colour values and restricted tonal ranges.
-Rich

Rich wrote:
Higher noise under what conditions?

All conditions comparable between large and small cameras.
If you have X photons with big camera A for a given f/stop
and exposure time (e.g. 1/400 second at f/8), then on a
small camera B for the same exposure and f/stop, you get
less photons per pixel. Signal to noise ratio is the square
root of the number of photons collected. The smaller camera
collects less photons so has worse signal to noise.
You might expose longer but smaller pixels have less capacity to
store electrons (converted photons) so the smaller camera simply
can never make up the difference.

If the shot has a sufficient
illumination level
and the ISO setting on the camera is low enough, you should see no more
noise
from most small pixel cameras than from a large pixel camera.

No. See above.

However, this only
applies to scenes with relatively small illumination "spreads." If
there are
deep shadows (essentially, underexposed areas) then they will
demonstrate
noise to some extent.

It has nothing to do with illumination spread. Because less
photons are collected by the small camera, the dynamic range
is less too.

The question is, what pixel size is needed to
keep noise
relatively unnoticeable in shots with wide illumination ranges?

That is a subjective question with a variable answer, depending
on the scene dynamic range and what is acceptable to the user.
The fact is that we have many many wonderful photographs from
film which has lower signal to noise than many point and shoot
digital cameras, and on slide film with very narrow dynamic range.
So P&S can still take great photos and in the hands of good
photographers, many great images can be obtained. But a larger
camera can generally do better, just like 35mm versus 4x5 film.

My guess is that no current camera can produce a completely noise free
image when all
areas are taken into account, because none of them have the pixel well
capacity and
dynamic range to handle scenes with wide-ranging illumination levels.

To get noise free, you need infinite photons, which is not possible.
But tens of thousands of photons per pixel makes very high
signal to noise images. The sweet spot in my opinion, are
cameras with 6 to 9 micron pixels. There is no reason a small P&S
camera could not have 8 micron size pixels with a pixel count
over 8 million.

Roger

Roger N. Clark (change username to rnclark) wrote:
There are often questions about smaller versus larger pixels
and the corresponding camera size. Many of us would like smaller
cameras that did just as good a job as larger ones. Is that
possible? No, at least in terms of signal-to-noise that can
be recorded. Here is why.

There is a common idea in photography that exposure doesn't
change between different size cameras when at the same f/ratio.
For example, the sunny f/16 rule says a good exposure for a daylight
scene is 1/ISO at f/16. Thus for ISO 100 film, you use a 1/100
second exposure on an 8x10 camera at f/16, a 4x5 camera at f/16,
a 35mm camera at f/16, an APS-C digital camera at f/16, down to the
smallest point and shoot camera at f/16 (assuming the small camera
goes to f/16). This leads people to think cameras scale
easily and still give the same image. But there is a fallacy
in this idea, and that is spatial resolution on the subject.
The smaller camera, even at the same f/ratio, has a smaller lens
which collects a smaller number of photons per unit time.
The smaller camera gets the same exposure time because the UNIT
AREA in the focal plane represents a larger angular size on the
subject.

The rate of arrival of photons in the focal plane of a lens
per unit area per unit time is proportional to the square of
the f-ratio. Corollary: if you keep f/ratio constant, and change
focal length then the photons per unit area in the focal plane is
constant but spatial resolution changes.

So how does this apply to making smaller cameras?

The problem is that if you scale a camera down, say 2x, the
aperture drops by 2x, the focal length drops by 2x (to give the
same field of view), the sensor size drops by 2x, and the pixel
size drops by 2x (to give the same spatial resolution
on the subject). It should be obvious by this point
that per unit time the aperture has collected only
1/4 the number of photons. Also, the smaller pixels each
collect 1/4 less photons since their area is
divided by 4 to keep spatial resolution constant.

Another way to look at the problem is aperture collects light, the
focal length spreads out the light, and the pixels are buckets that
collect the light in the focal plane. BUT THE TOTAL NUMBER OF PHOTONS
DELIVERED TO THE FOCAL PLANE IS ONLY DEPENDENT ON APERTURE
(ignoring transmission losses of the optics).

Back to the camera example: scale a camera down by 2x keeping
f/ratio and spatial resolution constant. You lose 4x the
photons entering the lens with the smaller camera, and since you
you must use 2x smaller pixels, the area is 4x less, so you LOSE
ANOTHER 4x photons/pixel. Thus, photons delivered to a pixel for a
given resolution on the subject goes as the 4th power of the
aperture (and camera size)! Decreasing your camera by 2x
means 16x less photons per pixel if you want to maintain field of
view and megapixel count!
I believe you are off on this one. For the same f/number the photon
flux at the sensor is the same. Take your case of scaling down a
camera by a factor of two, the lens collects 1/4 the photons but it
also spreads these photons out over 1/4the area that the larger camera
does. The number of photons per sensor is therefor scales as the area
of the sensor. So I believe you are looking at a 2nd power decrease
not 4th.

Scott

Scott W wrote:
Roger N. Clark (change username to rnclark) wrote:

There are often questions about smaller versus larger pixels
and the corresponding camera size. Many of us would like smaller
cameras that did just as good a job as larger ones. Is that
possible? No, at least in terms of signal-to-noise that can
be recorded. Here is why.

There is a common idea in photography that exposure doesn't
change between different size cameras when at the same f/ratio.
For example, the sunny f/16 rule says a good exposure for a daylight
scene is 1/ISO at f/16. Thus for ISO 100 film, you use a 1/100
second exposure on an 8x10 camera at f/16, a 4x5 camera at f/16,
a 35mm camera at f/16, an APS-C digital camera at f/16, down to the
smallest point and shoot camera at f/16 (assuming the small camera
goes to f/16). This leads people to think cameras scale
easily and still give the same image. But there is a fallacy
in this idea, and that is spatial resolution on the subject.
The smaller camera, even at the same f/ratio, has a smaller lens
which collects a smaller number of photons per unit time.
The smaller camera gets the same exposure time because the UNIT
AREA in the focal plane represents a larger angular size on the
subject.

The rate of arrival of photons in the focal plane of a lens
per unit area per unit time is proportional to the square of
the f-ratio. Corollary: if you keep f/ratio constant, and change
focal length then the photons per unit area in the focal plane is
constant but spatial resolution changes.

So how does this apply to making smaller cameras?

The problem is that if you scale a camera down, say 2x, the
aperture drops by 2x, the focal length drops by 2x (to give the
same field of view), the sensor size drops by 2x, and the pixel
size drops by 2x (to give the same spatial resolution
on the subject). It should be obvious by this point
that per unit time the aperture has collected only
1/4 the number of photons. Also, the smaller pixels each
collect 1/4 less photons since their area is
divided by 4 to keep spatial resolution constant.

Another way to look at the problem is aperture collects light, the
focal length spreads out the light, and the pixels are buckets that
collect the light in the focal plane. BUT THE TOTAL NUMBER OF PHOTONS
DELIVERED TO THE FOCAL PLANE IS ONLY DEPENDENT ON APERTURE
(ignoring transmission losses of the optics).

Back to the camera example: scale a camera down by 2x keeping
f/ratio and spatial resolution constant. You lose 4x the
photons entering the lens with the smaller camera, and since you
you must use 2x smaller pixels, the area is 4x less, so you LOSE
ANOTHER 4x photons/pixel. Thus, photons delivered to a pixel for a
given resolution on the subject goes as the 4th power of the
aperture (and camera size)! Decreasing your camera by 2x
means 16x less photons per pixel if you want to maintain field of
view and megapixel count!

I believe you are off on this one. For the same f/number the photon
flux at the sensor is the same. Take your case of scaling down a
camera by a factor of two, the lens collects 1/4 the photons but it
also spreads these photons out over 1/4the area that the larger camera
does. The number of photons per sensor is therefor scales as the area
of the sensor. So I believe you are looking at a 2nd power decrease
not 4th.

Scott

Scott,
You are correct. I forgot to include that for the smaller lens
at half the focal length, the area on the subject doubles, canceling
one of the squared terms. So, yes, I agree that the relation
scales as the square not as the 4th power.

So halving the camera size means pixels get 1/4 the photons.
A good example is the Canon 20D with 6.4 micron pixels and a
maximum signal at ISO 100 of 50,000 electrons, compared to
the Canon S60 with 2.8 micron pixels with a maximum signal
of about 11,000 electrons at ISO 100. The pixel size is
6.4^2 / 2.8^2 = 5.2x scaling, similar to the 50000/11000
= 4.5 scaling of maximum recorded signal.

Then, for photon noise limited systems, signal-to-noise ratio
scales as the square root of the camera size.

Roger

In article , says...


[SNIP]

Roger Clark
Photos, other digital info at: http://www.clarkvision.com

Without getting into photons/nanosecond striking the recording medium, I'll
give you a very rough set of observations in doing studio shooting over 30
years for the advertising industry.

Shot is set up for 4x5 and all is dialed in to, say f/16 on E-100 4x5.
Polaroids look good, and the E-6 confirms that all is excellent. Client
decides that 8x10 is needed, instead. Same lights, same power settings, same
film, but in 8x10. Whoa, it now looks like f/11 is the aperture that is
required, and the E-6 processing confirms it. Bellows factors? No, this is not
a macro shot. Finally, the client wants some 35mm, and with the same lighting
-dang, we're now shooting at f/22! Again, the E-6 line shows that this is
correct. I've seen it too many times. The exposures are approximate, and
sometimes the differences were in 2/3 /f, and not full /f-stops, but it was
closer to 1 per camera size difference. Six x six CM was always between 4x5
and 35mm.

Hunt

Roger N. Clark (change username to rnclark) wrote:
Scott W wrote:
Roger N. Clark (change username to rnclark) wrote:

There are often questions about smaller versus larger pixels
and the corresponding camera size. Many of us would like smaller
cameras that did just as good a job as larger ones. Is that
possible? No, at least in terms of signal-to-noise that can
be recorded. Here is why.

There is a common idea in photography that exposure doesn't
change between different size cameras when at the same f/ratio.
For example, the sunny f/16 rule says a good exposure for a daylight
scene is 1/ISO at f/16. Thus for ISO 100 film, you use a 1/100
second exposure on an 8x10 camera at f/16, a 4x5 camera at f/16,
a 35mm camera at f/16, an APS-C digital camera at f/16, down to the
smallest point and shoot camera at f/16 (assuming the small camera
goes to f/16). This leads people to think cameras scale
easily and still give the same image. But there is a fallacy
in this idea, and that is spatial resolution on the subject.
The smaller camera, even at the same f/ratio, has a smaller lens
which collects a smaller number of photons per unit time.
The smaller camera gets the same exposure time because the UNIT
AREA in the focal plane represents a larger angular size on the
subject.

The rate of arrival of photons in the focal plane of a lens
per unit area per unit time is proportional to the square of
the f-ratio. Corollary: if you keep f/ratio constant, and change
focal length then the photons per unit area in the focal plane is
constant but spatial resolution changes.

So how does this apply to making smaller cameras?

The problem is that if you scale a camera down, say 2x, the
aperture drops by 2x, the focal length drops by 2x (to give the
same field of view), the sensor size drops by 2x, and the pixel
size drops by 2x (to give the same spatial resolution
on the subject). It should be obvious by this point
that per unit time the aperture has collected only
1/4 the number of photons. Also, the smaller pixels each
collect 1/4 less photons since their area is
divided by 4 to keep spatial resolution constant.

Another way to look at the problem is aperture collects light, the
focal length spreads out the light, and the pixels are buckets that
collect the light in the focal plane. BUT THE TOTAL NUMBER OF PHOTONS
DELIVERED TO THE FOCAL PLANE IS ONLY DEPENDENT ON APERTURE
(ignoring transmission losses of the optics).

Back to the camera example: scale a camera down by 2x keeping
f/ratio and spatial resolution constant. You lose 4x the
photons entering the lens with the smaller camera, and since you
you must use 2x smaller pixels, the area is 4x less, so you LOSE
ANOTHER 4x photons/pixel. Thus, photons delivered to a pixel for a
given resolution on the subject goes as the 4th power of the
aperture (and camera size)! Decreasing your camera by 2x
means 16x less photons per pixel if you want to maintain field of
view and megapixel count!

I believe you are off on this one. For the same f/number the photon
flux at the sensor is the same. Take your case of scaling down a
camera by a factor of two, the lens collects 1/4 the photons but it
also spreads these photons out over 1/4the area that the larger camera
does. The number of photons per sensor is therefor scales as the area
of the sensor. So I believe you are looking at a 2nd power decrease
not 4th.

Scott

Scott,
You are correct. I forgot to include that for the smaller lens
at half the focal length, the area on the subject doubles, canceling
one of the squared terms. So, yes, I agree that the relation
scales as the square not as the 4th power.

So halving the camera size means pixels get 1/4 the photons.
A good example is the Canon 20D with 6.4 micron pixels and a
maximum signal at ISO 100 of 50,000 electrons, compared to
the Canon S60 with 2.8 micron pixels with a maximum signal
of about 11,000 electrons at ISO 100. The pixel size is
6.4^2 / 2.8^2 = 5.2x scaling, similar to the 50000/11000
= 4.5 scaling of maximum recorded signal.

Then, for photon noise limited systems, signal-to-noise ratio
scales as the square root of the camera size.
Would not the S/N ratio scale with the size of the camera. If I make
pixels with twice the linear dimensions I get 4 time the photons but
only 2 times the noise for a gain of 2 in S/N.

Scott

Roger N. Clark (change username to rnclark) wrote:
Scott W wrote:
Roger N. Clark (change username to rnclark) wrote:

There are often questions about smaller versus larger pixels
and the corresponding camera size. Many of us would like smaller
cameras that did just as good a job as larger ones. Is that
possible? No, at least in terms of signal-to-noise that can
be recorded. Here is why.

There is a common idea in photography that exposure doesn't
change between different size cameras when at the same f/ratio.
For example, the sunny f/16 rule says a good exposure for a daylight
scene is 1/ISO at f/16. Thus for ISO 100 film, you use a 1/100
second exposure on an 8x10 camera at f/16, a 4x5 camera at f/16,
a 35mm camera at f/16, an APS-C digital camera at f/16, down to the
smallest point and shoot camera at f/16 (assuming the small camera
goes to f/16). This leads people to think cameras scale
easily and still give the same image. But there is a fallacy
in this idea, and that is spatial resolution on the subject.
The smaller camera, even at the same f/ratio, has a smaller lens
which collects a smaller number of photons per unit time.
The smaller camera gets the same exposure time because the UNIT
AREA in the focal plane represents a larger angular size on the
subject.

The rate of arrival of photons in the focal plane of a lens
per unit area per unit time is proportional to the square of
the f-ratio. Corollary: if you keep f/ratio constant, and change
focal length then the photons per unit area in the focal plane is
constant but spatial resolution changes.

So how does this apply to making smaller cameras?

The problem is that if you scale a camera down, say 2x, the
aperture drops by 2x, the focal length drops by 2x (to give the
same field of view), the sensor size drops by 2x, and the pixel
size drops by 2x (to give the same spatial resolution
on the subject). It should be obvious by this point
that per unit time the aperture has collected only
1/4 the number of photons. Also, the smaller pixels each
collect 1/4 less photons since their area is
divided by 4 to keep spatial resolution constant.

Another way to look at the problem is aperture collects light, the
focal length spreads out the light, and the pixels are buckets that
collect the light in the focal plane. BUT THE TOTAL NUMBER OF PHOTONS
DELIVERED TO THE FOCAL PLANE IS ONLY DEPENDENT ON APERTURE
(ignoring transmission losses of the optics).

Back to the camera example: scale a camera down by 2x keeping
f/ratio and spatial resolution constant. You lose 4x the
photons entering the lens with the smaller camera, and since you
you must use 2x smaller pixels, the area is 4x less, so you LOSE
ANOTHER 4x photons/pixel. Thus, photons delivered to a pixel for a
given resolution on the subject goes as the 4th power of the
aperture (and camera size)! Decreasing your camera by 2x
means 16x less photons per pixel if you want to maintain field of
view and megapixel count!

I believe you are off on this one. For the same f/number the photon
flux at the sensor is the same. Take your case of scaling down a
camera by a factor of two, the lens collects 1/4 the photons but it
also spreads these photons out over 1/4the area that the larger camera
does. The number of photons per sensor is therefor scales as the area
of the sensor. So I believe you are looking at a 2nd power decrease
not 4th.

Scott

Scott,
You are correct. I forgot to include that for the smaller lens
at half the focal length, the area on the subject doubles, canceling
one of the squared terms. So, yes, I agree that the relation
scales as the square not as the 4th power.

So halving the camera size means pixels get 1/4 the photons.
A good example is the Canon 20D with 6.4 micron pixels and a
maximum signal at ISO 100 of 50,000 electrons, compared to
the Canon S60 with 2.8 micron pixels with a maximum signal
of about 11,000 electrons at ISO 100. The pixel size is
6.4^2 / 2.8^2 = 5.2x scaling, similar to the 50000/11000
= 4.5 scaling of maximum recorded signal.

Then, for photon noise limited systems, signal-to-noise ratio
scales as the square root of the camera size.
Would not the S/N ratio scale with the size of the camera. If I make
pixels with twice the linear dimensions I get 4 time the photons but
only 2 times the noise for a gain of 2 in S/N.

Scott

Hello Scott.

I have an issue with the following statement.
also spreads these photons out over 1/4the area that the larger camera
does.
Now I am not overly well versed in the compromises made in the design
photographic optics, but I can tell you that in difraction limited
astronomical optics the angular resolution increases ( diffraction disk
diameter decreases) in proportion to the diameter of the objective. This
means that if you double the objective (lens/mirror) diameter and maintain
the f-ratio, the diffraction disk remains a constant size. I recognise
that this is counter-intuitive but it is determined by wave diffraction
theory.
Do a Google on "f +ratio +diffraction +disk +size" and you will find that
this subject is well covered, particulary in photographic terms. I only
looked at this one and it seems to be quite useful.
http://www.cambridgeincolour.com/tut...hotography.htm

Regards
Ian



On Sun, 05 Feb 2006 18:49:15 -0800, Scott W wrote:

Roger N. Clark (change username to rnclark) wrote:
There are often questions about smaller versus larger pixels
and the corresponding camera size. Many of us would like smaller
cameras that did just as good a job as larger ones. Is that
possible? No, at least in terms of signal-to-noise that can
be recorded. Here is why.

There is a common idea in photography that exposure doesn't
change between different size cameras when at the same f/ratio.
For example, the sunny f/16 rule says a good exposure for a daylight
scene is 1/ISO at f/16. Thus for ISO 100 film, you use a 1/100
second exposure on an 8x10 camera at f/16, a 4x5 camera at f/16,
a 35mm camera at f/16, an APS-C digital camera at f/16, down to the
smallest point and shoot camera at f/16 (assuming the small camera
goes to f/16). This leads people to think cameras scale
easily and still give the same image. But there is a fallacy
in this idea, and that is spatial resolution on the subject.
The smaller camera, even at the same f/ratio, has a smaller lens
which collects a smaller number of photons per unit time.
The smaller camera gets the same exposure time because the UNIT
AREA in the focal plane represents a larger angular size on the
subject.

The rate of arrival of photons in the focal plane of a lens
per unit area per unit time is proportional to the square of
the f-ratio. Corollary: if you keep f/ratio constant, and change
focal length then the photons per unit area in the focal plane is
constant but spatial resolution changes.

So how does this apply to making smaller cameras?

The problem is that if you scale a camera down, say 2x, the
aperture drops by 2x, the focal length drops by 2x (to give the
same field of view), the sensor size drops by 2x, and the pixel
size drops by 2x (to give the same spatial resolution
on the subject). It should be obvious by this point
that per unit time the aperture has collected only
1/4 the number of photons. Also, the smaller pixels each
collect 1/4 less photons since their area is
divided by 4 to keep spatial resolution constant.

Another way to look at the problem is aperture collects light, the
focal length spreads out the light, and the pixels are buckets that
collect the light in the focal plane. BUT THE TOTAL NUMBER OF PHOTONS
DELIVERED TO THE FOCAL PLANE IS ONLY DEPENDENT ON APERTURE
(ignoring transmission losses of the optics).

Back to the camera example: scale a camera down by 2x keeping
f/ratio and spatial resolution constant. You lose 4x the
photons entering the lens with the smaller camera, and since you
you must use 2x smaller pixels, the area is 4x less, so you LOSE
ANOTHER 4x photons/pixel. Thus, photons delivered to a pixel for a
given resolution on the subject goes as the 4th power of the
aperture (and camera size)! Decreasing your camera by 2x
means 16x less photons per pixel if you want to maintain field of
view and megapixel count!
I believe you are off on this one. For the same f/number the photon
flux at the sensor is the same. Take your case of scaling down a
camera by a factor of two, the lens collects 1/4 the photons but it
also spreads these photons out over 1/4the area that the larger camera
does. The number of photons per sensor is therefor scales as the area
of the sensor. So I believe you are looking at a 2nd power decrease
not 4th.

Scott

Scott W wrote:

Roger N. Clark (change username to rnclark) wrote:

Scott W wrote:

Roger N. Clark (change username to rnclark) wrote:


There are often questions about smaller versus larger pixels
and the corresponding camera size. Many of us would like smaller
cameras that did just as good a job as larger ones. Is that
possible? No, at least in terms of signal-to-noise that can
be recorded. Here is why.

There is a common idea in photography that exposure doesn't
change between different size cameras when at the same f/ratio.
For example, the sunny f/16 rule says a good exposure for a daylight
scene is 1/ISO at f/16. Thus for ISO 100 film, you use a 1/100
second exposure on an 8x10 camera at f/16, a 4x5 camera at f/16,
a 35mm camera at f/16, an APS-C digital camera at f/16, down to the
smallest point and shoot camera at f/16 (assuming the small camera
goes to f/16). This leads people to think cameras scale
easily and still give the same image. But there is a fallacy
in this idea, and that is spatial resolution on the subject.
The smaller camera, even at the same f/ratio, has a smaller lens
which collects a smaller number of photons per unit time.
The smaller camera gets the same exposure time because the UNIT
AREA in the focal plane represents a larger angular size on the
subject.

The rate of arrival of photons in the focal plane of a lens
per unit area per unit time is proportional to the square of
the f-ratio. Corollary: if you keep f/ratio constant, and change
focal length then the photons per unit area in the focal plane is
constant but spatial resolution changes.

So how does this apply to making smaller cameras?

The problem is that if you scale a camera down, say 2x, the
aperture drops by 2x, the focal length drops by 2x (to give the
same field of view), the sensor size drops by 2x, and the pixel
size drops by 2x (to give the same spatial resolution
on the subject). It should be obvious by this point
that per unit time the aperture has collected only
1/4 the number of photons. Also, the smaller pixels each
collect 1/4 less photons since their area is
divided by 4 to keep spatial resolution constant.

Another way to look at the problem is aperture collects light, the
focal length spreads out the light, and the pixels are buckets that
collect the light in the focal plane. BUT THE TOTAL NUMBER OF PHOTONS
DELIVERED TO THE FOCAL PLANE IS ONLY DEPENDENT ON APERTURE
(ignoring transmission losses of the optics).

Back to the camera example: scale a camera down by 2x keeping
f/ratio and spatial resolution constant. You lose 4x the
photons entering the lens with the smaller camera, and since you
you must use 2x smaller pixels, the area is 4x less, so you LOSE
ANOTHER 4x photons/pixel. Thus, photons delivered to a pixel for a
given resolution on the subject goes as the 4th power of the
aperture (and camera size)! Decreasing your camera by 2x
means 16x less photons per pixel if you want to maintain field of
view and megapixel count!

I believe you are off on this one. For the same f/number the photon
flux at the sensor is the same. Take your case of scaling down a
camera by a factor of two, the lens collects 1/4 the photons but it
also spreads these photons out over 1/4the area that the larger camera
does. The number of photons per sensor is therefor scales as the area
of the sensor. So I believe you are looking at a 2nd power decrease
not 4th.

Scott


Scott,
You are correct. I forgot to include that for the smaller lens
at half the focal length, the area on the subject doubles, canceling
one of the squared terms. So, yes, I agree that the relation
scales as the square not as the 4th power.

So halving the camera size means pixels get 1/4 the photons.
A good example is the Canon 20D with 6.4 micron pixels and a
maximum signal at ISO 100 of 50,000 electrons, compared to
the Canon S60 with 2.8 micron pixels with a maximum signal
of about 11,000 electrons at ISO 100. The pixel size is
6.4^2 / 2.8^2 = 5.2x scaling, similar to the 50000/11000
= 4.5 scaling of maximum recorded signal.

Then, for photon noise limited systems, signal-to-noise ratio
scales as the square root of the camera size.

Would not the S/N ratio scale with the size of the camera. If I make
pixels with twice the linear dimensions I get 4 time the photons but
only 2 times the noise for a gain of 2 in S/N.

Scott

Scott,
I think you have clearly demonstrated that I should not do
math and watch the superbowl at the same time ;-).
Yes, you are correct.

Roger