2014-02-11

Canon G15 Flash and Flash Sync Performance

I’ve looked at the flash and flash sync performance of the Canon Powershot G15 with the built-in and with external flashes. My principal conclusions are:

  • The built-in flash has a guide number (GN) of about 8 meters in maximum setting and can sync up to 1/2000 of a second. The GN drops to 5 meters at medium setting and 1.7 meters at minimum setting.

  • External Canon Speedlite flashes can sync at up to 1/250 of a second in normal mode and, with much reduced energy, at up to 1/4000 of a second in high-speed sync mode.

  • External flashes that don’t use the Canon-specific pins (e.g., other-brand flashes or Canon flashes used with these pins electrically isolated) can sync at up to 1/4000 of a second, albeit probably with some loss of energy at the very highest speeds.

How did I come to these conclusions? I measured the guide numbers of the built-in flash of the G15 and an external Canon Speedlite 270EX II flash in different modes and at different shutter speeds.

Specifically, I used the G15 to take exposures of a clean, white, internal wall at a distance of about 2.5 meters. The exposures were taken at night to minimise the correction for ambient light. The camera was mounted on a tripod. I used manual mode, ISO 80, an equivalent focal length of 50 mm, and f/8. At 50 mm the pattern of illumination of both flashes is fairly even, so I don’t expect these results to depend significantly on the precise focal length. I took four sets of exposures at shutter speeds stepped by 1/3 of a stop from 1/100 of a second up to the fastest speed the camera would allow. The sets were:

  • With the built-in flash at maximum setting.

  • With the 270EX II attached to the hot shoe at 1/1 setting using normal (first-shutter) sync mode. This sync mode is selected in the “Shutter Sync” item in the “External Flash” menu.

  • With the 270EX II attached to the hot shoe at 1/1 setting using high-speed sync mode. This sync mode is selected in the “Shutter Sync” item in the “External Flash” menu.

  • With the 270EX II attached to the hot shoe, but with the four Canon-specific contacts on the hot shoe covered with insulating tape, as shown in the following image. I call this “dumb mode”.

    Canon hot shoe

    The photograph on the left shows the G15 hot shoe ready for normal or high-speed sync mode. The photograph on the right shows the G15 hot shoe ready for dumb mode, with the four Canon-specific contacts covered with insulating tape.

The 270EX II was extended to the “tele” position and aimed directly at the wall in front of the camera (i.e., not bounced).

In the exposure with the 270EX II, I had to use the nominal 3-stop ND filter. I calibrated this by comparing images at 1/16 setting with and without the ND filter, and determined that it was equivalent to 2.87 stops.

I measured the mean signal level in the green channels in each image, subtracted the mean signal level in the green channel in an ambient light exposure, and for the images taken with the 270EX II adjusted the signal level to compensate for the ND filter.

The 270EX II is specified to have a GN of 27 meters when used in normal mode at 50 mm effective focal length. I used this to scale the signal level in each case to a GN. If you prefer to think in stops rather than GN, remember that a factor of two in GN corresponds to two stops.

The GN as a function of flash, sync mode, and shutter speed is shown in the following plot.

Flash

GN for the G15 built-in flash and 270EX II as a function of sync mode and shutter speed

I interpret these results as follows.

  • The internal flash has an almost constant GN of about 8 meters, right out to the maximum speed allowed by the camera of 1/2000 of a second. This is about 3.5 stops fainter than the external 270EX II in normal mode.

    Although the camera is capable of speeds up to 1/4000 of a second (at least at f/6.3 or slower), when the built-in flash is used it does not allow speeds faster than 1/2000 of a second.

    There is a suggestion that at 1/2000 of a second the flash performance drops by about 1/10 of a stop (about 7%), which would imply a t.1 time of better than 1/2000 of a second.

    The camera manual states that flash range in wide angle (28 mm) is 7 meters. When the flash is in automatic mode, it typically selects ISO 320. The maximum range would correspond to the widest aperture of f/1.8. A GN of 8 meters with this ISO speed and aperture would give a nominal range of 8 meters, in close agreement with the manual.

    I also took images with the built-in flash set to the maximum, medium, and minimum level. These coarse levels are the only control that the camera provides over the flash energy in manual mode. They correspond to GN of about 8, 5, and 1.6 meters.

  • The external 270EX II flash works only up to 1/250 of a second in normal mode.

    This limitation would be expected for a camera with a standard curtain shutter, which can only simultaneously expose the whole detector at speeds of 1/250 of a second or slower. Nevertheless, it is difficult to understand for a camera such as the G15 with a leaf or electronic shutter capable of simultaneously exposing the complete detector even at the fastest speeds of 1/4000 of a second. We will return to this later.

  • The external 270EX II flash can work at shutters speeds speeds of up to 1/4000 of a second provided the camera selects high-speed sync mode.

    In this mode, the flash emits a rapid series of lower-energy pulses, and as a result produces a flash whose energy depends on the shutter speed.

    The 270EX II is specified to produce a GN of 13.5 meters at 1/250 of a second in high-speed sync mode. My measurements suggest it actually performs slightly better, achieving 13.6 meters at 1/320 second. This is about half of the GN in normal flash mode, which corresponds to a drop of about two stops.

    My measurements also confirm the linear drop in flash energy with shutter speed. This translates to the GN dropping as the square root of the shutter speed, so speeds of 1/500, 1/1000, 1/2000, and 1/4000 of a second have GN of 10.8, 7.6, 5.4, and 3.8 meters.

    Note in particular that at shutter speeds of 1/1000 of a second or faster the 270EX II is fainter than the built-in flash.

  • The external 270EX II flash in dumb mode (with the Canon-specific pins on the hot-shoe covered with insulating tape) works to 1/4000 of a second.

    In dumb mode, the camera doesn’t know that an external flash is attached (e.g., it does not show the “External Flash” menu). However, it still shorts the standard central pin to the trigger the flash.

    For reasons I do not understand, the brightness of the flash is reduced from a GN of about 27 meters for 1/1 in normal mode to about 21 meters, a drop of about 0.7 stops.

    The GN is more or less constant down to quite fast speeds. At 1/1000 the flash energy has dropped by about 10% and at 1/4000 is has dropped by about 50%. This suggests that the t.1 and t.5 times of the 270EX II are about 1/1000 and 1/4000 of a second.

    These data show that the G15 is quite capable of syncing to a flash well beyond 1/250 of a second. This confirms that there seems to be no technical reason to have limited the flash in normal mode to 1/250 of a second or slower. This restriction seems either to be an oversight or, perhaps, a deliberate reduction in capability for marketing reasons. I’m inclined to believe it’s an oversight, since fast sync is not a feature that is present on Canon’s more expensive DSLR cameras. Or perhaps Canon really want us to buy Fujis?

    Unfortunately, since the camera does not recognise the flash and since the 270EX II does not have manual controls beyond on/slave/off, there is no way to tune the exposure other than by adjusting the distance, the aperture, and the ISO. That doesn’t sound like a lot of fun.

  • I would imagine that other Canon Speedlites would perform similarly to the 270EX II, albeit with appropriately scaled GNs.

    For example, the top-of-the-line US$600 600EX-RT is specified to have a GN of 60 meters in normal mode and 31.8 meters in high-speed sync mode at 1/250 of a second. Both of these numbers are about two stops brighter than the 270EX II.

    The more expensive Canon Speedlite 320EX, 430EX II, 580EX II, and 600EX-RT external flashes do have manual controls to tune the flash setting, and would be much more practical in dumb mode.

  • I would imagine that non-Canon flashes would perform similarly to the Canon flashes in dumb mode. Most also provide manual control over the flash setting. If you’re interested in using an external flash at high speeds, this is probably the route to take. David Hobby and Zach Arias recommend the US$200 LumoPro LP180.

If you’re interested in confirming or extending these results, and don’t want to have to determine the mean signal levels, you can simply see what sort of aperture or ISO adjustment you need to give similarly exposed images under different conditions. Each change in GN by a factor of two corresponds to two stops. So, for example, a normal sync exposure with the external flash at f/8 and a high-speed sync exposure at f/2 and 1/1000 of a second should give similar exposures, provided the flash setting and ISO are equal.

Thanks to Rodney Bartlett for helping me understand how to get the G15 to work in high-speed sync mode.

2013-12-17

On “How to Take Good Photos for Under $1,000”

Stu Maschwitz has interesting advice on how to take good family photos for under US$1000. The long and short of it is buy a cheap DSLR and use it with a f/1.8 50-mm prime lens set to wide-open. This combination will give fast exposures (to freeze motion), excellent bright-light and low-light performance, and a deliciously shallow depth of field. In photographic terms, I can’t fault it.

So, you spend your $1000 (or a little less), and then it’s down to you. Maschwitz writes

Now that you’ve bought all this stuff, the only thing that remains is the habit of actually doing this stuff. Your iPhone camera is damnably good. Good enough that often you’ll be tempted not to schlep your big, heavy DSLR. If you succumb to this temptation, we’re right back to terrible photos.

How big is “big”? How heavy is “heavy”? Maschwitz’s suggested Canon DSLR and lens weights 610 g (about 1.3 lb). Before you zip over to amazon.com and start pressing buttons, visit your local camera store, and try out one of his suggested combinations. Think about carrying it with you everywhere you go with your family - good photographic opportunities can crop up when you least expect them. If you’ve left your DSLR at home or in your car, then it’s not going to take very good photographs. Having mulled this over, if you are prepared to carry a DSLR everywhere, then follow his advice.

If you’re not up for carrying a DSLR, what can you do? Are you condemned to “terrible photos” from smartphones? Well, no. Just follow the spirit of Maschwitz’s advice: buy the fastest camera you can afford that you are prepared to carry with you at all times.

Now, by “fastest”, I mean “fastest equivalent f-number at 75-mm equivalent focal length”; I explain these terms in my post on compact camera electro-optics. I’ve chosen an equivalent focal length of 75-mm since a 50-mm lens on a entry-level DSLR has an equivalent focal length of about 75 mm. That’s an excellent focal lengths for outdoor or individual portraits, and compresses the depth of focus quite nicely; Maschwitz didn’t choose it by accident.

To determine the equivalent f-number N’, you need to multiply the crop factor C and the real f-number at 75-mm equivalent focal length N. So, for example, for the Canon T5i DSLR (crop factor C of 1.6) and a f/1.8 50-mm lens (real f-number of 1.8), the effective f-number N’ is about 2.9. It takes a little bit of effort to dig up the necessary characteristics, but the reviews on dpreview.com are often helpful. As a short cut, they often show plots of equivalent f-number against equivalent focal length. For example, this figure from their review of the Canon G15.

So, let’s look at some the equivalent f-number for some cameras. For the iPhone, I’ve assumed a digital crop to increase the equivalent focal length from about 30 to about 75 mm.

CameraWeight (g)Equivalent f-number N’
Canon T5i DSLR with f/1.8 50-mm lens6103
Sony NEX-6 with f/1.8 50-mm lens5503
Sony NEX-6 with kit 16/50-mm lens4609
Canon G15/G1635012
Sony RX-10024011
iPhone 511045

You can see that in terms of image quality, the only thing that gets close to Maschwitz’s recommendation is a Sony NEX-6 with a f/1.8 prime lens. That’s not a surprise, since a NEX-6 is basically a DSLR without the mirror. Losing the mirror saves weight and bulk; it’s worth considering. Almost all of the major manufacturers have mirror-less cameras similar to the NEX-6.

Maschwitz disparages kit lenses. Comparing the NEX-6 with the kit 16-50 mm lens to the same body with a prime f/1.8 50-mm lens, you can see why. The kit lens is a lot slower, and the photographic performance is more like a compact camera without the advantage of lightness and compactness.

The Canon G15/G16 and Sony RX-100 are photographically significantly worse than the best options, with effective f-numbers of about 12 compared to 3, but have about half the weight and significantly less bulk. These will easily fit in a hand-bag, man-bag, satchel, or jacket pocket, and the RX-100 will even fit in a trouser pocket. These cameras are photographically worse than a DSLR, but they are so much better than even a good smartphone; an iPhone when cropped to an effective focal length of 75 mm has an effective f-number of 45.

Other cameras with similar performance to the Canon G15/G16 and Sony RX-100 are the Olympus ZX-2, the Panasonix LX7, and the Fujifilm X10. There are many other cameras with similar price and bulk, but most have slower lenses or smaller detectors and will not perform as well.

So, here’s my modification to Maschwitz’s advice:

  • If you’re prepared to carry a DSLR at all times, follow his advice. Perhaps substitute a mirror-less body for the DSLR.

  • If you’re not, buy a Sony RX-100, a Canon G15/G16, or one of the other similar cameras mentioned above, and otherwise follow his advice. These are much lighter, but will give much better images than your smartphone.

Related

2013-09-15

Canon G15 Gain and Read Noise

I've measured the gain and read noise of my Canon PowerShot G15 at different ISO settings.

For the dark exposures, I shut myself in a cupboard, switched off the lights, and took exposures of 1/4000 second at 28 mm equivalent focal length and f/8.

For the light exposures, I took out-of-focus exposures of the sheet of white paper illuminated by diffuse sunlight. I focused manually at infinity, and placed the paper about 10 cm in front of the lens. I used a 28 mm equivalent focal length and f/2.8. I adjusted the exposure time in step with the ISO setting, from 1/4 seconds at ISO 80/100 down to 1/500 second at ISO 12800. (I'd previously attempted to get light exposures with artificial light, but the fast exposures at high ISO suffered widely varying levels due to flicker. Incandescent, fluorescent, and LED lighting often flickers at twice the mains frequency of 50 or 60 Hz, and you can see this in exposures faster than 1/125 second.)

At each ISO setting I took four dark and four bright images. I then converted the raw files to FITS using Jatte and measured the gain and read noise using IRAF's findgain task. This measures the noise in the differences of a pair of dark and a pair of light images, and derives the gain and read noise from the standard noise model. I ended up with eight measurements for each ISO setting, since I had two sets of pairs and one measurement for each channel which allowed me to check for consistency. The results are shown in the following figure and table.

G15 gain and read noise
ISOGain (electrons/DN)Read Noise (electrons)Dynamic Range (Stops)
802.043.511.2
1001.603.310.9
2000.761.610.9
4000.381.210.3
8000.191.29.3
16000.091.28.3
32000.051.07.5
64000.021.16.2
128000.031.16.7

The decline in gain with ISO setting is approximately as expected, until an apparent turn up at ISO 12800. This odd behaviour at ISO 12800 is also seen in a plot of the mean rate (in DN/s) against ISO. If the gain declines as the inverse of the ISO, the mean rate in DN/s should increase linearly with ISO. This is seen from ISO 80 to ISO 6400 (the slope is 0.98 not 1, but that's not an important difference). However, the increase is sub-linear between ISO 6400 and ISO 12800.

G15 mean rate

On the other hand, the mean rate does still increase, which suggests that the gain at ISO 12800 is smaller than the gain at ISO 6400, not higher as suggested by findgain. I suspect the camera is applying noise reduction to the ISO 12800 image, which invalidates the standard noise model and leads to discrepancies between the gain determined from the relative mean rate and the gain determined from the noise properties.

The dynamic range is the saturation level (the gain multiplied by ADC saturation level of 4096 minus the bias level of 128) divided by the read noise, and the converted into stops (by taking the logarithm to base 2). As expected, the best dynamic range is at the slowest setting of ISO 80.

G15 dynamic range

There's little point in using an ISO faster than 400; the read noise doesn't fall significantly, but the dynamic range does. Rather than using ISO 1600, for example, it's probably better to use ISO 400 with 2 stops of exposure compensation. The only use I can think for higher ISOs is getting a brighter live image to frame a shot.

The full well at ISO 80 corresponds to a charge density of about 2300 electrons per square micron. This is about the level at which saturation effects begin to blight CMOS and CCD sensors, so the lack of a slower ISO is not surprising.

Compact Camera Electro-Optics

Compact cameras use smaller detectors and smaller lenses than 35-mm format or “full frame” cameras. How does this impact their electro-optical performance?

Theory

Crop Factor

The crop factor C of a detector is the inverse of its size relative to a 36 by 24 millimeter 35-mm format frame. The crop factors for compact cameras are greater than unity, since their detectors are smaller than a 35-mm frame.

Equivalent Focal Length: Field of View

Achieving the same field of view as a 35-mm camera requires a focal length that is smaller by a factor equal to the crop factor. That is, if the real focal length is f, the field of view will be the same as a 35-mm camera with an equivalent focal length f' given by

f' = Cf.

Equivalent f-Number: Sensitivity

The f-number determines the sensitivity of a camera. For an ideal camera, the sensitivity is proportional to the area of the aperture (a bigger aperture captures more light) and the solid angle of the field of view (a bigger field captures more light). Since the physical diameter of the aperture is f/N, from the definition of the f-number, its area is A = (π/4) (f/N)². The solid angle of the field of view is inversely proportional to f'². Thus, the sensitivity is proportional to

(f/f'N)².

Substituting for the equivalent focal length, we find that the sensitivity is proportional to

(1/CN)².

That is, if the real f-number is N, then the sensitivity will be the same as a 35-mm camera with an equivalent f-number of

N' = CN.

Incidentally, one can see here why the exposure time typically depends on the f-number but not directly on the focal length: the total amount of light reaching the detector is independent of the focal length but inversely proportional to the square of the f-number.

Equivalent f-Number: Depth of Field

The f-number also determines the depth of field, the range of distance over which the focus is acceptable. This is normally quantified as the range of distance over which the circle of confusion is acceptably small, where acceptable here is often taken to be c' = 30 microns for 35-mm cameras. However, for a camera with a smaller detector, the corresponding largest acceptable circle of confusion c will be smaller by the crop factor, giving c' = Cc.

The hyperfocal distance is a special case of the depth of field. When the lens is focused at the hyperfocal distance H, the depth of field extends from half the hyperfocal distance H/2 out to infinity. Essentially, the image is well focused at H and at the limit of adequate focus at H/2 and at infinity. We can easily determine the hyperfocal distance H using the formula

Hf²/Nc.

Writing this in terms of the equivalent focal distance f' and the equivalent f-number N', we have

Hf'²/CNc'f'²/N'c'.

That is, if the real focal distance is f and the real f-number is N, the hyperfocal distance will be the same a 35-mm camera with an equivalent focal distance f' = Cf and an equivalent f-number N' = CN.

We can see that wide-field lenses and narrow apertures give small hyperfocal distances and narrow-field lenses and wide apertures give large hyperfocal distances. The dependence on the focal length is greater than the dependence on the f-number.

The hyperfocal distance determines the depth of field in other circumstances. If a lens if focused at a distance s which lies beyond the hyperfocal distance H, then the depth of field extends from

H/(1+H/s)

to infinity. As a special case of this, if a lens is focused at infinity, then the depth of field extends from the hyperfocal distance to infinity. If a lens is focused at a distance s which lies within the hyperfocal distance, then the depth of field extends from

s/(1+s/H)

to

s/(1-s/H).

If s is small compared to H, the depth of field is distributed approximately equally on either side of s and the depth of field Δs is

Δs = 2s²/H.

The fractional depth of field Δs/s is then

Δs/s = 2s/H.

One can see that to get a deep depth of field, one wants a configuration with a short hyperfocal distance, and to get a shallow depth of field, one wants a configuration with a long hyperfocal distance.

Equivalent f-number: Circle of Confusion at Infinity

The equivalent f-number also determines by how much the background is out of focus when the camera is focused on a foreground object. Defocusing the background can be useful for drawing attention to a foreground object.

For an object in focus at a distance s which is finite but large compared to the focal length f, the diameter of the circle of confusion c of an object at infinity is

cf²/(Ns).

The corresponding angular diameter c/f is

c/ff/Ns = (f'/s)/N'.

So, for a given field of view (equivalent focal length f'), distant objects will be more out of focus when the effective f-number N' is smaller.

Diffraction Limit

Diffraction imposes a limit on the largest useful f-number. Diffraction will cause even a perfectly focused image to have a blur diameter of about , in which N is the f-number and λ is the wavelength of light. The wavelength of optical light ranges from about 400 nanometers (blue) to 700 nanometers (deep red). If the image is to be adequately sharp, then we require

< c.

Therefore, the largest useful f-number is c/λ. Substituting the crop factor C, we find the largest useful f-number is c'/Cλ. For c' of 30 µm and λ of 700 nm, this is about 43/C.

Optimal Signal-to-Noise Ratio

The CCD and CMOS sensors in digital cameras are limited to charge densities of about 2000 electrons per square micron. This means that large detectors can potentially accumulate more charge. The signal-to-noise ratio in image is inversely proportional to the square root of the total charge. Thus, the signal-to-noise ratio in an optimally exposed image increases as the inverse of the crop factor.

Examples

As examples, we will compare the performance of several representative cameras:

  • iPhone 5
  • Canon PowerShot S110
  • Canon PowerShot G15
  • Sony DSC-RX100
  • Canon PowerShot G1X

The rear camera of the iPhone 5 is representative of a fairly good smart phone camera. The S110 is representative of mid-range compact cameras. The G15, RX100, and G1X are representatives of high-end compact cameras. The S110 and G15 have identical detectors, and their comparison will highlight the importance of their different lenses.

This is not meant to be an exhaustive list of compact cameras. However, it illustrates some of the compromises that need to be made to gain compactness at different price points.

The first table gives the basic optical parameters of the cameras.

CameraiPhone 5S110G15RX100G1X
Detector Width (mm)4.577.447.4413.218.7
Detector Height (mm)3.435.585.588.814.0
Crop Factor C7.44.64.62.71.85
Min f (mm)4.15.26.110.415.1
Max f (mm)4.126.030.537.160.4
Min N at Min f2.42.01.81.82.8
Max N at Min f2.4881116
Min N at f' = 100 mm5.92.84.95.6
Max N at f' = 100 mm881116
Min N at Max f2.45.92.84.95.6
Max N at Max f2.4881116
Diffraction limit for N5.89.39.315.723.2

Note that, as expected, all of the cameras are working within their diffraction limit for the f-number.

Field of View

Let's consider the field of view, quantified by the equivalent focal distance f'.

CameraiPhone 5S110G15RX100G1X
Min f' (mm)3024282828
Max f' (mm)30120140100112

The iPhone 5 has a fixed wide-angle lens. The other cameras zoom from a wide-angle (24 or 28 mm) to a reasonably tight zoom (100 to 140 mm).

Sensitivity and Defocus at Infinity

Now let's consider sensitivity and the size of the circle of confusion at infinity, quantified by the minimum equivalent f-number N' at different focal lengths.

CameraiPhone 5S110G15RX100G1X
Min N' at f' = 28-30 mm17.89.28.34.95.2
Min N' at f' = 100 mm27.112.913.410.4

We can see that the compact cameras are 2 to 4 stops more sensitive than the iPhone 5 camera at similar effective focal lengths. That translates to better performance in low light and to shorter exposure times in general. (Image stabilization also gives the compact cameras an additional boost in low light.)

For wide-field imaging, with f' = 28 mm, the RX100 and G1X are about 1.5 stops more sensitive than both the S110 and the G15. However, when zoomed out to f' = 100 mm, the faster lens of the G15 (f/2.8 compared to f/4.9) compensates for the larger detector of the RX100, and the sensitivities of these two cameras are very similar. On the other hand, the slower lens of the S110 (f/5.9) causes it to be about 2 stops less sensitive than either the G15 or the RX100 when zoomed. The combination of large sensor and relatively fast lens gives the G1X an advantage over both the G15 and RX100 at f' = 100 mm.

In terms of blurring the background, the S110, G15, and RX100 do much better than the iPhone 5. The RX100 and G1X appear to be better than the S110 and G15 at an equivalent focal length of 28 mm, but in order to get significant defocus at infinity, one would have to be focused significantly within half of the hyperfocal distance of about 1 meter, which is not that common. The G15 and RX100 are similar at long equivalent focal lengths, with effective f-numbers of about 13. Both are much better than the S110, which has an effective f-number of about 27 and so about half as much blur. The G1X is again slightly better than the G15 and RX100. (The DP Review article of the G15 has a more detailed comparison of the equivalent f-number against equivalent focal length of the G15, G1X, and RX100.)

Depth of Field

Now let's consider control over the depth of field, quantified by the minimum and maximum hyperfocal distances H at different focal lengths.

CameraiPhone 5S110G15RX100G1X
Range of H at f' = 28-30 mm (m)1.70.7-2.80.7-3.20.9-5.50.9-5.0
Range of H at f' = 100 mm (m)9-129-2611-2511-32

The iPhone 5, having both a fixed focal length and a fixed aperture, offers no control of the depth of field. It is configured to give a close hyperfocal distance and a correspondingly wide depth of field.

The S110 has some control over the depth of field, but probably not as much as the G15, RX100, or G1X.

The performance of the G15, RX100, and G1X are similar. At f' = 28 mm, the RX100 and G1X can give a shallower field of view, but at f' = 100 mm the performance is almost identical.

Optimum Signal-to-Noise Ratio

CameraiPhone 5S110G15RX100G1X
1/C0.140.220.220.370.54

Recall that the optimum signal-to-noise ratio is proportional to the inverse of the crop factor C, so large detectors are better. Thus, the G1X is better than the RX100, the RX100 is better than the G15 and S110, and these are all better than the iPhone 5. From the iPhone 5 to the G1X the difference is a factor of four.

f/2.8 and Be There”

The classic configuration for 35-mm street photography, attributed to Arthur “Weegee” Fellig and imortalized in the phrase “f/8 and be there”, is a focal length of 35 millimeters, f/8, and the focus set to the hyperfocal distance of 5 meters (and so a depth of field from 2.5 meters to infinity). The closest equivalent configurations are shown in the following table.

CameraiPhone 5S110G15RX100G1X
f (mm)4.107.607.6012.818.9
N2.42.02.02.84.0
f' (mm)3017.8353535
N'17.89.29.27.67.4
H (m)1.74.44.45.45.5

The iPhone 5 can't really get close to the classic configuration. However, its depth of field is greater, which is probably desirable.

The S110, G15, RX100, and G1X can get quite close to the classic configuration, but notice that they have to reduce the f-number to f/2 to f/4. Thus, for these cameras the phrase is really “f/2.8 and be there,” but that's not half as snappy.

(I'm not actually advocating this configuration for modern compact cameras with autofocus and sensitive detectors. However, this exercise illustrates that the effective f-number depends on the crop factor.)

Summary

We can see that in many ways the Canon G15 and Sony RX100 are quite similar, and both significant improvements over the Canon S110.

Canon and Sony took different routes. For the G15, Canon used the same detector as the S110, but improved the lens significantly, increasing its brightness at maximum zoom to f/2.8 from f/5.9. For the RX100, Sony used a lens with similar performance to the S110, giving f/4.9 instead of f/5.9 at maximum zoom, but dramatically increased the size of the detector. The main advantage of the RX100 over the G15 is that its larger detector can give a better signal-to-noise ratio in optimally exposed images. The main advantage of the G15 over the RX100 is its longer zoom.

The even larger detector of the G1X lets it perform even better than the G15 and RX100, despite having a lens that's about as slow as the S110.