*15 September 2013*

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

*H* ≈ *f*²/*Nc*.

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

*H* ≈ *f’*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 Δ*s* is

*Δs* = 2*s*²/*H*.

The fractional depth of field *Δs*/*s* is then

*Δs*/*s* = 2*s*/*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

*c*_{∞} ≈ *f*²/(*Ns*).

The corresponding angular diameter *c*_{∞}/*f* is

*c*_{∞}/*f* ≈ *f*/*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 *Nλ*, 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

*Nλ* < *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.

Camera | iPhone 5 | S110 | G15 | RX100 | G1X |
---|---|---|---|---|---|

Detector Width (mm) | 4.57 | 7.44 | 7.44 | 13.2 | 18.7 |

Detector Height (mm) | 3.43 | 5.58 | 5.58 | 8.8 | 14.0 |

Crop Factor C | 7.4 | 4.6 | 4.6 | 2.7 | 1.85 |

Min f (mm) | 4.1 | 5.2 | 6.1 | 10.4 | 15.1 |

Max f (mm) | 4.1 | 26.0 | 30.5 | 37.1 | 60.4 |

Min N at Min f | 2.4 | 2.0 | 1.8 | 1.8 | 2.8 |

Max N at Min f | 2.4 | 8 | 8 | 11 | 16 |

Min N at f’ = 100 mm | 5.9 | 2.8 | 4.9 | 5.6 | |

Max N at f’ = 100 mm | 8 | 8 | 11 | 16 | |

Min N at Max f | 2.4 | 5.9 | 2.8 | 4.9 | 5.6 |

Max N at Max f | 2.4 | 8 | 8 | 11 | 16 |

Diffraction limit for N | 5.8 | 9.3 | 9.3 | 15.7 | 23.2 |

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’*.

Camera | iPhone 5 | S110 | G15 | RX100 | G1X |
---|---|---|---|---|---|

Min f’ (mm) | 30 | 24 | 28 | 28 | 28 |

Max f’ (mm) | 30 | 120 | 140 | 100 | 112 |

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.

Camera | iPhone 5 | S110 | G15 | RX100 | G1X |
---|---|---|---|---|---|

Min N’ at f’ = 28-30 mm | 17.8 | 9.2 | 8.3 | 4.9 | 5.2 |

Min N’ at f’ = 100 mm | 27.1 | 12.9 | 13.4 | 10.4 |

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.

Camera | iPhone 5 | S110 | G15 | RX100 | G1X |
---|---|---|---|---|---|

Range of H at f’ = 28-30 mm (m) | 1.7 | 0.7-2.8 | 0.7-3.2 | 0.9-5.5 | 0.9-5.0 |

Range of H at f’ = 100 mm (m) | 9-12 | 9-26 | 11-25 | 11-32 |

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

Camera | iPhone 5 | S110 | G15 | RX100 | G1X |
---|---|---|---|---|---|

1/C | 0.14 | 0.22 | 0.22 | 0.37 | 0.54 |

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.

Camera | iPhone 5 | S110 | G15 | RX100 | G1X |
---|---|---|---|---|---|

f (mm) | 4.10 | 7.60 | 7.60 | 12.8 | 18.9 |

N | 2.4 | 2.0 | 2.0 | 2.8 | 4.0 |

f’ (mm) | 30 | 17.8 | 35 | 35 | 35 |

N’ | 17.8 | 9.2 | 9.2 | 7.6 | 7.4 |

H (m) | 1.7 | 4.4 | 4.4 | 5.4 | 5.5 |

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.

© 2013 Alan Watson Forster. All rights reserved.