How can dynamic-range be larger than sensor bit-depth?

Question

Found something that confused me and so I thought the crowd here can probably answer this one since its camera-related and technical at the same time.

How can dynamic-range be larger than sensor bit-depth?

Someone sent me the DXOMark results for the Pentax K-5 which shows 14.1 EV of dynamic-range at its lowest ISO. However, given that the sensor is 14-bits, this does not fit with my intuition... It seems strange that a linear device like a CMOS-sensor can capture more DR than it has bits. Would it have a sparse dynamic-range, skipping EVs in the middle?

Answer 1

Cambridge in Colour has a very good article on this. If the sensor has a linear A/D converter, the bit depth would cap dynamic range at at 14 EVs as a theoretical limit. However, if it is non-linear, then the bit depth doesn't necessarily correlate. From that, I think we can determine that the sensor in the K-5 doesn't have a linear A/D converter.

I can say, from personal experience, that this sensor definitely has enormous dynamic range. I managed to recover an image that was close to 8 stops underexposed on the K-5.

Answer 2

How can dynamic-range be larger than sensor bit-depth?

Dynamic range is the logarithm of the ratio between the brightest and and the darkest intensities on the linear part of the sensibility curve. There may be other definitions, but in general it is derived from the ratio of two intensities, objective physical properties of the scene. It is a real number.

Bit-depth is the number of bits per channel used to quantize the continuous variable. More bit-depth gives more distinct shades of gray in between. It is purely a question of how an image is represented in computer memory.

Dynamic range reflects how much contrast the sensor can register. The bit depth reflects how many distinct colors the camera can “give names” to. Or into how many pieces the camera can divide the range. If a camera were a ruler, then the dynamic range would be the (logarithm of the) length of the ruler, and the bit depth would be the (logarithm of the) number of marks along its edge. And you can divide the length into as many pieces as you like. Similarly, the bit depth does not have to be the same as the dynamic range.

If dynamic range is S EV, and bit depth is n, then it means that the camera can register scenes with contrast at least as large as

(Actually a little more if you use also the non-linear part of the sensor response curve). And you can theoretically distinguish

shades of gray.

I own a compact camera which can write 12-bit RAW. Inspite of the high bit-depth, its dynamic range is very modest. You can imaging an opposite situation, when the sensor can register a high contrast scene, without over- and underexposure, but if the bit depth is low, that scene will be represented with few intermediate colors.

Answer 3

Firstly to be clear, dynamic range has an inverse relationship to noise - low noise (all else equal) leads to a greater dynamic range. Noise comes primarily from the sensor electronics (read noise, dark current noise), from the discrete nature of light (photon/shot noise) and from conversion from analogue to digital (quantisation noise).

DXO mark dynamic range scores are based on the difference between the light intensity required to saturate the sensor and the light intensity at which the SNR hits 1:1 (i.e. the point at which the signal equals the noise)

You would expect that in the absence of shot noise and readout noise that the DR of a sensor with a linear response would equal the bit depth. Given the K-5's score in the presence of these sources of noise indicates to me that the image pipeline has a moderate degree of nonlinearity (all sensors have some inherent nonlinearity), probably engineered that way to increase the dynamic range.

Nonlinearity helps escape the bit depth limit, bit what you gain in gradations in the shadows you lose elsewhere in the tonecurve (albeit probably somewhere less important). There is no such thing as a free lunch!

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With regard to the K-5, it's class leading at low ISO sensitivity, which is determined mainly by read noise. It's really great to see the manufacturers turning their attention to this area and it thoroughly deserves the attention, however dynamic range at higher ISO sensitivities is dominated by photon noise which is only countered by capturing more light, so large sensors will always have an advantage here. As some people predominantly shoot ISO400 and above so it's worth bearing this in mind!

Answer 4

The "dynamic range" (DR) is not an absolute characteristic.

The most coarse definition of DR is "ratio between the brightest and darkest gray intensities which sensor may record just fine".

The DR of a digital sensor is derived from two measurements:

1. clipping intensity [for the most sensitive channel] at a given colour temperature (DxO is most probably using D65);
2. intensity which produces boundary amount of noise (i.e. if it is any darker the noise is unacceptable).

Then, you have two ways of computing DR of digital image.

• Dumb way is using pixel data to compute the noise ("screen" measurements on DxO site). If you compute the DR of the linear sensor with X bit ADC this way it can in no way be bigger than X EV.
• Smart way (which is the only possible way of comparing photos from cameras with different resolution) is taking the resolution into account when computing the noise ("print" measurements on DxO). The DR is not limited by ADC this way, one can potentially make a camera with a bigger sensor and the same ADC and it will have bigger percepted dynamic range.

So, you won't find any camera the "screen" DR of which expressed in EVs exceeds the ADC resolution expressed in bits.

Comments on other answers:

From that, I think we can determine that the sensor in the K-5 doesn't have a linear A/D converter.

There is not a single digital sensor with nonlinear A/D conversion developed. Every tonal conversion which camera does (including the special output modes of cinema cameras and Sony A7 series in particular) is done using the discrete data.

The Kodak DCS Pro 14n has dual slope ADC operation mode in which the output is piecewise linear.

Given the K-5's score in the presence of these sources of noise indicates to me that the image pipeline has a moderate degree of nonlinearity

K-5 has perfectly flat response (as any other camera with probably the only exclusion being Kodak DCS Pro). I have measured it myself.

Note: DxO Labs are not resizing or printing anything for "print" measurements, they rather use resollution coefficient in the formulae. Sidenote: in this post "linear" is not "logarythmic".