# DxOMark measurements for lenses and camera sensors

## How do sensors see colors?

Color sensors usually have three different types of photosites that are sensitive to light in different bands of wavelengths. Although there are other technologies, the overwhelming majority of color sensors use a Bayer color filter array. Each block of 4 contiguous photosites contains one photosite sensitive to low wavelengths (blue), one photosite sensitive to high wavelengths (red), and two identical photosites sensitive to medium wavelengths (green).

After analog-to-digital conversion, the signal for each channel is represented by a certain number of bits, usually 14 or 12 for a reflex camera, and 10 for a cameraphone; thus a chip can encode colors on 42 bits, 36 bits, or 30 bits in RAW format.

Since each channel has its own noise level, it actually distinguishes far fewer than the theoretical 10, 12, or 14 bits of RAW data. The sum of the tonal range on each channel gives a rough idea of the color sensitivity of a sensor, and experiments show that this sum is already fewer than 24 bits.

Moreover, the R, G, and B channels of a sensor are a
consequence of its design. They are characterized by their spectral
sensitivities, giving the proportion of incident photons that eventually reach
the sensor for a given wavelength (called *quantum efficiency*). The figure below
gives an example of spectral sensitivities of a sensor. Note that the photosites
are frequently sensitive to near-infrared, so an infrared filter must be added
to the system to filter out these wavelengths. In particular, note that these
curves are not the color-matching functions _{}.

The RAW values in the R, G, and B channels on a uniform surface (ignoring noise for the time being) are not the sRGB values of this surface. In other words, the color space of the sensor is not the sRGB color space or any usual color space. A first consequence of these spectral sensitivities is that a theoretical constant spectrum (every wavelength of a theoretical constant spectrum has the same intensity) would yield three values of R, G, and B that are usually different. As silicon is weakly sensitive to low wavelengths, B is often much smaller than G (by a factor 2 or more is not unusual).

## Definitions

Color sensitivity is the number of reliably distinguishable colors up to noise. Roughly speaking, two colors are considered as distinguishable if their difference is larger than the noise. In this respect, color sensitivity is the generalization of color to the notion of tonal range.

To extend this idea for color data defining color sensitivity:

Let be
the noise covariance matrix at the value (*r, g, b*).

For ,
let be
the eigenvalues of this covariance matrix.

We call **Color Sensitivity** the number derived from

.

The above formula shows that the determinant of the noise
covariance matrix is the volume of the incertitude ellipsoid, in which the
difference of two colors is most likely due to noise. However, because digital
images are encoded using integer values, the dimensions of the incertitude ellipsoid
are quantized. Hence the integrand in the equation above can be seen as the
density of distinguishable colors around the point (*r, g, b*), and takes
quantization into account. The integral itself can then be interpreted as the
total number of colors that can be distinguished by the sensor.

As with tonal range, color sensitivity is a number with no
unit, so instead we consider _{}, which represents the number
of bits necessary to encode all distinguishable color values.

## SMI: Sensitivity Metamerism Index

The sensitivity metamerism index (SMI) is defined in the ISO standard 17321 and describes the ability of a camera to reproduce accurate colors. Digital processing permits changing color rendering at will, but whether the camera can or cannot exactly and accurately reproduce the scene colors is intrinsic to the sensor response and independent of the raw converter.

The underlying physics is that a sensor can distinguish exactly the same colors as the average human eye, if and only if the spectral responses of the sensor can be obtained by a linear combination of the eye cone responses. These conditions are called Luther-Ives conditions, and in practice, these never occur. There are objects that a sensor sees as having certain colors, while the eye sees the same objects differently, and the reverse is also true.

SMI is an index quantifying this property, and is represented by a number lower than 100 (negative values are possible). A value equal to 100 is perfect color accuracy, and is only attained when Luther-Ives conditions hold (which, as previously stated, never happens in practice). A value of 50 is the difference in color between a daylight illuminant and an illuminant generated by fluorescent tubes, which is considered a moderate error.

More precisely, SMI is defined as

,

where is the average CIELAB error observed on a set of various colors. In our experiments, we used the 18 colored patches of a Gretag McBeth color checker, as ISO 17321 recommends. The SMI varies depending on the illuminant.

In practice, the SMI for DSLRs ranges between 75 and 85, and is not very discriminating. It is different for low-end cameras (such as camera phones), which typically have a SMI of about 40. For this reason, we give this measurement as an indication but do not integrate it in DxO Mark.

## Relative sensitivities and white balance scale

All the standard RGB color spaces (such as sRGB, Adobe RGB, Prophoto) assume that a neutral reflectance object (i.e., an object that reflects every wavelength equally) has equal values on the three channels. However, the RAW values on the sensor depend on the spectrum of the light illuminating the neutral object, and on the spectral responses of the sensor. The three channels are usually different. The relative sensitivity of both the red and blue channels is the ratio between the value of the red (or blue) channel and the green channel.

The SMI depends both on the illuminant and the spectral responses of the sensor. Typically, with a tungsten illuminant, the blue sensitivity is very low, due to the weak sensitivity of silicon in short wavelengths and the lack of short wavelengths in tungsten light.

For design reasons, relative sensitivities are almost always lower than 1.

White balance scales are the digital gains applied on each channel to compensate for the lack of sensitivity in the red and blue channels. After applying the white balance, red, green and blue should be equal on a neutral object.

White balance scales are simply the inverse of the relative sensitivity. However, we think the redundancy is interesting and expresses two different factors: On the one hand, relative sensitivity is an objective characteristic of the sensor; on the other hand, white balance scale is what image science engineers apply to compensate for the differences in sensitivity.

Large white balance scales have a major influence on image quality since they directly impact noise, and this is taken into account in the color sensitivity measurement.

## Channel decomposition and color matrix

Assume that the three channels of a sensor have equal sensitivities (or, equivalently, that white balance has been applied). In practice, the primaries of the sRGB color space do not correspond with the primaries of the sensor. (See the explanation for Luther-Ives condition in the section SMI above.) For each sensor channel, the channel decomposition describes the linear combination of the primaries of the sRGB color space that best fits the sensor channel. Typically, the red sensor channel is expected to contain mostly red, a bit of green, and almost no blue. A decomposition of the red channel close to (1,0,0) shows that the red channel of the sensor is very “pure.”

The color matrix gives the coefficients of the linear combination of the sensor channels which have to be applied to compensate for the sensor channels’ lack of purity as compared to the sRGB primaries. As with the white balance scales and the relative sensitivities, the channel decomposition and the color matrix give exactly the same information in a complementary manner. The channel decomposition coefficient can be portrayed as a 3x3 matrix, which means that the color matrix is then simply the inverse of the channel decomposition coefficient matrix.

The channel decomposition is a low-level description of the sensor spectral response, while the color matrix is what engineers use to make the sensor react as though it had sRGB primaries.

A color matrix with large singular values yields a dramatic amplification of noise, and this is taken into account in the color sensitivity measurement.

## Full-color sensitivity

The noise on the sensor is processed by a RAW converter to lead to the final image. In particular, RAW converters apply a color rendering which maps the RAW values to RGB values in the final image (for instance, sRGB values). Color rendering includes at least white balancing and a color matrix, as described above. These steps yield an amplification of noise. For each triple RGB in the final image, noise can be predicted from the RAW noise characteristics, the white balance scales, and the color matrix coefficients. The noise on the three color channels is described by a three-dimensional covariance matrix, which is not easy to represent. In the full-color sensitivity tab, we chose to illustrate the noise in the (a,b) plane of the CIE Lab color space. For different values of luminance and different ISO sensitivities, the noise at given colors is represented by an ellipse which shows the colors that are likely to be generated in the image instead of the real color (displayed at the ellipse center). The orientation and the size of the ellipses show the coloration of noise and its amplitude.

## Further readings for the DxOMark measurements for lenses and camera sensors

To provide photographers with a broader perspective about mobiles, lenses and cameras, here are links to articles, reviews, and analyses of photographic equipment produced by DxOMark, renown websites, magazines or blogs.

On DxOMark, we evaluate and rank many types of digital cameras with image sensors that vary widely in pixel count, pixel size, and digital signal processing. To ensure that sensor performance comparisons between cameras are fair, it is very important both to test under identical shooting conditions and to take viewing conditions into account. |