Radiometric Calibration from Noise Distributions
Yasuyuki Matsushita
Stephen Lin
Microsoft Research Asia
Beijing, P.R. China, 100080
Email: {yasumat,stevelin}@microsoft.com
Radiometric response function

Abstract
A method is proposed for estimating radiometric response functions from noise observations. From the statistical properties of noise sources, the noise distribution
for each scene radiance value is shown to be symmetric fora radiometrically calibrated camera. However, due to thenon-linearity of camera response functions, the observed
noise distributions become skewed in an uncalibrated camera. In this paper, we capitalize on these asymmetric profiles of measured noise distributions to estimate radiometric response functions. Unlike prior approaches, the proposed method is not sensitive to noise level, and is therefore
particularly useful when the noise level is high. Also, theproposed method does not require registered input images
taken with different exposures; only statistical noise distributions at multiple intensity levels are used. Real-world experiments demonstrate the effectiveness of the proposed approach in comparison to standard calibration techniques.

1. Introduction
Many computer vision algorithms rely on the assumptionthat image intensities are linearly related to the image irradiance recorded at the camera sensor. Since most cameras
non-linearly alter irradiance values for purposes such as dynamic range compression, this assumption generally does
not hold. It is therefore important to calibrate the response
function of a camera, so that the non-linear mapping can beinverted and subsequent algorithms can assume linearity ofintensity observations.
As stated by the late Rolf Landauer, a founder of mesoscopic physics, noise is a signal [10]. Fluctuations of a measurement in time, or the distribution of fluctuations, can offer information that cannot be found in the time-averaged
value. This observation is exploited in noise thermometry, an established technique for measuring temperature inhigh-pressure environments based on the level of electric
noise. Likewise, image noise can provide valuable information about the imaging process, and we show in this paperthat the noise distribution in observed images reveals the

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observation
Figure 1. The symmetric profile of a noise distribution in the image irradiance domain becomes skewed by the non-linearity of theradiometric response function. The profile of the measured noise
distribution becomes asymmetric accordingly.

shape of the radiometric response function. Because of thesymmetrically random nature of noise sources in the imaging process, asymmetry of noise distributions in a measured
image is the result of a non-linear transformation that occurs in the camera due to the radiometric response function.
Based on this observation, our method estimates the inverse
response function as the function that maps the asymmetric
distribution of noise in the observation domain to a symmetric distribution in the irradiance domain. This idea is
illustrated in Figure 1.
The primary contributions of our work are twofold. First,
it introduces the use of image noise as a signal for estimating radiometric response functions. Noise has widely been
considered a nuisance in computer vision, but here we derive an important benefit from noise observations. Second,
the proposed radiometric calibration method is effective athigh noise levels, which can substantially degrade previous
radiometric calibration techniques. Our method examines
only the symmetry of noise distributions, which is not affected by noise level. Since noise is inevitable in real im-PhotonCamera irradiance+
Photon noiseElectron

Voltage

Electronic
shot noise
SensorDark current

Thermal noise

OutputAmplifier

Demosaicking

Read-out noise

Demosaicking
noise

Bits

Response function

Quantization

Observation

Quantization
noise

Figure 2. Imaging process and noise sources of digital cameras.

ages, the proposed method has wide applicability.
Since the noise characteristics of a camera typically varywith respect to irradiance value, our technique takes as input the noise distributions for different image irradiances.
This data can be collected by capturing multiple images ofa fixed scene, or by combining intensity data from uniform
scene regions in a single image, as done in [13]. With theproposed approach, calibration results that match those ofstandard techniques can be computed from less data, bycapitalizing on a new source of calibration information.

1.1. BackgroundRadiometric calibration aims to estimate the response
function f of a camera. The radiometric response function
f maps irradiance I that is captured at the sensor to the image intensity M that is read from the camera:
M = f (I).

(1)

For vision algorithms that require irradiance values I rather
than measured intensity M as input, the inverse response
function g = f −1 needs to be determined so that measured
intensities can be made linear with respect to irradiances.
Since response functions f are monotonic, they are invertible. As in most previous radiometric calibration methods,
we assume in this work that the sensor response does notchange over the image.

1.2. Prior work
Several methods have been proposed to estimate thecamera response function. Most of them require as input animage sequence taken with varying exposures from a fixedcamera. With known ratios among exposure levels, Mannand Picard [15] compute a parametric response function in
the form of a gamma curve, and Debevec and Malik [4]obtain a nonparametric response function using a smoothness constraint. With only approximate knowledge of relative exposure levels, Mitsunaga and Nayar [16] iteratively
solve for a response function based on the assumption thatit has a polynomial form. Other iterative estimation methods include that of Tsin et al. [22], which estimates nonparametric responses using a statistical model of the CCD

imaging process, and of Pal et al. [20], which utilizes probabilistic imaging models and prior models of response functions to compute response functions that can differ from image to image.
A few prior methods allow some camera movement or
scene motion, but still require changes in exposure level.
Mann [14] presents a method for response function estimation from image sequences taken by a rotating and zooming
camera. Kim and Pollefeys [9] compute point correspondences to allow free movement of the camera and some motion in the scene. Grossberg and Nayar [6] avoid reliance on
spatial correspondences by relating histogrammed intensity
values between two images of different exposure.
In many applications such as those for web cameras,
multiple images at different exposures cannot be obtained
for radiometric calibration. Some previous techniques havebeen presented without the need to make adjustments incamera exposure settings. Nayar and Mitsunaga [17] place
an optical filter with spatially varying transmittance on theimaging array, which effectively leads to different exposures for neighboring pixels. The response function canthen be computed from neighboring pixels that have different exposures but equal scene brightness. Farid [5] proposesa gamma correction technique based on the frequencydomain correlations that it introduces to an image. This approach requires assumptions on the statistics of scene radiance, and the radiometric responses of many cameras differ
significantly from a gamma function.
A more general approach based on edge information insingle-image input was proposed by Lin et al. for color images [11] and for grayscale images [12]. The former obtains
information about the radiometric response function fromcolor distributions of local edge regions. Due to blending ofdistinct region colors, irradiance colors from edge regions
should form linear distributions in color space. But becauseof nonlinear radiometric response functions, measured edge
colors actually compose nonlinear distributions that are directly related to the response function. With a prior modelof response functions compiled by Grossberg and Nayar [7],the inverse radiometric response is computed as the function that maps the nonlinear distributions of measured edge
colors into linear distributions. For grayscale images, a 1D

Imaging process

Measurement

Response fRadiometric calibration

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Figure 3. Illustration of our method. Symmetric noise profiles are skewed by the radiometric response function in the imaging process.
Radiometric calibration is performed by estimating a function that transforms the skewed noise distributions into symmetric forms.

analogue of the 3D color method is presented in [12].
Our work may be considered as complementary to themethods based on edge information. Since the edgebased methods rely on statistical distributions from possibly
sparse data, they may be susceptible to high levels of image
noise. In contrast, our technique utilizes noise as an information source. Our method is also able to use non-edge regions in an image for radiometric calibration. In principle,
a single-image technique could be developed that combines
the edge processing of [11] together with our proposed approach applied to uniform regions in the image.

2. Transformation of noise distributions
Figure 2 outlines the imaging process and sources ofnoise. Our method is built upon the assumption that thenoise distributions from the various noise sources are statistically symmetric. In this section, we first review the properties of the different types of noise in the imaging pipeline,
and confirm the validity of this assumption. Then we describe the transformation of these symmetric noise distributions by nonlinear camera response functions into asymmetric distributions in measured images.

2.1. Symmetry of imaging noise
In the imaging process, radiance from the scene is first
focused by the lens onto the sensor. This scene radiancefluctuates over time due to the discrete nature of photons.
These temporal fluctuations are called photon noise, or photon shot noise. Photon noise obeys the Poisson law [3, 1],
and a Poisson distribution is well-approximated by a Gaussian distribution when the number of arrivals is large [21].
Since the number of photon arrivals at an sensor element
can be considered large under ordinary imaging conditions,the distribution of photon noise can reasonably be approximated as Gaussian. Photon noise is therefore symmetric.
The incoming rays are then captured by the imaging sensor and converted to an electrical signal. In the process of
conversion, the signal is contaminated by dark current, shotnoise, and thermal noise. Dark current adds a constant offset to the signal with zero-mean fluctuations. This fluctu-

ation is called dark current noise, or thermal noise. Dark
current noise arises from thermally generated charges in thesensor. Due to manufacturing variances, some pixels in thesensor have greater dark current than others. These pixels
are referred to as hot pixels, and are a source of fixed pattern noise. Since dark current noise occurs in the form of
discrete electrons, the noise process obeys the Poisson law.
Electronic shot noise, which is different from photon shotnoise, occurs in the electronic circuitry due to random fluctuations of current in an electrical conductor. These fluctuations similarly arise from the fact that current is carriedby discrete electrons. These sources of noise at the sensor
all obey the Poisson process. Since the number of electrons
involved per pixel is large, their noise distributions are also
well-approximated by a Gaussian.
An output amplifier sequentially transforms the electrons
from each pixel into a voltage. The amplifier adds read-out
noise, whose distribution is zero-mean and independent of
the number of captured electrons. Read-out noise can betreated as having a symmetric distribution since it resultsfrom the random effects of coupled sources: that countingof electrons is not a perfectly repeatable process, and thatspurious electrons exist within the amplifier.
In the case of color cameras, a demosaicking process
converts the raw sensor signal into a three-channel color
image by interpolating the red, green, and blue pixels in
the imaging array. This process can affect the noise distributions for interpolated pixel values, since they are being
estimated from other noisy signals. Since interpolation iscomputed among measurements with symmetric noise distributions, we regard the noise distributions of demosaickedcolor values also to be statistically symmetric.
At this point, the demosaicked irradiance value is thentransformed to the measured intensity by the camera’s radiometric response function f . In this process, the symmetric profiles of the noise distributions becomes distorted by
the non-linearity of the camera response function, as further
described in Section 2.2.
Finally the signal is quantized, and quantization noise isintroduced. Since quantization noise results from the rounding of signal amplitudes to discrete levels, it can be consid-

Log of normalized frequency

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Figure 4. A noise profile measured using a Canon EOS 20D (ISO1600), and the corresponding noise profile after calibration withthe ground truth response curve. The noise profiles are scaled horizontally for better visualization.

ered as a zero mean random variable with a uniform probability distribution [19].

2.2. Asymmetry of measured noise
As reviewed above, all noise sources in a digital camera can be considered statistically symmetric. For a camera
with a linear response function, its observed noise distributions should therefore be symmetric. On the other hand,
non-linearity of the radiometric response function will leadto asymmetry in the observed noise distributions, as illustrated in Figure 3. For typical response functions, the nonlinear mapping results in greater condensation of the distribution towards higher intensities and broadening towards
lower intensities. This leads to measured noise distributions
that are negatively skewed.
The correct inverse response function g should transformthe asymmetric profile of the measured noise distribution toa symmetric profile as illustrated in Figure 3 (right). In Figure 4, we show a real-world example of how a measured
asymmetric noise profile is projected using the ground truth
response function into a symmetric distribution in the irradiance domain. Our work aims to find such a function forradiometric calibration.

3. Calibration algorithm
In presenting our calibration algorithm, we first describe
how we evaluate the degree of symmetry of a noise profile. Based on this symmetry measure, we then derive anenergy function that estimates the inverse response function
g that best transforms measured noise profiles to symmetricforms. For cases where the available noise data is insufficient to avoid solution ambiguities, we utilize a prior modelof inverse response functions to regulate the solution.

3.1. Degree of symmetry
There exist various possible approaches to measuring thesymmetry of a statistical distribution. Since the mean, median and mode are all the same in a symmetric distribution,

d−
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Figure 5. Evaluation of the degree of distribution symmetry.

one basic symmetry metric is Pearson’s skewness coefficients S1 = 3(x̄ − x̃)/σ and S2 = 3(x̄ − x̌)/σ, with distribution mean x̄, mode x̃, median x̌, and standard deviation
σ [8]. Values of these two coefficients that are closer to zero
indicate greater distributional symmetry.
In this work, we evaluate symmetry in a manner that issimilar to the Bowley skewness [8], which computes distances among interquartile ranges to effectively obtain ameasure of distribution balance around its median. In oursymmetry metric, we compute the distance from the distribution mode to the profile of the distribution at multiple
different heights as shown in Figure 5. At each i-th heightin the distribution q, the distance to the profile on the left+d−q,i and on the right dq,i are respectively computed. The
degree of symmetry of the distribution q is then evaluated
as
n  −
+ 2
1  dq,i − dq,i,(2)
S(q) = −+n i=1 d−q,i + dq,iwhere n is the number of heights to be evaluated. A larger
value of n yields a more accurate result at the expense of
greater computation time, which is proportional to n. The
denominator normalizes each term by the distribution width
at the given height. Larger values of S indicate greater symmetry.
While the inverse of any skewness metric could in principle be used in place of Equation (2), we utilize this measure because of its effectiveness in optimization. In contrast, we have empirically found the Bowley skewness andPearson’s skewness coefficients to be less sensitive to theslight changes in distribution structure that occur in an iterative optimization procedure.

3.2. Solution method
With this symmetry measure, our method computes theinverse response function g from the set of collected noise
profiles Ω by maximizing the following energy function:
1 
1 
S(g; q) =S(g(q)),(3)
E(g; Ω) =
|Ω|
|Ω|
q∈Ω

q∈Ω

where |Ω| represents the number of noise distributions inthe set Ω. This function evaluates the degree of symmetry

of noise profiles that are projected to the irradiance domainby the inverse response function g.
While argmaxg E(g) gives the optimal estimate of theinverse response function g, it is computationally difficult
to solve for a non-parametric inverse response function, because of the large number of intensity levels (e.g., 256 for 8bit images). To facilitate optimization, we utilize a parametric model based on Grossberg and Nayar’s principal components analysis (PCA) on the DoRF database of real-world
response functions [7]. The response functions in the DoRF
database are first inverted so that principal components ofinverse response functions can be computed. With theseprincipal components, an inverse response function g canbe represented asg = g0 + Hc,(4)

where g0 is the mean inverse response, H is the matrix
whose columns are composed of the first N eigenvectors,
and c is an N -dimensional vector of PCA coefficients. In
this work we set N = 5 as done in [7, 12]. With thisrepresentation of inverse response functions, the problem
is transformed into estimating the N coefficients of c:ĉ = argmax E(g; g = g0 + Hc).
c(5)

Normalized irradiance

In cases where the projected noise profiles g(∀q ∈ Ω) do
not cover the entire range of irradiance values, there may
exist multiple solutions that result in symmetry of noise distributions. To avoid this ambiguity, additional constraints
on the inverse response function, such as smoothness andmonotonicity, could be used. In this paper, we use prior datafrom the DoRF database to regulate the solution as donein [12].
Using the eigenvectors of Equation (4), we computethe PCA coefficients of each inverse response function in
DoRF. We then construct a prior model on inverse responses
by fitting a multivariate Gaussian mixture model to this setof PCA coefficients:
p(g) =K

αi N (g; µi , Σi ),

(6)

i=1

where N represents a normal distribution with mean µi
and covariance matrix Σi . We empirically set K = 5,
and the mixture model is obtained using the cross-entropy
method [2].
We also model the likelihood of the degree of symmetryas1(7)
p(Ω|g) = exp(−λE(g; Ω)),
Z
where Z is the normalization factor, and λ is a regularization coefficient, which is empirically set to 104 .

Normalized irradiance

(a)

(b)

Figure 6. Processing of incomplete noise distributions. (a) Upper
portion of noise profile is used for evaluating the degree of symmetry. (b) Noise profile is discarded due to insufficient symmetrydata.

The optimal coefficients ĉ that define the inverse response function g are solved in the following MAP problem:ĉ = argmax p(g(c)|Ω) = argmax p(Ω|g(c))p(g(c)).
c

c

(8)

Inserting Equations (6) and (7) into the logarithmic form ofEquation (8), we obtainĉ = argmin λE(g(c); Ω) − log p(g(c)).
c3.3. Solution using prior model

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The optimized coefficients ĉ yield our estimate of the inverse response function as ĝ = g0 + Hĉ.

4. Experiments
To evaluate our method, we performed experiments on
various real-world cameras and scenes. In our implementation, optimization is performed using the Nelder-Meadsimplex method [18] with multiple initial values. The degree of symmetry in Equation (2) is evaluated with n = 20
for all examples in this paper. The evaluation is performedin the irradiance domain as described in Equation (3). At
the lower and upper bounds of the intensity range where acomplete noise profile cannot be observed, we use only theavailable partial profile in computing the degree of symmetry, as illustrated in Figure 6.
In our experiments, we first examine cases where noise
distributions are recovered over the entire range of intensity levels. We then consider the case of incomplete data,
which utilizes the prior model of inverse response functions
to determine a solution.
Results with complete data In this experiment, we efficiently measure noise distributions at different intensity
levels by capturing a static scene with a fixed video camera. With this setup, the intensity fluctuations at a pixel overtime are caused only by noise. By combining the intensity
observations of a pixel into a histogram, noise distributions

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CameraJVC GR-D230AH
Sony DCR-TRV9E
Sony DSR-PD190P
PointGrey DragonFly

RMSE
0.02240.02470.03320.0265

Disparity
0.04950.05220.06730.0478

Table 1. RMSE and disparity of estimated inverse response functions in terms of normalized irradiance.
Inverse response function
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Mitsunaga & Nayar
Our method

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Figure 7. Measured noise distributions (Sony DSR-PD190P). The
logarithm of normalized frequency values are used for visualization purposes. Each vertical slice represents a noise distribution at
the corresponding intensity level.

are directly measured for each pixel. Furthermore, thesecollected noise distributions from the image set are merged
together according to intensity level for greater statisticalstability. The mode of each measured noise distribution istaken as its true intensity.
We used four different video cameras for experiments: JVC GR-D230AH, Sony DCR-TRV9E, Sony DSRPD190P, and PointGrey DragonFly. Each input dataset iscollected by capturing a static scene for 100 frames (about 3
secs at 30 fps). In this experiment, we ensure that noise distributions are acquired over the full range of intensity levels.
This can often be achieved simply by capturing two image
sequences of a scene at different exposure levels, e.g., one
bright scene and one dark scene, and then merging them together to form a single dataset. Figure 7 shows a measured
noise distribution for the Sony DSR-PD190P camera. Eachcolumn of the chart represents the noise distribution of thecorresponding intensity level.
Representative results of the proposed method are shownin Figure 8. In the figure, comparisons of our estimated inverse response curves to those of standard radiometric calibration methods are shown in the left column. The center
column displays the measured noise profiles and the resulting symmetric noise profiles after applying the estimated inverse response functions. For better visualization, the noise
profiles are displayed at every ten intensity levels startingfrom intensity level 5. In actual computation, the noise profiles for all intensity levels are used to estimate the inverse
response functions.
For a quantitative comparison of our method to standard
techniques, we evaluated the RMSE (root-mean-squared er-

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Figure 9. Result of calibration from a single image using our
method. The scene is photographed at an ISO 800 gain level.

ror) in terms of normalized irradiance and disparity, which
represents the average value among the dataset of the maximum difference between our function and the comparison
curve. In computing disparity, four sequences for each camera are used. The result is summarized in Table 1. For
the Sony DSR-PD190P and PointGrey DragonFly cameras,
the comparison curve is generated by averaging the curves
obtained by the method in [16] and by fitting to Macbeth
color checker measurements, which were interpolated using a fifth-degree polynomial. For the JVC GR-D230AH
and Sony DCR-TRV9E cameras, the inverse response function obtained using the Macbeth color checker is used as
the comparison curve. The low RMSE and disparity values
demonstrate an agreement between our calibration method
and the standard techniques, which utilize additional datasuch as multiple exposure settings or known relative scene
colors.
Results with incomplete data When the observed noise
distributions do not span the entire range of intensity levels, we use the prior model on inverse response functions toconstrain the solution as described in Section 3.3. We conduct this experiment using a single input image that does
not include a full range of intensities. For simplicity in validating our method, we photographed a scene composed ofrelatively flat, uniform surfaces, and then manually selected
image regions from which to obtain noise distributions. A
more sophisticated algorithm such as in [13] could alternatively be used to automate this process.
Figure 9 shows a result from a single input image using
the prior. The RMSE and disparity are 0.0337 and 0.096 re-spectively using five datasets, where the ground truth curveis computed as the mean curve between those obtained using a Macbeth color chart and the method of Mitsunaga &Nayar [16]. The relatively large disparity value likely results from the limited number of observations per noise distribution, which degrades the accuracy of the distributions
in a statistical sense. Our solution nevertheless exhibits aclose match to the ground truth curve.

5. Discussion
Symmetry assumption Our method is based on the assumption of symmetric noise distributions from the various sources. As we reviewed in Section 2, the assumption is valid under typical conditions where the numbers ofphotons and electrons in the sensor are sufficiently large.
However, this assumption breaks down in low-light conditions, where photon and electron counts become small.
In low-light conditions, photon noise and electronic shotnoise distributions deviate from Gaussian approximation,and hence the symmetry assumption breaks down. The proposed method is therefore not intended for low-light images/videos.
Statistical validity of noise distributions For the measured noise distributions to be statistically valid, a sufficiently large number of samples per intensity level needs
to be collected. One indicator of data sufficiency is thesmoothness of the measured noise profiles. Since noise distributions are expected to be smooth, a coarse profile suggests inadequate sampling, and more image data is needed
for greater reliability in radiometric calibration.
Future work In this work, we presented a radiometriccalibration technique that takes advantage of the image
noise that normally degrades computer vision algorithms.
Its tolerance of high noise levels and effectiveness without
multiple exposures are its primary benefits over previous
algorithms. Because of these particular advantages, an interesting direction for future work is in employing our technique in a conjunctive manner with other radiometric calibration methods, to reduce sensitivity to noise and to resolve
solution ambiguity in cases there would otherwise be little
calibration data.

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Normalized irradiance

Normalized intensity(a) JVC GR-D230AH (red channel)
Inverse response function

Observed noise profiles

1

1

Color chart
Our method

0.9

0.8
0.6

0.8

Normalized frequency

Normalized irradiance

0.4
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Normalized intensity
Projected noise profiles
1
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Normalized irradiance

Normalized intensity(b) Sony DCR-TRV9E (green channel)
Inverse response function

Observed noise profiles

1

1

Color chart
Mitsunaga & Nayar
Our method

0.9

0.8
0.6

0.8

Normalized frequency

Normalized irradiance

0.4
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Normalized intensity
Projected noise profiles
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Normalized irradiance

Normalized intensity(c) Sony DSR-PD190P (blue channel)
Inverse response function

Observed noise profiles

1

1

Color chart
Mitsunaga & Nayar
Our method

0.9

0.8
0.6

0.8

Normalized frequency

Normalized irradiance

0.4
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Normalized intensity
Projected noise profiles
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Normalized intensity

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Normalized irradiance(d) PointGrey DragonFly (green channel)
Figure 8. Results of our radiometric calibration method. Left column: comparison of inverse response functions. Middle column: observed
noise profiles and transformed noise profiles. Right column: an image from the corresponding test image sequence. Noise profiles are
displayed at every ten intensity levels for visualization purposes. From top to bottom, the results of different cameras are shown: (a) JVC
GR-D230AH, (b) Sony DCR-TRV9E, (c) Sony DSR-PD190P, and (d) PointGrey DragonFly.