Radiometric Calibration Using Temporal Irradiance Mixtures
Bennett Wilburn
Microsoft Research Asia
Beijing, China

Hui Xu∗TsingHua University
Beijing, China

Yasuyuki Matsushita
Microsoft Research Asia
Beijing, China

bwilburn@microsoft.com

xu-h06@mails.tsinghua.edu.cn

yasumat@microsoft.com

Abstract

The response function f is represented as
M = f (I),We propose a new method for sampling camera response
functions: temporally mixing two uncalibrated irradiances
within a single camera exposure. Calibration methods rely
on some known relationship between irradiance at the camera image plane and measured pixel intensities. Prior approaches use a color checker chart with known reﬂectances,
registered images with different exposure ratios, or even theirradiance distribution along edges in images. We show that
temporally blending irradiances allows us to densely sample the camera response function with known relative irradiances. Our ﬁrst method computes the camera response
curve using temporal mixtures of two pixel intensities onan uncalibrated computer display. The second approachmakes use of temporal irradiance mixtures caused by motion blur. Both methods require only one input image, although more images can be used for improved robustness
to noise or to cover more of the response curve. We showthat our methods compute accurate response functions fora variety of cameras.

(1)

where I represents the irradiance at sensor, and M is themeasured pixel intensity. Conventionally the axes are normalized to measurement and irradiance ranges in [0, 1].
This function is assumed to be constant throughout the image. Imaging and computer vision algorithms are concernedwith irradiance, so we generally solve for the inverse radiometric response function, g = f −1 (M ).
We propose a radiometric calibration approach based on
temporally mixing two different irradiances within a single camera exposure. Cameras accumulate incoming lightfor the duration of their exposure, and scene brightness at
any given pixel may vary due to motion blur or illumination changes. We constrain the incident brightness to be
one of only two values. For example, we could use motion
blur across the boundary between two uniformly colored
regions. Although we do not know the accumulated irradiance at each pixel, we do know that if the motion is linear,
the brightness in the blurred region will vary linearly along
the direction of motion. For a camera with a non-linear
response, however, the measured pixel intensities will notvary linearly. Similarly, we can use a simple illumination
change, like a rapidly blinking light, to create brightness
mixtures. In this case, the irradiance at each pixel is a linear
blend of the irradiances with the light on and off. Linearly
increasing the portion of the exposure that the light is on
creates a linearly increasing series of accumulated irradiances. Again, this linearity does not hold for the measured
pixel intensities if the response function is nonlinear. Our
method exploits these nonlinear intensity measurements toestimate the response functions.
In the rest of this section, we describe how our approachcomplements prior work in radiometric calibration. Section 2 explains the general idea of temporal irradiance mixing. In sections 3 and 4, we present two speciﬁc methods
to use this idea for radiometric calibration. Section 5 shows
results with various cameras validating our techniques, andsection 6 discusses the implications of our work and av-

1. Introduction
Many computer vision algorithms assume that image
pixel intensities are linearly related to scene irradiance. For
most cameras, however, this is not the case. Camera manufacturers often deliberately engineer nonlinear response
functions into their cameras in order to match ﬁlm characteristics or to account for nonlinear characteristics of computer displays and the human visual system. Even if the
response is intended to be linear, the analog circuitry in theimage sensors themselves introduces small nonlinearities.
Thus, the camera’s radiometric response function, whichmaps scene irradiance to measured pixel intensities, is generally a nonlinear function. Therefore, we must estimate theresponse function in order to linearize the image intensities
and improve the performance of vision algorithms.
∗ This work was done while Hui Xu was an intern at Microsoft Research
Asia.

1

enues for future research.

1.1. Prior WorkRadiometric calibration methods require means to collect samples of the radiometric response function with someknown relationship. For example, one approach is to use animage of a uniformly illuminated Macbeth color chart [1],which has color patches with known reﬂectances. Mannand Picard [9] calibrate using multiple registered images ofa static scene with different known exposure times. The ratio of the irradiance at the same pixel in different images
is equal to the ratio of the exposure times of the images.
They use this relationship to solve for a parametric response
function of the form M = α + βI γ . Debevec and Malik [2] also use multiple images with different exposures,
but solve for a smooth non-parametric function. Mitsunaga
and Nayar’s method [11] requires only rough estimates ofthe exposure times and iteratively computes a polynomial
inverse response function and more accurate estimates ofthe exposure times. Tsin et al. [13] use a statistical modelof the CCD image formation process to iteratively estimate
non-parametric inverse response functions.
Several methods have been developed that use multiple
exposures, but do not require precise registration. Grossberg and Nayar [3] use the relationship between the intensity histograms of two scenes imaged with different exposures, because intensity histograms are relatively unaffectedby small changes in the scene. Kim and Pollefeys [5] compute point correspondences between images. Mann [8] estimates response functions from a rotating and zooming camera.
Lin et al. [6] compute the radiometric response function from RGB distributions along color edges in a single
image. They use a prior model of response functions compiled by Grossberg and Nayar [4] to compute the radiometric response function as the one which maps nonlinear measured RGB intensity distributions to linear ones. Lin andZhang [7] adapt this idea to grayscale images by measuring the nonuniformity of edge intensity histograms. Recently, Matsushita and Lin [10] proposed a method to compute camera response functions from noise distributions.
Analyzing noise distributions requires much input data, although a single image can be used under very carefully controlled conditions.
Our work differs from the prior art in that we use temporal color mixtures to directly sample the response function.
We do not need special equipment or multiple registeredimages with precise exposure control. Our methods require
only a single image, do not rely on statistical priors over the
response function, and work with color or grayscale images.
In some ways our approach complements the work of Lin etal. [6] in that we use temporal mixtures, while their methods use spatial mixtures. Although they mention defocus

and motion blur as sources of linear irradiance blends, they
do not test this idea. Moreover, calibration from temporalcolor mixtures frees us from reliance on indirect, statisticalmethods and priors over camera response functions.

2. Calibration Using Temporal Irradiance
Mixtures
Following the notation of Mitsunaga and Nayar [11], we
can describe the relationship between image irradiance I
and scene radiance L as
 2
π dcos4 (φ),(2)E(t) = L(t)4 hwhere h is the lens focal length, d is the aperture diameter,and φ is the angle subtended by the principal ray from theimaging axis. We will simplify this to E = kL(t), wherek = π4 ( hd )2 cos4 (φ). We assume, for now, that k is constant.
Image sensors and ﬁlm are integrating devices, so the total
integrated irradiance for an exposure that lasts from time t0
to time t1 is

I=

t1

kL(t)dt.

(3)

t0

Prior methods assume that L(t) is constant. Let us assume, instead, L(t) can be one of only two values, L0 or L1 .
Then the integrated irradiance at the sensor becomes
I = αI0 + (1 − α)I1 ,(4)

where 0 ≤ α ≤ 1 is the portion of time that the scene
radiance is L0 . I0 and I1 are the accumulated irradiances
corresponding to a constant scene radiance of L0 or L1 , ret
spectively, for the entire exposure, i.e., I0 = t01 kL0 (t)dt
tand I1 = t01 kL1 (t)dt. We can generate any convex combination of I0 and I1 by varying α. We propose to use theseblends to sample the response function for radiometric calibration. Our method can be applied to grayscale or color
images; for color images, we can use the same mixtures
to sample and independently calibrate the three color channels.

2.1. Scale and Offset Ambiguities
The irradiances in our mixtures are unknown, so ourmeasurements have two ambiguities. Figure 1 shows thisgraphically. We are blending a low irradiance Ilo and a highirradiance Ihi , corresponding to measured intensities Mlo
and Mhi , respectively. Because Ilo is unknown, there may
be a global offset to the curve. We alleviate this by making
Ilo as close to zero as possible. Ihi is also unknown, leading
to a scale ambiguity for the curve in the measured region.
Often, we are able to set Ilo below the noise ﬂoor and Ihi
above the saturation level, enabling us to capture the entire
response curve. Regardless, even with both ambiguities, it
is still possible to linearize the measured intensities.

1.0

Irradiance

I hi

Scaling ambiguity

I lo

Offset ambiguity

M lo

Measurement

M hi 1.0

A
B

Irradiance

Irradiance

Figure 1. Scale and offset ambiguities for radiometric calibration.
The response curve computed from our method suffers from two
ambiguities. Ilo is unknown, so there may be a global offset to thecurve. We alleviate this by making Ilo as close to zero as possible.
Ihi is also unknown, leading to a scale ambiguity for the curve inthe measured region.

j=0 i=0k=0

We add a monotonicity constraint that forces the ﬁrst
derivative of the inverse response function to be nonnegative. The scale and offset of the ﬁnal curve is still ambiguous, so we normalize
it by setting a0 to zero and addinga constraint that k ak = 1.

3. Display-Based Calibration Method
Scaling for A
Offset for B

Measurement

mixture, for j ∈ (0, ..., n − 1), is lj . Each mixture is represented by a set of measurement-irradiance pairs, (Mij , Iij ),for i ∈ (0, ..., lj − 1). Mij is the pixel value for observation i in mixture j. Iij is the relative irradiance for thatmeasurement, up to the scale and offset ambiguity. Thus, it
could correspond to α from equation 4. Or, for evenly distributed samples between two irradiances (a linear ramp),
the Iij could simply be 0,1,2,...,(lj − 1), i.e., Iij = i.
We want to compute an inverse response function and aset of irradiance scales and offsets that aligns these measurements. We denote the offsets by rj and the scales by
sj . We assume the inverse response function g(M ) can bemodeled by a polynomial of degree d with coefﬁcients ak ,dki.e., g(M ) =k=0 ak M . We solve for a, r and s thatminimize the following error function:

2
lj
d
n−1


k
ak Mij.
(5)
si Iij + rj −E=Measurement

Figure 2. Curve merging. On the left, we see an example with two
measured irradiance blends. Each curve has an offset and scaleambiguity. To merge the data from both curves, we must computethe scale and offset that causes the two curves to coincide. To
do this, we simultaneously solve for a camera inverse response
function g(M ) and scale/offset values for each mixture that bestalign the data to g(M ). The desired result is shown on the right.

2.2. Combining Multiple Irradiance Blends
We may want to use data from many different irradiances
mixtures to improve our estimate of the response function.
For example, we could increase the range of our measurements by using multiple mixtures that span a broader rangeof irradiances. Across a motion-blurred edge, we can use
many lines of pixel data to improve robustness to noise. In
either case, we estimate the low and high irradiances fromeach mixture in order to combine the data. Figure 2 shows
this process graphically.
Mathematically, the problem is as follows. We observe nirradiance mixtures. The number of observations in the j-th

If the radiance of a computer monitor were linearly related to the displayed pixel value, we could easily measure acamera’s response curve by displaying different known radiances and recording the intensities measured by the camera for a ﬁxed exposure time. Most displays, however, havea non-linear relationship between the display pixel value
and emitted intensities. Using temporal color mixtures, we
can use an uncalibrated display to generate a linear rampof irradiances. We take advantage of the high frame rate ofcomputer displays to show several different images during
a single camera exposure. For some fraction α of the exposure, we display pixel value Plo , and for the remainder
we display Phi . As in equation 4, this allows us to makeknown linear combinations of the irradiances Ilo and Ihi
corresponding to the two selected pixel values.
For calibration, we choose Plo to be completely dark,
and Phi high enough to saturate the image sensor. If we candisplay N images in one camera exposure, then we can use
a sequence of exposures to generate N +1 samples of the response function that are evenly spaced in irradiance. Fitting
a curve through the measured values produces a continuous
representation of the camera response function.
As described, this method requires one image for each
irradiance measurement. We can accelerate the data capture by dividing the displayed image into a grid of patches,and using each patch to display a different linear combination of radiances. In this way, we can calibrate the cameraresponse curve using just one image. Figure 3 shows anexample of this process. For cameras with very nonlinear
responses, one irradiance blend might poorly sample somepart of the response curve. In these cases, we extend themethod again by simultaneously displaying multiple grids
to produce multiple blends. We then combine the data fromthe blends as described in section 2.2. In this way, we can
add more measurements to poorly sampled regions.
Implementation. We chose to implement our displaybased calibration using a regular LCD computer monitor.
Provided the camera exposure duration can be set to a multiple of the display time for the entire image sequence, we
do not need to synchronize the camera and display. We simply continuously repeat the sequence of images for the grid
pattern (see ﬁgure 3).
LCD monitors might have other characteristics that
could interfere with our measurements: unequal on and off
transition times for the LCD material, backlight ﬂicker, orinversion (switching the polarity of the applied LCD voltage at each pixel every frame to prevent damage to the LCD
material). To minimize these effects, we use a relatively
long exposure time of one second, display each image in thesequence for an even number of frames (four), and ensurethat each pixel only switches its display value once during
the sequence.
One phenomena we observed when implementing our
grid-based method was that the brightness of each patch
was heavily inﬂuenced by bright surrounding regions. We
found that separating the patches by a black border eliminated this effect. We use 30×30 pixel patches with 30 pixel
black borders between all patches. To prevent artifacts due
to non-uniform illumination of the display or vignetting in
the camera, we made sure that the overall size of the displayed pattern was small relative to the display and camera
image dimensions.

4. Calibration from Motion BlurOur display-based method requires a display with known
frame rate and a camera with accurate control of the exposure duration. For situations which preclude that method,
we offer a second approach: calibration from motion blur.
Figure 4 shows a simple way to use motion blur for radiometric calibration. We create a target with a step edge
between two uniformly colored regions, place the target
fronto-parallel to the camera, then take a picture of the target while moving the camera so the direction of blur isacross the edge. Of course, one could also move the target.
Pixels in the blurred region along the edge will measure a
blend of the two uniform irradiances. If the target is uniformly illuminated and the motion is constant and parallelto the plane of the target, the irradiance at the camera will
increase (or decrease) linearly from pixel to pixel across theedge. This produces a series of irradiances at the sensors
that are linearly spaced between the irradiances for the two
uniform regions.
By choosing one region to be black and the other to bebright enough to saturate the camera, we can capture theentire range of the camera response. Moving the camera
quickly enough (or exposing for long enough) to blur overmany pixels will produce many samples of the response
function. We can combine the blurred values from manylines to estimate the camera response function as describedin section 2.2.
Implementation. Some care must be taken to calibrate
the camera radiometric response using motion blur. The
relative motion between the camera and the target must be
constant, roughly parallel to the plane of the target, and withsome component across the edge. The target must also be
uniformly illuminated. Because the linear irradiance blend
is distributed across many pixels in the image, radiometricfalloff must be negligible over the motion blurred region.
The direction of camera motion need not be exactly perpendicular to the edge between the two colored regions, aslong as the edge is straight and the uniform regions are large
enough that the blurred region sees only the two intended irradiances. Also, we carefully focus the camera to prevent
adding additional blur to the irradiance proﬁle.
We manually specify the motion-blurred region using avery simple user interface shown in ﬁgure 4 (c). We automatically locate the monotonically decreasing or increasing
ramps of grayscale values in each scanline, and combine
data from the speciﬁed rows to produce the response function. If we use a symmetric target like that in the ﬁgure, we
can verify the accuracy of the calibration; if the motion is
not linear or the illumination varies too greatly, the response
curves calculated from the left and right edges of the black
region will not match. On the other hand, if these conditions are satisﬁed, we expect the two curves to agree. We
visually inspect the curves; for a more quantitative method,
one could also measure the RMSE between them.

5. Experimental Results
In this section, we evaluate our results against for several
cameras. We use the online implementation of Mitsunaga
and Nayar’s method[11] (which we will call “MitsunagaNayar”) as a reliable reference, and also compute our own
ground truth data.
Ground Truth Data. We compute ground truth data using multiple exposures of a Macbeth color chart and a technique very similar to Mitsunaga and Nayar’s method. For agiven color patch, the relative exposure time of each image is the relative irradiance of the patch in that image.

IrradianceDisplay patterns

Time

Exposure time

Observed image

Measurement

Intensity

Figure 3. Radiometric calibration by temporally blending two intensities on a computer monitor. In this example, we display a sequence ofnine images during a single camera exposure. For each patch, the bright and dark values are shown for different portions of the exposure.
This creates a grid of patches in the camera image that have linearly spaced irradiances. Plotting the irradiances versus their measured
values in the image produces a sampled version of the inverse response curve.

Scanline

(a)

(b)

(c)

Figure 4. Calibrating from motion blur. (a) An example of motion blur with constant velocity. (b) The pixel intensity for one row of pixels
crossing the blurred target. This is a sampled version of the camera response curve. (c) An example of a user interface for specifying the
motion blurred region.

This gives us one series of measurements with known irradiances for each patch. We then simultaneously solve forthe relative irradiances of the patches (i.e. the irradianceof one patch relative to another patch) and a single polynomial that ﬁts all of the scaled measurements. The relative
irradiance is not simply the patch reﬂectance; other factors
such as veiling glare in the camera, brightness offsets, andnon-uniform illumination also affect this ratio.
Display-Based Calibration. For the display-based calibration method, we used a DELL 1907FPt monitor and displayed two grid patterns in each image. The low displayed
pixel value was always 0, and the high one was 220 (out of255) for one grid and 100 for the other. The two grids were
each three rows of ﬁve patches, and displayed one above
the other on the screen. We used 60Hz displays and a one
second exposure for all cameras, so each pattern in the gridsequence was displayed for four frames. We optimized using the Nelder-Mead simplex method [12]. Figure 5 shows
an example image of our pattern, taken with the Canon EOS
D20.
Calibration from Motion Blur. To verify the correctnessof our motion blur approach, we used a controlled setup to
create a very high-contrast edge and guarantee linear mo-

(a)

(b)

Figure 5. Example inputs for our calibration method. (a) Two 5x3
grids of irradiance blends. (b) A static image of the light trap used
for our motion blur experiments (top), and an image taken while
the camera was moving (bottom).

tion. We used a light trap as our target. A light trap is alarge cavity with black interior walls and a small opening.
Light entering the opening is “trapped” inside because it ismostly absorbed by the walls, so the hole emits very little
light. For our experiments, the hole is rectangular, and theoutside of the trap is white, creating a very high-contrast
white-black-white edge, shown in ﬁgure 5. We made thelight trap ourselves from an ordinary cardboard box. Care

Table 1. Mean RMSE, variance, and disparity of the estimatedinverse response functions in the normalized input domain. Ten
datasets were used for each method. For Mitsunaga-Nayar, weremoved two out of ten estimates that were obvious outliers.

1
Ground truth
Dark Æ bright ramp
Bright Æ dark ramp

0.9
0.8

Irradiance

0.7
0.6

RMSE
Variance
Disparity

Display
0.012
0.050
0.017

Motion blur
0.012
0.052
0.029

0.5

Mitsunaga & Nayar
0.006
0.061
0.017

0.4

RMSE: 0.006
Disparity: 0.011
Difference: 0.009

0.3
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0.1
0

0

0.1

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0.5

0.6

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0.8

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1

0.8

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1

Measurement

(a)

(b)
1
Ground truth
Dark Æ bright ramp
Bright Æ dark ramp

0.9
0.8
0.7

Irradiancemust be taken to make the edge of the hole straight and thin(so light does not reﬂect off the sides). We used sheets ofwhite paper to make the border. To ensure linear motion, we
mounted each camera on a linear translation stage. The exposure time for the cameras was roughly one second. Stilland blurred images of the light trap are shown in ﬁgure 5.

0.6

RMSE: 0.015
Disparity: 0.027
Difference: 0.028

0.5
0.4
0.3

Results. Figure 6 shows ground truth and our results forthe green channel response of three cameras: a Canon EOS
20D, a Nikon D70, and a PointGrey Research Dragonﬂy.
We chose the Canon and Nikon for their popularity, andthe Dragonﬂy because its response (nearly linear) differs
greatly from the other two. The plots were normalized tothe domain and range of our measured pixel intensities. A
typical range was 15 to 240 (out of 255). Our results agree
well with those obtained from the other methods.
Some of these calibration methods might be expected to
produce varying results for different input images. To investigate this, we used each method ten times with different inputs. Table 5 shows a comparison of the mean RMSE, variance, and disparity (maximum error) for ten trials of eachmethods. The RMSE of our methods are roughly twice thatof Mitsunaga-Nayar, while our variance is slightly less. For
the Mitsunaga-Nayar method, however, we had to remove
two curves that were obvious outliers.
Motion Blur by Hand. Our motion blur calibration
method is not limited to lab settings. In ﬁgure 7, we show
two examples of motion-blur based calibration where wemanually created the motion. For this experiment, we tapeda rectangular piece of black cardboard to the inside of awindow. When photographed from the inside, this createsa very high-contrast edge. We then photographed the cardboard while waving the camera sideways by hand. If the
exposure time for the camera is short enough, we can expect the motion to be roughly linear. We used a 1/30 second exposure. Because the target has two symmetric edges,
we can verify the results by comparing the response curves
computed from the blurred regions on each side of the cardboard.
Figure 7 (a) shows one attempt. Figure 7 (b) shows thatthe inverse response curves obtained from the left and rightsides of the target. These coincide very well, indicating

0.2
0.1
0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Measurement

(c)

(d)

Figure 7. Calibration from manual motion blur. (a) An image of apiece of cardboard against a window, taken with the Canon EOS
20D while the camera was waved sideways by hand. (b) The inverse response curves computed from the left and right sides of theblur region match each other, showing that this calibration result
is good. The difference between the two curves, measured as the
RMSE error, is under 1%. Both curves ﬁt the ground truth well.
(c) A second motion blur example. (d) This time, the curves computed from both sides do not match each other (an RMSE errorof 2.8%), indicating the motion was not linear. These are the best
and worst results from six trials. We used a 1/30 second exposure
time for all images.

that the data are good. As we see, the computed curve also
matches the ground truth. Figure 7 (c) and (d) show an image and plot from a less successful attempt. The curves do
not match each other or the ground truth. These are the best
and worst results from six trials.

6. Discussion and Conclusion
We have presented the general idea and two speciﬁc
methods for using temporal color mixtures to compute
camera radiometric response functions. Our display-based
method is well-suited for lab environments. For outdoor
settings or with cameras whose exposure time cannot be set,
calibration from motion blur is recommended. The strengthof these techniques lies in their ability to directly sample theresponse function. We can easily verify that we have covered the full measurement range, and plotting the measured
data provides immediate feedback about the shape of the re-11

Ground truth
Mitsunaga & Nayar
Display
Motion Blur0.9

Irradiance

0.8

1Ground truth
Mitsunaga & Nayar
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Motion Blur

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1

Measurement(a) Canon EOS 20D(b) Nikon D70(c) PointGrey Dragonﬂy

Figure 6. Our calibration results for the green channel of three different cameras, compared to ground truth and Mitsunaga-Nayar. Allplots were normalized to the domain and range of our measured pixel intensities. Our results agree well with those obtained from the othermethods.

sponse curve and the accuracy of the computed ﬁt. This is
not true for methods based on analyzing edges or noise distributions. Moreover, we do not rely on statistical priors [4]over the camera response function. This might allow us tocalibrate cameras not represented well by existing priors.
One limitation of our method is the offset ambiguity incomputed response curve; in practice, we minimize this byobserving irradiances as near to zero as possible. Regardless, the scale and offset ambiguities do not prevent us fromlinearizing the intensities. We assume control over the exposure time for our display-based calibration method, but
one could imagine ways to calibrate without this constraint.
As one example, we could take a series of images of a single high-speed LED ﬂashing with varying duty cycles. One
interesting feature of our display-based method using multiple irradiance blends is that it computes relative irradiances
for different displayed pixel values. This could be extended
to simultaneously calibrate the monitor and the camera.
We veriﬁed the correctness of the motion blur approach
using a controlled setup and further demonstrated the applicability of our method to a hand-held camera in a relaxedenvironment. Conceivably, our method could even be used
to calibrate videos or images for which the camera is nolonger available.

References[1] Y.-C. Chang and J. F. Reid. Rgb calibration for color image analysis in machine vision. IEEE Transactions on Image
Processing, 5(10):1414–1422, 1996.
[2] P. E. Debevec and J. Malik. Recovering high dynamic range
radiance maps from photographs. In ACM SIGGRAPH,
pages 369–378, 1997.
[3] M. D. Grossberg and S. K. Nayar. Determining the camera
response from images: What is knowable? IEEE Trans.

on Pattern Analysis and Machine Intelligence, 25(11):1455–
1467, 2003.
[4] M. D. Grossberg and S. K. Nayer. What is the space of camera response functions? In IEEE Conference on Computer
Vision and Pattern Recognition (CVPR), volume 2, pages
602–609, 2003.
[5] S. J. Kim and M. Pollefeys. Radiometric alignment of image
sequences. In IEEE Conf. on Computer Vision and Pattern
Recognition (CVPR), volume 1, pages 645–651, 2004.
[6] S. Lin, J. Gu, S. Yamazaki, and H. Y. Shum. Radiometric calibration from a single image. In IEEE Conference
on Computer Vision and Pattern Recognition (CVPR), volume 2, pages 938–945, 2004.
[7] S. Lin and L. Zhang. Determining the radiometric response
function from a single grayscale image. In IEEE Conference
on Computer Vision and Pattern Recognition (CVPR), volume 2, pages 66–73, 2005.
[8] S. Mann. Comparametric imaging: Estimating both the unknown response and the unknown set of exposures in a plurality of differently exposed images. In IEEE Conference
on Computer Vision and Pattern Recognition (CVPR), volume 1, pages 842–849, 2001.
[9] S. Mann and R. Picard. Being ‘undigital’ with digital cameras: extending dynamic range by combining differently exposed pictures. In IS&T, 48th annual conference, pages 422–
428, 1995.
[10] Y. Matsushita and S. Lin. Radiometric calibration from noise
distributions. In IEEE Conference on Computer Vision and
Pattern Recognition (CVPR), 2007.
[11] T. Mitsunaga and S. K. Nayar. Radiometric self-calibration.
In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), volume 2, pages 374–380, 1999.
[12] J. Nelder and R. Mead. A simplex method for function minimization. Computer Journal, 7:308–313, 1965.
[13] Y. Tsin, V. Ramesh, and T. Kanade. Statistical calibration of
ccd imaging process. In International Conference of Computer Vision (ICCV), volume 1, pages 480–487, 2001.