Estimating demosaicing algorithms using image noise variance
Jun Takamatsu
Nara Institute of Science and Technology
MSR-IJARC fellow
8916-5, Takayama-cho, Ikoma, Nara, Japan

Yasuyuki Matsushita
Microsoft Research Asia
5F, Beijing Sigma Center, No. 49, Zhichun road,
Haidian district, Beijing, China

j-taka@is.naist.jp

yasumat@microsoft.com

Tsukasa Ogasawara
Nara Institute of Science and Technology
8916-5, Takayama-cho, Ikoma, Nara, Japan

Katsushi Ikeuchi
University of Tokyo
4-6-1, Komaba, Meguro-ku, Tokyo, Japanogasawar@is.naist.jp

ki@cvl.iis.u-tokyo.ac.jp

Abstract
We propose a method for estimating demosaicing algorithms from image noise variance. We show that the noise
variance in interpolated pixels becomes smaller than that of
directly observed pixels without interpolation. Our method
capitalizes on the spatial variation of image noise variancein demosaiced images to estimate the color filter array patterns and demosaicing algorithms. We verify the effectiveness of the proposed method using various images demosaiced with different demosaicing algorithms extensively.

1. Introduction

Figure 1. A color image (left) and the corresponding raw image (right). A demosaicing algorithm produces a color image froma raw image. Our method can invert the process of demosaicing.

Many consumer digital cameras are equipped with asquare grid of photo-sensors overlaid with a color filter array (CFA). The color filters selectively allow light according
to wavelength range to produce color information. Some
digital cameras employ three separate sensors, each sensor
taking a separate measurement of red, green, and blue by
splitting light through a prism assembly. In single-sensor
cameras, almost all of them use a Bayer filter on which eachtwo-by-two submosaic contains two green, one blue, andone red filters. The raw image data captured by a sensor
with a CFA is converted to a color image by a demosaicing
algorithm. This process is illustrated in Figure 1 from rightto left.
Precise understanding of imaging process is importantfor many computer vision algorithms that require accurate
knowledge of irradiance. For the task of photometric calibration, there has been plenty of studies on estimation ofcamera response functions, vignetting, etc. Knowledge of
the CFA pattern and the demosaicing algorithm is important

as well to understand true irradiance; however, not manystudies have been done in this direction. Since information on the CFA pattern and the demosaicing algorithm are
typically not available from camera manufacturers, development of an estimation algorithm is important.
In this paper, we develop a method for automatically determining CFA patterns and demosaicing algorithms. We
refer to a CFA pattern to indicate the arrangement of thesubmosaic color filter pattern at a particular location on thesensor. We use a physical property of the image noise variance, i.e., it becomes smaller after interpolation, to determine the interpolated pixels. After CFA estimation, ourmethod further estimates the demosaicing algorithm using
the distribution of interpolation weights. The overview ofthe proposed method is illustrated in Figure 2.
This paper has two major contributions. First, we show1

InputDemosaicing trace !
& CFA pattern

Demosaicing algorithm

R G
G B
Pattern of image noise variances!

Algorithm A

Histograms of !
Fisher’s linear discriminantinterpolation patterns & nearest neighbor search

Figure 2. Overview of the proposed method. Our method takes registered color images or a single color image as input. Our method
consists of three stages: 1) detection of the demosaicing trace, 2) estimation of the CFA pattern, and 3) estimation of the demosaicing
algorithm.

how the image noise variance is skewed by the demosaicing
process; the noise variance of interpolated pixels becomes
smaller than that of the directly observed pixels. Second,
we develop a method to automatically identify CFA patterns
and demosaicing algorithms from the distribution of noise
variances.

2. Prior work
Demosaicing aims to produce high-quality color images
while avoiding the introduction of false color artifacts (e.g.,chromatic aliases, zippering, etc.) at low computationalcost. There have been many studies on demosaicing algorithms. A recent survey by Liu et al. covers a wide variety
of state-of-the-art demosaicing algorithms [12].
Recent demosaicing algorithms are far more complex
than straightforward interpolation methods such as nearestneighbor/bilinear interpolation methods. Since the Bayerpattern has more green pixels than either red or blue pixels,
many demosaicing algorithms first interpolate the green colors for better edge preservation, and later resample the red
and the blue colors. For edge preservation, Chang et al. [2]proposed to interpolate pixel values based on the magnitudeof the image gradient. Hirakawa and Parks [7] selected thebest value among the pre-computed candidates based on thecriterion of image naturalness. Tsai and Song proposed anefficient selection method that avoids heavy computation of
the candidate colors [20].
Estimation of the demosaicing algorithms has beenstudied in the context of digital forensics. Popescu andFarid [15, 16] used the EM algorithm for identifying a demosaicing algorithm. Bayram et al. proposed a method forclassifying digital camera models from information about
camera-specific interpolation [1]. Gallagher proposed amethod for detecting linear or cubic interpolation using
the periodicity of the second order derivatives of the interpolated images [5]. Gallagher and Chen presented amethod for distinguishing natural images from photo realistic computer-graphics images using demosaicing traces [6].

Our work is also related to image noise analysis inphysics-based computer vision. Image noise has been actively used for vision algorithms in previous approaches.
For example, Liu et al. [11] developed a method for estimating noise level function from a single image and used
it for efficient image denoising. Matsushita and Lin [13]used image noise for defining accurate intensity similarity
measure. Hwang et al. [9] proposed a noise-robust edge
detection algorithm based on noise observations. Treibitz
and Schechner [19] theoretically proved the recovery limits
in point-wise degradation considering intensity-dependentimage noise effects.
In another stream of image noise analysis in computer
vision, there are methods that perform estimation only fromnoise observations. Matsushita and Lin [14], and Takamatsu et al. [17, 18] have shown that image noise provides
sufficient information for estimating radiometric response
functions. Like their methods that use the image noise as
signal, our method estimates the trace of the demosaicing
and the CFA pattern only from noise observations.

3. Demosaicing and image noiseThis section describes the relationship between image
noise variance and color interpolation in a demosaicing process. Specifically, we show that the noise variance of interpolated pixels tends to become smaller than that of the directly observed pixels. In the following, we call the directly
observed pixel value as observed intensity to differentiate itfrom interpolated intensity.
In the demosaicing process, the intensity II (p) of an interpolated pixel p is obtained by combining the observed
intensities IO (q) of the neighboring pixels q for each color
channel. This process can be formulated as Eq. (1), where
Rp represents the set of the observed pixels that are locatednear pixel p.
X
II (p) =w(q; p)IO (q).
(1)
q∈Rp

Figure 3. Visualization of image noise variances of the G-channel(right) computed from registered images (left). Variances of pixels
whose values are obtained by interpolation tend to be smaller. The
visible checker pattern corresponds to the Bayer pattern.

Note that subscripts I and O denote interpolated and observed pixels, respectively.
Let us now consider how image noise variance is alteredthrough the demosaicing process. Here we assume that theobserved image noise is spatially independent. An intensity I(p) in a demosaiced image can be described by the
˜noise-free intensity I(p)and the image noise N (p) as
˜I(p) = I(p)+ N (p).
Substituting Eq. (2) into Eq. (1), we obtain
X
NI (p) =w(q; p)NO (q),

(2)

(3)

2
where σ̄O(p) is the average variance of the neighboring observed pixels Rp . From this result, it is naturally expectedthat the variance at the interpolated pixel becomes smaller.
Figure 3 shows the visualization of image noise variance of
the G-channel that is computed from registered color images. In the figure, a checker-board pattern can be clearly
seen, which corresponds to the Bayer pattern.
We use the decreasing tendency of the noise variance of
the interpolated pixels to determine the CFA pattern. Also,
this tendency weakens when either 1) weights of interpolation, or 2) the noise variance distribution in the neighboring
observed pixels are very biased. However, these two conditions seldom occur in practice.

4. Estimation method
Our estimation method consists of the following steps:
1) detection of the demosaicing trace, 2) estimation of the
CFA pattern, and 3) estimation of the demosaicing algorithm. To obtain image noise variance, multiple registeredimages with fixed camera parameters and view position areusually used. The image noise variance is obtained fromthe fluctuated values at the corresponding pixels across theimages. While a large number of images is preferred statistically, our method fortunately works well with just therough estimates of the variance. Therefore, it only requires
a few images (five images in this paper). In this section, we
will describe the algorithms of the above steps. We assume
that the possible candidates for CFA patterns are known inadvance because the types of CFA patterns (e.g. Bayer filters) are limited in practice.

q∈Rpbecause I˜I is canceled out by the weighted sum of I˜O in
Eq. (3). From Eq. (3), the image noise variance can be described as
X
σI2 (p) =w(q; p)w(r; p)cov(NO (q), NO (r))
q,r∈Rp=X2w(q; p)2 σO
(q),

(4)

q∈Rp
2
where σI2 (p) and σO(q) represent the variances of NI (p)and NO (q), respectively. And cov(NO (q), NO (r)) represents the covariance between NO (q) and NO (r). Becauseof the spatial independence of the noise distributions,cov(NO (q), NO (r)) = 0, q 6= r.
Let us take a simple example to illustrate. Consider thecase of bilinear interpolation, w(q; p) = 1/n for all q
where n is the number of elements in the set Rp . Substituting w(q; p) = 1/n into Eq. (4), we obtain
σI2 (p) =1 2
σ̄ (p),
n O

4.1. Detection of the demosaicing trace
Consider a 1D sequence of noise variances where thevariances of the interpolated and the observed pixels alternately appear. Applying the discrete Fourier transform(DFT) to the sequence, the DFT magnitude is maximized
when the frequency ω equals to π, because the interpolated
and the observed variances appear by turns. This DFT property is also used by Gallagher and Chen [6], but we use
noise variance as input instead of the derivatives of the color
images. Given a hypothesized CFA pattern, our methodassigns labels (interpolated or observed) to pixels. Then,
we apply DFT as mentioned above and evaluate the magnitude. When the hypothesized CFA pattern is correct, theDFT magnitude becomes large. We test for every possible
hypothesis.
To construct the 1D sequence of variances, we take theaverage of the variances in the interpolated pixels along the
diagonal path of submosaics. The same procedure is performed on the observed pixels. Using many diagonal paths,
we obtain a set of average variances of the interpolated andthe observed pixels. Finally, a long 1D sequence of vari-ances is obtained by arranging them. The reason why wesample variances diagonally is to avoid JPEG compression
artifacts. Since JPEG compression is applied to each 8 × 8
pixel block, horizontal or vertical arrangement generates
other peak frequencies [5].
By applying DFT to the 1D sequence {xk }, the Fourier
series {fj } in the frequency domain is obtained as
fj =

m−1
X

2πi

xk e− m jk (j = 0, · · · , m − 1),

(5)

k=0

the criterion C1 becomes small as JPEG compression ratio
becomes high. Therefore, if the difference in C1 is larger
than a predefined threshold (we used 0.5 in this paper), we
consider JPEG compression ratio to be low enough and use
C3 criterion. Otherwise, we use C1 criterion.

4.3. Estimation of the demosaicing algorithm
To estimate the demosaicing algorithm, we use the histogram of the interpolation weights w(q; p). Representing
Eq. (1) in a vector form, we obtain

where m is the length of the sequence. Similar to themethod of [6], we define the criterion C1 for determining
whether there exists a trace of demosaicing as|fmid |
C1 =,|fm/2 |

(6)

where fm/2 is the m/2-th element of the Fourier series. Its
norm |fm/2 | is the amplitude of (m/2)-Hz wave, and fmid
is the element whose amplitude is the median of all the amplitudes {|fj |}.
This value C1 becomes smaller if the hypothesized CFA
pattern is plausible. If the criterion C1 is larger than a predefined threshold τ for all possible CFA patterns, we consider
that as the absence of demosaicing trace.

4.2. Estimation of the CFA pattern

2
σ̄O
,
σ̄I2

(7)

2
where σ̄I2 and σ̄Oare the average variances of all the interpolated pixels and that of observed pixels, respectively. The
larger C2 is, the more likely the hypothesized CFA pattern
is. This criterion C2 can be computed in each color channel. We use all color channels to make the criterion robust
by summing them up as

C3 = C2R + C2G + C2B ,(9)at each interpolated pixel p. We use notation Rp to represent the set of pixels that are used for the interpolationdef(Rp = {q1 , . . . , qn }), where n is the number of theneighboring observed pixels. The observed pixel intensities IO and interpolation weights w are represented in aTT=vector form as IT
O = (IO (q1 ), . . . , IO (qn )) and w
T(w(q1 ; p), . . . , w(qn ; p)) . We can obtain many samples
for Eq. (9) from many locations in the image coordinates.
For each p, we can create a set of equations of Eq. (9) using
multiple registered images. With conditionsXw(q; p) = 1, andq∈Rp

The method described in Section 4.1 is effective for finding CFA patterns as well as trace of demosaicing. However,periodicity can be observed even when an incorrect CFA
pattern is assumed. To solve this problem, we use an additional criterion for estimating the CFA pattern.
We define a simple criterion C2 for the estimation thatassesses the plausibility of the hypothesized CFA pattern as
C2 =IT
O w = II (p)(8)

where C2R , C2B , and C2G are the C2 criteria in the RGB
channels.
One weakness of this criterion C3 is that it is more sensitive to JPEG compression artifacts than the criterion C1 .
JPEG compression contaminates the pixel values regardlessof observed and interpolated pixels. As a result, the difference between the maximum and the second maximum of∀p, q, 0 ≤ w(q; p)(≤ 1),(10)

we estimate the interpolation weights w by solving the linear system of equations in a least-squares manner.
We use Fisher’s linear discriminant (FLD) [4] andnearest-neighbor search for the classification of the demosaicing algorithms. Given the input feature, the classifier
finds the nearest class in the FLD subspace.
To generate the training dataset, we first calculate interpolation weight vectors w from images demosaiced by acertain demosaicing algorithm. Then, the weight vector w
is computed at every interpolated pixel location. Dependingon the pixel location, the length of weight vectors w varies.
Next, we create a histogram of the weight vectors w foreach length of the weight vectors. Finally, these histograms
are normalized and concatenated to obtain a single vector
form of the training data. The same procedure is applied for
creating other training data representing a different demosaicing algorithm.
Note that the order of elements in the weight vector does
not have much meaning from the view point of the interpolation method. For example, the weight vector (0.4, 0.6)
is regarded to be the same as the weight vector (0.6, 0.4).
Therefore, we just sort the vector elements in ascending order.

R G R G R

R G

G R

G B G B G
R G R G R

G B

B G

Pattern 1

Pattern 2

B G

G B

G B G B G
R G R G R

G R

R G

Pattern 3

Pattern 4

Figure 4. Bayer pattern and possible two-by-two submosaic patterns.

4.4. Estimation from a single image
The accurate estimation of image noise variance requires
multiple registered images of the same scene. It is often too
demanding, because such a dataset is not available in practice. To relax the condition, we assume that the irradianceof the neighboring pixels is similar, and that variance canbe computed using the group of neighboring pixels. A similar assumption was also used in previous work (e.g., [17]),although in a different context. Since the characteristics ofnoise variances in interpolated and observed pixels are different, we create the pixel groups by using only the sameclass of pixels (either interpolated or observed).
In the single-image case, when the number of observed
pixels which are used for interpolation, i.e., elements in the
set RP , is more than three, the available constraints are
insufficient to uniquely determine interpolation weights w
at pixel p. This is because we only have two constraints,
Eqs. (9) and (10). In the multiple-image case, this does not
become a problem since we can create more equations frommultiple registered images. To resolve this ill-posedness inthe single-image case, we use an additional condition to regulate the solution. To avoid excessive deviation, we define
the condition that the sum of squared weights is minimized:
min wT w.

5. Experiments
In this section, we evaluate the proposed method in two
scenarios: the multiple-image case and the single-imagecase. We also assess the robustness of the algorithm against
JPEG compression.

5.1. Multiple-image case
Setup In this experiment, we assume four local types of
the Bayer pattern as candidate CFA pattern as shown in Figure 4, like most of the demosaicing algorithms [12]. We
use the neighbor set Rp , which consists of four neighboring pixels in the G-channel and two or four pixels in R/B-

G-channel

: interpolated pixel

R- & B-channels

: observed pixel: neighborhoods

Figure 5. Definition of neighborhoods in interpolation. The circle
indicates the interpolated pixel and the gray block indicates theobserved pixel. The arrow shows the neighbor relationship.

G R

B
G
R

B G

Figure 6. Construction of non-interpolated images by downsampling using only the observed pixels.

channels for each p (see Figure 5). The threshold τ is empirically set to τ = 0.1 throughout the experiment.
To prepare the ground truth dataset, we captured fiveregistered RAW images for each of 18 different scenes(four scenes taken by EOS kiss digital original version,seven scenes by EOS kiss digital N, seven scenes by EOS
20D). The captured RAW images are converted to color images by six different demosaicing algorithms implemented
in dcraw [3] and RAW THERAPEE [8]. These are bilinear, variable number of gradients (VNG) [2], patterned
pixel grouping (PPG) [10], and adaptive homogeneitydirected (AHD) [7] algorithms implemented in dcraw,
and Horváth’s AHD (EAHD) and Heterogeneity-Projection
Hard-Decision (HPHD) [20] algorithms of RAW THERAPEE.
We also created color images without demosaicing interpolation by directly down-sampling RAW images as shownin Figure 6. In down-sampling, every 2×2 submosaic of the
RAW image produces only one color pixel. We include thisas one of the demosaicing algorithm. In the following, thisdata is referred to as non-interpolated, since demosaicing
algorithms generally use some interpolation. Therefore wehave seven demosaicing algorithms in total. After demosaicing, we applied JPEG compression to the demosaiced
images for assessing the robustness against JPEG compression. Figure 7 shows example of the images used for theFirst projection axis – Second projection axis
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EAHD

HPHG

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Bilinear

VNG

PPG

AHD

Second projection axis – Third projection axis
"%Figure 7. Example of images used for the experiment.

Third

&#+%&#*%experiment. We use the half of all the images as a training
set, and the other half as a testing set.
When creating histograms of the weight vectors w, we
set the size of the histogram bin to be one tenth of 1/n
where n is the number of observed neighbors for interpolation, because each element in w spans in the range of[0, 1/n]. Using FLD, the histograms of w were projected
into a compact 4-D subspace.
Demosaicing trace The first row in Table 1 shows theaccuracy of the detection of the demosaicing trace by the
proposed method. The accuracy is evaluated with different
JPEG compression qualities. True positive indicates the rateof the correct answer for demosaiced inputs. On the other
hand, True negative is the rate of the correct answer for noninterpolated inputs. When JPEG quality is greater than 90,
the accuracy of the proposed method is also high (over 95percent). Because JPEG compression tends to uniformly increase noise variance in the entire image, the periodicity ofnoise variances used for detection is relatively maintained.
For this reason, performance degradation of the proposed
method against JPEG compression is not significant.
CFA pattern The second row in Table 1 shows the estimation accuracy of CFA patterns against JPEG compression. Note that we did not use the image set of which
the demosaicing trace could not be detected. As describedabove, JPEG compression contaminates the pixel values regardless of observed and interpolated pixels, and therefore
it smooths out the difference between the observed and theinterpolated pixels. As a result, the information of CFA pattern is buried as the compression ratio becomes higher.
Once the CFA pattern is estimated, we can produce anirradiance image before demosaicing within an 8-bit accuracy. Figure 4 shows an example of reversing the demosaicing process. The Bayer pattern image (right) is computedfrom the input image (left) using our method.

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HPHG

Figure 8. Plots of vectors which represent histogram of interpolation weight, in the feature space. They are visualized by projection
onto the 2-D plane using Fisher’s linear discriminant analysis [4].
Different demosaicing algorithms form distinct clusters in the features.

Demosaicing algorithm The third column in Table 1
shows the estimation accuracy of the demosaicing algorithm against JPEG compression. Our method performs accurately when the compression ratio is low. However, it isaffected by the JPEG quality; accuracy decreases with lossof quality. This is due to the fact that the compression contaminates the pixel values randomly, which makes it difficult to robustly estimate the interpolation weights.
Figure 8 shows the 2-D plots of all the image sets in the
FLD space. Since the bilinear interpolation is very distinct
from other demosaicing algorithms, the FLD’s first projection separates the bilinear interpolation compared to others. The other distributions also show clear distinction fromeach other in the higher-order projection.

5.2. Single image case
For the single-image case, we used 32 images for theexperiment. The right-hand side of Table 1 shows the accuracy of all the steps in the single-image case. The robustness against JPEG compression is very similar to that of themultiple-image case.

Table 1. Quantitative evaluation of our method in the multiple-image case and single-image case. From top to bottom, results of 1) detectionof the demosaicing trace, 2) estimation of the CFA pattern, and 3) estimation of the demosaicing algorithm are shown. From left to right,
the accuracy is evaluated with various JPEG qualities (100 indicates no compression).

Demosaicing trace
CFA pattern
Demosaicing algorithm

JPEG quality
True positive [%]True negative [%]Accuracy [%]Accuracy [%]

100
100
100
95.8
89.8

Multiple images
98
95
90
100 99.1 96.3
100 100 100
92.6 73.1 63.0
78.5 70.5 53.8

Table 2. Accuracy of the estimation of demosaicing algorithms.
Comparison between the proposed method and Popescu andFarid’s method [16] are shown.

JPEG quality
98
95
90
88.6 79.6 63.1
73.4 69.3 64.1

5.3. Comparison

90
90.1
75.0
83.3
75.8

0.8

We compare our method with previous approaches. In
this comparison, we use the same dataset used for the
single-image case (Section 5.2).

0.6
0.4
Proposed method

0.2

Gallagher's method
0
0

0.2

0.4

0.6

0.8

1

True negative rate

Detection of demosaicing trace We compared the proposed method with Gallagher and Chen’s method [6]. Figure 9 shows ROC curves of the two methods in the cases
where JPEG quality is 70 and 90. This result shows the
proposed method is slightly superior to the method [6], eventhough our method only uses noise information.

JPEG quality: 90

1
0.8

True positive rate

Estimation of demosaicing algorithm We compared the
proposed method with Popescu and Farid’s method [16].
Popescu and Farid’s method assumes that the input to theiralgorithm is a demosaiced image but does not perform thedetection of demosaicing trace. Therefore, we used only
demosaiced images as input to these two methods. Because
their method estimates the algorithm without the knowledgeof the CFA pattern, we regarded the failure of estimating
CFA patterns as the failure of estimating the demosaicing
algorithm in our method, i.e., the recognition accuracy iscomputed as the product of the accuracy of the CFA patternrecognition and the accuracy of the demosaicing algorithm
recognition.
Table 2 shows the result with various degrees of JPEG
compression. Our method performs well when the JPEG
compression artifact is not significant. As the compression artifact becomes stronger, the accuracy of both methods goes down.

Single image
98
95
100 94.8
100 96.9
94.3 89.6
94.0 88.9

JPEG quality: 70

1

True positive rate

The proposed method
Popescu and Farid

100
94.6
91.5

100
100
100
98.4
96.2

0.6
0.4
Proposed method

0.2

Gallagher's method
0
0

0.2

0.4

0.6

0.8

1

True negative rate
Figure 9. Detection rate of demosaicing traces. ROC curves of the
proposed method and Gallagher and Chen’s method [6] are shownfor the cases of JPEG quality 70 and 90.

6. Discussions
In this paper, we showed the relationship between image noise variance and demosaicing. We have developed analgorithm for estimating CFA patterns and demosaicing algorithms. Extensive quantitative evaluation was performed
to verify the effectiveness of the proposed method. Nevertheless, our method has some limitations, and there areseveral avenues for future work.
One limitation of our method is that the accuracy goes
down when the image is processed after demosaicing, e.g.,
image compression and other image filtering. These image
processing operations significantly alter the noise distribution in unpredictable manner. It is very likely that explicitly
accounting for these factors can increase the applicabilityof the proposed method. We are investigating the possibility of deciphering such post-processing operations using
the observation of image noise. Another direction for future
work is to apply the proposed method as a pre-processing
stage for various vision tasks. Because interpolated pixels
are essentially synthetically produced pixels, identifying the
pixels that really receive irradiance (not by interpolation) is
important for physics-based vision methods.

Acknowledgement
This work is supported by Microsoft Institute forJapanese Academic Research Collaboration (MS-IJARC).

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