An intensity similarity measure in low-light
conditions
François Alter1 , Yasuyuki Matsushita2 , and Xiaoou Tang2
1

CMLA, École Normale Supérieure de Cachan
2
Microsoft Research Asia
francois.alter@mines.org, {yasumat,xitang}@microsoft.com
Abstract. In low-light conditions, it is known that Poisson noise andquantization noise become dominant sources of noise. While intensity difference is usually measured by Euclidean distance, it often breaks down
due to an unnegligible amount of uncertainty in observations caused bynoise. In this paper, we develop a new noise model based upon Poissonnoise and quantization noise. We then propose a new intensity similarity
function built upon the proposed noise model. The similarity measure
is derived by maximum likelihood estimation based on the nature ofPoisson noise and quantization process in digital imaging systems, andit deals with the uncertainty embedded in observations. The proposed
intensity similarity measure is useful in many computer vision applications which involve intensity diﬀerencing, e.g., block matching, opticalﬂow, and image alignment. We veriﬁed the correctness of the proposednoise model by comparisons with real-world noise data and conﬁrmed
superior robustness of the proposed similarity measure compared withthe standard Euclidean norm.

1

Introduction

Noise is inevitable in any imaging device. A digital imaging system consists ofan optical system followed by a photodetector and associated electrical ﬁlters.
The photodetector converts the incident optical intensity to a detector current,or photons to electrons. During the process, the true signals are contaminated bymany diﬀerent sources of noise. In fact, the true signal itself has ﬂuctuations intime due to the discrete nature of photons; the arrival of photons is not a steadystream and obeys the Poisson law [1]. It implies that no matter how accurately
a computer vision experiment is performed, temporal ﬂuctuation in intensity exists. The ﬂuctuation becomes signiﬁcant especially in low-light conditions where
the number of incoming photons is limited, i.e., photon-limited conditions. In
photon-limited conditions, quantization noise also becomes dominant due to thelack of intensity resolution in the limited dynamic range. Besides these noise
sources, reset noise, dark current noise and read-out noise also become signiﬁcant, and it is known that they can also be approximated by the Poisson noisemodel. Read-out noise is sometimes modelled by Gaussian noise; however, Gaussian noise with variance σ 2 and mean σ 2 is nearly identical to Poisson noise withThis work is done while the ﬁrst author was visiting Microsoft Research Asia.

Quantizationnoise

Photon noise

Real-world scene

Optical System

Photodetector

Observation

Fig. 1. Sources of noise in imaging process in low-light conditions. In low-light conditions, photon noise and quantization noise become dominant. In addition to them, there
exist reset noise, dark current noise and readout noise that also cannot be ignored. It
is known that they can be approximated by a Poisson distribution.

mean σ 2 if σ is suﬃciently large. Read-out noise satisﬁes this condition when
operating a camera at room temperature and with a high read-out frequency.
Important low-light vision applications include night vision, medical imaging, underwater imaging, microscopic imaging and astronomical imaging. Evenin daily situations, Poisson noise often becomes signiﬁcant in high-speed imaging. In these situations, uncertainty in observations increases signiﬁcantly due
to Poisson noise and quantization noise. Particular operations under these conditions, such as template matching [8, 6, 12] and edge detection [5, 4, 2, 10], havebeen widely studied for many applications in ﬁelds such as object recognition andmotion estimation. Image restoration is also one of the central problems since
photon-limited images are usually severely degraded. Statistical methods [15, 14,
11, 13, 7], such as maximum likelihood (ML) estimation, are found to be eﬀective
since they can account for the special properties of the Poisson distribution. Allof these techniques are found useful in low-light conditions; however, one fundamental question still remains open; what is the similarity between two intensity
observations with uncertainty?
Intensity distance is often measured in many computer vision algorithms, and
it is usually computed by Euclidean distance. Let k, l be two intensity measurements. The Euclidean distance dE (k, l) is given by d2E (k, l) = (k−l)2 . It is correct
for measuring the intensity distance between two signals when intensity noise
is negligible, or a non-biased distribution of noise is assumed. However, they do
not hold in photon-limited conditions where a signiﬁcant amount of biased noiseis added to the signal. Therefore, it is important to establish a new intensity
similarity function which deals with the uncertainty embedded in observations.
In this paper, we describe a new noise model for low-light conditions andpropose a new intensity similarity measure based upon the noise model. This
paper has two primary contributions.

– Poisson-quantization noise model: A realistic noise model in low-light
conditions is derived. The new noise model is built upon two inevitable noise
sources: Poisson noise and quantization noise. We call the combined model ofthese two noise sources the Poisson-quantization noise, or PQ-noise in short.
The proposed noise model is able to account for the uncertainty caused by
the nature of photon arrival and digitizing process. The correctness of the
proposed noise model is conﬁrmed by experiments with real-world data.
– A new intensity similarity measure: A new intensity similarity measure
is proposed based on our noise model, which deals with the uncertainty
caused by PQ-noise. The proposed similarity measure is useful in many
computer vision applications which involve intensity diﬀerencing, e.g., blockmatching, stereo, optical ﬂow and image alignment. The key advantage ofthe similarity function is that it can easily take place of existing intensity distance functions based on Euclidean distance. We compare the performanceof the new intensity similarity function with the Euclidean distance functionin order to verify the robustness of the proposed method against noise.
The outline of the paper is as follows: In Section 2, we brieﬂy review the
Poisson noise model and quantization noise model, and derive the PQ-noisemodel. Section 3 formalizes the intensity similarity function which measuresthe likelihood of two observations. The correctness and eﬀectiveness of the PQnoise model and the intensity similarity measure are veriﬁed with experiments
described in Section 4.

2

Poisson-quantization noise model

In this section, we ﬁrst brieﬂy review Poisson noise and quantization noise in
Sections 2.1 and 2.2. We then formalize the Poisson-quantization noise model in
Section 2.3.
2.1

Poisson noise model

Poisson noise is modelled by a Poisson distribution deﬁned as follows.
Definition 2.1: A Poisson distribution [9] with parameter λ is defined for allk ∈ IN by the probabilityp(k, λ) =

λk −λ
e ,
k!

(1)

with the mean E and variance V defined as follows.
E(λ) =∞


kp(k, λ) = λ,

k=0

2.2

V (λ) =∞
k 2 p(k, λ) − E 2 = λ.

(2)

k=0

Quantization noise model

Quantization noise is the uncertainty caused by rounding observation amplitudes
to discrete levels which occurs due to the ﬁnite amplitude resolution of any digitalsystem. In analog-to-digital conversion, the signal is assumed to lie within apredeﬁned range. Suppose the minimum number of electrons which is necessary
to raise one level of observed intensity is q, and eq is the quantization noise. The
count of electrons is proportional to the number of photons by the factor of thephoton-electron conversion eﬃciency (the quantum eﬃciency of the sensor). A
simple model of quantization noise can be described as
N N 
−
,
(3)
eq =
q
q
where N is the number of electrons which are generated by the measurement.
For a more detailed quantization error analysis in computer vision, readers arereferred to [3].
2.3

Poisson-quantization noise modelNow we formalize the PQ-noise model, which is the combination of Poisson noiseand quantization noise.
Definition 2.2: A Poisson distribution with parameter λ and quantization
Q = {q0 = 0, . . . , qk , . . . , qn+1 = ∞} is defined for all k ∈ {0, . . . , n} by theprobability p(k, λ, Q) as
qk+1 −1p(k, λ, Q) =

 λi
e−λ .
i!
i=q

(4)

k

The quantization parameter qk represents the minimum number of electrons
which produces intensity level k.
Suppose the simple case where the quantization interval is constant, i.e.,
qk = kq, and the observations are far from saturation. The quantization interval
is deﬁned as the range of input values assigned to the same output level. With
this simple model, we ﬁrst observe the diﬀerent behavior of PQ-noise from that of
the Poisson noise model. We later relax the assumption to ﬁt to a more realistic
model. In this condition, the mean E(λ, q) and variance V (λ, q) of the PQ-noisemodel are given by3 :
2πik
q

−1)
λ 1
1
1  eλ(eE(λ, q) = − ++2πik ,q
2 2q
q
1 − e− q
q−1

(5)

k=1

2πik

2πik

2πik
q−1 2πik
q−1
q −1)
q −1)
λ
1
1
2λ  e q eλ(e2  e− q eλ(eV (λ, q) = +
−
−
+
2πik
2πik
q
12 12q 2
q2
q2
1 − e− q(1 − e− q )2
k=1
k=1

2
2πik
2πik
q−1
q−1
q −1)
q −1)
1  eλ(e1  eλ(e .
+(6)2πik −
2πikqq2
1 − e− q
1 − e− q

k=1

3

k=1

The derivation of the mean and variance is detailed in Appendix A.

0.6

10
Constant Interval PQ-noise (q=100)
Poisson noise scaled by 1/100

9

7

0.4
Variance

6
Mean

Constant Interval PQ-noise (q=100)
Poisson noise scaled by 1/100

0.5

8

5
4

0.3
0.2

3
2

0.1

1
0

0
0

200

400

600

800

1000

0

5

10

15

λ

20

25

30

35

40

45

Mean

Fig. 2. The mean and variance properties of PQ-noise. Due to quantization noise,strong oscillations are observed. Left: Evolution of the mean with respect to λ withq = 100. Right: Evolution of the variance with respect to the mean with q = 100.

As seen in the above equations and in Fig. 2, oscillation with an exponen2π
tial decay e−λ(1−cos( q )) is observed in both mean and variance. The minimum
q21
intensity level of the linear range corresponds to λ ∝ 1−cos(
≈ 2π
2π
2.
)
q

Fig. 2 illustrates the diﬀerence between the ordinary Poisson noise model and
the PQ-noise model. As shown in the ﬁgures, PQ-noise has oscillations due tothe quantization noise component.
In practice, q1 does not equal to q due to the shift of the function causedby the oﬀset voltage. Here we consider the more realistic model of the PQ-noisemodel with the following conditions: q0 = 0 and qk = q1 + (k − 1)q. It is possible
to derive E(λ, q) and V (λ, q) for this condition by the same derivation used
for Eqs. (5) and (6). When E(λ, q) is reasonably high, i.e., E(λ, q)  2πq 2 , thefollowing relationship4 between E(λ, q) and V (λ, q) holds in the linear range.
V (E, q, q1 ) =q 2 + 12q1 − 6q − 7
E
+
.
q
12q 2

(7)

This model has two unknown parameters q and q1 . These unknown parameters
can be calibrated by ﬁtting the observed noise data to Eq. (7).

3

Derivation of Intensity Similarity Measure

Given two intensity observations k and l, what can we tell about the similarity
between them? Usually, the similarity is measured by Euclidean distance withan assumption that k and l are true signals, or the noise model is non-biased.
However, they do not hold in low-light conditions. We develop a new intensity
similarity measure which is based upon the probability that two intensity observations come from the same source intensity. In this section, we ﬁrst derive
the intensity similarity measure for the Poisson noise case in order to make thederivation clear. We then develop the intensity similarity measure for the PQnoise model. In fact, the Poisson noise model can be considered as the special
case of the general PQ-model with the quantization parameter q = q1 = 1.
4

The derivation of Eq. (7) is detailed in Appendix B.

3.1

Poisson noise caseIn the Poisson noise model, the case where two observed intensities arise fromthe same intensity distribution is equivalent to their sharing the same parameter
λ. If we assume that two intensity observations have the same parameter λ, theprobability of obtaining two observations k and l is
P (k, l, λ) = p(k, λ)p(l, λ) =λk+l −2λ
e
.
k!l!

(8)

This is obviously not suﬃcient to produce the actual probability because of theunknown parameter λ. However, the best case where λ maximizes the probability
gives the measure which maximizes the similarity between k and l. Indeed, thisapproach corresponds to the ML estimation of the observation of the pair (k, l).
Therefore, the optimal λ̂ can be obtained by putting the ﬁrst derivative ∂P∂λ = 0,and we obtain
λ̂ =

k+l
,
2

(9)

which maximizes the probability deﬁned in Eq. (8). In this way, the intensity
similarity function can be deﬁned ask + ld(k, l) = − ln P (k, l, λ̂) = (k + l) 1 − ln2

+ ln(k!) + ln(l!).

(10)

Note that this similarity measure does not agree with the exact deﬁnition of
distance because d(k, k) > 0 if k > 0, but it produces the similarity between two
observations.
Fig. 3 shows the intensity similarity function deﬁned in Eq. (10). The function has a similarity to the l2 norm when two observed intensity levels arehigh. In fact, when k and l are suﬃciently big, d(k, l) has a connection to thesquared l2 norm. This can be shown by rewriting Eq. (10) with the approximation ln(k!) ≈ k ln(k) − k:k+l2 k+l2f(k − l)22 (k − l)≈.
≈42(k + l)d(k, l) ≈ k ln(k) + l ln(l) − (k + l) ln= f (k) + f (l) − 2f

3.2

k+l
2

(11)

Poisson-quantization noise case

Using the derivation in the previous section, we formalize the intensity similarity
function for the PQ-noise model. We ﬁrst deﬁne the joint probability P ; theprobability of observing k and l having parameters λk and λl respectively with
the quantization Q by
P (k, l, λk , λl , Q) = p(k, λk , Q)p(l, λl , Q).

(12)

x 10

4

Intensity similarity measure d(k,l)

2

1.5

1

0.5

0
300
200
100
Observation l

0

0

50

100

150

200

250

300

Observation k

Fig. 3. The intensity similarity function deﬁned in Eq. (10).

We are seeking λ̂ (= λk = λl ) which maximizes the probability P . This isequivalent to maximizing the probability that two intensity observations sharethe same intensity source λ̂. We denote P (k, l, λ, Q) = P (k, l, λk , λl , Q) when
λk = λl . The optimal λ̂ always exists since the logged probability − ln P isconvex.5
1. If k = l, the maximum of P (k, k, λ, Q) is given by61

λ̂ = (qk ...(qk+1 − 1)) qk+1 −qk −1 .

(13)

2. If k = l, we can obtain the optimal λ̂ by minimizing the convex function− ln(P ). Here we describe a simple algorithm for ﬁnding the optimal λ̂ witha dichotomic search over the ﬁrst derivative of P .
Algorithm for ﬁnding the optimal λ̂
Input are k, l, qk , qk+1 , ql , ql+1 and niter .
Set λmin = 0 and λmax suﬃciently big, and n = 0,
While n < niter and P  = 0
domax
and n ← n + 1.
Set λ ← λmin +λ2∂ ln(P (k,l,λ,Q))
Compute P = −∂λ
If P  < 0 set λmin ← λ.
If P  > 0 set λmax ← λ.
done
Set λ̂ ← λ.
5
6

The convexity is proved in Appendix C.
It can be derived from the ﬁrst derivation of the function described in Appendix C.

The sign of P  can be determined by computing the sign of
λqk+1 −1
λqk −1−(qk+1 − 1)! (qk − 1)!

ql+1 −1

 λi
+
i!
i=q
l

λql+1 −1
λql −1−(ql+1 − 1)! (ql − 1)!

qk+1 −1

 λi
.
i!
i=q
k

To ﬁnd the optimal λ̂, other descent methods such as gradient descent,
Newton-Raphson, etc. can also be used alternatively.
In the above way, the optimal λ̂ is determined. The intensity similarity function is ﬁnally determined by plugging in the optimal λ̂ into the following function
using Eqs. (4) and (12):d(k, l, Q) = min{− ln(P (k, l, λ, Q))} = − ln(P (k, l, λ̂, Q))λ= − ln e−2λ̂qk+1 −1

 λ̂i
i!
i=q
k

4

(14)

ql+1 −1

 λ̂j
j!
j=q

.

l

Experiments

In order to conﬁrm our theoretical results, we performed experiments with realworld noise datasets. Our interests are 1) verifying the correctness of the proposed PQ-noise model and 2) conﬁrming the superiority of the proposed intensity
similarity measure over the standard l2 norm.
In order to obtain datasets, we mounted a video camera at a ﬁxed positionand captured an image sequence of a static scene in a low-light condition. Therefore, the only ﬂuctuation in an image sequence is caused by noise. The images
captured under the severe low-light conditions are almost totally black to human eyes, but they still contain intensity information. Fig. 4 shows one of such
scenes used for the experiment. We captured raw image sequences by a Point
Grey DragonﬂyTM camera, and the intensities observed in the green channel areused for the entire experiment. For the illumination source, a DC light source isused to avoid intensity oscillations. We also used a small aperture and a short
shutter speed to produce a low-light environment.
PQ-noise model To verify the correctness of the PQ-noise model, we compared the mean-variance distribution of the real-world data with our analytic
model described in Sec. 3.2. For the experiment, we have captured 1000 images of a static scene in a low-light condition. The unknowns q and q1 are both
estimated by least squares ﬁtting to the linear range of the PQ-noise model described in Eq. (7). Fig. 5 shows the plot of observations and the analytic modelwith estimated q and q1 . As shown in the ﬁgure, our analytic model well ﬁts theactual observations, especially in the low intensity levels where the oscillation isobserved clearly. The root mean-square error of observations from the theoretic
curve is 0.0090 in Fig. 5.
Intensity similarity measure To evaluate the robustness of the proposed
similarity measure against noise, block matching is applied to the image sequences; if the block stays at the original position, the measure is not aﬀected

Fig. 4. One of the scenes used for the experiment. Left: The original input image in a
low-light condition. Right: The left image is linearly scaled by 60.
Observed noise
Theoretic model
0.5

Variance

0.4

0.3

0.2

0.1

0

0

5

10

15

20

25

Mean

Fig. 5. Evolution of the variance with the mean. The dotted line is the theoreticalresult with the calibrated parameters, q = 67 and q1 = 168. The dots are the measured
noise which is obtained from 1000 images of a static scene under a low-light condition.

by noise. The same test is performed using the l1 and l2 norms over the samedatasets, and we compared the outcomes of these norms with that of the proposed intensity similarity measure. We denote the l1 and l2 norm measure andour intensity similarity measure described in Eq. (14) as dl1 , dl2 and dP Q respectively. The parameters for dP Q , i.e., q and q1 , are calibrated beforehand by
curve ﬁtting as done in the previous experiment. The parameters q and q1 are
estimated as q = 67 and q1 = 168 in our experiment.
In the left image of Fig. 6, the performance growth obtained by our similarity
measure in comparison with the l1 and l2 norm is shown. Our major interestis the comparison with the l2 norm; however, a comparison with the l1 norm isalso given since the l1 norm is often used in practice because of its simplicity. In
the left graph, 0% indicates the same performance as the l1 or l2 norm. The perN (d)−N (d )formance growth from the l2 norm, for example, is computed by c PNQc (d 2 )c l2
lwhere Nc (dP Q ) and Nc (dl2 ) are the number of correct matches with dP Q and dl2
respectively. The same computation is applied for the l1 norm case to obtain theperformance improvement analysis as well. We can observe that our similarity

200
Logged performance growth with our method (ln(%))

Performance growth with our method (%)

6

Performance growth from L2
Performance growth from L1

180
160
140
120
100
80
60
40
20
0

0

5

10
15
Size of the block m (m x m)

20

25100.0%
4

Exposure time t
Exposure time 2t
Exposure time 3t
Exposure time 4t
Exposure time 40t

10.0%

2

0

1.0%

-2

-4

-6

0.01%

0

5

10
15
Size of the block m (m x m)

20

25

Fig. 6. Performance evaluation the proposed intensity similarity measure. Left: Performance growth of block matching using our intensity similarity measure in comparison
with the l1 and l2 norms. Right: Variation of the performance growth in comparison with l2 norm under diﬀerent low-light conditions. The logarithm is taken for avisualization purpose on the right graph.

measure signiﬁcantly exceeds the l1 and l2 measures, especially when the block
size is small. In fact, when the number of pixels in a block is large, the averagingof error reduces the biased property of the Poisson-quantization distribution.
Therefore, the l2 norm which is adapted to Gaussian noise becomes eﬀective as
well.
In order to analyze the performance variation in diﬀerent low-light conditions,
we performed the same experiment over image sequences with diﬀerent exposure
settings. The dataset is obtained by capturing the same scene with changing
exposure time, i.e., t, 2t, 3t, 4t and 40t where t = 15ms. For these datasets, theobserved intensity values have a range of [0, 7], [0, 12], [0, 18], [0, 24] and [0, 235]
respectively. The last setting 40t is not considered a low-light condition, but wetested with this setting as well in order to see the behavior of our similarity
measure in such a condition. The right graph of Fig. 6 shows the performance
growth in comparison with the l2 norm using the diﬀerent exposure settings.
As shown in the graph, the proposed similarity measure is eﬀective especiallyin severe low-light conditions. It can be seen that it also works for the ordinary
condition (the 40t setting), although the performance improvement from the l2
norm is almost zero.

5

Conclusion

In this work, we have proposed a new intensity similarity measure which isuseful for low-light conditions where Poisson noise and quantization noise become signiﬁcant. The intensity similarity measure is derived from the Poissonquantization noise model which we develop as the combination of the Poissonnoise model and the quantization noise model.
The correctness of the proposed PQ-noise model is veriﬁed by comparison
with real-world noise data. The proposed intensity similarity measure is robust

against Poisson-quantization noise, and is therefore eﬀective in low-light conditions. The robustness is compared with the l2 norm using block matching,
and we conﬁrmed that the proposed method largely exceeds the performanceof the l2 norm especially when the block size is small. Our intensity similarity
measure is capable of achieving more accurate matching, especially in situations
where large blocks cannot be used. The proposed noise model and intensity similarity measure are useful for many computer vision applications which involve
intensity/image matching in photon-limited conditions.

References
1. N. Campbell. Discontinuities in light emission. Proc. Cambridge Phil. Soc., 15:310–
328, 1909.
2. M. Das and J. Anand. Robust edge detection in noisy images using an adaptive stochastic gradient technique. In Proc. of International Conference on Image
Processing, volume 2, pages 23–26, 1995.
3. B. Kamgar-Parsi and B. Kamgar-Parsi. Evaluation of quantization error in computer vision. IEEE Trans. Pattern Anal. Mach. Intell., 11(9):929–940, 1989.
4. A. Kundu. Robust edge detection. In Proc. of Computer Vision and Pattern
Recognition, pages 11–18, 1989.
5. L. Liu, B.G. Schunck, and C.R. Meyer. Multi-dimensional robust edge detection.
In Proc. of Computer Vision and Pattern Recognition, pages 698–699, 1991.
6. S.B. Lowen and M.C Teich. Power-law shot noise. IEEE Trans. Information
Theory, 36:1302–1318, 1990.
7. H. Lu, Y. Kim, and J.M.M. Anderson. Improved poisson intensity estimation:
Denoising application using poisson data.IEEE Trans. on Image Proc., 13(8), 2004.
8. G.M. Morris. Scene matching using photon limited images. Journal of Opt. Soc.
Am. A, pages 482–488, 1984.
9. A. Papoulis. Probability, Random Variables, and Stochastic Processes, pages 554–
576. McGraw-Hill, New York, 2nd edition, 1984.
10. P. Qiu and S.M. Bhandarkar. An edge detection technique using local smoothing
and statistical hypothesis testing. Pattern Recognition Lett., 17:849–872, 1996.
11. W.H. Richardson. Bayesian-based iterative method of image restoration. Journalof Opt. Soc. of Am., 62:55–59, 1972.
12. R.E. Sequira, J. Gubner, and B.E.A. Saleh. Image detection under low-level illumination. IEEE Trans. on Image Processing, 2(1):18–26, 1993.
13. K.E. Timmermann and R.D. Nowak. Multiscale modeling and estimation of poisson processes with application to photon-limited imaging. IEEE Trans. on Inform.
Theory, 45:846–862, 1999.
14. G.M.P. van Kempen, H.T.M. van der Voort, J.G.J. Bauman, and K.C. Strasters.
Comparing maximum likelihood estimation and constrained tikhonov-miller
restoration. IEEE Engineering in Medicine and Biology Magazine, 15:76–83, 1996.
15. Y. Vardi, L.A. Shepp, and L. Kaufman. A statistical model for positron emission
tomography. Journal of Amer. Statist. Assoc., 80:8–37, 1985.

Appendix A
We derive the mean and variance for a PQ-distribution described in Eqs. (5)and (6). We begin with the following theorem about quantization.

Theorem 1. Let X ∈ Z be a discrete random variable which has a characteristic
function φ(t) = E(eitX ). Its quantized version Xq can be defined by Xq =  Xq .
The characteristic function φq of Xq is given by
φq (t) =q−1t+2πk
1 − e−it  φ( q )t+2πk .
q
1 − e−i qk=0

Proof. Let pn be pn = P (X = n). The characteristic function φq of Xq is∞φq (t) =∞npn ei q t =

n=−∞

n

n

n

pn ei q t ei( q t− q t) .

n=−∞

Z
n

→C
is q-periodic, therefore it can be writtennn→ ei( q t− q t)
q−1

2πikn
by a trigonometric polynomial
ak e q . Using a discrete Fourier transfor-

The function f :k=0mation, we obtain:

q−1
q−1
q−1
1
1  −i qj t − 2πijk
1  −j i(t+2πk)− 2πijkqqq
ak =f (j)e
=
e
e
=
e
q j=0
q j=0
q j=0

=

1 − e−i(t+2πk)q(1 − e−i

t+2πk
q

=

)

1 − e−itq(1 − e−i

t+2πk
q

)

.

Therefore,φq (t) =

∞

n=−∞

=

q−1


φ

k=0

n

pn ei q t

q−1


ak e

2πikn
q

=

pn eint+2πkqk=0 n=−∞

k=0

t + 2πk
q

q−1 
∞


1−e

1 − e−itq(1 − e−i

t+2πk
q

)

−it

q(1 − e−i

t+2πk
q

)

.

QED
Corollary 1. The mean Eq of Xq and variance Vq of Xq can be written by themean E, the variance V , and the characteristic function φ of X:
Eq =

q−1
2πk
1
1
1  φ( q )
E
− +
+
2πik
q
2 2q
q
1 − e− q
k=1

2πik

2πik

q−1
q−1 −
2πk
2πk
1
1
V
2E  e q φ( q )2  e q φ( q )
+
−
Vq =
−
+
2πik
2πik
− q
− q
q
12 12q 2
q2
q2)2
k=1 1 − ek=1 (1 − e


2
q−1
q−1
2πk
2πk
1  φ( q )1  φ( q )
+
−
.
2πik
2πik
q
q2
1 − e− q
1 − e− qk=1

k=1

These formulas are given by the computation of the derivatives of φq , using thefact that Eq = −iφq (0) and Vq = −φq (0) − Eq2 .
Eqs. (5) and (6) are the result of the previous formulas with a Poisson distribuittion which has the characteristic function φ(t) = eλ(e −1) .
Remark 1. We can extend the previous result to the continuous random variablecase. Let X be a random variable in R, and φX (t) its characteristic function.
Let X be the quantized version of X. Using the Fourier series of the 1-periodic∞e−i(t+2πn) − 1 2πinxi, wefunction deﬁned over R, i.e., f (x) = eit(x−x) =et + 2πnn=−∞can show that the characteristic function φX of X isφX (t) =∞e−i(t+2πn) − 1
φX (t + 2πn),t + 2πnn=−∞iwhere the summation has to be done by grouping terms of n and −n together
if it is not convergent. The mean EX can be simply derived by
EX = EX −1  φX (2πn)
+
.
2
2πn
n=0

This applies to any kind of distribution, for example, the mean of a Gaussian
distribution can be derived as follows. Let X be a random variable following theσ2 t2Gaussian law N (µ, σ 2 ). The characteristic function becomes φX (t) = eiµt− 2 .
Therefore,
2
2 2
2
2 2
∞
1  cos(2πnµ)e−σ 2π n
1  eiµ2πn−σ 2π n=µ− +EX = µ − +
2
2πn
2 n=1
πn

n=0

As the series on the right is decreasing exponentially with n, this gives a practical
way to compute EX .

Appendix B
We derive the PQ-noise model in the linear range described in Eq. (7). Withan assumption that observed intensities are far enough from saturation, theapproximation n = ∞ can be used. We use the previous theorem described inAppendix A and divide the problem into two cases.
1. If q1 ≤ q, let X(λ) be a discrete random variable on the shifted Poisson
distribution deﬁned by∀k ≥ q1 − q, P (X(λ) = k) =λk+q1 −qe−λ .
(k + q1 − q)!

In this case,  X(λ)
has the wanted PQ-distribution. It is straightforward toq
see that EX(λ) = λ + q1 − q, and VX(λ) = λ.

2. If q1 > q, let X(λ) be a discrete random variable which satisﬁes8
q1 −q jX λ −λ>>>e ,<P (X(λ) = 0) =
j!
j=0
>
>
λk+q1 −q>
:P (X(λ) = k) =e−λ .
(k + q1 − q)!

∀k ≥ 1

Then again,  X(λ)
has the wanted PQ-distribution. When λ is big, theqapproximation P (X = 0) ≈ 0 holds. Therefore, EX(λ) ≈ λ + q1 − q, andVX(λ) ≈ λ are deduced.
Using Corollary 1 and the remark, which shows lim φX(λ) (2kπ) = lim φX(λ) (2kπ) =
λ→∞

λ→∞

1
1
− 12 + 2q
and V (λ, q, q1 ) =0, we are able to deduce that E(λ, q, q1 ) = λ+qq
λ1
1q2 + 12 − 12q2 in the linear range. Therefore,

V (λ, q, q1 ) =qE(λ, q, q1 ) − q1 + q2 − 121
q 2 + 12q1 − 6q − 7
1
E(λ, q, q1 )
−
+
+
=
.
2
2
q
12
12q
q
12q 2

Appendix C
We show that the probability function − ln P in Eq. (12) is convex.
n

λi −λ
e
Proposition: f : λ → −ln
is convex.
i!
i=m
Proof.
−f  (λ) =

n


λi−1
λi −λ
λn
λm−1
−
e
−
(i − 1)!
i!
n!
(m − 1)!
=
 λi −λ
 λi
i! e
i!

i=m
n


f  (λ) = i=m

λn
λn−1
λm−2
λi
λm−1
λi−1
−
−
−
(n − 1)! (m − 2)! i!
n!
(m − 1)! (i − 1)!
 λi 2( i! )n
λn+i−1

= i=m

n!i!

Therefore f is convex.
QED(n − i) +λm+i−2(i − m + 1)
(m − 1)!i!

 i( λi! )2

> 0.