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Blur invariants: A novel representation in the wavelet domain
Pattern Recognition 43 (2010) 3950–3957
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Pattern Recognition journal homepage: www.elsevier.com/locate/pr
Blur invariants: A novel representation in the wavelet domain Iman Makaremi , Majid Ahmadi Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON, Canada N9B 3P4
a r t i c l e in f o
a b s t r a c t
Article history: Received 7 April 2010 Received in revised form 21 June 2010 Accepted 15 July 2010
Blur invariants in the wavelet domain are proposed for the first time in this paper. Wavelet domain blur invariants take advantage of several benefits that this domain provides, i.e. different alternatives for wavelet function and analysis in different scales. It is not required to model the blur system in order to extract the invariants. It will be shown how the space domain blur invariants are a special case of the proposed invariants. It will also be explained how the proposed invariants would not have the null space that their special case in the spatial domain have which limits their discriminative power. The performance of these invariants will be demonstrated through experiments, and compared to its counterpart which is defined in the spatial domain. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Blur invariant moment Direct analysis Feature extraction Wavelet transform
1. Introduction Signal acquisition is the first step in all signal processing tasks, and is always accompanied with different sources of degradation. The effect of some of the degradations is considerably high, which could vastly affect expected outcomes. Blur, as one of the degradations that is classified as radiometric degradation and can significantly reduce the accuracy, has been studied for a few decades. The general model that is used for the observed signal is gðxÞ ¼ Hf ðxÞ þ nðxÞ
ð1Þ
where g, f, and n are the observed signal, the actual signal, and the additive noise, respectively, and H is the blur operator. If it is assumed that the blur operator is linear space-invariant, the above observation model can be simplified to gðxÞ ¼ f ðxÞ hðxÞ þ nðxÞ,
ð2Þ
where hðxÞ is point spread function (PSF) of the system. There are two proposed approaches for resolving this problem: blind restoration and direct analysis. The purpose in blind restoration methods is to identify the blur system model and extract the actual signal. Therefore, a prior information is required in order to set constraints and choose proper parameters. There are numerous proposed methods in literature for this type of approach [1,2], and their main application is in signal restoration. The drawback is, however, that they are computationally expensive, and that the Corresponding author.
E-mail addresses: [email protected], [email protected] (I. Makaremi), [email protected] (M. Ahmadi). 0031-3203/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.patcog.2010.07.020
problem is usually ill-posed. Also, in many applications it is not necessary to restore the actual signal. The second set of approaches try to extract properties in the degraded signal that are invariant to blur. Such methods do not go through the process of identifying the blur system. Flusser et al. [3] proposed the first direct analysis method in the spatial domain that is based on ordinary and central moments. The only assumption was that the blur system is centrally symmetric and energy-preserving. Afterward, they modified the method and made it invariant to geometric distortions as well [4,5]. Wee and Paramesran [6] and Zhang et al. [7] employed Legendre moments in order to define their invariants in the spatial domain. Metari and Deschˆenes [8] exploited the Mellin transform in order to extract blur invariant descriptors. They also showed that the proposed descriptors are also invariant to some of the geometric distortions. The invariants proposed by Ji and Zhu [9] are based on Zernike moments. Along with the blur invariant features defined in the spatial domain, there are some other methods that are developed in the Fourier domain. Flusser and Suk [10] proposed their Fourier based invariants based on the tangent of the phase of signals, and showed that they are invariant to blur. They also made it invariant to some other geometric distortions and generalized them for N-dimensional signals [11]. Ojansivu and Heikkila¨ [12], however, showed that Flusser’s invariants are sensitive to noise because of their use of tangent operator. They proposed another representation of invariants in the Fourier domain. Afterward, they made them invariant to affine transform as well [13]. In this paper, we propose blur invariant moments in the wavelet domain for the first time. The different alternatives that exist for wavelet functions and the benefit of analyzing signals at different scales in the wavelet domain are the vivid advantages
I. Makaremi, M. Ahmadi / Pattern Recognition 43 (2010) 3950–3957
that the proposed invariants have over those defined in other domains. The idea will be developed for 1D signals, and the effect of blur and noise will be studied and compared to a similar technique. There will be a discussion on how to expect better results with this new proposed method. The proposed invariants could be used in analyzing EEG, ECG, and speech signals and those acquired in single point ultrasound measurements. The rest of the paper is organized as following. Section 2 provides the basic and complementary definitions that are required in this paper. In Section 3, ordinary and central moments are defined in the wavelet domain, and the relation between the moments in the wavelet domain with those in the spatial domain is extracted. Also, it is shown how the moments of a degraded signal are related to the moments of its actual source both in the wavelet domain. Section 4 presents the blur invariant moments in the wavelet domain and how Flusser’s blur invariant moments [10] are a special case of the proposed invariants. The proofs are also provided. Section 5 is devoted to the experimental results through which the performance of the proposed invariants is evaluated, compared to Flusser’s invariants, and their robustness and discriminative power are discussed. Also, there will be a discussion on the effect of wavelet functions on the quality of the invariants. Finally, section 6 concludes the paper.
2. Basic definitions and notations
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Definition 7. The wavelet transform of signal f(x) at shift u and scale s with wavelet function cðxÞ is [15] Z þ1 xu c 1 pffiffi f ðxÞc dx: ð8Þ f^ ðs,uÞ ¼ s s 1
where c ðÞ is the complex conjugate of cðÞ. 3. Moments in the wavelet domain First, it will be shown that how (2) would be presented in the wavelet domain. Also for the sake of simplicity, noise is not shown in the equations. However, its effect on the invariants will be shown in Section 5. The wavelet transform of signal g(x), the blurred signal, at shift u and scale s with wavelet function cðxÞ is Z þ1 xu 1 c pffiffi gðxÞc dx: ð9Þ g^ ðs,uÞ ¼ s s 1 Replacing g(x) by its equivalent in (2) gives Z þ1 Z þ1 xu 1 c pffiffi hðyÞf ðxyÞc dy dx: g^ ðs,uÞ ¼ s s 1 1 Substituting z for x y, the above equation changes to Z þ1 Z þ1 1 zðuyÞ c pffiffi f ðzÞc dz dy hðyÞ g^ ðs,uÞ ¼ s s 1 1 Z þ1 c ¼ hðyÞf^ ðs,uyÞ dy
ð10Þ
ð11Þ
1
In this section, some basic terms are defined and explained. Definition 1. The pth order ordinary geometric moment of signal f(x) in the spatial domain is defined by [14] Z þ1 mfp ¼ xp f ðxÞ dx: ð3Þ
which shows that the wavelet transform of g(x) is the convolution of h(x) by the wavelet transform of f(x): c
c g^ ðs,uÞ ¼ f^ ðs,uÞ hðuÞ:
ð12Þ
1
3.1. Relation between moments in the spatial and wavelet domains Definition 2. The centroid of signal f(x) is [14] cf ¼
mf1 mf0
ð4Þ
Definition 3. The pth order central moment of signal f(x) in the spatial domain is defined by [14] Z þ1 mfp ¼ ðxcf Þp f ðxÞ dx: ð5Þ 1
For (4) to hold and Definition 3 to be valid, f(x) is required to have a nonzero first moment. In the case that the moments of f(x) are zero up to a certain order, the following definitions are proposed. Definition 4. If the moments of signal f(x) are zero up to order M 1, its centroid is defined as
Bf ¼
mfM þ 1 ðM þ 1ÞmfM
ð6Þ
Definition 5. If the moments of signal f(x) are zero up to order M 1, its pth order central moment (p ZM) in the spatial domain is defined by Z þ1 mfp ¼ ðxBf Þp f ðxÞ dx: ð7Þ 1
Definitions 2 and 3 are special cases of Definitions 4 and 5, respectively, in which M ¼0. Definition 6. If the moments of wavelet function cðxÞ is zero up to order Mc 1, it is said that cðxÞ has Mc vanishing moments [15].
c Before extracting the relation between the moments of g^ ðu,sÞ c and those of f^ ðu,sÞ and h(u), it is shown how the moments of f(x) in the wavelet domain are related to those in the spatial domain. c Rewriting Definition 3 for f^ ðu,sÞ, the wavelet transform of f(x) with wavelet function cðÞ at shift u and scale s, we have the following: Z þ1 Z þ1 Z þ1 c 1 ^c xu dx du up f^ ðs,uÞ du ¼ pffiffi up f ðxÞc mpf ðsÞ ¼ s s 1 1 1
¼
pffiffi s
Z
þ1
1
Z
þ1
ðxsyÞp f ðxÞc ðyÞ dx dy:
ð13Þ
1
In (13), y is substituted for (ðxuÞ=sÞ. Considering the fact that P ða þbÞp ¼ pk ¼ 0 pk ak bpk , the relation between the ordinary moments of signal f(x) in the wavelet domain and the spatial domain can be extracted: p X p k þ 1=2 ^c c mpf ðsÞ ¼ ð1Þk mfpk mk : ð14Þ s k k¼0 Assuming that the wavelet function has Mc vanishing moments, (14) can be modified to p X p k þ 1=2 ^c c mpf ðsÞ ¼ ð1Þk mfpk mk : ð15Þ s k k¼M c
It can be concluded from (15) that the transform of a signal with a wavelet vanishing moments are zero up to Definition 7 should be used in order
moments of the wavelet function which has Mc order Mc 1. Therefore, to calculate the central
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I. Makaremi, M. Ahmadi / Pattern Recognition 43 (2010) 3950–3957
are of almost the same order of magnitude. Comparison between the moments calculated directly from the signal’s wavelet transform and those derived from (17) indicates that they match accurately in implementation as well.
moments in the wavelet domain: Z þ1 c ^c ^c mpf ðsÞ ¼ ðuBf Þp f^ ðs,uÞ du 1
Z þ1 Z þ1 1 ^c xu dx du ðuBf Þp f ðxÞc ¼ pffiffi s s 1 1 Z Z pffiffi þ 1 þ 1 ¼ s ððxcf ÞsðyBc ÞÞp f ðxÞc ðyÞ dx dy 1
3.2. Relation between the moments of original and degraded signals ð16Þ
c The relation between the pth order central moment of g^ ðs,uÞ
1 ^c
where y is substituted for ððxuÞ=sÞ, and Bf is replaced by its equivalent cf sBc . The later can be trivially shown. From (16), the relation between the central moments of f(x) in the wavelet domain and the spatial domain can be extracted: p X p k þ 1=2 ^c c mpf ðsÞ ¼ ð1Þk mfpk mk : ð17Þ s k k¼M
c and the central moments of f^ ðs,uÞ and h(x) can be extracted by c rewriting (7) for g^ ðs,uÞ, using (12), and considering the fact that c
^c
Bg^ ¼ Bf þ ch . c
mgp^ ðsÞ ¼
Z
c
Z
þ1
þ1
¼
The credibility of Eq. (17) is verified in an example as follows. Example 1. In this example, it is shown how (17) holds. The signal that is used in this example is illustrated in Fig. 1. It is an EEG sample from the database generated by Andrzejak et al. [16]. The wavelet transform of this signal is calculated at scale 1 and 4 (s¼1,4) with Coilflet of order 1 as the wavelet function [17], which has two vanishing moments. For comparison, the first ten c central moments of f^ ðu,sÞ are calculated from both the signal’s wavelet transform and the derived equation in (17). As it is illustrated in Table 1, the first two moments of the wavelet c function are extremely small. Feasibly, f^ ðu,sÞ’s first two moments
1
¼
c
ðuBg^ Þp g^ ðs,uÞ du
1
Z
c
þ1
c
c
ðuBg^ Þp f^ ðs,uÞhðvuÞ dv du
1
p X p
k
k¼0
^c
mkf ðsÞmhpk :
ð18Þ
If the employed wavelet function has Mc vanishing moments, by q q þ Mc l M ¼ l þ M c , the above
defining q þMc ¼ p, m q ðsÞ ¼ mp ðsÞ, and
c
equation can be simplified into (19). q X c q ^c m gq^ ðsÞ ¼ m fl ðsÞ mhql : l Mc l¼0
ð19Þ c
It shows that the central moments of g^ ðs,uÞ of a certain order are c
related to the central moments of f^ ðs,uÞ and h(x) of the same and lower orders.
200 150
4. Blur invariant features in the wavelet domain
100
In order to extract blur invariant features, it is necessary to look for combinations of moments such that those of h(x) would be canceled out. It is assumed that h(x), the PSF, is symmetric and energy-preserving. When h(x) is symmetric, i.e. h(x)¼h( x), its odd order moments are zero. And because of the energy-preserving R þ1 property, the integral 1 hðxÞ dx, which is mh0 , equals one.
50 0 −50 −100 −150
0
50
100
150
200
250
300
350
400
450
500
Fig. 1. An EEG signal. The results in Table 1 are based on this signal.
Theorem 1. If q is odd, then Cq ðsÞ, which is defined as follows, is invariant to blur in the wavelet domain with wavelet function cðÞ. q ðq1Þ=2 q2l X 1 M 2l c Cq2l ðsÞm 2l ðsÞ Cq ðsÞ ¼ m q ðsÞ ð20Þ m 0 ðsÞ l ¼ 1 0 M c
Table 1 The first ten central moments of Coiflet of order 1 and the signal’s wavelet transform at scale 1 (columns 3 and 4 from left) and 4 (columns 5 and 6 from left) are shown. The third and fifth columns (DT) include the central moments calculated directly from the signal’s wavelet transform, and the fourth and sixth columns (DV) are induced from (17). p
^c
^c
^c
^c
mcp
mpf ð1Þ DT
mpf ð1Þ DV
mpf ð4Þ DT
mpf ð4Þ DV
1.14e 16 6.94e 17 1.16e + 00 1.17e 15 1.70e + 00 2.83e + 00 3.50e + 00 3.61e + 01 9.49e + 01 3.85e + 02
1.59e 13 1.05e 11 2.46e + 03 5.45e 06 5.94e + 08 1.38e + 11 9.21e + 13 2.58e + 16 1.24e + 19 3.98e + 21
0.00e + 00 0.00e + 00 2.46e +03 3.96e 10 5.94e +08 1.38e +11 9.21e +13 2.58e +16 1.24e +19 3.98e +21
3.91e 13 9.73e 11 4.46e+ 05 6.73e 05 1.08e+ 11 2.49e+ 13 1.67e+ 16 4.68e+ 18 2.25e+ 21 7.22e+ 23
0.00e + 00 0.00e + 00 4.46e + 05 7.48e 08 1.08e + 11 2.49e + 13 1.67e + 16 4.68e + 18 2.25e + 21 7.22e + 23
^
^
c
Proof. When q¼1, C1g ðsÞ ¼ C1f ðsÞ ¼ 0. ^c ^c For q¼3, it can be trivially shown that C3g ðsÞ ¼ C3f ðsÞ. ^c
c
^ g C3g ðsÞ ¼ m 3 ðsÞ
¼
3 X 3 l¼0
0 1 2 3 4 5 6 7 8 9
c
l
^c
m lf ðsÞmh3l Mc
^c ^c f3 ðsÞ ¼ C3f ðsÞ:
¼m
ð21Þ
If (20) is valid for 1,3, . . . ,q2, it can be proved that it is valid for q as well: q ðq1Þ=2 q2l M X c c c c 1 g^ g^ g^ 2l c Cq2l ðsÞm 2l ðsÞ Cqg^ ðsÞ ¼ m q ðsÞ c g^ 0 Mc m 0 ðsÞ l ¼ 1
I. Makaremi, M. Ahmadi / Pattern Recognition 43 (2010) 3950–3957
c q X q þMc f^ m qk ðsÞmhk k k¼0 q c ðq1Þ=2 2l c q2l X X 2l þ Mc 1 M f^ f^ 2l c Cq2l c ðsÞ m 2lk ðsÞmhk k f^ 0 k ¼ 0 l ¼ 1 M m 0 ðsÞ c c q c X qþ Mc f^ f^ ¼ Cq ðsÞ þ m qk ðsÞmhk k k¼1 q ðq1Þ=2 X q2l Mc f^ c X2l 2l þ Mc f^ c 1 2l c Cq2l ðsÞ m 2lk ðsÞmhk k¼1 ^ k 0 Mc m f0 ðsÞ l ¼ 1 c q X qþ Mc ^c f^ ¼ Cqf ðsÞ þ m qk ðsÞmhk k k¼1 q q2l ðq1Þ=2 2l c c X X 2l þMc 1 M f^ f^ 2l c Cq2l c ðsÞm 2lk ðsÞmhk ^ k 0 Mc m f0 ðsÞ l ¼ 1 k ¼ 1 q X qþ Mc ^f c ^f c ¼ Cq ðsÞ þ m qk ðsÞmhk k k¼1 qk q2l ðq1Þ=2 q1 c c X X q þMc 1 M f^ f^ 2lk c Cq2l c ðsÞm 2lk ðsÞmhk ^f k 0 Mc m 0 ðsÞ k ¼ 1 l ¼ ½ðk þ 1Þ=2 ! q1 X qþ Mc q þ Mc ^c ^c ¼ Cqf ðsÞ þ m 0f ðsÞmhq mhk q k k¼1 0 1 qk ðq1Þ=2 c c c q2l M X 1 B f^ C f^ c f^ 2lk Cq2l ðsÞm 2lk ðsÞA @m qk ðsÞ c f^ m 0 ðsÞ l ¼ ½ðk þ 1Þ=2 0 Mc ! q1 c X q þ Mc q þMc ^c ^ m 0f ðsÞmhq þ mhk Jq,k ðsÞ: ¼ Cqf ðsÞ þ q k k¼1 ¼
ð22Þ When k is odd, mhk is zero. Hence, the second term in (22) becomes zero. The only term in (22) that is dependent on the moments of h is Jq,k ðsÞ. In order to prove that the proposed invariant is independent of PSF, it is therefore necessary to show that when k is even, Jq,k ðsÞ ¼ 0. qk ðq1Þ=2 c c c q2l X 1 M f^ f^ f^ 2lk c Cq2l ðsÞm 2lk ðsÞ Jq,k ðsÞ ¼ m qk ðsÞ c ^f m 0 ðsÞ l ¼ k=2 0 Mc qk ðqk1Þ=2 c c qk2l M X ^f c 1 c f^ f^ 2l ¼ m qk ðsÞ c Cqk2l ðsÞm 2l ðsÞ f^ 0 Mc m 0 ðsÞ l ¼ 0 qk ðqk1Þ=2 c c qk2l M X ^f c 1 f^ c f^ 2l Cqk2l ðsÞm 2l ðsÞ ¼ m qk ðsÞ c f^ 0 Mc m 0 ðsÞ l ¼ 1
1 ^
c
m 0f ðsÞ
qk qk M c
0
not a wavelet function anymore. Therefore, p ¼ q þMc ¼ q and c
mp ¼ m q . By setting s ¼1 at all time, it is trivial to show that f^ ðx,1Þ is equivalent to f(x), since the signal is convolved by the Dirac delta function. From here, (20) is rewritten with the new assumptions. p ðp1Þ=2 X p2l 1 f 2l 0 Cp2l Cpf ¼ mp m2l
m0
¼ mp
1
m0
0 0
l¼1
ðp1Þ=2 X l¼1
p Cf m : p2l p2l 2l
ð24Þ
which is the definition of Flusser’s blur invariants [10].
&
5. Experimental results In this section, the performance of the proposed invariants is demonstrated by applying it on 1D signals, and are compared to Flusser’s invariants [10]. Two different types of wavelet functions were employed to demonstrate the performance of the invariants under the change of this parameter. The first wavelet function is Coiflet of order 1 (Mc ¼ 2) [17]. There is no closed-form representation for this orthogonal wavelet. The two filters that represent it are presented in Table 2. The second wavelet function belongs to the crude wavelet class. The wavelet function that is employed here is a hyperbolic kernel, which, unlike Coiflet, has an explicit expression (25). This wavelet function is comprehensively studied in [18–20], and different properties are extracted. For the experiments of this paper, n and b were set to 4 and 1, respectively. This wavelet function has two vanishing moments as well.
cn, b ðxÞ ¼ nb2 sechn ðbxÞðnðn þ 1Þsech2 ðbxÞÞ
ð25Þ
In two experiments, different aspects of the proposed blur invariants were studied. In the first experiment, artificial signals were generated and the features’ invariance to blur and robustness to noise were studied. In the second experiment, three EEG signals were used in order to explore the invariance to blur and discriminative power of the proposed features. 5.1. Experiment I For this experiment, 1000 1D signals with the length of 200 and in the range of [ 1 1] were artificially generated. Fig. 2 shows one of the signals. The signals were blurred by calculating N-point neighborhood averaging with a neighborhood of 5, 7, 9, 11, 13, 15, 17, 19, 21, and 23. In order to study the effect of noise on the performance of the invariants, the signals were also contaminated with additive noise. The added noise was Gaussian with zero mean and the standard deviation of 0 (no noise), 0.001, 0.002, 0.008, 0.02, 0.04, and 0.08. The wavelet transform of the signals
c f^
^c
f Cqk ðsÞm 0 ðsÞ
0 Mc
c
c f^ f^ ðsÞCqk ðsÞ ¼ ¼ Cqk
0:
ð23Þ
Table 2 Decomposition filters for coiflet of order 1. n
Since Jq,k ðsÞ is zero when k is odd, the only remaining term in (22) ^c
3953
^
^
is Cqf ðsÞ. Therefore, Cqg ðsÞ ¼ Cqf ðsÞ.
&
Corollary 1. Flusser’s invariants [10] are a special case of (20). Proof. To obtain Flusser’s invariants, it is assumed that cðÞ is the Dirac delta function. In this case, Mc ¼ 0 which implies that cðÞ is
1 2 3 4 5 6
h[n]
g[n]
0.015655728135465 0.072732619512854 0.384864846864203 0.852572020212255 0.337897662457809 0.072732619512854
0.072732619512854 0.337897662457809 0.852572020212255 0.384864846864203 0.072732619512854 0.015655728135465
h[n] and g[n] are the lowpass and highpass filters, respectively.
3954
I. Makaremi, M. Ahmadi / Pattern Recognition 43 (2010) 3950–3957
were obtained with the wavelet functions described above at three scales, s¼2, 4, and 8. The invariants for all of the original and blurred signals were calculated up to order 11 (q¼9, p ¼ q þMc ¼ 11). However, the invariants of order 3 (q¼ 1, p ¼ q þMc ¼ 3) are not reported since they are zero. Then, based on the following equation, the similarity between the original signals and their blurred versions was measured C X C O p p R¼ ð26Þ CpO X where CO p and Cp are the pth order invariants of the actual and blurred signals, respectively. The closer CXp is to CO p , the smaller R becomes.
1
0.5
0
−0.5
−1 0
50
100
150
200
Fig. 2. One of the artificial signals that was used in experiment I.
p=5
Flusser
p=7
Coiflet, s = 4
Coiflet, s = 8
0.14
0.14
0.14
0.09
0.09
0.09
0.05
0.05
0.05
5
11
17
23
5
11
17
23
0.2
0.2
0.2
0.13
0.13
0.13
0.07
0.07
0.07
5
p=9
Figs. 3 and 4 show the medians of the similarity measures for invariants obtained with Coiflet and hyperbolic wavelet functions, respectively. Both figure show invariants of three orders: 5, 7, and 9, and consists of nine plots: three representing the results acquired with Flusser’s invariants, and the other six results with the proposed invariants at two scales: 4 and 8. Every plot has several graphs representing different levels of noise. The x-axis represents the number of neighborhoods in averaging, N, and the y-axis shows the similarity measure, R. The results acquired with Flusser’s invariants are repeated in both figures for the sake of a simple comparison. Also, Table 3 presents the 11th order invariants of one the signals for different levels of noises. The first noticeable property of the invariants is that by increasing the order of the invariants R increases as well. The reason of this rise is because the invariants also increase exponentially with respect to their order. By changing the noise level, the similarity changes from 0% to 30% in Flusser’s invariants and 25% in the wavelet based invariants, which is 5% less than the one in the spatial domain (Figs. 3 and 4). The wavelet based invariants show a generally better robustness than Flusser’s invariants in this experiment. However, it is impossible to make a general statement about the robustness of invariants in the wavelet domain than those in the spatial domain. The reason is that there are various wavelet functions available with different properties, which requires a comprehensive study of its own. A slight increase in similarity is observable by the increase of N in Flusser’s invariants. It is also noticeable in the proposed invariants at scale 4, but it becomes less evident for the invariants at scale 8, specially for invariants with Coiflet. The similarity measure is also slightly less for these invariants. Since a higher scale is equivalent to a lower resolution, the effect of blur and
11
17
23
5
11
17
23
0.26
0.26
0.26
0.18
0.18
0.18
0.09
0.09
0.09
5
11
17
23
5
11
17
23
5
11
17
23
5
11
17
23
5
11
17
23
Fig. 3. Median of similarity measures calculated for Flusser’s invariants and the proposed invariants with Coiflet wavelet of order 1 at scales 4 and 8. Every plot shows the effect of different levels of noise. The x-axis represents N, which is the number of neighborhood for averaging, and the y-axis shows the similarity measure R.
I. Makaremi, M. Ahmadi / Pattern Recognition 43 (2010) 3950–3957
p=5
Flusser
p=7
Hyperbolic, s = 8
0.14
0.14
0.14
0.09
0.09
0.09
0.05
0.05
0.05
5
11
17
23
5
11
17
23
0.2
0.2
0.2
0.13
0.13
0.13
0.07
0.07
0.07
5
p=9
Hyperbolic, s = 4
11
17
23
5
11
17
23
0.26
0.26
0.26
0.18
0.18
0.18
0.09
0.09
0.09
5
11
17
23
5
11
17
3955
23
5
11
17
23
5
11
17
23
5
11
17
23
Fig. 4. Median of similarity measures calculated for Flusser’s invariants and the proposed invariants with Hyperbolic wavelet of order 4 at scales 4 and 8. Every plot shows the effect of different levels of noise. The x-axis represents N, which is the number of neighborhood for averaging, and the y-axis shows the similarity measure R.
Table 3 Eleventh order invariants of an artificial signal that was used in experiment I. Noise
0.000 0.001 0.002 0.008 0.020 0.040 0.080
Flusser
Coiflet (s ¼2)
Hyperbolic (s ¼2)
N ¼7
N ¼ 11
N ¼7
N ¼ 11
N ¼7
N¼ 11
1.26e+ 31 1.26e+ 31 1.26e+ 31 1.33e+ 31 1.12e+ 31 1.59e+ 31 9.81e+ 30
1.26e + 31 1.26e + 31 1.27e + 31 1.32e + 31 1.13e + 31 1.57e + 31 9.81e + 30
5.51e+ 30 5.50e+ 30 5.52e+ 30 5.79e+ 30 4.88e+ 30 6.92e+ 30 4.28e+ 30
5.51e + 30 5.50e + 30 5.53e + 30 5.74e + 30 4.92e + 30 6.84e + 30 4.28e + 30
1.59e +32 1.59e +32 1.59e +32 1.69e +32 1.37e +32 2.12e +32 1.16e +32
1.59e+ 32 1.59e+ 32 1.60e+ 32 1.68e+ 32 1.38e+ 32 2.09e+ 32 1.16e+ 32
The invariants are shown for different levels of noise variances and two numbers of neighborhoods in averaging. The scale, s, is 2.
noise is less significant. However, the risk of losing the properties of signals also increases. Therefore, a proper selection of involving scales is substantial. Comparing the results obtained by Flusser’s invariants and those of the wavelet based invariants indicates that the later is generally performing better. Employing different wavelet functions also did not have any major influence on the performance of the invariants, but provide a different range of values (Table 3). As it is mentioned above, it is possible that with a different choice of wavelet function contrary results would be obtained. The application that the proposed invariants are supposed to be used in could have a significant influence on the choice of the wavelet function, since different types of wavelet functions are sensitive to different properties in signals [15]. It might be even useful to employ multiple wavelet functions in order to extract different moment invariants for a comprehensive analysis.
5.2. Experiment II An important aspect that should be studied in features is their discriminative power. Flusser and Suk showed that if signals are centrally symmetric, their invariants are equal to zero [5], which means that it is impossible to distinguish between different centrally symmetric signals. The same statement is true about the proposed invariants of this paper only in a case where the signal and the wavelet function are both centrally symmetric. In f^
this case, mpc is zero when p is odd, which can be easily inferred from (17). The odd order moments of a signal in the wavelet domain is the summation of odd order moments of the signal in the spatial domain multiplied by even order moments of the wavelet function and even order moments of the signal in the spatial domain multiplied by odd order moments of the wavelet f^
function. Since the odd order central moments are zero, Cp c is zero as well. However, if the selected wavelet function is not f^
centrally symmetric, mpc would not become zero. The reason is that the odd order moments of the wavelet function in (17) are f^
not zero anymore, which bears out that Cp c is nonzero as well. Therefore, the limitedness of discriminative power which is addressed by Flusser and Suk [5] is not an issue in the wavelet based invariants. In order to further study the discriminative power of the invariants, three EEG signals (Fig. 5) were used from the database which is generated by Andrzejak et al. [16]. Similar to experiment I, the signals were blurred by calculating N-point neighborhood averaging with a neighborhood of 15, 30, and 45, and their wavelet transforms were obtained with the two mentioned wavelet functions at scale 8. The seventh and 11th order
EEG Sig #3
EEG Sig #2
EEG Sig #1
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200 0 −200
0
50
100
150
200
250
300
350
400
450
500
0
50
100
150
200
250
300
350
400
450
500
200 0 −200
Real-world applications mostly deal with finite-extend signals. Candocia [21] showed that in order to have perfectly invariant features, it is necessary to have the full result of the convolution of the blur system with the signal without any modifications at boundaries. The reason that the presented results have a prefect accuracy is following Candocia’s conclusion. However, such accuracy is not perfectly achievable in practice. Therefore, a margin of error should be considered when they are employed in real-world problems, where there is absolutely no information available about the original signal and PSF.
200
6. Discussion and conclusion
0 −200
0
50
100
150
200
250
300
350
400
450
500
Fig. 5. Three EEG signals that were used in Experiment II.
Table 4 Invariants of EEG signals shown in Fig. 5 with Coiflet of order 1 at scale 2. N¼5
N¼ 13
N ¼21
Signal 1
p¼7 p¼9 p ¼ 11
2.5723e + 19 4.5788e + 25 1.2458e + 32
2.5723e + 19 4.5788e + 25 1.2458e + 32
2.5723e +19 4.5788e +25 1.2458e +32
Signal 2
p¼7 p¼9 p ¼ 11
2.3007e +19 1.2791e + 25 9.4448e + 30
2.3007e + 19 1.2791e + 25 9.4448e + 30
2.3007e+ 19 1.2791e +25 9.4448e +30
Signal 3
p¼7 p¼9 p ¼ 11
1.0092e +20 2.3239e + 26 7.7957e + 32
1.0092e + 20 2.3239e + 26 7.7957e + 32
1.0092e+ 20 2.3239e +26 7.7957e +32
N is the number of neighborhood for averaging and p is the order of invariants.
Table 5 Invariants of EEG signals shown in Fig. 5 with Hyperbolic wavelet of order 4 at scale 2.
Signal 1
Signal 2
Signal 3
p¼7 p¼9 p ¼ 11 p¼7 p¼9 p ¼ 11 p¼7 p¼9 p ¼ 11
N¼5
N¼ 13
N ¼21
5.8943e + 19 1.0492e + 26 2.8547e + 32 5.2720e + 19 2.9310e + 25 2.1643e + 31 2.3126e + 20 5.3253e + 26 1.7864e + 33
5.8944e + 19 1.0492e + 26 2.8548e + 32 5.2720e + 19 2.9310e + 25 2.1643e + 31 2.3126e + 20 5.3253e + 26 1.7864e + 33
5.8944e +19 1.0492e+ 26 2.8548e +32 5.2720e+ 19 2.9310e+ 25 2.1643e +31 2.3126e +20 5.3252e +26 1.7864e +33
N is the number of neighborhood for averaging and p is the order of invariants.
invariants of these three signals with Coiflet of order 1 and hyperbolic wavelet are presented in Tables 4 and 5, respectively. It is clear that the change of N does not have any effect on the values of invariants of the signals. The seventh order invariants of signals 1 and 2 are almost the same. However, this similarity is not present in the ninth and 11th order invariants of these two signals. Also, the invariants of signal 3 are totally different from the other two. The other observable fact is that both wavelets are performing equally well in this experiment. Therefore, it is better to use multiple orders of invariants in order to achieve a better discrimination.
A new technique has been proposed for extracting blur invariant features from degraded signals. In this technique, blur invariants are defined in the wavelet domain. Defining the invariants in the wavelet domain provides the opportunity to take advantage of different wavelet functions and extracting features at different scales. It has been proved that blur invariants in the spatial domain are a special case of the proposed invariants. Also, the limitedness of discriminative power of spatial domain invariants does not exist for those in the wavelet domain. In this paper, the performance and robustness of the proposed invariants have been examined through an experiment. In this experiment which was held on 1000 artificially generated signals, the average showed the performance of the proposed invariants is better than that of its counterpart in the spatial domain. In another experiment, the discriminative power of the wavelet based invariants were checked with EEG signal. The results indicated a good power in discriminating between different signals. However, obtaining better results with the proposed invariants depends on the choice of the wavelet function. Scale might also have an impact on effectiveness of the invariants. Since a higher scale is equivalent to a lower resolution, in specific applications these invariants might not be beneficial. It is possible that based on the application and the type of the wavelet function, final results are not as good as what can be obtained with a simpler invariant technique. Therefore, an insightful selection of wavelet functions and the highest engaged scale seems to be unavoidable. Studying different types of wavelet functions, categorizing them based on their properties and their effect on the quality of the invariants, and evaluating effectiveness of the invariants at different scales in recognition problems is the next step that is being taken in order to enlighten different aspects of using this novel definition for blur invariants.
Acknowledgments The authors wish to thank Professor Jan Flusser for excellent comments and suggestions. The authors also thank NSERC for its financial support of the project. References [1] M.I. Sezan, A.M. Tekalp, Survey of recent developments in digital image restoration, Optical Engineering 29 (1990) 393–404. [2] D. Kundur, D. Hatzinakos, Blind image deconvolution, Signal Processing Magazine, IEEE 13 (1996) 43–64. [3] J. Flusser, T. Suk, S. Saic, Image features invariant with respect to blur, Pattern Recognition 28 (1995) 1723–1732. [4] T. Suk, J. Flusser, Features invariant simultaneously to convolution and affine transformation, in: W. Skarbek (Ed.), CAIP, Lecture Notes in Computer Science, vol. 2124, Springer2001, pp. 183–190. [5] J. Flusser, T. Suk, Degraded image analysis: an invariant approach, IEEE Transactions on Pattern Analysis and Machine Intelligence 20 (1998) 590–603.
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[6] C.-Y. Wee, R. Paramesran, Derivation of blur-invariant features using orthogonal legendre moments, Computer Vision, IET 1 (2007) 66–77. [7] H. Zhang, H. Shu, G. Han, G. Coatrieux, L. Luo, J. Coatrieux, Blurred image recognition by legendre moment invariants, IEEE Transactions on Image Processing 19 (2010) 596–611. [8] S. Metari, F. Deschenes, New classes of radiometric and combined radiometric-geometric invariant descriptors, IEEE Transactions on Image Processing 17 (2008) 991–1006. [9] H. Ji, H. Zhu, Degraded image analysis using zernike moment invariants, in: IEEE International Conference on Acoustics, Speech and Signal Processing 2009, ICASSP 2009, 2009, pp. 1941–1944. [10] J. Flusser, T. Suk, Classification of degraded signals by the method of invariants, Signal Processing 60 (1997) 243–249. [11] J. Flusser, J. Boldys, B. Zitova, Moment forms invariant to rotation and blur in arbitrary number of dimensions, IEEE Transactions on Pattern Analysis and Machine Intelligence 25 (2003) 234–246. ¨ Object recognition using frequency domain blur [12] V. Ojansivu, J. Heikkila, invariant features, in: B.K. Ersbøll, K.S. Pedersen (Eds.), SCIA, Lecture Notes in Computer Science, vol. 4522, Springer2007, pp. 243–252. ¨ A method for blur and similarity transform invariant [13] V. Ojansivu, J. Heikkila, object recognition, in: R. Cucchiara (Ed.), ICIAP, IEEE Computer Society2007, pp. 583–588.
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[14] A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill Book Company, New York, 1965. [15] S. Mallat, A Wavelet Tour of Signal Processing, Wavelet Analysis & Its Applications, second ed., Academic Press, 1999. [16] R.G. Andrzejak, K. Lehnertz, F. Mormann, C. Rieke, P. David, C.E. Elger, Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: dependence on recording region and brain state, Physical Review E 64 (2001) 061907. [17] I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Math, 1992, doi:10.2277/0898712742. [18] K.N. Le, K.P. Dabke, G.K. Egan, Hyperbolic wavelet power spectra of nonstationary signals, Optical Engineering 42 (2003) 3017–3037. [19] K.N. Le, K.P. Dabke, G.K. Egan, Hyperbolic wavelet family, Review of Scientific Instruments 75 (2004) 4678–4693. [20] K.N. Le, K.P. Dabke, G.K. Egan, On mathematical derivations of auto-term functions and signal-to-noise ratios of the Choi–Williams, first- and nthorder hyperbolic kernels, Digital Signal Processing 16 (2006) 84–104. [21] F.M. Candocia, Moment relations and blur invariant conditions for finiteextent signals in one, two and n-dimensions, Pattern Recognition Letters 25 (2004) 437–447.
Iman Makaremi obtained his B.Sc. in Electrical Engineering from the University of Tehran, Iran in 2005. He finished his masters in Electrical Engineering at the KN Toosi University of Technology, Tehran, Iran in 2007 in the area of Intelligent Fault Detection. Currently, he is working towards his Ph.D. in Electrical Engineering at the University of Windsor, Canada. His research interest includes geometric and radiometric invariant features and their application in face recognition.
Majid Ahmadi received the B.Sc. degree in Electrical Engineering from Arya Mehr University, Tehran, Iran, in 1971 and the Ph.D. degree in Electrical Engineering from Imperial College of London University, London, UK, in 1977. He has been with the Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON, Canada, since 1980, currently as a Professor and Associate Director of the Research Center for Integrated Microsystems. His research interests include digital signal processing, machine vision, pattern recognition, neural network architectures, applications, and VLSI implementation. He has coauthored the book Digital Filtering in oneand two-Dimensions; Design and Applications (New York: Plenum, 1989) and has published over 300 articles in this area. He is the Associate Editor for the Journal of Pattern Recognition, the Journal of Circuits, Systems and Computers, and the International Journal of Computers and Electrical Engineering. Dr. Ahmadi was the IEEE-CAS representative on the Neural Network Council and the Chair of the IEEE Circuits and Systems Neural Systems Applications Technical Committee. He was a recipient of an Honorable Mention award from the Editorial Board of the Journal of Pattern Recognition in 1992 and received the Distinctive Contributed Paper award from the MultipleValued Logic Conference Technical Committee and the IEEE Computer Society in 2000. He is a Fellow of the Institution of Electrical Engineering.
Contents lists available at ScienceDirect
Pattern Recognition journal homepage: www.elsevier.com/locate/pr
Blur invariants: A novel representation in the wavelet domain Iman Makaremi , Majid Ahmadi Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON, Canada N9B 3P4
a r t i c l e in f o
a b s t r a c t
Article history: Received 7 April 2010 Received in revised form 21 June 2010 Accepted 15 July 2010
Blur invariants in the wavelet domain are proposed for the first time in this paper. Wavelet domain blur invariants take advantage of several benefits that this domain provides, i.e. different alternatives for wavelet function and analysis in different scales. It is not required to model the blur system in order to extract the invariants. It will be shown how the space domain blur invariants are a special case of the proposed invariants. It will also be explained how the proposed invariants would not have the null space that their special case in the spatial domain have which limits their discriminative power. The performance of these invariants will be demonstrated through experiments, and compared to its counterpart which is defined in the spatial domain. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Blur invariant moment Direct analysis Feature extraction Wavelet transform
1. Introduction Signal acquisition is the first step in all signal processing tasks, and is always accompanied with different sources of degradation. The effect of some of the degradations is considerably high, which could vastly affect expected outcomes. Blur, as one of the degradations that is classified as radiometric degradation and can significantly reduce the accuracy, has been studied for a few decades. The general model that is used for the observed signal is gðxÞ ¼ Hf ðxÞ þ nðxÞ
ð1Þ
where g, f, and n are the observed signal, the actual signal, and the additive noise, respectively, and H is the blur operator. If it is assumed that the blur operator is linear space-invariant, the above observation model can be simplified to gðxÞ ¼ f ðxÞ hðxÞ þ nðxÞ,
ð2Þ
where hðxÞ is point spread function (PSF) of the system. There are two proposed approaches for resolving this problem: blind restoration and direct analysis. The purpose in blind restoration methods is to identify the blur system model and extract the actual signal. Therefore, a prior information is required in order to set constraints and choose proper parameters. There are numerous proposed methods in literature for this type of approach [1,2], and their main application is in signal restoration. The drawback is, however, that they are computationally expensive, and that the Corresponding author.
E-mail addresses: [email protected], [email protected] (I. Makaremi), [email protected] (M. Ahmadi). 0031-3203/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.patcog.2010.07.020
problem is usually ill-posed. Also, in many applications it is not necessary to restore the actual signal. The second set of approaches try to extract properties in the degraded signal that are invariant to blur. Such methods do not go through the process of identifying the blur system. Flusser et al. [3] proposed the first direct analysis method in the spatial domain that is based on ordinary and central moments. The only assumption was that the blur system is centrally symmetric and energy-preserving. Afterward, they modified the method and made it invariant to geometric distortions as well [4,5]. Wee and Paramesran [6] and Zhang et al. [7] employed Legendre moments in order to define their invariants in the spatial domain. Metari and Deschˆenes [8] exploited the Mellin transform in order to extract blur invariant descriptors. They also showed that the proposed descriptors are also invariant to some of the geometric distortions. The invariants proposed by Ji and Zhu [9] are based on Zernike moments. Along with the blur invariant features defined in the spatial domain, there are some other methods that are developed in the Fourier domain. Flusser and Suk [10] proposed their Fourier based invariants based on the tangent of the phase of signals, and showed that they are invariant to blur. They also made it invariant to some other geometric distortions and generalized them for N-dimensional signals [11]. Ojansivu and Heikkila¨ [12], however, showed that Flusser’s invariants are sensitive to noise because of their use of tangent operator. They proposed another representation of invariants in the Fourier domain. Afterward, they made them invariant to affine transform as well [13]. In this paper, we propose blur invariant moments in the wavelet domain for the first time. The different alternatives that exist for wavelet functions and the benefit of analyzing signals at different scales in the wavelet domain are the vivid advantages
I. Makaremi, M. Ahmadi / Pattern Recognition 43 (2010) 3950–3957
that the proposed invariants have over those defined in other domains. The idea will be developed for 1D signals, and the effect of blur and noise will be studied and compared to a similar technique. There will be a discussion on how to expect better results with this new proposed method. The proposed invariants could be used in analyzing EEG, ECG, and speech signals and those acquired in single point ultrasound measurements. The rest of the paper is organized as following. Section 2 provides the basic and complementary definitions that are required in this paper. In Section 3, ordinary and central moments are defined in the wavelet domain, and the relation between the moments in the wavelet domain with those in the spatial domain is extracted. Also, it is shown how the moments of a degraded signal are related to the moments of its actual source both in the wavelet domain. Section 4 presents the blur invariant moments in the wavelet domain and how Flusser’s blur invariant moments [10] are a special case of the proposed invariants. The proofs are also provided. Section 5 is devoted to the experimental results through which the performance of the proposed invariants is evaluated, compared to Flusser’s invariants, and their robustness and discriminative power are discussed. Also, there will be a discussion on the effect of wavelet functions on the quality of the invariants. Finally, section 6 concludes the paper.
2. Basic definitions and notations
3951
Definition 7. The wavelet transform of signal f(x) at shift u and scale s with wavelet function cðxÞ is [15] Z þ1 xu c 1 pffiffi f ðxÞc dx: ð8Þ f^ ðs,uÞ ¼ s s 1
where c ðÞ is the complex conjugate of cðÞ. 3. Moments in the wavelet domain First, it will be shown that how (2) would be presented in the wavelet domain. Also for the sake of simplicity, noise is not shown in the equations. However, its effect on the invariants will be shown in Section 5. The wavelet transform of signal g(x), the blurred signal, at shift u and scale s with wavelet function cðxÞ is Z þ1 xu 1 c pffiffi gðxÞc dx: ð9Þ g^ ðs,uÞ ¼ s s 1 Replacing g(x) by its equivalent in (2) gives Z þ1 Z þ1 xu 1 c pffiffi hðyÞf ðxyÞc dy dx: g^ ðs,uÞ ¼ s s 1 1 Substituting z for x y, the above equation changes to Z þ1 Z þ1 1 zðuyÞ c pffiffi f ðzÞc dz dy hðyÞ g^ ðs,uÞ ¼ s s 1 1 Z þ1 c ¼ hðyÞf^ ðs,uyÞ dy
ð10Þ
ð11Þ
1
In this section, some basic terms are defined and explained. Definition 1. The pth order ordinary geometric moment of signal f(x) in the spatial domain is defined by [14] Z þ1 mfp ¼ xp f ðxÞ dx: ð3Þ
which shows that the wavelet transform of g(x) is the convolution of h(x) by the wavelet transform of f(x): c
c g^ ðs,uÞ ¼ f^ ðs,uÞ hðuÞ:
ð12Þ
1
3.1. Relation between moments in the spatial and wavelet domains Definition 2. The centroid of signal f(x) is [14] cf ¼
mf1 mf0
ð4Þ
Definition 3. The pth order central moment of signal f(x) in the spatial domain is defined by [14] Z þ1 mfp ¼ ðxcf Þp f ðxÞ dx: ð5Þ 1
For (4) to hold and Definition 3 to be valid, f(x) is required to have a nonzero first moment. In the case that the moments of f(x) are zero up to a certain order, the following definitions are proposed. Definition 4. If the moments of signal f(x) are zero up to order M 1, its centroid is defined as
Bf ¼
mfM þ 1 ðM þ 1ÞmfM
ð6Þ
Definition 5. If the moments of signal f(x) are zero up to order M 1, its pth order central moment (p ZM) in the spatial domain is defined by Z þ1 mfp ¼ ðxBf Þp f ðxÞ dx: ð7Þ 1
Definitions 2 and 3 are special cases of Definitions 4 and 5, respectively, in which M ¼0. Definition 6. If the moments of wavelet function cðxÞ is zero up to order Mc 1, it is said that cðxÞ has Mc vanishing moments [15].
c Before extracting the relation between the moments of g^ ðu,sÞ c and those of f^ ðu,sÞ and h(u), it is shown how the moments of f(x) in the wavelet domain are related to those in the spatial domain. c Rewriting Definition 3 for f^ ðu,sÞ, the wavelet transform of f(x) with wavelet function cðÞ at shift u and scale s, we have the following: Z þ1 Z þ1 Z þ1 c 1 ^c xu dx du up f^ ðs,uÞ du ¼ pffiffi up f ðxÞc mpf ðsÞ ¼ s s 1 1 1
¼
pffiffi s
Z
þ1
1
Z
þ1
ðxsyÞp f ðxÞc ðyÞ dx dy:
ð13Þ
1
In (13), y is substituted for (ðxuÞ=sÞ. Considering the fact that P ða þbÞp ¼ pk ¼ 0 pk ak bpk , the relation between the ordinary moments of signal f(x) in the wavelet domain and the spatial domain can be extracted: p X p k þ 1=2 ^c c mpf ðsÞ ¼ ð1Þk mfpk mk : ð14Þ s k k¼0 Assuming that the wavelet function has Mc vanishing moments, (14) can be modified to p X p k þ 1=2 ^c c mpf ðsÞ ¼ ð1Þk mfpk mk : ð15Þ s k k¼M c
It can be concluded from (15) that the transform of a signal with a wavelet vanishing moments are zero up to Definition 7 should be used in order
moments of the wavelet function which has Mc order Mc 1. Therefore, to calculate the central
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are of almost the same order of magnitude. Comparison between the moments calculated directly from the signal’s wavelet transform and those derived from (17) indicates that they match accurately in implementation as well.
moments in the wavelet domain: Z þ1 c ^c ^c mpf ðsÞ ¼ ðuBf Þp f^ ðs,uÞ du 1
Z þ1 Z þ1 1 ^c xu dx du ðuBf Þp f ðxÞc ¼ pffiffi s s 1 1 Z Z pffiffi þ 1 þ 1 ¼ s ððxcf ÞsðyBc ÞÞp f ðxÞc ðyÞ dx dy 1
3.2. Relation between the moments of original and degraded signals ð16Þ
c The relation between the pth order central moment of g^ ðs,uÞ
1 ^c
where y is substituted for ððxuÞ=sÞ, and Bf is replaced by its equivalent cf sBc . The later can be trivially shown. From (16), the relation between the central moments of f(x) in the wavelet domain and the spatial domain can be extracted: p X p k þ 1=2 ^c c mpf ðsÞ ¼ ð1Þk mfpk mk : ð17Þ s k k¼M
c and the central moments of f^ ðs,uÞ and h(x) can be extracted by c rewriting (7) for g^ ðs,uÞ, using (12), and considering the fact that c
^c
Bg^ ¼ Bf þ ch . c
mgp^ ðsÞ ¼
Z
c
Z
þ1
þ1
¼
The credibility of Eq. (17) is verified in an example as follows. Example 1. In this example, it is shown how (17) holds. The signal that is used in this example is illustrated in Fig. 1. It is an EEG sample from the database generated by Andrzejak et al. [16]. The wavelet transform of this signal is calculated at scale 1 and 4 (s¼1,4) with Coilflet of order 1 as the wavelet function [17], which has two vanishing moments. For comparison, the first ten c central moments of f^ ðu,sÞ are calculated from both the signal’s wavelet transform and the derived equation in (17). As it is illustrated in Table 1, the first two moments of the wavelet c function are extremely small. Feasibly, f^ ðu,sÞ’s first two moments
1
¼
c
ðuBg^ Þp g^ ðs,uÞ du
1
Z
c
þ1
c
c
ðuBg^ Þp f^ ðs,uÞhðvuÞ dv du
1
p X p
k
k¼0
^c
mkf ðsÞmhpk :
ð18Þ
If the employed wavelet function has Mc vanishing moments, by q q þ Mc l M ¼ l þ M c , the above
defining q þMc ¼ p, m q ðsÞ ¼ mp ðsÞ, and
c
equation can be simplified into (19). q X c q ^c m gq^ ðsÞ ¼ m fl ðsÞ mhql : l Mc l¼0
ð19Þ c
It shows that the central moments of g^ ðs,uÞ of a certain order are c
related to the central moments of f^ ðs,uÞ and h(x) of the same and lower orders.
200 150
4. Blur invariant features in the wavelet domain
100
In order to extract blur invariant features, it is necessary to look for combinations of moments such that those of h(x) would be canceled out. It is assumed that h(x), the PSF, is symmetric and energy-preserving. When h(x) is symmetric, i.e. h(x)¼h( x), its odd order moments are zero. And because of the energy-preserving R þ1 property, the integral 1 hðxÞ dx, which is mh0 , equals one.
50 0 −50 −100 −150
0
50
100
150
200
250
300
350
400
450
500
Fig. 1. An EEG signal. The results in Table 1 are based on this signal.
Theorem 1. If q is odd, then Cq ðsÞ, which is defined as follows, is invariant to blur in the wavelet domain with wavelet function cðÞ. q ðq1Þ=2 q2l X 1 M 2l c Cq2l ðsÞm 2l ðsÞ Cq ðsÞ ¼ m q ðsÞ ð20Þ m 0 ðsÞ l ¼ 1 0 M c
Table 1 The first ten central moments of Coiflet of order 1 and the signal’s wavelet transform at scale 1 (columns 3 and 4 from left) and 4 (columns 5 and 6 from left) are shown. The third and fifth columns (DT) include the central moments calculated directly from the signal’s wavelet transform, and the fourth and sixth columns (DV) are induced from (17). p
^c
^c
^c
^c
mcp
mpf ð1Þ DT
mpf ð1Þ DV
mpf ð4Þ DT
mpf ð4Þ DV
1.14e 16 6.94e 17 1.16e + 00 1.17e 15 1.70e + 00 2.83e + 00 3.50e + 00 3.61e + 01 9.49e + 01 3.85e + 02
1.59e 13 1.05e 11 2.46e + 03 5.45e 06 5.94e + 08 1.38e + 11 9.21e + 13 2.58e + 16 1.24e + 19 3.98e + 21
0.00e + 00 0.00e + 00 2.46e +03 3.96e 10 5.94e +08 1.38e +11 9.21e +13 2.58e +16 1.24e +19 3.98e +21
3.91e 13 9.73e 11 4.46e+ 05 6.73e 05 1.08e+ 11 2.49e+ 13 1.67e+ 16 4.68e+ 18 2.25e+ 21 7.22e+ 23
0.00e + 00 0.00e + 00 4.46e + 05 7.48e 08 1.08e + 11 2.49e + 13 1.67e + 16 4.68e + 18 2.25e + 21 7.22e + 23
^
^
c
Proof. When q¼1, C1g ðsÞ ¼ C1f ðsÞ ¼ 0. ^c ^c For q¼3, it can be trivially shown that C3g ðsÞ ¼ C3f ðsÞ. ^c
c
^ g C3g ðsÞ ¼ m 3 ðsÞ
¼
3 X 3 l¼0
0 1 2 3 4 5 6 7 8 9
c
l
^c
m lf ðsÞmh3l Mc
^c ^c f3 ðsÞ ¼ C3f ðsÞ:
¼m
ð21Þ
If (20) is valid for 1,3, . . . ,q2, it can be proved that it is valid for q as well: q ðq1Þ=2 q2l M X c c c c 1 g^ g^ g^ 2l c Cq2l ðsÞm 2l ðsÞ Cqg^ ðsÞ ¼ m q ðsÞ c g^ 0 Mc m 0 ðsÞ l ¼ 1
I. Makaremi, M. Ahmadi / Pattern Recognition 43 (2010) 3950–3957
c q X q þMc f^ m qk ðsÞmhk k k¼0 q c ðq1Þ=2 2l c q2l X X 2l þ Mc 1 M f^ f^ 2l c Cq2l c ðsÞ m 2lk ðsÞmhk k f^ 0 k ¼ 0 l ¼ 1 M m 0 ðsÞ c c q c X qþ Mc f^ f^ ¼ Cq ðsÞ þ m qk ðsÞmhk k k¼1 q ðq1Þ=2 X q2l Mc f^ c X2l 2l þ Mc f^ c 1 2l c Cq2l ðsÞ m 2lk ðsÞmhk k¼1 ^ k 0 Mc m f0 ðsÞ l ¼ 1 c q X qþ Mc ^c f^ ¼ Cqf ðsÞ þ m qk ðsÞmhk k k¼1 q q2l ðq1Þ=2 2l c c X X 2l þMc 1 M f^ f^ 2l c Cq2l c ðsÞm 2lk ðsÞmhk ^ k 0 Mc m f0 ðsÞ l ¼ 1 k ¼ 1 q X qþ Mc ^f c ^f c ¼ Cq ðsÞ þ m qk ðsÞmhk k k¼1 qk q2l ðq1Þ=2 q1 c c X X q þMc 1 M f^ f^ 2lk c Cq2l c ðsÞm 2lk ðsÞmhk ^f k 0 Mc m 0 ðsÞ k ¼ 1 l ¼ ½ðk þ 1Þ=2 ! q1 X qþ Mc q þ Mc ^c ^c ¼ Cqf ðsÞ þ m 0f ðsÞmhq mhk q k k¼1 0 1 qk ðq1Þ=2 c c c q2l M X 1 B f^ C f^ c f^ 2lk Cq2l ðsÞm 2lk ðsÞA @m qk ðsÞ c f^ m 0 ðsÞ l ¼ ½ðk þ 1Þ=2 0 Mc ! q1 c X q þ Mc q þMc ^c ^ m 0f ðsÞmhq þ mhk Jq,k ðsÞ: ¼ Cqf ðsÞ þ q k k¼1 ¼
ð22Þ When k is odd, mhk is zero. Hence, the second term in (22) becomes zero. The only term in (22) that is dependent on the moments of h is Jq,k ðsÞ. In order to prove that the proposed invariant is independent of PSF, it is therefore necessary to show that when k is even, Jq,k ðsÞ ¼ 0. qk ðq1Þ=2 c c c q2l X 1 M f^ f^ f^ 2lk c Cq2l ðsÞm 2lk ðsÞ Jq,k ðsÞ ¼ m qk ðsÞ c ^f m 0 ðsÞ l ¼ k=2 0 Mc qk ðqk1Þ=2 c c qk2l M X ^f c 1 c f^ f^ 2l ¼ m qk ðsÞ c Cqk2l ðsÞm 2l ðsÞ f^ 0 Mc m 0 ðsÞ l ¼ 0 qk ðqk1Þ=2 c c qk2l M X ^f c 1 f^ c f^ 2l Cqk2l ðsÞm 2l ðsÞ ¼ m qk ðsÞ c f^ 0 Mc m 0 ðsÞ l ¼ 1
1 ^
c
m 0f ðsÞ
qk qk M c
0
not a wavelet function anymore. Therefore, p ¼ q þMc ¼ q and c
mp ¼ m q . By setting s ¼1 at all time, it is trivial to show that f^ ðx,1Þ is equivalent to f(x), since the signal is convolved by the Dirac delta function. From here, (20) is rewritten with the new assumptions. p ðp1Þ=2 X p2l 1 f 2l 0 Cp2l Cpf ¼ mp m2l
m0
¼ mp
1
m0
0 0
l¼1
ðp1Þ=2 X l¼1
p Cf m : p2l p2l 2l
ð24Þ
which is the definition of Flusser’s blur invariants [10].
&
5. Experimental results In this section, the performance of the proposed invariants is demonstrated by applying it on 1D signals, and are compared to Flusser’s invariants [10]. Two different types of wavelet functions were employed to demonstrate the performance of the invariants under the change of this parameter. The first wavelet function is Coiflet of order 1 (Mc ¼ 2) [17]. There is no closed-form representation for this orthogonal wavelet. The two filters that represent it are presented in Table 2. The second wavelet function belongs to the crude wavelet class. The wavelet function that is employed here is a hyperbolic kernel, which, unlike Coiflet, has an explicit expression (25). This wavelet function is comprehensively studied in [18–20], and different properties are extracted. For the experiments of this paper, n and b were set to 4 and 1, respectively. This wavelet function has two vanishing moments as well.
cn, b ðxÞ ¼ nb2 sechn ðbxÞðnðn þ 1Þsech2 ðbxÞÞ
ð25Þ
In two experiments, different aspects of the proposed blur invariants were studied. In the first experiment, artificial signals were generated and the features’ invariance to blur and robustness to noise were studied. In the second experiment, three EEG signals were used in order to explore the invariance to blur and discriminative power of the proposed features. 5.1. Experiment I For this experiment, 1000 1D signals with the length of 200 and in the range of [ 1 1] were artificially generated. Fig. 2 shows one of the signals. The signals were blurred by calculating N-point neighborhood averaging with a neighborhood of 5, 7, 9, 11, 13, 15, 17, 19, 21, and 23. In order to study the effect of noise on the performance of the invariants, the signals were also contaminated with additive noise. The added noise was Gaussian with zero mean and the standard deviation of 0 (no noise), 0.001, 0.002, 0.008, 0.02, 0.04, and 0.08. The wavelet transform of the signals
c f^
^c
f Cqk ðsÞm 0 ðsÞ
0 Mc
c
c f^ f^ ðsÞCqk ðsÞ ¼ ¼ Cqk
0:
ð23Þ
Table 2 Decomposition filters for coiflet of order 1. n
Since Jq,k ðsÞ is zero when k is odd, the only remaining term in (22) ^c
3953
^
^
is Cqf ðsÞ. Therefore, Cqg ðsÞ ¼ Cqf ðsÞ.
&
Corollary 1. Flusser’s invariants [10] are a special case of (20). Proof. To obtain Flusser’s invariants, it is assumed that cðÞ is the Dirac delta function. In this case, Mc ¼ 0 which implies that cðÞ is
1 2 3 4 5 6
h[n]
g[n]
0.015655728135465 0.072732619512854 0.384864846864203 0.852572020212255 0.337897662457809 0.072732619512854
0.072732619512854 0.337897662457809 0.852572020212255 0.384864846864203 0.072732619512854 0.015655728135465
h[n] and g[n] are the lowpass and highpass filters, respectively.
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I. Makaremi, M. Ahmadi / Pattern Recognition 43 (2010) 3950–3957
were obtained with the wavelet functions described above at three scales, s¼2, 4, and 8. The invariants for all of the original and blurred signals were calculated up to order 11 (q¼9, p ¼ q þMc ¼ 11). However, the invariants of order 3 (q¼ 1, p ¼ q þMc ¼ 3) are not reported since they are zero. Then, based on the following equation, the similarity between the original signals and their blurred versions was measured C X C O p p R¼ ð26Þ CpO X where CO p and Cp are the pth order invariants of the actual and blurred signals, respectively. The closer CXp is to CO p , the smaller R becomes.
1
0.5
0
−0.5
−1 0
50
100
150
200
Fig. 2. One of the artificial signals that was used in experiment I.
p=5
Flusser
p=7
Coiflet, s = 4
Coiflet, s = 8
0.14
0.14
0.14
0.09
0.09
0.09
0.05
0.05
0.05
5
11
17
23
5
11
17
23
0.2
0.2
0.2
0.13
0.13
0.13
0.07
0.07
0.07
5
p=9
Figs. 3 and 4 show the medians of the similarity measures for invariants obtained with Coiflet and hyperbolic wavelet functions, respectively. Both figure show invariants of three orders: 5, 7, and 9, and consists of nine plots: three representing the results acquired with Flusser’s invariants, and the other six results with the proposed invariants at two scales: 4 and 8. Every plot has several graphs representing different levels of noise. The x-axis represents the number of neighborhoods in averaging, N, and the y-axis shows the similarity measure, R. The results acquired with Flusser’s invariants are repeated in both figures for the sake of a simple comparison. Also, Table 3 presents the 11th order invariants of one the signals for different levels of noises. The first noticeable property of the invariants is that by increasing the order of the invariants R increases as well. The reason of this rise is because the invariants also increase exponentially with respect to their order. By changing the noise level, the similarity changes from 0% to 30% in Flusser’s invariants and 25% in the wavelet based invariants, which is 5% less than the one in the spatial domain (Figs. 3 and 4). The wavelet based invariants show a generally better robustness than Flusser’s invariants in this experiment. However, it is impossible to make a general statement about the robustness of invariants in the wavelet domain than those in the spatial domain. The reason is that there are various wavelet functions available with different properties, which requires a comprehensive study of its own. A slight increase in similarity is observable by the increase of N in Flusser’s invariants. It is also noticeable in the proposed invariants at scale 4, but it becomes less evident for the invariants at scale 8, specially for invariants with Coiflet. The similarity measure is also slightly less for these invariants. Since a higher scale is equivalent to a lower resolution, the effect of blur and
11
17
23
5
11
17
23
0.26
0.26
0.26
0.18
0.18
0.18
0.09
0.09
0.09
5
11
17
23
5
11
17
23
5
11
17
23
5
11
17
23
5
11
17
23
Fig. 3. Median of similarity measures calculated for Flusser’s invariants and the proposed invariants with Coiflet wavelet of order 1 at scales 4 and 8. Every plot shows the effect of different levels of noise. The x-axis represents N, which is the number of neighborhood for averaging, and the y-axis shows the similarity measure R.
I. Makaremi, M. Ahmadi / Pattern Recognition 43 (2010) 3950–3957
p=5
Flusser
p=7
Hyperbolic, s = 8
0.14
0.14
0.14
0.09
0.09
0.09
0.05
0.05
0.05
5
11
17
23
5
11
17
23
0.2
0.2
0.2
0.13
0.13
0.13
0.07
0.07
0.07
5
p=9
Hyperbolic, s = 4
11
17
23
5
11
17
23
0.26
0.26
0.26
0.18
0.18
0.18
0.09
0.09
0.09
5
11
17
23
5
11
17
3955
23
5
11
17
23
5
11
17
23
5
11
17
23
Fig. 4. Median of similarity measures calculated for Flusser’s invariants and the proposed invariants with Hyperbolic wavelet of order 4 at scales 4 and 8. Every plot shows the effect of different levels of noise. The x-axis represents N, which is the number of neighborhood for averaging, and the y-axis shows the similarity measure R.
Table 3 Eleventh order invariants of an artificial signal that was used in experiment I. Noise
0.000 0.001 0.002 0.008 0.020 0.040 0.080
Flusser
Coiflet (s ¼2)
Hyperbolic (s ¼2)
N ¼7
N ¼ 11
N ¼7
N ¼ 11
N ¼7
N¼ 11
1.26e+ 31 1.26e+ 31 1.26e+ 31 1.33e+ 31 1.12e+ 31 1.59e+ 31 9.81e+ 30
1.26e + 31 1.26e + 31 1.27e + 31 1.32e + 31 1.13e + 31 1.57e + 31 9.81e + 30
5.51e+ 30 5.50e+ 30 5.52e+ 30 5.79e+ 30 4.88e+ 30 6.92e+ 30 4.28e+ 30
5.51e + 30 5.50e + 30 5.53e + 30 5.74e + 30 4.92e + 30 6.84e + 30 4.28e + 30
1.59e +32 1.59e +32 1.59e +32 1.69e +32 1.37e +32 2.12e +32 1.16e +32
1.59e+ 32 1.59e+ 32 1.60e+ 32 1.68e+ 32 1.38e+ 32 2.09e+ 32 1.16e+ 32
The invariants are shown for different levels of noise variances and two numbers of neighborhoods in averaging. The scale, s, is 2.
noise is less significant. However, the risk of losing the properties of signals also increases. Therefore, a proper selection of involving scales is substantial. Comparing the results obtained by Flusser’s invariants and those of the wavelet based invariants indicates that the later is generally performing better. Employing different wavelet functions also did not have any major influence on the performance of the invariants, but provide a different range of values (Table 3). As it is mentioned above, it is possible that with a different choice of wavelet function contrary results would be obtained. The application that the proposed invariants are supposed to be used in could have a significant influence on the choice of the wavelet function, since different types of wavelet functions are sensitive to different properties in signals [15]. It might be even useful to employ multiple wavelet functions in order to extract different moment invariants for a comprehensive analysis.
5.2. Experiment II An important aspect that should be studied in features is their discriminative power. Flusser and Suk showed that if signals are centrally symmetric, their invariants are equal to zero [5], which means that it is impossible to distinguish between different centrally symmetric signals. The same statement is true about the proposed invariants of this paper only in a case where the signal and the wavelet function are both centrally symmetric. In f^
this case, mpc is zero when p is odd, which can be easily inferred from (17). The odd order moments of a signal in the wavelet domain is the summation of odd order moments of the signal in the spatial domain multiplied by even order moments of the wavelet function and even order moments of the signal in the spatial domain multiplied by odd order moments of the wavelet f^
function. Since the odd order central moments are zero, Cp c is zero as well. However, if the selected wavelet function is not f^
centrally symmetric, mpc would not become zero. The reason is that the odd order moments of the wavelet function in (17) are f^
not zero anymore, which bears out that Cp c is nonzero as well. Therefore, the limitedness of discriminative power which is addressed by Flusser and Suk [5] is not an issue in the wavelet based invariants. In order to further study the discriminative power of the invariants, three EEG signals (Fig. 5) were used from the database which is generated by Andrzejak et al. [16]. Similar to experiment I, the signals were blurred by calculating N-point neighborhood averaging with a neighborhood of 15, 30, and 45, and their wavelet transforms were obtained with the two mentioned wavelet functions at scale 8. The seventh and 11th order
EEG Sig #3
EEG Sig #2
EEG Sig #1
3956
I. Makaremi, M. Ahmadi / Pattern Recognition 43 (2010) 3950–3957
200 0 −200
0
50
100
150
200
250
300
350
400
450
500
0
50
100
150
200
250
300
350
400
450
500
200 0 −200
Real-world applications mostly deal with finite-extend signals. Candocia [21] showed that in order to have perfectly invariant features, it is necessary to have the full result of the convolution of the blur system with the signal without any modifications at boundaries. The reason that the presented results have a prefect accuracy is following Candocia’s conclusion. However, such accuracy is not perfectly achievable in practice. Therefore, a margin of error should be considered when they are employed in real-world problems, where there is absolutely no information available about the original signal and PSF.
200
6. Discussion and conclusion
0 −200
0
50
100
150
200
250
300
350
400
450
500
Fig. 5. Three EEG signals that were used in Experiment II.
Table 4 Invariants of EEG signals shown in Fig. 5 with Coiflet of order 1 at scale 2. N¼5
N¼ 13
N ¼21
Signal 1
p¼7 p¼9 p ¼ 11
2.5723e + 19 4.5788e + 25 1.2458e + 32
2.5723e + 19 4.5788e + 25 1.2458e + 32
2.5723e +19 4.5788e +25 1.2458e +32
Signal 2
p¼7 p¼9 p ¼ 11
2.3007e +19 1.2791e + 25 9.4448e + 30
2.3007e + 19 1.2791e + 25 9.4448e + 30
2.3007e+ 19 1.2791e +25 9.4448e +30
Signal 3
p¼7 p¼9 p ¼ 11
1.0092e +20 2.3239e + 26 7.7957e + 32
1.0092e + 20 2.3239e + 26 7.7957e + 32
1.0092e+ 20 2.3239e +26 7.7957e +32
N is the number of neighborhood for averaging and p is the order of invariants.
Table 5 Invariants of EEG signals shown in Fig. 5 with Hyperbolic wavelet of order 4 at scale 2.
Signal 1
Signal 2
Signal 3
p¼7 p¼9 p ¼ 11 p¼7 p¼9 p ¼ 11 p¼7 p¼9 p ¼ 11
N¼5
N¼ 13
N ¼21
5.8943e + 19 1.0492e + 26 2.8547e + 32 5.2720e + 19 2.9310e + 25 2.1643e + 31 2.3126e + 20 5.3253e + 26 1.7864e + 33
5.8944e + 19 1.0492e + 26 2.8548e + 32 5.2720e + 19 2.9310e + 25 2.1643e + 31 2.3126e + 20 5.3253e + 26 1.7864e + 33
5.8944e +19 1.0492e+ 26 2.8548e +32 5.2720e+ 19 2.9310e+ 25 2.1643e +31 2.3126e +20 5.3252e +26 1.7864e +33
N is the number of neighborhood for averaging and p is the order of invariants.
invariants of these three signals with Coiflet of order 1 and hyperbolic wavelet are presented in Tables 4 and 5, respectively. It is clear that the change of N does not have any effect on the values of invariants of the signals. The seventh order invariants of signals 1 and 2 are almost the same. However, this similarity is not present in the ninth and 11th order invariants of these two signals. Also, the invariants of signal 3 are totally different from the other two. The other observable fact is that both wavelets are performing equally well in this experiment. Therefore, it is better to use multiple orders of invariants in order to achieve a better discrimination.
A new technique has been proposed for extracting blur invariant features from degraded signals. In this technique, blur invariants are defined in the wavelet domain. Defining the invariants in the wavelet domain provides the opportunity to take advantage of different wavelet functions and extracting features at different scales. It has been proved that blur invariants in the spatial domain are a special case of the proposed invariants. Also, the limitedness of discriminative power of spatial domain invariants does not exist for those in the wavelet domain. In this paper, the performance and robustness of the proposed invariants have been examined through an experiment. In this experiment which was held on 1000 artificially generated signals, the average showed the performance of the proposed invariants is better than that of its counterpart in the spatial domain. In another experiment, the discriminative power of the wavelet based invariants were checked with EEG signal. The results indicated a good power in discriminating between different signals. However, obtaining better results with the proposed invariants depends on the choice of the wavelet function. Scale might also have an impact on effectiveness of the invariants. Since a higher scale is equivalent to a lower resolution, in specific applications these invariants might not be beneficial. It is possible that based on the application and the type of the wavelet function, final results are not as good as what can be obtained with a simpler invariant technique. Therefore, an insightful selection of wavelet functions and the highest engaged scale seems to be unavoidable. Studying different types of wavelet functions, categorizing them based on their properties and their effect on the quality of the invariants, and evaluating effectiveness of the invariants at different scales in recognition problems is the next step that is being taken in order to enlighten different aspects of using this novel definition for blur invariants.
Acknowledgments The authors wish to thank Professor Jan Flusser for excellent comments and suggestions. The authors also thank NSERC for its financial support of the project. References [1] M.I. Sezan, A.M. Tekalp, Survey of recent developments in digital image restoration, Optical Engineering 29 (1990) 393–404. [2] D. Kundur, D. Hatzinakos, Blind image deconvolution, Signal Processing Magazine, IEEE 13 (1996) 43–64. [3] J. Flusser, T. Suk, S. Saic, Image features invariant with respect to blur, Pattern Recognition 28 (1995) 1723–1732. [4] T. Suk, J. Flusser, Features invariant simultaneously to convolution and affine transformation, in: W. Skarbek (Ed.), CAIP, Lecture Notes in Computer Science, vol. 2124, Springer2001, pp. 183–190. [5] J. Flusser, T. Suk, Degraded image analysis: an invariant approach, IEEE Transactions on Pattern Analysis and Machine Intelligence 20 (1998) 590–603.
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[6] C.-Y. Wee, R. Paramesran, Derivation of blur-invariant features using orthogonal legendre moments, Computer Vision, IET 1 (2007) 66–77. [7] H. Zhang, H. Shu, G. Han, G. Coatrieux, L. Luo, J. Coatrieux, Blurred image recognition by legendre moment invariants, IEEE Transactions on Image Processing 19 (2010) 596–611. [8] S. Metari, F. Deschenes, New classes of radiometric and combined radiometric-geometric invariant descriptors, IEEE Transactions on Image Processing 17 (2008) 991–1006. [9] H. Ji, H. Zhu, Degraded image analysis using zernike moment invariants, in: IEEE International Conference on Acoustics, Speech and Signal Processing 2009, ICASSP 2009, 2009, pp. 1941–1944. [10] J. Flusser, T. Suk, Classification of degraded signals by the method of invariants, Signal Processing 60 (1997) 243–249. [11] J. Flusser, J. Boldys, B. Zitova, Moment forms invariant to rotation and blur in arbitrary number of dimensions, IEEE Transactions on Pattern Analysis and Machine Intelligence 25 (2003) 234–246. ¨ Object recognition using frequency domain blur [12] V. Ojansivu, J. Heikkila, invariant features, in: B.K. Ersbøll, K.S. Pedersen (Eds.), SCIA, Lecture Notes in Computer Science, vol. 4522, Springer2007, pp. 243–252. ¨ A method for blur and similarity transform invariant [13] V. Ojansivu, J. Heikkila, object recognition, in: R. Cucchiara (Ed.), ICIAP, IEEE Computer Society2007, pp. 583–588.
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[14] A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill Book Company, New York, 1965. [15] S. Mallat, A Wavelet Tour of Signal Processing, Wavelet Analysis & Its Applications, second ed., Academic Press, 1999. [16] R.G. Andrzejak, K. Lehnertz, F. Mormann, C. Rieke, P. David, C.E. Elger, Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: dependence on recording region and brain state, Physical Review E 64 (2001) 061907. [17] I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Math, 1992, doi:10.2277/0898712742. [18] K.N. Le, K.P. Dabke, G.K. Egan, Hyperbolic wavelet power spectra of nonstationary signals, Optical Engineering 42 (2003) 3017–3037. [19] K.N. Le, K.P. Dabke, G.K. Egan, Hyperbolic wavelet family, Review of Scientific Instruments 75 (2004) 4678–4693. [20] K.N. Le, K.P. Dabke, G.K. Egan, On mathematical derivations of auto-term functions and signal-to-noise ratios of the Choi–Williams, first- and nthorder hyperbolic kernels, Digital Signal Processing 16 (2006) 84–104. [21] F.M. Candocia, Moment relations and blur invariant conditions for finiteextent signals in one, two and n-dimensions, Pattern Recognition Letters 25 (2004) 437–447.
Iman Makaremi obtained his B.Sc. in Electrical Engineering from the University of Tehran, Iran in 2005. He finished his masters in Electrical Engineering at the KN Toosi University of Technology, Tehran, Iran in 2007 in the area of Intelligent Fault Detection. Currently, he is working towards his Ph.D. in Electrical Engineering at the University of Windsor, Canada. His research interest includes geometric and radiometric invariant features and their application in face recognition.
Majid Ahmadi received the B.Sc. degree in Electrical Engineering from Arya Mehr University, Tehran, Iran, in 1971 and the Ph.D. degree in Electrical Engineering from Imperial College of London University, London, UK, in 1977. He has been with the Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON, Canada, since 1980, currently as a Professor and Associate Director of the Research Center for Integrated Microsystems. His research interests include digital signal processing, machine vision, pattern recognition, neural network architectures, applications, and VLSI implementation. He has coauthored the book Digital Filtering in oneand two-Dimensions; Design and Applications (New York: Plenum, 1989) and has published over 300 articles in this area. He is the Associate Editor for the Journal of Pattern Recognition, the Journal of Circuits, Systems and Computers, and the International Journal of Computers and Electrical Engineering. Dr. Ahmadi was the IEEE-CAS representative on the Neural Network Council and the Chair of the IEEE Circuits and Systems Neural Systems Applications Technical Committee. He was a recipient of an Honorable Mention award from the Editorial Board of the Journal of Pattern Recognition in 1992 and received the Distinctive Contributed Paper award from the MultipleValued Logic Conference Technical Committee and the IEEE Computer Society in 2000. He is a Fellow of the Institution of Electrical Engineering.