CHAPTER 3. Derivation of Moment Invariants. Huazhong Shu, Limin Luo and Jean Louis Coatrieux

Transcription

1 CHAPTER 3 Derivation o Moment Invariants Huazhong Shu, Limin Luo and Jean Louis Coatrieux In most computer vision applications, the extraction o key image eatures, whatever the transormations applied and the image degradations observed, is o major importance. A large diversity o approaches has been reported in the literature. This chapter concentrates on one particular processing rame: the moment-based methods. It provides a survey o methods proposed so ar or the derivation o moment invariants to geometric transorms and blurring eects. Theoretical ormulations and some selected examples dealing with very dierent problems are given. Huazhong Shu, Limin Luo Laboratory o Image Science and Technology, School o Computer Science and Engineering, Southeast University, 196 Nanjing, China Centre de Recherche en Inormation Biomédicale Sino-rançais (CRIBs, France shu.list@seu.edu.cn Jean Louis Coatrieux Laboratory o Image Science and Technology, School o Computer Science and Engineering, Southeast University, 196 Nanjing, China INSERM U64, 34 Rennes, France Laboratoire Traitement du Signal et de l'image, Université de Rennes I, 34 Rennes, France Centre de Recherche en Inormation Biomédicale Sino-rançais (CRIBs, France jean-louis.coatrieux@univ-rennes1.r Editor: G.A. Papakostas, Moments and Moment Invariants - Theory and Applications DOI: 1.179/gcsr.vol1.ch3, GCSR Vol. 1, c Science Gate Publishing 14 7

2 8 H. Shu et al. 3.1 Introduction Nowadays, gray level and color images play a central role in many research elds going rom biology, medicine, chemistry, physics up to archeology, geology and others. In the sole engineering area, to take a ew examples, robot control, remote sensing, road trac analysis, enhanced virtual reality, compression and watermarking are concerned. All these elds required tools or extracting, quantiying and interpreting the inormation they convey. Such tools and methods reer to image processing or computer vision at large. They can be classied in a general way according to low-level methods (bottom-up approaches operating on edges or regions or high-level techniques (top-down approaches driven by models o objects and scenes. These image analyses depend on the applications to ace. Image acquisition is a rst and critical step determining the quality o the input data and thereore the perormance o the analysis that will be carried out. A number o dierent physics are available to get the observations required: or instance optical (including inrared camera, X-ray and acoustic sensors. Specic procedures must be designed to generate D and 3D images beore considering any analysis: a good example is ound with Computer Tomography (CT Scanner in medicine, where reconstruction rom projection data has to be perormed rst. Then, and depending on the nature o the data and the targets, key steps should be addressed among which object boundary detection, object segmentation, matching and registration, pattern recognition, shape modeling, texture labeling, motion tracking, classication, etc. Even i major progresses have been recognized over the past decades by the emergence o very diverse methodological rames (or instance, level-set and graph-cut techniques aimed at object segmentation, signicant diculties remain when the images are too noisy or degraded, the object contrasts too low, when we have to ace to deormable objects or to important occlusions. In this very dense landscape o methods and problems [34, 3], the description o object invariants or point-o-interest detection and matching, registration, image orgery detection, image indexing-retrieval and more widely or pattern recognition, is o high signicance. They rst have to deal with geometric transormations such as translation, scale and rotation, and more generally to ane transormation. Image blurring is another major concern. Blur may occur due to wrong ocus, object/camera motion. The perormance o any computer vision system, whatever their target application, relies on the specic eature extraction technique used. A popular class o invariant eatures is based on the moment techniques. As noted by Ghorbel et al. [, 14], the most important properties to assess by the image descriptors are: (1 invariance against some geometrical transormations; ( stability to noise, to blur, to non-rigid and small local deormations; and (3 completeness. The objective o this chapter is to present a comprehensive survey o the state-o-the-art on the moment invariants to geometric transormations and to blur in pattern recognition. Proos o theorems are reerred to already published papers. Some illustrations on dierent applications will be also provided along the chapter.

3 Derivation o Moment Invariants 9 3. Derivation o Moment Invariants to Geometric Transormations In the past decades, the construction o moment invariants and their application to pattern recognition have been extensively investigated. They can be classied into three categories: (1 Normalization techniques [17, 1]; ( Indirect methods [, 11]; (3 Direct approaches. It is well known that the normalization process can be used to achieve the invariance. However, such a process may lead to inaccuracy since the normalization o the images requires re-sampling and re-quantiying. The indirect methods make use o geometric moments or complex moments to achieve the invariance, but they are time expensive due to the long time allocated to compute the polynomial coecients when orthogonal moments are concerned. In order to improve the accuracy and to speed up the computational eciency, many direct algorithms have been reported in the literature. Chong et al. [3] proposed a method based on the properties o pseudo-zernike polynomial to derive the scale invariants o pseudo- Zernike moments. A similar approach was then used to construct both translation and scale invariants o Legendre moments [4]. The problem o scale and translation invariants o Tchebiche moments has been investigated by Zhu et al. [43]. Discrete orthogonal moments such as Tchebiche moments yield better perormance than the continuous orthogonal moments, but the rotation invariants are dicult to derive. To overcome this shortcoming, Mukundan [19] introduced the radial Tchebiche moments, which are dened in polar coordinate system, to achieve the rotation invariance. It was shown that the methods reported in [3, 4, 43, 19] perorm better than the classical approaches such as image normalization and indirect methods. However, it seems dicult to obtain the completeness property by the above mentioned methods since no explicit ormulation is derived or moment invariants. A set o invariant descriptors is said to be complete i it satises the ollowing property: two objects have the same shape i and only i they have the same set o invariants. A number o studies have been conducted on completeness. Flusser et al. proposed a complete set o rotation invariants by normalizing the complex moments [6, 7]. The construction o a complete set o similarity (translation, scale and rotation invariant descriptors by means o some linear combinations o complex moments has been addressed by Ghorbel et al. [14]. This rst part o the chapter will review the ways to construct a complete set o orthogonal moments dened in polar coordinate system, and how to derive a set o moment invariants with respect to ane transormation Derivation o a Complete Set o Orthogonal FourierMellin Moment Invariants It is well known that the moments dened in polar coordinate system including the complex moments, Zernike moments, pseudo-zernike moments [4], orthogonal Fourier- Mellin moments [8], BesselFourier moments [36], can easily achieve the rotation invariance by taking the module o the moments. However, the moment magnitudes do not generate a complete set o invariants. In this section, we present a general

4 6 H. Shu et al. scheme to derive a complete set o radial orthogonal moments with respect to similarity transormation. We take the orthogonal FourierMellin moments (OFMMs as an example or illustration [39]. The D OFMM, Zpq, o order p with repetition q o an image intensity unction (r, θ is dened as [8] Z pq = p + 1 π ˆ πˆ1 where Q p (r is a set o radial polynomials given by Q p (r e jqθ (r, θ rdrdθ, q p, (3.1 with Q p (r = c p,k r k, (3. k= c p,k = ( 1 p+k (p + k + 1! (p k!k! (k + 1! (3.3 Since OFMMs are dened in terms o polar coordinates (r, θ with r 1, the computation o OFMMs requires a linear transormation o the image coordinates to a suitable domain inside a unit circle. Here, we use the mapping transormation proposed by Chong et al. [3], which is shown in Fig.(3.1. Based on this transormation, we have the ollowing discrete approximation o Eq.(3.1: Z pq = p + 1 π N 1 i= N 1 j= Q p (r ij e jqθij (i, j, (3.4 where the image coordinate transormation to the interior o the unit circle is given by ( r ij = (c 1 i + c + (c 1 j + c, θ ij = tan 1 c1 j + c (3. c 1 i + c with c 1 = N 1, c = 1. We now describe a general approach to derive a complete set o OFMM invariants. We use the same method to achieve the translation invariance as described in [14]. That is, the origin o the coordinate system is located at the center o mass o the object to achieve the translation invariance. This center o mass, (x c, y c, can be computed rom the rst geometric moments o the object as ollows x c = m 1 m, y c = m 1 m, (3.6 where m pqare the (p + q-th order geometric moments dened by m pq = ˆ ˆ x p y q (x, y dxdy. (3.7

5 Derivation o Moment Invariants 61 Figure 3.1: Mapping image inside the unit circle. Let U m (r = (Q (r, Q 1 (r,..., Q m (r T and M m (r = ( r, r 1,..., r m T be two vectors, where the superscript T indicates the transposition, we have U m (r = C m M m (r, (3.8 where C m = (c i,j, with j i m, is a (m + 1 (m + 1 lower triangular matrix whose element c i,j is given by Eq.(3.3. Since all the diagonal elements o C m, c l,l = (l+1! l!(l+1!, are not zero, the matrix C m is non-singular, thus M m (r = (C m 1 U m (r = D m U m (r, (3.9 where D m = (d i,,j, with j i m, is the inverse matrix o C m. It is also a (m + 1 (m + 1 lower triangular matrix. The computation o the elements o D m is given in the ollowing Proposition. Proposition 1. For the lower triangular matrix C m whose elements c i,j are dened by Eq.(3.3, the elements o the inverse matrix D m are given by d i,j = (j + i! (i + 1! (i + 1! (i + j +!. (3.1 The proo o Proposition 1 can be ound in [39]. Let and g be two images display the same pattern but with distinct orientation β and scale λ, i.e., g (r, θ = (r/λ, θ β. The OFMM o the image intensity unction g (r, θ is dened as Z g pq = p + 1 π ˆ πˆ1 = λ e jqβ p + 1 π Q p (r e jqθ g (r, θ rdrdθ ˆ πˆ1 Q p (λr e jqθ (r, θ rdrdθ, (3.11

6 6 H. Shu et al. Letting U m (λr = (Q (λr, Q 1 (λr,..., Q m (λr T, ( M m (λr = 1, (λr 1,..., (λr m T, it can be seen rom Eq.(3.8 that On the other hand, U m (λr = C m M m (λr. (3.1 M m (λr = diag ( 1, λ 1,..., λ m ( 1, r 1,..., r m T = diag ( 1, λ 1,..., λ m M m (r. Substituting Eq.(3.13 and Eq.(3.9 into Eq.(3.1, we obtain (3.13 U m (λr = C m diag ( 1, λ 1,..., λ m D m U m (r. (3.14 By expanding Eq.(3.14, we have Q p (λr = Q k (r λ l c p,l d l,k. (3.1 k= l=k With the help o Eq.(3.1, Eq.(3.11 can be rewritten as Z g pq = λ e jqβ p + 1 π = λ e jqβ p + 1 π = e jqβ k= = e jqβ k= ˆ πˆ1 ˆ πˆ1 Q p (λr e jqθ (r, θ rdrdθ ( Q k (r k= λ l c p,l d l,k e jqθ (r, θ rdrdθ l=k p + 1 k + 1 k + 1 ˆ λ l c p,l d l,k π l=k ( p + 1 λ l+ c p,l d l,k Z k + 1 pq. l=k πˆ1 Q k (r e jqθ (r, θ rdrdθ (3.16 The above equation shows that the D scaled and rotated OFMMs, Zpq g, can be expressed as a linear combination o the original OFMMs Zpq with k p. By using this relationship, we can construct a complete set o both rotation and scale invariants Ipq, which is described in the ollowing theorem.

7 Derivation o Moment Invariants 63 Theorem 1. For a given integer q and any positive integer p, let ( Ipq = e jqθ p + 1 Γ (l+ c p,l d l,k Z k + 1 pq, (3.17 k= l=k ( with θ = arg Z 11 and Γ = Z. Then I pq, is invariant to both image rotation and scaling. The reader can reer to [39]or the proo o Theorem 1. Equation 3.17 can be expressed in matrix orm as I q I 1q... I pq ( =e jqθ diag (1,,..., p + 1 C p diag ( D p diag 1, 1,..., 1 p + 1 Z q Z 1q... Z pq. Γ, Γ 3,..., Γ (p+ (3.18 It is easy to veriy that the set o invariants is complete by rewriting Eq.(3.18 as Z q Z 1q... Z pq ( =e jqθ diag (1,,..., p + 1 C p diag ( D p diag 1, 1,..., 1 p + 1 I q I 1q... I pq Γ, Γ 3,..., Γ (p+. (3.19 Thus, we have Z pq = k= ( e jqθ p + 1 Γ (l+ c p,l d l,k I k + 1 pq. (3. l=k

8 64 H. Shu et al. The above equation shows that the set o invariants is complete. Note that the above approach is general enough to be extended to any other radial orthogonal moments. The only dierence is that or a given matrix C p whose elements correspond to the coecients o a set o radial polynomials up to order p, we need to nd its inverse matrix, D p. 3.. Derivation o Ane Invariants by Orthogonal Legendre Moments Ane moment invariants (AMIs are useul eatures o an image since they are invariant to general linear transormations o an image. The AMIs were introduced independently by Reiss [] and Flusser and Suk [9]. Since then, they have been utilized as pattern eatures in a number o applications such as pattern recognition, pattern matching, image registration and contour shape estimation. The construction o ane moment invariants has been also extensively investigated. The existing methods can be generally classied into two categories: (1 direct method, and ( image normalization. Among the direct methods, Reiss [] and Flusser and Suk [9] derived the AMIs based on the theory o algebraic invariants and tensor techniques. Suk and Flusser [3] used a graph method to construct the ane moment invariants. Liu et al. [18] proposed an automatic method or generating the ane invariants. Normalization is an alternative approach to derive the moment invariants. An ane normalization approach was rst introduced by Rothe et al. [6]. In their work, two dierent ane decompositions were used. The rst known as XSR decomposition consists o two skews, anisotropic scaling and rotation. The second is the XYS and consists o two skews and anisotropic scaling. Zhang et al. [41] perormed a study o these ane decompositions and pointed out that both decompositions lead to some ambiguities. More details on these decompositions will be given below. Pei and Lin [] presented an ane normalization or asymmetrical object and Suk and Flusser [33] dealt with symmetrical object. Zhang and Wu [4] extended the method to Legendre moments. Shen and Ip [7] used the generalized complex moments and analyzed their behavior in recognition o symmetrical objects. Almost all the existing methods to derive the ane invariants are based on geometric moments and complex moments. Since the kernel unctions o both geometric moments and complex moments are not orthogonal, this leads to inormation redundancy when they are used to represent the image. This motivates us to use the orthogonal moments in the construction o ane moment invariants. We present here a new method to derive a set o ane invariants based on the orthogonal Legendre moments. The D (p + q-th order Legendre moment o an image unction (x, y is dened as [3] ˆ1 L ( pq = ˆ1 1 1 P p (x P q (x (x, y dxdy, p, q =, 1,,..., (3.1 where P p (x is the p-th order orthonormal Legendre polynomials given by

9 Derivation o Moment Invariants 6 with P p (x = c p,k x k, (3. k= p+1 ( 1 p k (p+k! c p,k = p ( p k!(, p k = even p+k!k!. (3.3, p k odd It can be deduced rom Eq.(3. that x p = d p,k P k (x, (3.4 k= where D M = (d p,k, k p M, is the inverse matrix o the lower triangular matrix C M = (c p,k. The elements o D M are given by [3] 3k p k+1 d p,k = ( p k! p!k! (p k/ j=1 (k+j+1 (k!, p k = even. (3., p k odd The orthogonality property leads to the ollowing inverse moment transorm (x, y = i= j= P i (x P j (x L ( ij. (3.6 I only the moments o order up to (M, M are computed, Eq.(3.6 is approximated by (x, y M M i= j= P i (x P j (x L ( ij. (3.7 In the next, we will establish a relationship between the Legendre moments o an ane transormed image and those o the original image. The ane transormation can be represented by [6] ( ( ( x x x y = A +, (3.8 y y ( a11 a where A = 1 is called the homogeneous ane transormation matrix. a 1 a The translation invariance can be achieved by locating the origin o the coordinate system to the center o mass o the object, that is, L ( 1 = L( 1 =. Thus, (x, y can be ignored and only the matrix A is taken into consideration in the ollowing. The D (p + q-th order Legendre moment o the ane transormed image (x, y is dened by

10 66 H. Shu et al. ˆ1 L (g p,q = ˆ1 1 1 ˆ1 = det (A P p (x P q (y (x, y dx dy ˆ1 1 1 P p (a 11 x + a 1 y P q (a 1 x + a y (x, y dxdy, (3.9 where det (A denotes the determinant o the matrix A. In the ollowing, we discuss the way to express the Legendre moments o the ane transormed image dened by Eq.(3.9 in terms o Legendre moments o the original image. By replacing the variable x by a 11 x + a 1 y in Eq.(3., we have Similarly P p (a 11 x + a 1 y = = P q (a 1 x + a y = c p,m (a 11 x + a 1 y m m= m= s= n= t= m ( m s n ( n t Substituting Eq.(3.3 and Eq.(3.31 into Eq.(3.9 yields c p,m a s 11a m s 1 x s y m s. (3.3 c q,n a t 1a n t xt y n t. (3.31 ˆ1 L (g p,q = det (A Using Eq.(3.4, we have ˆ1 m q 1 1 m= s= n= t= n ( m s ( n t c p,m c q,n a s 11a m s 1 a t 1a n t xs+t y m+n s t (x, y dxdy. (3.3 s+t x s+t = d s+t,i P i (x, y m+n s t = i= m+n s t j= Substitution o Eq.(3.33 into Eq.(3.3 leads to d m+n s t,j P j (y. (3.33 L (g p,q = det (A q m n s+t m= n= s= t= i= m+n s t j= ( m s ( n t (a 11 s (a 1 m s (a 1 t (a n t c p,m c q,n d s+t,i d m+n s t,j L ( ij. (3.34

11 Derivation o Moment Invariants 67 Equation 3.34 shows that the Legendre moments o the transormed image can be expressed as a linear combination o those o the original image. The direct use o Eq.(3.34 leads to a complex non-linear systems o equations. To avoid this, the homogeneous ane transormation matrix A is usually decomposed into a product o simple matrices. The decompositions mentioned above, known as XSR and XYS decompositions [6, 41], can be used. The XSR decomposition decomposes the ane matrix A into an x-shearing, an anisotropic scaling and a rotation matrix as ollows [ a11 a 1 a 1 a ] [ cos θ sin θ = sin θ cos θ ] [ λ µ ] [ 1 ρ 1 ], (3.3 where λ, µ and ρ are real numbers, and θ is the rotation angle between and π. The XYS decomposition relies on decomposing the ane matrix A into an x- shearing, an y-shearing and an anisotropic scaling matrix, that is [ a11 a 1 a 1 a ] = [ a δ ] [ 1 γ 1 ] [ 1 β 1 ], (3.36 where a, β, δ and γ are real numbers. We adopt here the XYS decomposition. The same reasoning can be applied to the XSR decomposition. Using Eq.(3.34, we can derive a set o ane Legendre moment invariants (ALMIs based on the ollowing theorems. Theorem. For given integers p and q, let I xsh pq = q m s m= n= s= i= m+n s j= ( m s β m s c p,m c q,n d s,i d m+n s,j L ij, (3.37 then I xsh pq are invariant to x-shearing. The proo o Theorem is available in [37]. Theorem 3. Let I ysh pq = q n m+t n t m= n= t= i= j= ( n t γ t c p,m c q,n d m+t,i d n t,j L ij, (3.38 then I ysh pq are invariant to y-shearing. The proo o Theorem 3 is very similar to that o Theorem as pointed out in [37]. Theorem 4. Let I αs pq = q m m= n= i= j= n α m+1 δ n+1 c p,m c q,n d m,i d n,j L ij, (3.39

12 68 H. Shu et al. then I αs pq are invariant to anisotropic scaling. Reer also to [37] or the proo o Theorem 4. the ollowing theorem. Notice that we can easily derive Theorem. The Legendre moments o an image can be expressed as a linear combination o their invariants as ollows: L pq = L pq = L pq = q m s m= n= s= i= q n m+t m+n s j= m= n= t= i= j= q m n m= n= i= j= n t ( n t ( m s ( β m s c p,m c q,n d s,i d m+n s,j Iij xsh (3.4, ( γ t c p,m c q,n d m+t,i d n t,j I ysh ij, (3.41 α (m+1 δ (n+1 c p,m c q,n d m,i d n,j I αs ij. (3.4 The above equations show that the set o invariants is complete. From this standpoint, by combining Ipq xsh, Iysh pq, Iαs pq, that are respectively invariant to x-shearing, y-shearing and anisotropic scaling, we can obtain a set o ALMIs. For an image (x, y, we use the ollowing process: Step 1: x-shearing Legendre moment invariants are calculated by Eq.(3.37, where the Legendre moments L ij are computed with Eq.(3.1. Step : The combined invariants with respect to x-shearing and y-shearing are calculated by Eq.(3.38 where the Legendre moments on the right-hand side o Eq.(3.38 are replaced by computed in Step 1. Step 3: The ane Legendre moment invariants are calculated by Eq.(3.39 where the Legendre moments on the right-hand side o Eq.(3.39 are replaced by computed in Step. Because the parameters β, γ, α and δ in Eqs.(3.37-(3.39 are image dependent, they must be estimated. Considering an ane transorm and its XYS decomposition, by setting I3 xsh = in Eq.(3.37, we have 1 1 L 3 β 3 + L 1 β + L 1 β + L 3 =. ( The parameter β can then be determined by solving Eq.(3.43. From Eq.(3.38, we have I ysh 11 = γ Letting I ysh 11 =, we obtain ( L + L + L 11. (3.44 L 11 γ = L +. (3.4 L

13 Derivation o Moment Invariants 69 Setting I as = I as = 1, we have b + 4a b b + 4a α =, δ = b a a, (3.46 where a = V 3 U, b = L V U, a = U 3 V, b = L U V, U = L + L, V = L + L. (3.47 The parameters β g, γ g, α g and δ g associated with the transormed image g (x, y can also be estimated according to Eqs.(3.43, (3.4 and (3.46. It can be veried that the parameters provided by the above method satisy the ollowing relationships: β = β g + β, γ = γ g + γ, α = α α g and δ = δ δ g, where α, β, γ and δ are the coecients o the ane transorm applied to. Based on these relationships, the conditions given in Theorems to 4 are satised. It is worth noting that other choice o parameters can also be made to keep the invariance o Eqs.(3.37-(3.39 to image transormation. To illustrate this section, a result extracted rom a watermarking application has been selected [37]. Image watermarking aimed at responding to copyright protection concerns. To be ecient, such watermarking must be robust against a variety o attacks among which geometric distortions. In [37], ane invariants derived rom Legendre moments were used. Watermark embedding and detection were directly perormed on this set o invariants. Moreover, these moments were exploited or estimating the geometric distortion parameters in order to permit watermark extraction. Figure 3..a shows the watermark used. It was embedded in our standard images urther attacked by ane transormations (Fig.(3..b to (3..e. The size o the watermark image in that case is equal to the initial sizes o embedding images. The extracted watermarks in these respective images are depicted in Fig.(3.. to (3..i and it can easily be seen that they are correctly recovered even i a certain loss in quality is observed. 3.3 Derivation o Moment Invariants to Blur, and Combined Invariants to Geometric Transormation and to Blur Because the real sensing systems are usually imperect and the environmental conditions are changing over time, the acquired images oten provide a degraded version o the true scene. An important class o degradations we are aced with in practice is image blurring, which can be caused by diraction, lens aberration, wrong ocus, and atmospheric turbulence. Blurring can be usually described by a convolution o an unknown original image with a space invariant point spread unction (PSF. In pattern recognition, or instance, two options have been widely explored either through a two-step approach by restoring the image and then applying recognition methods, or by designing a direct one-step solution, ree o blurring eects. In the ormer case, the

14 7 H. Shu et al. Figure 3.: (a Logo used as watermark. (b-(e Watermarked images under ane transormation. (-(i Extracted watermark rom (b-(e. With permission. point spread unction, most oten unknown in real applications, should be estimated. In the latter case, nding a set o invariants that are not aected by blurring is the key problem. The pioneering work in this eld was perormed by Flusser et al. [1] who derived invariants to convolution with an arbitrary centrosymmetric PSF. These invariants have been successully used in template matching o satellite images, in pattern recognition, in blurred digit and character recognition, in normalizing blurred images into canonical orms, and in ocus/deocus quantitative measurement. More recently, Flusser and Zitova introduced the combined blur-rotation invariants [1] and reported their successul application to satellite image registration and camera motion estimation. Suk and Flusser urther proposed a set o combined invariants which are invariant to ane transorm and to blur [31]. The extension o blur invariants to N-dimensions has also been investigated [8, 1]. All the existing methods to derive the blur invariants are based on geometric moments or complex moments. However, both geometric moments and complex moments contain redundant inormation and are sensitive to noise especially when high-order moments are concerned. This is due to the act that the kernel polynomials are not orthogonal. Since the orthogonal moments are better than other types o moments in terms o inormation redundancy, and are more robust to noise, it could be expected that the use o orthogonal moments in the construction o blur invariant provides better recognition results. The second part o this chapter is aimed at showing how to construct a set o blur invariants by means o orthogonal moments.

15 Derivation o Moment Invariants Derivation o Invariants to Blur by Legendre Moments We rst review the theory o blur invariants o geometric moments reported in [3, 37], and then we present some basic denitions o Legendre moments. A. Blur Invariants o Geometric Moments The D geometric central moment o order (p + q, with image intensity unction (x, y, is dened as ˆ1 µ ( pq = ˆ1 1 1 ( x x ( c p ( y y ( c q (x, y dxdy, (3.48 where, without loss o generality, ( we assume that the image unction (x, y is dened on the square [ 1, 1] [ 1, 1]., y c ( denotes the centroid o (x, y, which is x ( c dened by Eq.(3.6. Let g (x, y be a blurred version o the original image (x, y. classically described by the convolution The blurring is g (x, y = ( h (x, y, (3.49 where h (x, y is the PSF o the imaging system, and denotes the linear convolution. Assuming that the PSF, h (x, y, is a centrally symmetric image unction and the imaging system is energy-preserving, that is h (x, y = h ( x, y, (3. ˆ1 ˆ1 1 1 h (x, y dxdy = 1. (3.1 As noted by Flusser et al. [1], the assumption o centrally symmetry is not a signiicant limitation o practical utilization o the method. Most real sensors and imaging systems have PSFs with certain degrees o symmetry. In many cases they have even higher symmetry than central, such as axial or radial symmetry. Thus, the central symmetry assumption is general enough to describe almost all practical situations. Lemma 1 [1]. The centroid o the blurred image g (x, y is related to the centroid o the original image (x, y and that o the PSF h (x, y as x (g c y c (g = x ( c = y c ( + x (h c, + y c (h, (3. In particular, i h (x, y is centrally symmetric, then x (h c case, we have x (g c = x ( c, y c (g = y c (. = y (h c =. In such a

16 7 H. Shu et al. B. Blur Invariants o Legendre Moments The D (p + q-th order Legendre central moment o image intensity unction (x, y, is dened as ˆ1 L ( pq = ˆ1 1 1 ( ( P p x x ( P q c y y ( c (x, y dxdy, p, q =, 1,,..., (3.3 where P p (x is the p-th order orthonormal Legendre polynomials given by Eq.(3.. In the ollowing, we rst establish a relationship between the Legendre moments o the blurred image and those o the original image and the PSF. We then derive a set o blur moment invariants. The D normalized Legendre moments o a blurred image, g(x, y, are dened by ˆ1 L (g p,q = = = = ˆ1 1 1 ˆ1 ˆ1 1 1 ˆ1 ˆ1 1 1 ˆ1 ˆ1 1 1 P p (x P q (y g (x, y dxdy P p (x P q (y ( h (x, y dxdy P p (x P q (y h (a, b ˆ1 ˆ1 1 1 ˆ ˆ h (a, b (x a, y b dadb dxdy P p (x + a P q (y + b (x, y dxdy dadb. (3.4 The ollowing theorem reveals the relationship between the Legendre moments o the blurred image and those o the original image and the PSF. Theorem 6. Let (x, y be the original image unction and the PSF h (x, y be an arbitrary image unction, and g (x, y be a blurred version o (x, y, then the relations and L (g p,q = q p i q j p s L (h s,t q t L ( i,j i= j= s= t= (3. k=i m=k+s l=j n=l+t c p,m c q,n d k,i d m k,s d l,j d n l,t q ( m k ( n l L (g p,q = q i= j= L ( i,j p i q j s= t= L (h p s s,t c p,m c q,n d k,i d m k,s d l,j d n l,t. q t q k=i m=k+s l=j n=l+t ( m k ( n l (3.6

17 Derivation o Moment Invariants 73 hold or every p and q. We urther have the ollowing result. Theorem 7. I h (x, y satises the conditions o central symmetry, then (a L (h p,q = L (h p,q or every p and q; (b L (h p,q = i (p + q is odd. The proo o the above two theorems can be ound in [38]. With the help o Theorems 6 and 7, we are now ready to construct a set o blur invariants o Legendre moments through the ollowing theorem. Theorem 8. Let (x, y be an image unction. Let us dene the ollowing unction I ( : N N R. I (p + q is even then I (p, q ( =. i (p + q is odd then I (p, q ( =L ( p,q 1 L (, p s q t q i= j= <i+j<p+q q k=i m=k+s l=j n=l+t ( m k p i q j I (i, j ( ( n l s= t= L ( s,t c p,m c q,n d k,i d m k,s d l,j d n l,t. (3.7 Thereore, I (p, q is invariant to centrally symmetric blur or any p and q. The number (p + q is called the order o the invariant. The proo o Theorem 8 is also given in [38]. Using the Legendre central moments instead o Legendre moments, we can obtain a set o invariants to translation and to blur which are ormally similar to I (p, q (. Theorem 9. Let (x, y be an image unction. Let us dene the ollowing unction I ( : N N R. I (p + q is even then I (p, q ( =. i (p + q is odd then

18 74 H. Shu et al. Figure 3.3: Eight objects selected rom the Coil-1 image database o Columbia University. With permission. Figure 3.4: Some examples o the blurred images corrupted by various types o noise. With permission. I (p, q ( =L ( p,q 1 L (, p s q t q i= j= <i+j<p+q q k=i m=k+s l=j n=l+t ( m k p i q j I (i, j ( ( n l s= t= L ( s,t c p,m c q,n d k,i d m k,s d l,j d n l,t. (3.8 Thus I (p, q (, is invariant to centrally symmetric blur and to translation or any p and q. It should be noted that I (p, q in Eq.(3.8 deals with translation o both the image and the PSF. Based on Eq.(3.8, we can construct a set o blur and translation invariants o Legendre moments and express them in explicit orm. The invariants o the third, th and seventh orders are listed in Appendix. Partial results o the method reported in [38] or blurred image recognition by Legendre moment invariants will serve as an example o what can be expected in terms o perormance. The original images were selected rom the Coil-1 image database o Columbia University (size 16 16, which are shown in Fig.(3.3. They were blurred by means o averaging blur, out-o-ocus blur, Gaussian blur and motion blur with dierent mask sizes and corrupted by additive Gaussian noise or salt-and-pepper noise (see Fig.(3.4 or some samples. Table 3.1 provides a comparison between geometric moments invariants (GMI, complex moment invariants (CMI and Legendre moment invariants (LMI. Only the highest levels o noise are considered here. This table shows that all moments behave well in noise-ree situations. Relatively poor classication rates are observed when the noise level is high. LMIs perom better

19 Derivation o Moment Invariants 7 Table 3.1: The recognition rates o the GMI, CMI and LMI in object recognition (Fig.4. With permission. GMI CMI LMI Noise-ree 1% 1% 1% Additive white noise with STD= % 8% 96.% Additive white noise with STD= 6.4%.6% 74.79% Additive salt-and-pepper noise with noise density = % 6.46% 79.37% Additive multiplicative noise with noise density =. 9% 81.88% 9.63% Computation time 9.4s 44.14s 9.8s than GMIs and CMIs. The classication rates are higher than 9% but show insucient perormance in some cases. The computational times are also indicated. The implementations were done in MATLAB 6. on a PC P4.4 GHZ, 1 M RAM. More detailed experimentations and results can be ound in [38] Combined Invariants to Similarity and to Blur by Zernike Moments We have proposed in the previous section an approach based on the orthogonal Legendre moments to derive a set o blur invariants. It has been shown in [38] that they are more robust to noise and have better discriminative power than the existing methods. However, one weak point o Legendre moment descriptors is that they are only invariant to translation, but not invariant under image rotation and scaling. Zhu et al. [4] and Ji and Zhu [1] proposed the use o the Zernike moments to construct a set o combined blur-rotation invariants. Unortunately, there are two limitations to their methods: (1 only a Gaussian blur has been taken into account, which is a special case o PSF having circularly symmetry; ( only a subset o Zernike moments o order p with repetition p, Z p,p, has been used in the derivation o invariants. Since Z p,p corresponds to the radial moment D p,p or the complex moment C,p i neglecting the normalization actor, the set o invariants constructed by Zhu et al. is a subset o that proposed by Flusser [13]. Here, we propose a new method to derive a set o combined geometric-blur invariants based on orthogonal Zernike moments. We urther assume that the applied PSF is circularly symmetric. The reasons or such a choice o PSF are as ollows [13]: (1 the majority o the PSFs occurring in real situations exhibit a circular symmetry; ( since the PSFs having circular symmetry are a subset o centrosymmetric unctions, it could be expected that we can derive some new invariants. In act, the previously reported convolution invariants with centrosymmetric PSF include only the odd order moments. Flusser and Zitova [13] have shown that there exist even order moment invariants with circularly symmetric PSF. The radial moment o order p with repetition q o image intensity unction (r, θ is dened as

20 76 H. Shu et al. D ( p,q = ˆ πˆ1 r p e ĵqθ (r, θ rdrdθ, (3.9 with ĵ = 1, r 1, p, q =, ±1, ±,...The Zernike moment o order p with repetition q o (r, θ is dened as [3] Z ( p,q = p + 1 π ˆ πˆ1 R p,q (r e ĵqθ (r, θ rdrdθ, p, q p, p q being even, where R p,q (ris the real-valued radial polynomial given by R p,q (r = (p q / k= k! ( 1 k (p k!! ( p+ q k ( p q k (3.6 r p k. (3.61! The above equation shows that the radial polynomial R p,q (r is symmetric with q, that is, R p, q (r = R p,q (r, or q. Thus, we can consider the case where q. Letting p = q + l in Eq.(3.61 with l, and substituting it into Eq.(3.6 yields Z ( q+l,q = q + l + 1 π where ˆ πˆ1 [ l ( 1 k k= k= ] (q + l k! k! (q + l k! (l k! rq+(l k e ĵqθ (r, θ rdrdθ = q + l + 1 ˆπ ˆ1 [ l ] ( 1 l k (q + l + k! π k! (q + k! (l k! rq+k = e ĵqθ (r, θ rdrdθ l k= ( 1 l k q + l + 1 π e ĵqθ (r, θ rdrdθ l = c q l,k D( q+k,q, k= ˆ (q + l + k! k! (q + k! (l k! πˆ1 r q+k (3.6 c q l,k = ( 1l k q + l + 1 π (q + l + k! k! (q + k! (l k! (3.63 Let be a rotated version o, i. e. (r, θ = (r, θ β, where β is the angle o rotation, and let Z ( q+l,q be the Zernike moments o. It can be seen rom Eq.(3.61 that

21 Derivation o Moment Invariants 77 Z ( q+l,q = e ĵqβ Z ( q+l,q. (3.64 Let g (x, y be a blurred version o the original image, and h (x, y be the PSF o the imaging system. We assume that the PSF, h (x, y, is a circularly symmetric image unction, and that the imaging system is energy-preserving, which leads to h (x, y = h (r, θ = h (r, (3.6 ˆ ˆ h (x, y dxdy = 1. (3.66 Under the assumption o Eq.(3.6, the Zernike moments oh (r, θ equal those o any rotated image h. Combining this act with Eq.(3.64, we get Z (h q+l,q = Z(h q+l,q = e ĵqβ Z (h q+l,q. (3.67 Equation 3.67 is veried i and only i either Z (h q+l,q = or q =. Thus, an important property o circularly symmetric unctions can be stated as ollows. Proposition. I q and h (r, θ is a circularly symmetric image unction, then = or any non-negative integer l. Z (h q+l,q The proo o Proposition can be ound in []. We can now establish the relationship between the Zernike moments o the blurred image and those o the original image and the PSF. To that end, we rst consider the radial moments. Applying Eq.(3.9 to blurred image g (x, y, we have with D (g q+k,q = = ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = T (a, b = ˆ ( q+k ( k x ĵy x + ĵy g (x, y dxdy ( q+k ( k x ĵy x + ĵy h (a, b (x a, y b dadb dxdy h (a, b T (a, b dadb ˆ (( x + ĵy (( ( q+k x ĵy + a ĵb ( k + a + ĵb (x, y dxdy. (3.68

22 78 H. Shu et al. Finally, Eq.(3.68 take the orm q+k D (g q+k,q = k m= n= ( q + k m ( k n D ( m+n,m nd (h q+k m n,q+n m. (3.69 Applying Eq.(3.6 to blurred image g (x, y = g (r, θ and using Eq.(3.69, we obtain Z (g q+l,q = l q+k k k= m= n= ( q + k m ( k n c q l,k D( m+n,m nd (h q+k m n,q+n m. (3.7 From Eq.(3.6, the radial moments can also be expressed as a series o Zernike moments D ( q+l.q = l k= d q l,k Z( q+k,q, (3.71 where D q l = ( d q i,j, j i l, is the inverse matrix o C q l = ( c q i,j. Both C q l and D q l are lower triangular matrices o size (l + 1 (l + 1, the elements o C q l are given by Eq.(3.63. The elements o D q l are given by [9] d q i,j = From Eq.(3.71, we have i! (q + i!π, j i l. (3.7 (i j! (q + i + j + 1! D ( m+n,m n = n i= k n D (h q+k m n,q+n m = j= d m n n,i Z ( m n+i,m n, (3.73 d q+n m k n,j Z (h q+n m+j,q+n m, (3.74 By introducing Eq.(3.73 and Eq.(3.74 into Eq.(3.7, we obtain Z (g q+l,q = l q+k k n k n k= m= n= i= j= c q l,k dm n n,i ( q + k m ( k n d q+n m k n,j Z ( m n+i,m n Z(h q+n m+j,q+n m. Based on Eq.(3.7, we have the ollowing theorem. (3.7 Theorem 1. Let (r, θ be the original image unction and the PSF h (r, θ be circularly symmetric, g (r, θ be a blurred version o (r, θ, then the ollowing relation

23 Derivation o Moment Invariants 79 Z (g q+l,q = l i= l i Z ( q+i,q j= Z (h j,a (q, l, i, j, (3.76 stands or any q and l, where the coecients A (q, l, i, j are given by A (q, l, i, j = l k j k=i+j n=i ( q + k q + n ( k n c q l,k dq n,i d k n,j. (3.77 Its proo can also be ound in []. Based on Theorem 1, it becomes possible to construct a set o blur invariants o Zernike moments through the ollowing theorem. Theorem 11. Let (r, θ be an image unction. Let us dene the ollowing unction I ( : N N R. I (q + l, q ( = Z ( q+l,q 1 Z (, π l 1 l i I (q + i, q ( i= j= Z ( j,,a (q, l, i, j. (3.78 Then I (q + l, q ( is invariant to circularly symmetric blur or any q and l. The number p = q + l is called the order o the invariant. The proo o Theorem 11 has been reported in []. Some remarks deserve to be made. Remark 1. By using the symmetric property o R p,q (r with q, it can be easily proven that I ( q + l, q ( or q < is also invariant to convolution. Remark. It can be deduced rom Eq.(3.78 that I (l, ( = ( 1 l (l + 1 Z (,. Thus, only I (, ( = Z (, will be used as invariant or the case q =. Remark 3. I we use the Zernike central moments Z ( q+l,q instead o Zernike moments in Eq.(3.78, then we can obtain a set o invariants I (q + l, q ( that is invariant to both translation and to blur. Based on Theorem 11, we can construct a set o blur invariants o Zernike moments with arbitrary order and express them in explicit orm. The invariants up to sixth order are listed in the Appendix. Lemma. Let be a rotated version o, i.e., (r, θ = (r, θ β, where β denotes the rotation angle, then the ollowing relation holds or any q and l I (q + l, q ( = e ĵqβ I (q + l, q (. (3.79

24 8 H. Shu et al. Lemma 3. Let (r, θ be an image unction. It holds or any q and l that I (q + l, q ( = I (q + l, q (, (3.8 where the superscript denotes the complex conjugate. The proo o the two lemmas can be ound in []. In the ollowing, we construct a set o combined geometric-blur invariants. As already stated, the translation invariance can be achieved by using the central Zernike moments. Equation 3.79 shows that the magnitude o I (q + l, q ( is invariant to rotation. However, the magnitudes do not yield a complete set o the invariants. Herein, we provide a way to build up such a set. Let and be two images having the same content but distinct orientation β and scale λ, that is, (r, θ = (r/λ, θ β, the Zernike moment o the transormed image is given by Z ( q+l,q = q + l + 1 π ˆ πˆ1 ĵqβ q + l + 1 = e π R q+l,q (r e ĵqθ (r/λ, θ β rdrdθ λ ˆ πˆ1 R q+l,q (λr e ĵqθ (r, θ rdrdθ. (3.81 Using Eq.(3.6 and Eq.(3.71 we have Z ( q+l,q = e ĵqβ = e ĵqβ = e ĵqβ l k= l λ q+k+ c q l,k D( q+k,q k k= m= l l m= k=m Thereore, we have the ollowing theorem: λ q+k+ c q l,k dq k,m Z( q+m,q λ q+k+ c q l,k dq k,m Z( q+m,q. (3.8 Theorem 1. Let L ( q+l,q = e ĵqθ l l m= k=m Γ (q+k+ c q l,k dq k,m Z( q+m,q (3.83 ( with θ = arg Z ( 1,1 and Γ = Z (,. Then L( q+l,q is invariant to both image rotation and scaling or any non-negative integers q and l.

25 Derivation o Moment Invariants 81 Figure 3.: Images o the outdoor scene. (a The original image, (b The transormed and blurred image, (c The matched templates using CMIs, (d The matched templates using the proposed ZMIs With permission. Remark 4. Many other choices o θ and Γ are possible. In act, they can be chosen in such a way that we Γ = λγ, θ = θ β where (r, θ = (r/λ, θ β is the transormed image o. However, it is preerable to use the lower order moments because they are less sensitive to noise than the( higher order ones. I the central moments are used, θ can be chosen as θ = arg Z 3,1. Theorem 13. For any q and l, let SI (q + l, q ( = e ĵqθ l l m= k=m Γ (q+k+ c q l,k dq k,m I (q + m, q(, (3.84 where I (q + m, q ( is dened in Eq.(3.78. Then SI (q + l, q ( is both invariant to convolution and to image scaling and rotation. The proos o Theorems 1 and 13 are given in []. The combined invariants up to sixth order are listed in Appendix. For illustration purpose o this section, we take one o the examples depicted in []. It relies on a set o combined geometric-blur invariants derived rom Zernike moments and aims at localizing templates in a real outdoor scene displayed in Fig.(3.. The initial images were acquired by a standard digital camera submitted to a rotation and an out-o-ocus blur. Nine templates were extracted in the reerence image (numbered rom 1 to 9 in the let image. They were then searched in the transormed and blurred image by using a template matching procedure. The results show that the ZMIs lead to nd all templates when CMIs ail to only detect part o them. 3.4 Conclusion Although orthogonal moments and their respective invariants have attracted much interest all over the world these last years with both impressive theoretical contributions

26 8 H. Shu et al. and multiple applications, it is expected that this trend will continue and will open new paths in computer vision at large. The complexity o situations to handle (dense scenes, non-geometric/unstructured objects, large deormations, occlusions, projective transormations, etc, the strong competition with other methodological approaches (dierential invariants, multi-scale methods, shape-based algorithms, shape rom texture, etc will continuously challenge the moment-based techniques. I most o the papers recently published are comparing moment-based methods (Zernike to Legendre and the like, more comparisons with dierent rames are required in order to assess when and where they bring a real superiority. Criteria such that robustness (to noise, deormations, blur, and so on, accuracy in localization (i.e. critical in matching and registration, distinctiveness (or discrimination purpose and others can be considered or that. Let us recall that moment invariants do solve only partially the computer vision problems. They have to be integrated in the ull image processing scheme (rom image acquisition to image interpretation. Specic constraints (large data sets to deal with, real-time or almost real-time processing must also be taken into consideration to address practical problems: accelerated algorithms deserve interest to reduce the computation load. Following [16], special attention should be paid to a proper implementation o moment-based algorithms to preserve their orthogonality properties. Acknowledgements The authors are indebted to their master and PhD students Hui Zhang, Beijing Chen, Guanyu Yang, Jiasong Wu, Yining Hu and Hongqing Zhu or their contributions in the development o moment-based methods and the experiments they conducted to assess their interest in various applications. These works have been made possible with the support o numerous grants among which the National Basic Research Program o China under Grants 11CB7794 and 1CB733, the National Natural Science Foundation o China under Grants , , , , and , the Ministry o Education o China under Grants and 19136, the Key Laboratory o Computer Network and Inormation Integration (Southeast University, Ministry o Education, the Qing Lan project, the Natural Science Foundation o Jiangsu Province under Grants BK139 and BK1743, and Jiangsu Province under Grant DZXX-31. They are also part o the collaboration established or years between the Chinese (LIST and French (LTSI laboratories through the joint international research center, CRIBs.

27 Derivation o Moment Invariants 83 Appendix A The expressions given below provide to the interested readers all the elements to replicate our method and to apply it to other examples. A1. List o Legendre moment invariants up to the seventh order Third Order I (3, = L 3 I (, 1 = L 1 I (1, = L 1 I (, 3 = L 3 Fith Order I (, = L 3 77 L L L 3 L I (4, 1 = L 41 7 L 1 1 ( 7L 1 L + 1 L 11 L 3 L I (3, = L 3 1 L1 6 L 3 1 L ( 1 L L L 11 L 1 + L L 3 3 I (, 3 = L 3 1 L1 6 ( L 3 1 L L L L 11 L 1 + L L 3 3 I (1, 4 = L 14 7 L 1 1 L ( 7L 1 L + 1 L 11 L 3 I (, = L 3 77 L L L 3 L Seventh Order I (7, =L L I (, 1 6 L 4 L L I (, 3 L ( 19 1 L L 3 4

28 84 H. Shu et al. I (6, 1 =L L I (4, 1 1 ( 4 13 L L 1 L L 11 L L 4 L L I (4, 1 + L 11 I (, 3 L 31 L 3 I (, =L L ( L L 31 L L I (3, I (, L L L 4 L L 11 L 1 6 L L L L 3 L L L I (3, 16 L 11 I (4, 1 + L I (, I (4, 3 =L L 8 L ( 7 L L 3 + L 4 L L I (, 3 I (4, L 11 L L 31 L L L L L L L 1 + 7L I (, L 11L L 13 L L 11 I (3, + L I (4, 1 3 I (3, 4 =L L 8 L ( 7 L L 3 + L 4 L L I (3, I (1, L 11 L L 13 L L L L L L L 1 + 7L I (, L 11L L 31 L L 11 I (, 3 + L I (1, 4 3

29 Derivation o Moment Invariants 8 I (, =L L ( L 3 L I (, 3 I (, L L L 4 L L 11 L L 13 L L L L L L L L I (, L 11 I (1, 4 + L I (, I (1, 6 =L L I (1, 4 1 ( 4 13 L L 1 L L 11 L L 4 L L I (1, 4 + L 11 I (, 3 L 13 L 3 I (, 7 =L L I (, 1 6 L 4 L L I (, 3 L ( 19 1 L L 3 4 A. List o Zernike moment blur invariants up to the sixth order Zero Order First Order Second Order I (, = Z, I (1, 1 = Z 1,1 I (, = Z, Third Order I (3, 1 = Z 3,1 6I (1, 1 I (1, 1 Z, /Z, I (3, 3 = Z 3,3

30 86 H. Shu et al. Fourth Order I (4, = Z 4, 1I (, 1I (, Z, /Z, I (4, 4 = Z 4,4 Fith Order I (, 1 = Z,1 4I (1, 1 1I (3, 1 [ 3I (1, 1 Z, +3I (1, 1 Z 4, + I (3, 1 Z, ]/Z, I (, 3 = Z,3 1I (3, 3 I (3, 3 Z, /Z, I (, = Z, Sixth Order I (6, = Z 6, 1I (, 1I (4, [ 14I (, Z, /3 +7I (, Z 4, + 7I (4, Z, ]/Z, I (6, 4 = Z 6,4 1I (4, 4 7I (4, 4 Z, /Z, I (6, 6 = Z 6,6 A3. List o combined Zernike moment invariants up to the sixth order Second Order SI (, = 3Γ I (, + 3Γ 4I (, + Γ 4I (, SI (, = e ĵθ Γ 4 I (, Third Order [ ] SI (3, 1 = e ĵθ 4Γ 3 I (1, 1 + 4Γ I (1, 1 + Γ I (3, 1 = e ĵθ Γ I (3, 1 SI (3, 3 = e ĵθ Γ I (3, 3

31 Derivation o Moment Invariants 87 Fourth Order SI (4, = Γ I (, 1Γ 4I (, + 1Γ 6I (, Γ 4I (, +Γ 6 I (, + Γ 6 I (4, [ ] SI (4, = e ĵθ Γ 4 I (, + Γ 6 I (, + Γ 6 I (4, SI (4, 4 = e 4ĵθ Γ 6 I (4, 4 Fith Order Sixth Order [ SI (, 1 = e ĵθ 9Γ 3 I (1, 1 4Γ I (1, 1 + 1Γ 7 I (1, 1 ] 6Γ I (3, 1 + Γ 7 I (, 1 [ ] = e ĵθ 6Γ I (3, 1 + 6Γ 7 I (3, 1 + Γ 7 I (, 1 [ ] SI (, 3 = e ĵθ 6Γ I (3, 3 + 6Γ 7 I (3, 3 + Γ 7 I (, 3 SI (, = e ĵθ Γ 7 I (, SI (6, = 7Γ I (, + 4Γ 4I (, 7Γ 6I (, +3Γ 8 I (, + 14Γ 4I (, = 3Γ 6 I (, + 1Γ 8I (, 7Γ 6I (4, +7Γ 8 I (4, + Γ 8 I (6, [ SI (6, = e ĵθ 14Γ 4 I (, 36Γ 6 I (, + 1Γ 8 I (, ] 7Γ 6 I (4, + 7Γ 8 I (4, + Γ 8 I (6, [ ] SI (6, 4 = e 4ĵθ 7Γ 6 I (4, 4 + 7Γ 8 I (4, 4 + Γ 8 I (6, 4 SI (6, 6 = e 6ĵθ Γ 8 I (6, 6 Reerences [1] F.M. Candocia. Moment relations and blur invariant conditions or nite-extent signals in one, two and N-dimensions. Pattern Recognition Letters, (4: , 4.

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