- Email: [email protected]

PII: DOI: Reference:

S0031-3203(15)00349-0 http://dx.doi.org/10.1016/j.patcog.2015.09.018 PR5522

To appear in: Pattern Recognition Received date: 6 December 2014 Revised date: 8 July 2015 Accepted date: 8 September 2015 Cite this article as: Hongqing Zhu, Yan Yang, Zhiguo Gui, Yu Zhu and Zhihua Chen, Image analysis by generalized Chebyshev-Fourier and generalized pseudo Jacobi-Fourier moments, Pattern Recognition, http://dx.doi.org/10.1016/j.patcog.2015.09.018 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Image analysis by generalized Chebyshev-Fourier and generalized pseudo Jacobi-Fourier moments Hongqing Zhua,∗, Yan Yanga , Zhiguo Guib , Yu Zhua , Zhihua Chena a

School of Information Science and Engineering, East China University of Science and Technology, Shanghai 200237, China b National Key Laboratory for Electronic Measurement Technology, North University of China, Taiyuan 030051, China

Abstract In this paper, we present two new sets, named the generalized ChebyshevFourier radial polynomials and the generalized pseudo Jacobi-Fourier radial polynomials, which are orthogonal over the unit circle. These generalized radial polynomials are then scaled to deﬁne two new types of continuous orthogonal moments, which are invariant to rotation. The classical ChebyshevFourier and pseudo Jacobi-Fourier moments are the particular cases of the proposed moments with parameter α = 0. The relationships among the proposed two generalized radial polynomials and Jacobi polynomials, shift Jacobi polynomials, and the hypergeometric functions are derived in detail, and some interesting properties are discussed. Two recursive methods are developed for computing radial polynomials so that it is possible to improve computation speed and to avoid numerical instability. Simulation results are provided to validate the proposed moment functions and to compare their ∗

School of Information Science and Engineering, East China University of Science and Technology, Shanghai 200237, China Email address: [email protected] (Hongqing Zhu)

Preprint submitted to Pattern Recognition

September 23, 2015

performance with previous works. Keywords: Generalized radial polynomial, Jacobi polynomial, recurrence relation, rotation invariant, Chebyshev-Fourier moment, Pseudo Jacobi-Fourier moment 1. Introduction In past decades, a number of studies have been conducted on moments and moment invariants because of their abilities to extract an image’s features. As a result, image moments have been widely used in image analysis [1], pattern recognition [2], image patches [3], image watermarking [4], etc. The most popular continuous orthogonal moments among the family of moments are Legendre moments [5], Zernike moments [5], pseudo-Zernike moments [6], and Fourier-Mellin moments [7]. Other continuous orthogonal moments recently reported in the literatures are Chebyshev-Fourier moments (CHFM) [8], pseudo Jacobi-Fourier moments (PJFM) [9], Gaussian-Hermite moments [10], etc. Practical implementations of continuous moments require approximating the integration with the summation. Additionally, this implementation will also require a suitable transformation from the square domain into the deﬁnition domain of moments. It will inevitably lead to numerical errors so that the accurate representation of image features is aﬀected even though the basis set is strictly orthogonal in its domain. Consequentially, to reduce numerical error, considerable attention has been given to the study of discrete orthogonal moments, such as Tchebichef moments [11], Krawtchouk moments [12], Hahn moments [13, 14], dual Hahn moments [15], and Racah moments [16] because the numerical approximations and image coordinate 2

transformations are not necessary for the computation of discrete moments. However, the computation of discrete moments does not mean that they can completely replace the continuous ones. Actually, discrete moment invariants, particularly rotation invariants, are relatively diﬃcult to determine because of the extremely complicated basis functions of these discrete orthogonal moments [17]. These impede the extensive applications of discrete orthogonal moments in pattern recognition tasks. In current research, we have developed the generalized Chebyshev-Fourier radial polynomials (G-CHFP) and generalized pseudo Jacobi-Fourier radial polynomials (G-PJFP). Then, the two radial polynomials are normalized to construct two new descriptors, the generalized Chebyshev-Fourier moments (G-CHFM) and the generalized pseudo Jacobi-Fourier moments (G-PJFM). The new sets of the moment are orthogonal over the unit circle with a free parameter α > −1; hence, the proposed moment sets have an inherent property of rotation invariance. The classical CHFM and PJFM are special cases of the proposed moments with α = 0. Unlike the classical CHFM and PJFM, the distribution of zeroes of the proposed radial polynomials can be controlled by the parameter α, so it is possible to choose appropriate values of α for a speciﬁc emphasis over the unit circle. Such a speciﬁc distribution can lead to the intentional distribution of the essential discriminative information in the spatial domain [1, 18], making the proposed moments more suitable for particular applications. This paper also details the computation aspect of the generalized radial polynomials, and presents a method which is called the n-recursive strategy to reduce the time complexity of computing the radial polynomial terms. The proposed n-recursive method can eliminate the fac-

3

torial terms involved in the computation of the coeﬃcients of the generalized radial polynomials. This improves the numerical stability when the order of moments is large. In addition, the paper investigates the explicit formulae of the generalized radial polynomials, as well as the relations of them with Jacobi polynomials, shift Jacobi polynomials, and hypergeometric functions, respectively. Experimental results show that the G-CHFM and G-PJFM perform very well in terms of image reconstruction capabilities and invariant recognition accuracies in noise-free or noisy cases. Therefore, the proposed moments are potentially useful as feature descriptors in image analysis. This paper is organized as follows. A brief review of CHFM and PJFM are given in section 2. Section 3 provides the derivation of two unit circle-based generalized radial polynomials, and discusses the computational aspects of these polynomials. In section 3, we also establish the relationships between the proposed generalized radial polynomials and Jacobi polynomials, shift Jacobi polynomials, hypergeometric functions, etc. Section 4 presents GCHFM and G-PJFM, and discusses some properties of the moment functions. Section 5 provides the experiments to verify the proposed descriptors. Section 6 concludes the paper. 2. Two classical orthogonal moments on the unit circle 2.1. Chebyshev-Fourier moments The 2D classical Chebyshev-Fourier moments [8], Anm , of order n with repetition m are deﬁned using polar coordinates (r, θ) inside the unit circle as Anm =

2π 1 0

0

Cn (r) exp(−jmθ)f (r, θ)rdrdθ, 4

(1)

Here, Cn (r) are the real-valued Chebyshev-Fourier radial polynomials deﬁned as Cn (r) = [

(n+2)/2 64(1 − r) 1 (−1)k (n − k)! 4 ] (4r − 2)n−2k . π2r k!(n − 2k)! k=0

(2)

2.2. Pseudo Jacobi-Fourier moments The PJFM [9] of an image function f (r, θ) of order n and repetition m over a unit circle are deﬁned by Bnm =

2π 1 0

0

Jn (r) exp(−jmθ)f (r, θ)rdrdθ.

(3)

The real-valued pseudo Jacobi-Fourier radial polynomials Jn (r) are expressed as Jn (r) = [

n 2(n + 2)(r − r 2 ) 1 (−1)n+k (n + k + 3)! k ]2 r . (n + 3)(n + 1) k=0 k!(n − k)!(k + 2)!

(4)

3. Generalized radial polynomials 3.1. Generalized radial polynomials: The Jacobi form Motivated by the work of W¨ unsche [19], it is not diﬃcult to ﬁnd that the classical Chebyshev-Fourier (2) and pseudo Jacobi-Fourier radial polynomials (4) have the following Jacobi representation, such as Cn (r) =

(n + 1)! 4(1 − r) 1 ( 12 , 12 ) [ ] 4 Pn (2r − 1), (n + 12 )! r

(5)

and 1

Jn (r) = [2(n + 1)(n + 2)(n + 3)(r − r 2 )] 2

5

n! P (1,2) (2r − 1), (n + 1)! n

(6)

¯

respectively. where Pn(¯α,β) (u) denotes the Jacobi polynomials with respect to the real parameters α ¯ > −1 and β¯ > −1 [20]. ¯

Pn(¯α,β) (u) = 2 F1 (a, b; c; x)

¯ (−1)n (n + β)! u+1 ¯ + β¯ + 1; β¯ + 1; ). 2 F1 (−n, n + α ¯ 2 n!β!

(7)

is the hypergeometric function deﬁned by ∞

(a)k (b)k xk · , 2 F1 (a, b; c; x) = (c)k k! k=0

(8)

and (a)k is the Pochhammer symbol deﬁned as (a)k = a(a + 1)(a + 2) . . . (a + k − 1),

with (a)0 = 1.

(9)

Following expressions (5)-(6), for given free parameter α > −1, this paper deﬁnes the following generalized Chebyshev-Fourier radial polynomials with a pair of complex conjugate variables (z ≡ x + jy ≡ r exp(jθ), z ∗ ≡ x − jy ≡ r exp(−jθ)) as Cnα (r) =

(n + 1)! 4(1 − r) 1 (α+ 12 , 12 ) ] 4 Pn [ (2r − 1), r (n + 12 )!

(10)

and generalized pseudo Jacobi-Fourier radial polynomials as 1

Jnα (r) = [2(n + 1)(n + 2)(n + 3)(r − r 2 )] 2

1 Pn(α+1,2) (2r − 1). n+1

(11)

The classical Chebyshev-Fourier radial polynomial and pseudo Jacobi-Fourier radial polynomial is a special case of the G-CHFP and G-PJFP, respectively; also, the free parameter α corresponding to (10) and (11) is zero. Because of 6

the interesting properties of Jacobi polynomials, representing G-CHFP and G-PJFP using the shifted Jacobi polynomials, hypergeometric function, and so on, is made convenient. The above two equations (10) and (11) will be needed in constructing these relations. 3.2. Generalized radial polynomials: The shifted Jacobi form Shifted Jacobi polynomials are also related to Jacobi polynomials as follows [21, 22]. (n,m)

Pk

(r) = (−1)k

Γ(k + m + 1) r+1 Gk (n + m + 1, m + 1, ), k!Γ(m + 1) 2

(12)

where Gamma functions Γ(n) = (n − 1)!, for n ∈ N, Gk has the following explicit deﬁnition Gk (n, m, r) =

k (−1)s (n + k + s − 1)! s k!(m − 1)! r. (n + k − 1)! s=0 (k − s)!s!(m + s − 1)!

(13)

Thus, using the deﬁnitions (10)-(11) and above relations between Jacobi and shift Jacobi polynomials (12), we can rewrite these generalized radial polynomials G-CHFP and G-PJFP in the following shift Jacobi polynomials form: Cnα (r) = (−1)n [

64(1 − r) 1 3 ] 4 (n + 1)Gn (α + 2, , r), 2 π r 2

(14)

and Jnα (r) = (−1)n [

(n + 1)(n + 2)3 (n + 3)(r − r 2 ) 1 ] 2 Gn (α + 4, 3, r). 2

7

(15)

Normalized Generalized Chebyshev−Fourier Radial Polynomials with α = 0

Normalized Generalized Chebyshev−Fourier Radial Polynomials with α = 5

5

3

Polynomial Results

2

1 0 −1

1 0 −1

1 0 −1

−2

−2

−2

−3

−3

−3

−4

−4

−5

0

0.2

0.6

0.4

0.8

−5

1

−4 0

0.2

0.6

0.4

0.8

−5

1

0

0.2

0.6

0.4

0.8

1

Radius r

Radius r

Normalized Generalized Chebyshev−Fourier Radial Polynomials with α = 15 5 n=0 n=1 4 n=2 n =3 3 n=4 n=5 2

Normalized Generalized Chebyshev−Fourier Radial Polynomials with α = 20 5 n=0 n=1 4 n=2 n=3 3 n=4 n=5 2

Normalized Generalized Chebyshev−Fourier Radial Polynomials with α = 25 5 n=0 n=1 4 n=2 n=3 3 n=4 n=5 2

1 0 −1

Polynomial Results

Radius r

Polynomial Results

Polynomial Results

2

n=0 n=1 n=2 n=3 n=4 n=5

4

Polynomial Results

3

Polynomial Results

Normalized Generalized Chebyshev−Fourier Radial Polynomials with α = 10 5 n=0 n=1 4 n=2 n=3 3 n=4 n=5 2

5 n=0 n=1 n=2 n=3 n=4 n=5

4

1 0 −1

1 0 −1

−2

−2

−2

−3

−3

−3

−4

−4

−5

0

0.2

0.4

0.6

0.8

−5

1

−4 0

0.2

0.4

Radius r

0.6

0.8

−5

1

0

0.2

0.4

Radius r

0.6

0.8

1

Radius r

0

−1

−2

0

−1

0

0.2

0.4

0.6

0.8

−3

1

Normalized Generalized Pseudo Jacobi−Fourier Radial Polynomials with α = 10 3 n=0 n=1 n=2 2 n=3 n=4 n=5 1

0

−1

−2

−2

0

0.2

0.4

0.6

0.8

−3

1

0

0.2

0.4

0.6

0.8

1

Radius r

Radius r

Normalized Generalized Pseudo Jacobi−Fourier Radial Polynomials with α = 15 3 n=0 n=1 n=2 2 n=3 n=4 n=5 1

Normalized Generalized Pseudo Jacobi−Fourier Radial Polynomials with α = 20 3 n=0 n=1 n=2 2 n=3 n=4 n=5 1

Normalized Generalized Pseudo Jacobi−Fourier Radial Polynomials with α = 25 3 n=0 n=1 n=2 2 n=3 n=4 n=5 1

0

−1

0

−1

−2

−2

−3

−3

0

0.2

0.4

0.6

0.8

1

Polynomial Results

Radius r

Polynomial Results

Polynomial Results

−3

Normalized Generalized Pseudo Jacobi−Fourier Radial Polynomials with α = 5 3 n=0 n=1 n=2 2 n=3 n=4 n=5 1

Polynomial Results

Normalized Generalized Pseudo Jacobi−Fourier Radial Polynomials with α = 0 3 n=0 n=1 n=2 2 n=3 n=4 n=5 1

Polynomial Results

Polynomial Results

Figure 1: Normalized generalized Chebyshev-Fourier radial polynomials.

0

−1

−2

0

0.2

0.4

0.6 Radius r

Radius r

0.8

1

−3

0

0.2

0.4

0.6 Radius r

Figure 2: Normalized generalized pseudo Jacobi-Fourier radial polynomials.

8

0.8

1

3.3. Generalized radial polynomials: The hypergeometric form To derive G-CHFP and G-PJFP using hypergeometric form function, we need the above relations (10) and (11), as well as the relations of Jacobi polynomials and the hypergeometric function (7) of this paper; thus, the Cnα (r) can be written in the hypergeometric form as follows Cnα (r) = (−1)n [

64(1 − r) 1 3 ] 4 (n + 1)2 F1 (−n, n + α + 2; ; r). 2 π r 2

(16)

Similarly, the proposed Jnα (r) are interrelated to the hypergeometric function by the following relations: Jnα (r) = (−1)n [

1 (n + 1)(n + 2)3 (n + 3) (r − r 2 )] 2 2

(17)

×2 F1 (−n, n + α + 4; 3; r). 3.4. Generalized radial polynomials: The explicit expression Now, we are interested in ﬁnding an explicit form for the computation of the G-CHFP and G-PJFP. By using the relation of the proposed radial polynomials with the hypergeometric function (16) and (17), alternately, we derive the G-CHFP, with the explicit form as follows Cnα (r) =

n (−1)k (2n + α + 1 − k)! n−k (n + 1)! 4(1 − r) 1 [ ]4 r . 1 (n + α + 1)! r k=0 k!(n − k)!(n + 2 − k)!

(18)

The explicit expression for G-PJFP is also obtained by 1

Jnα (r) = [2(n + 1)(n + 2)(n + 3)(r − r 2 )] 2 ×

n

(−1)k (2n + α + 3 − k)! n−k r . k=0 k!(n − k)!(n + 2 − k)! 9

(n + α + 4)(−α−1) n+1

(19)

3.5. Orthogonality It is well known that the classical Jacobi polynomials with the real parameter (α ¯ > −1, β¯ > −1) satisfy the orthogonality relation with respect to ¯

the weight function (1 − u)α¯ (1 + u)β on[-1,1] of the form [20] 1 −1

α ¯

(1 − u) (1 + u)

β¯

¯ ¯ (¯ α,β) Pl (u)Pn(¯α,β) (u)du

¯

¯ β+1 2α+ = 2n + α ¯ + β¯ + 1

¯ (n + α ¯ )!(n + β)! × ¯ δln , n!(n + α ¯ + β)!

(20)

where δln denotes the Kronecker symbol. Applying (10)-(11) and the above orthogonality of Jacobi polynomials, it is not diﬃcult to derive the orthogonality relation of the proposed G-CHFP on the unit radius 0 ≤ r ≤ 1 by 1 0

(1 − r)α rCnα (r)Clα (r)dr =

2(n + α + 12 )!Γ2 (n + 2) δln , (21) (2n + α + 2)n!(n + α + 1)!(n + 12 )!

and the G-PJFP over the unit circle as follows 1 0

(1 − r)α rJnα (r)Jlα (r)dr =

2(n + 2)2 (n + 3) δln . (22) (2n + α + 4)(n + α + 3)(n + α + 2)

The above two equations show that r(1 − r)α is the weight function of the orthogonal relation on the unit circle. In this paper, we deﬁne the so called normalized G-CHFP and G-PJFP as (2n + α + 2)n!(n + α + 1)!(n + 12 )!(1 − r)α , C˜nα (r) = Cnα (r) 4π(n + α + 12 )!Γ2 (n + 2)

10

(23)

and (2n + α + 4)(n + α + 3)(n + α + 2)(1 − r)α , J˜nα (r) = Jnα (r) 4π(n + 3)(n + 2)2

(24)

respectively. Figs. 1 and 2 show the plots for the certain orders of polynomials C˜nα (r) and J˜nα (r) with diﬀerent choices of free parameter values α. If the normalized generalized radial polynomials C˜nα (r) and J˜nα (r) are used as radial kernels, we deﬁne the following generalized Chebyshev-Fourier polynomials α C¯nm (r, θ), which are orthogonal and satisfy

1 2π 0

α α C¯nm (r, θ)[C¯pq (r, θ)]∗ rdrdθ = δnp δmq ,

(25)

α C¯nm (r, θ) = C˜nα (r) exp(jmθ).

(26)

0

with

α (r, θ) Similarly, the proposed generalized pseudo Jacobi-Fourier polynomials J¯nm

are also orthogonal and satisfy 1 2π 0

0

α α J¯nm (r, θ)[J¯pq (r, θ)]∗ rdrdθ = δnp δmq ,

(27)

α J¯nm (r, θ) = J˜nα (r) exp(jmθ).

(28)

with

3.6. Recurrence relations with respect to order n Because of the high order of factorial terms involved in the radial polynomials, the computation process of radial polynomials will suﬀer from a high computation cost, especial at high orders of moments. To improve speed 11

performance, this subsection develops two recursive methods with respect to order n (called the n-recursive method) for calculation of the generalized radial polynomials Cnα (r), and Jnα (r), respectively. Another attractive feature of this n-recursive method is that it can also provide numerical stability at high order moments. The Jacobi polynomials satisfy the following three-term recurrence relation [20] ¯ ¯ ¯ + β¯ − 1) 2n(n + α ¯ + β)(2n +α ¯ + β¯ − 2)Pn(¯α,β) (u) = (2n + α ¯ (¯ α,β)

¯ × ((2n + α ¯ + β)(2n +α ¯ + β¯ − 2)u + α ¯ 2 − β¯2 )Pn−1 (u)

(29)

¯ (¯ α,β) ¯ n−2 (u). − 2(n + α ¯ − 1)(n + β¯ − 1)(2n + α ¯ + β)P

Combining (5) and (29), the generalized Chebyshev-Fourier radial polynomials are computed recursively as follows α α (r) + M3 Cn−2 (r), Cnα (r) = (M1 r + M2 )Cn−1

(30)

with initial value C0α (r) = [

64(1 − r) 1 ]4, π2r

and C1α (r) = [

8(α + 3) 4(1 − r) 1 r − 4][ ]4 , 3 π2r

(31)

where the coeﬃcients M1 , M2 , and M3 are given by (n + 1)(2n + α)(2n + α + 1) , n(n + 12 )(n + α + 1)

(32)

α(α + 1) − (2n + α + 1)(2n + α − 1) M1 , 2(2n + α + 1)(2n + α − 1)

(33)

M1 =

M2 =

12

M3 = −

n(n + α − 12 ) M1 . (2n + α)(2n + α − 1)

(34)

Consequently, according to (6) and (29), the generalized pseudo JacobiFourier radial polynomials are computed recursively by α α Jnα (r) = (N1 r + N2 )Jn−1 (r) + N3 Jn−2 (r),

(35)

with initial value 1

J0α (r) = [12(r − r 2 )] 2 ,

1

and J1α (r) = [12(r − r 2 )] 2 [(α + 5)r − 3],

(36)

where the coeﬃcients N1 , N2 , and N3 are given by (2n + α + 3)(2n + α + 2) n + 3 1 ( )2 , (n + 1)(n + α + 3) n

(37)

(α + 1)2 − 4 − (2n + α + 3)(2n + α + 1) N1 , 2(2n + α + 1)(2n + α + 3)

(38)

N1 =

N2 =

N3 = −

n+2 1 (n + α)(n − 1)(n + 1) ( ) 2 N1 . n(2n + α + 1)(2n + α + 2) n − 1

(39)

4. Generalized Chebyshev-Fourier and pseudo Jacobi-Fourier moments In this section, the deﬁnition of G-CHFM and G-PJFM, which are two sets of moments based on the G-CHFP and G-PJFP, respectively, are ﬁrst provided, and then we discuss their invariance to rotation transforms.

13

4.1. Generalized continuous orthogonal moments: G-CHFM and G-PJFM α of order n and A general expression for the orthogonal moments Znm

repetition m for a given image f (r, θ) over the unit circle are built on the following deﬁnition α = Znm

2π 1 0

0

¯ α (r, θ)]∗ rdrdθ. f (r, θ)[D nm

(40)

¯ α (r, θ) as C¯ α (r, θ), and J¯α (r, θ), one can obtain By choosing polynomials D nm nm nm G-CHFM, and G-PJFM, respectively. The reconstructed image f (r, θ) can be computed using the inﬁnite series and takes the following form f (r, θ) =

∞ ∞ n=0 m=0

α ¯α Dnm (r, θ). Znm

(41)

¯ α (r, θ)} allows us to In fact, the orthogonality and completeness of sets {D nm represent any square integrable function f (r, θ) via a limit to the number of terms in the series expansion n = 0, 1, . . . , M. fˆ(r, θ) =

M

M

n=0 m=−M

α ¯α Dnm (r, θ), Znm

(42)

where the orthogonal moments of orders up to M are used. This is actually the basic equation used in image reconstruction via the generalized moments. The integration formula given in (40) does not have an analytical solution, and it cannot be applied directly to a digital image. In general, for a given N × N image deﬁned in a discrete domain f (s, t), to compute its G-CHFM (or G-PJFM), one has to map f (s, t) to the unit circle domain. Thus, Eq.(40)

14

Figure 3: Test images (64 × 64); left: binary image “Logo”; right: grey-level image “Flower”.

is approximated in the discrete domain as α Znm =

N −1 N −1 2 C˜ α (rst )e−jmθst f (s, t), (N − 1)2 s=0 t=0 n

(43)

where the image coordinate transformation to the interior of the unit circle is given by

c1 t + c2 (c1 s + c2 )2 + (c1 t + c2 )2 , θst = tan−1 ( ), c1 s + c2 √ 2 1 c1 = , c2 = − √ . N −1 2

rst =

(44)

There is no perfect mapping of the square grids onto a unit circle. The geometric error might occur when mapping a discrete image from the square domain onto a circular domain. Some outstanding works about how to reduce the geometric error were done by the research group of Singh and Walia [23, 24]. 4.2. Rotation invariance The present G-CHFM (or G-PJFM) of patterns have an inherent property of rotation invariance. If we let the G-CHFM of an original image α α and Znm , respectively, where, f (r, θ) and its rotated version f rot (r, θ) be Znm

f rot (r cos θ, r sin θ) = f (r cos(θ − φ), r sin(θ − φ)), and φ is a rotation angle obtained by clockwise rotation of f (r, θ), then the new set of G-CHFM 15

Table 1: Reconstructed results using G-CHFM moments with diﬀerent parameter α

Order

5

10

20

40

64

Classical CHFM G-CHFM α=5 G-CHFM α = 15 G-CHFM α = 25 α α coeﬃcients, {Znm }, are related to the old set, {Znm }, by α Znm

= =

2π 1 0

0

2π 1 0

0

α Vˆnm (r)e−jmθ f rot (r cos θ, r sin θ)rdrdθ α Vˆnm (r)e−jm(θ +φ) f (r cos θ , r sin θ )rdrdθ

(45)

α = e−jmφ Znm ,

where θ = θ − φ; it can be seen immediately that the magnitude operator eﬀectively cancels out the exponential factor, e−jmφ . Therefore, the rotation 16

Table 2: Reconstructed results using G-PJFM moments with diﬀerent parameter α

Order

5

10

20

40

64

Classical PJFM G-PJFM α=5 G-PJFM α = 15 G-PJFM α = 25 invariants from G-CHFM can also be directly achieved by the magnitude of α ; thus, they are useful features for rotation invariant pattern recognition. Znm

5. Experimental results and discussions In this section, numerical experiments are conducted to evaluate the theoretical framework presented in the previous sections. We also evaluate the classical CHFM [8] and PJFM [9] for comparison. The experiments have 17

MSE Curve of G−CHFM with Different α

MSE Curve of G−CHFM with Different α

0.9

1

−3

0.045 0.8 0.04 0.035 0.6 0.03 0.5 0.025 0.4 0.02 0.3 0.015 40

45

50

65

60

55

α=0 α=5 α = 15 α = 25

4.2

0.6 4 0.5 3.8

0.4

3.6 40

45

0

10

20

30 40 Moment Order

50

0

60

0

10

20

30 40 Moment Order

MSE Curve of G−PJFM with Different α

−3

4.6

0.035

0.4

0.02 0.015

0.3

0.01 40

0.2

45

50

55

60

α=0 α=5 α = 15 α = 25

4.4

0.7 Reconstruction Error

0.03 0.025

x 10

0.8

α=0 α=5 α = 15 α = 25

0.7

0.5

60

MSE Curve of G−PJFM with Different α 0.9

0.04

0.6

50

(b)

0.8

Reconstruction Error

65

60

55

0.1

(a)

0.6

4.2

0.5

4

0.4

3.8

0.3 3.6 40

65

45

50

55

60

65

0.2

0.1 0

50

0.2

0.1 0

0.7

0.3

0.2

x 10

4.4

0.8

Reconstruction Error

Reconstruction Error

0.9

α=0 α=5 α = 15 α = 25

0.7

4.6

0.1

0

10

20

30 40 Moment Order

50

0

60

0

10

20

(c)

30 40 Moment Order

50

60

(d)

Figure 4: Mean relative errors variation for the test images with respect to the parameter α, (a) G-CHFM for image “Logo”; (b) G-CHFM for image “Flower”; (c) G-PJFM for image “Logo”; (d) G-PJFM for image “Flower”.

been developed in Matlab 7.11.0 (2010b) in a Windows environment on a 3.4-GHz Intel(R) Core(TM) i7 processor with 4 GB RAM. 5.1. Image representation capability for test images In the ﬁrst experiment, the image representation capability of the GCHFM and G-PJFM are shown. For this reason, a binary image “Logo” and a grey-level image “Flower” (64×64, as shown in Fig. 3) are used to compare 18

the performance of the proposed methods with others. The G-CHFM and G-PJFM of each test image at order n = 5, 10, 20, 40, 64 are ﬁrst calculated, where the n-recursive method is adopted for their radial polynomials. Tables 1 and 2 present the reconstructed images with an increasing number of moments using G-CHFM and G-PJFM, respectively. This simulation adopted the state-of-the-art Gaussian quadrature method [24] to enhance numerical stability. It can be noted that the quality of the reconstructed images is closer to the original test quality when the moments order reaches a certain value. In addition, the reconstructed images using the proposed moments with a small value for parameter α show more visual resemblance to the original image in the early orders. To objectively measure the reconstructed results, this paper employs the mean square error (MSE), deﬁned as MSE =

−1 N −1 N 1 |f (i, j) − fˆ(i, j)|2 N × N i=0 j=0

(46)

to characterize the reconstructed error. f (i, j) and fˆ(i, j) denote the intensity functions of the test image and the reconstructed image of N × N size, respectively. The MSE results of the test images by G-CHFM and G-PJFM with α = 0, 5, 15, 25 are depicted in Fig. 4. Fig. 4 indicates the reconstructed images using the low-order CHFM and PJFM give the better reconstructed results. However, with increasing order, the decreasing trend of MSE with respect to moment orders becomes more pronounced for the G-CHFM and G-PJFM descriptors. Speciﬁcally, the reconstructed results obtained with large values of α produce the best results among all other choices. This may be related to the distribution zeroes of radial polynomials. Because, with the

19

Figure 5: Some sample images taken from the COREL photography database. Table 3: Some rotation transformed images with salt-and-pepper noise (5%)

Flower 8 Flower 10 increase of parameter α, the uniform distribution of zeroes of G-CHFM and G-PJFM is moving toward the middle of the deﬁnition region so that more emphasis of the moments will focus on a range surrounding the centre of the image (see Figs. 1 and 2). Therefore, the feature close to the image centre does receive enough attention. 5.2. Invariant pattern recognition This subsection ﬁrst investigates the numerical stability of the invariant descriptors proposed in the previous section, namely their invariance to rotation and their robustness to additive noise. Then, we test the classiﬁcation behaviour of these descriptors with regard to diﬀerent types of noise. Rather arbitrarily, two images were chosen from the COREL photography database [25] and are corrupted by salt-and-pepper noise with density 5%. Some rotated versions of these images are arbitrarily generated by a computer, and they are shown in Table 3. Because the magnitudes of GCHFM or G-PJFM are a rotation invariant feature of the underlying image, in the current experiment, we have decided to apply the following feature 20

α α α α α α α vector V = {|Z10 |, |Z01 |, |Z00 |, |Z12 |, |Z21 |, |Z33 |, |Z13 |}, containing 7 rotation α is the G-CHFM or G-PJFM deinvariants with orders from 0 to 3, where Zpq

ﬁned by (40). Tables 4 and 5 recorded several module values of the proposed G-CHFM (G-PJFM) and the conventional CHFM (PJFM) for some rotation versions of images which are shown in Table 3. The paper takes τ /μ% as a measure of stability of the invariants. It denotes the percentage of the distance the rotation invariants distribute apart from their corresponding mean. τ and μ, respectively represent the standard deviation and the mean of the rotation invariants computed from diﬀerent rotated images. From Tables 4 and 5, we found the G-CHFM (G-PJFM) or CHFM (PJFM), has diﬀerent values for diﬀerent images and is almost unchanged for the same image. By examining the values of τ /μ% in Tables 4 and 5 it can be deduced that the rates τ /μ% of the G-CHFM (G-PJFM) are more lower, indicating that the corresponding invariants are very stable under diﬀerent kinds of rotation transforms. To investigate the numerical behaviour of the proposed invariant descriptors under other noise, we repeated the above experiment with two other images (see Table 6), and each added zero mean Gaussian white noise with variance σ 2 = 0.01. Tables 7 and 8 illustrate the variation of the feature vector of the same object under diﬀerent rotation angles. From the experimental results listed, it is evident that the eﬀects of Gaussian noise and rotation transforms are negligible because of the invariance property. The values of the proposed G-CHFM (G-PJFM) invariants are fairly stable with respect to Gaussian noise, which again indicates good robustness of the rotation invariants. According the values of τ /μ% in Tables 7 and 8, it also

21

Table 4: G-CHFM and CHFM rotation invariants of the same object under diﬀerent rotation angles (salt-and-pepper noise with intensity 5%) 25 G-CHFM invariants |Z10 |

25 |Z01 |

25 |Z00 |

25 25 25 25 |Z12 | |Z21 | |Z33 | |Z13 |

Flower 8

42.4911 0.8268 33.4820 1.6888 3.4681 2.1202 0.5083 42.4482 0.8597 33.5384 1.6213 3.4550 2.1216 0.5063 42.3441 0.8762 33.8216 1.6563 3.4544 2.1488 0.5026 42.4458 0.8308 33.1053 1.6850 3.4278 2.1219 0.5147 42.4408 0.8120 33.1610 1.6355 3.4330 2.1218 0.5113

τ /μ

0.0013 0.0311 0.0088 0.0179 0.0049 0.0058 0.0091

Flower 10

61.4594 2.1266 45.0116 1.1282 5.0910 4.3163 0.9622 61.1356 2.1025 45.8359 1.1283 5.0629 4.3428 0.9670 61.4128 2.1935 45.8590 1.1130 5.0824 4.3574 0.9698 61.3054 2.1355 45.7082 1.1019 5.0371 4.3401 0.9702 61.1684 2.1128 45.1245 1.1355 5.0453 4.3455 0.9621

τ /μ

0.0023 0.0166 0.0090 0.0122 0.0046 0.0035 0.0041

CHFM invariants

0 |Z10 |

0 |Z01 |

0 |Z00 |

0 0 0 0 |Z12 | |Z21 | |Z33 | |Z13 |

Flower 8

44.6177 1.9567 57.9437 0.5222 0.7302 1.1989 0.8206 44.0579 1.8290 58.6326 0.6026 0.6500 1.1012 0.8424 44.0305 1.8765 58.9700 0.5541 0.6116 1.1214 0.8097 43.2972 1.8307 58.6476 0.5666 0.6348 1.2099 0.8081 44.7441 1.7448 59.3660 0.5517 0.6868 1.1171 0.8474

τ /μ

0.0130 0.0419 0.0089 0.0520 0.0703 0.0441 0.0222

Flower 10

66.6574 8.5917 82.3026 0.6979 4.6392 1.2804 0.3601 66.9635 8.4423 82.1365 0.6558 4.5845 1.3394 0.3766 67.2126 8.0813 82.9069 0.6818 4.6464 1.2570 0.3688 66.9623 8.1310 82.9792 0.6156 4.6826 1.2585 0.3795 67.0951 8.4614 82.3055 0.6608 4.5463 1.2535 0.3529

τ /μ

0.0031 0.0267 0.0047 0.0470 0.0117 0.0282 0.0303

indicates that the proposed descriptors are more robust to Gaussian white noise than the conventional CHFM (PJFM) descriptors. The classiﬁcation experiments are conducted on ten randomly selected ﬂower images, as shown in Fig. 5, from the COREL photography database [25]. The testing set is generated by adding salt-and-pepper noise or Gaussian white noise. This is followed by rotating each image with angles from 0 to 358 at an increment of 2, forming a set of 1800 images. The range of pixel values is [0, 255]. In the experiment, we use these seven rotation invariants 22

Table 5: G-PJFM and PJFM rotation invariants of the same object under diﬀerent rotation angles (salt-and-pepper noise with intensity 5%) 25 G-PJFM invariants |Z10 |

25 |Z01 |

25 |Z00 |

25 25 25 25 |Z12 | |Z21 | |Z33 | |Z13 |

Flower 8

36.6709 1.8580 47.5178 1.6576 0.8251 2.7228 1.0736 36.9134 1.8584 47.4311 1.6048 0.8212 2.7623 1.0761 36.8515 1.8812 47.6040 1.6160 0.8242 2.7417 1.0782 36.3153 1.8207 47.7230 1.6402 0.8564 2.7641 1.0871 36.7229 1.8495 47.7949 1.6747 0.8325 2.7863 1.0793

τ /μ

0.0064 0.0118 0.0031 0.0181 0.0172 0.0088 0.0047

Flower 10

58.2309 2.5611 67.4580 0.7707 7.0168 4.1151 0.5742 58.5283 2.5286 67.6851 0.7731 6.9100 4.1529 0.5360 58.7372 2.5676 67.1527 0.7539 6.9590 4.1850 0.5387 58.9352 2.5164 67.1127 0.7422 6.9904 4.1816 0.5584 58.2116 2.5091 67.4221 0.7522 6.9250 4.1709 0.5730

τ /μ

0.0054 0.0104 0.0035 0.0173 0.0094 0.0069 0.0327

PJFM invariants

0 |Z10 |

0 |Z01 |

0 |Z00 |

0 0 0 0 |Z12 | |Z21 | |Z33 | |Z13 |

Flower 8

54.1242 1.2750 45.8506 1.4800 2.4361 1.2354 1.5713 54.0190 1.1648 45.8450 1.3801 2.2950 1.2462 1.5761 54.2003 1.1257 46.1705 1.4637 2.2184 1.3021 1.5190 54.6310 1.1427 45.9327 1.4926 2.2701 1.2447 1.5712 54.8425 1.1633 46.2387 1.4482 2.2892 1.2649 1.5208

τ /μ

0.0065 0.0499 0.0040 0.0303 0.0352 0.0211 0.0187

Flower 10

81.2921 7.6866 62.8261 0.5706 0.8246 1.5964 0.8876 81.2437 6.8954 62.5053 0.5791 0.8096 1.5273 0.7974 81.8569 7.1855 63.2536 0.5900 0.8072 1.5182 0.9821 81.5962 7.0056 63.3878 0.5903 0.8873 1.5290 0.8751 81.2065 7.4455 63.5906 0.6467 0.8240 1.5084 0.8993

τ /μ

0.0034 0.0446 0.0070 0.0502 0.0394 0.0227 0.0742

as the feature vector. The feature vector is tested against the features of the training set using the minimum Euclidean distance classiﬁer. Table 9 demonstrates the classiﬁcation results. Here the parameter α is set from α = 0 to 25. We can observe from this table that 100% recognition results are obtained for noise-free images. In addition, the feature vector constructed from proposed G-CHFM or G-PJFM invariants are much closer for the training images. The rate of accurate classiﬁcation is over 87% (α = 25) and this proves the robustness of the invariant descriptors of the proposed moments. 23

Table 6: Some rotation transformed images with zero mean Gaussian white noise (σ 2 = 0.01)

Flower 2 Flower 4 However, the classical Zernike moment invariants [5], CHFM- or PJFM-based invariants show moderately large diﬀerences in the values of distance. The last experiment evaluates the performance of proposed invariants in the noisy environment using the MNIST database [26] as a real-world problem. The MNIST database of handwritten digits consists of 10,000 testing and 60,000 training images. For convenience, only 2500 testing and 500 training images are used in the current recognition task. Each image is resized to ﬁt in a 28 × 28 pixel box before performing the experiment. Part of the images of the testing set is shown in Fig. 6. The feature vector based on the proposed G-CHFM and G-PJFM with diﬀerent values of parameter α is used to classify these images. The recognition accuracies are compared to Zernike, CHFM, and PJFM descriptors. The recognition results of the classiﬁer are depicted in Table 10. As can be observed from this table, the Zernike moment invariants are less sensitive to noise than CHFM- or PJFMbased invariants. Table 10 demonstrates, again, the good performance of the feature vector based on the proposed G-CHFM or G-PJFM in the noisy environment as well as the noise-free environment.

24

Table 7: G-CHFM and CHFM rotation invariants of the same object under diﬀerent rotation angles (zero mean Gaussian white noise σ 2 = 0.01) 25 G-CHFM invariants |Z10 |

25 |Z01 |

25 |Z00 |

25 25 25 25 |Z12 | |Z21 | |Z33 | |Z13 |

Flower 2

26.0898 0.1656 18.8183 0.8210 0.3615 1.1294 0.7085 26.6146 0.1662 18.8892 0.8697 0.3719 1.1162 0.7260 26.2651 0.1624 18.1356 0.8510 0.3638 1.1637 0.7019 26.8234 0.1652 18.9190 0.8706 0.3753 1.1511 0.7279 26.4285 0.1617 18.8246 0.8521 0.3852 1.1408 0.7271

τ /μ

0.0109 0.0123 0.0184 0.0236 0.0256 0.0162 0.0170

Flower 4

53.2898 2.5351 38.9606 1.4369 6.1309 0.8516 1.1944 53.8136 2.5300 38.7681 1.4368 6.1818 0.8776 1.1349 53.8210 2.4754 38.9444 1.4932 6.1529 0.8774 1.1349 53.9934 2.5728 38.8028 1.4156 6.1316 0.8628 1.1720 53.6906 2.5761 38.8696 1.4387 6.1384 0.8305 1.1598

τ /μ

0.0049 0.0161 0.0022 0.0201 0.0033 0.0230 0.0219

CHFM invariants

0 |Z10 |

0 |Z01 |

0 |Z00 |

0 0 0 0 |Z12 | |Z21 | |Z33 | |Z13 |

Flower 2

28.8813 0.4299 45.3755 1.2621 0.4021 0.4619 0.4311 28.6460 0.4779 45.7410 1.2864 0.4811 0.3886 0.4111 28.6516 0.4627 44.2067 1.2863 0.4841 0.4322 0.4768 28.9680 0.5377 45.5571 1.0275 0.4344 0.4575 0.4588 28.3095 0.4757 44.1241 1.2741 0.4668 0.4177 0.4801

τ /μ

0.0089 0.0820 0.0192 0.0914 0.0770 0.0698 0.0661

Flower 4

56.4718 7.2531 68.9015 1.1378 1.3192 0.8919 1.0177 56.8851 6.8367 68.5645 1.3261 1.3272 0.8515 1.0834 56.3406 6.8499 69.2267 1.3616 1.4848 0.8824 1.0585 57.1457 7.1099 68.3873 1.3290 1.4525 0.7667 1.0574 57.1433 7.1227 69.0682 1.3875 1.3599 0.8320 1.0594

τ /μ

0.0066 0.0261 0.0051 0.0754 0.0543 0.0590 0.0224

Figure 6: Some test images from the MNIST handwritten digits database.

6. Conclusions Unit circle-based continuous orthogonal moments are very useful tools for image analysis. In this paper, two new descriptors, named G-CHFM and G-PJFM, were proposed for image reconstruction and pattern classiﬁcation. 25

Table 8: G-PJFM and PJFM rotation invariants of the same object under diﬀerent rotation angles (zero mean Gaussian white noise σ 2 = 0.01) 25 G-PJFM invariants |Z10 |

25 |Z01 |

25 |Z00 |

25 25 25 25 |Z12 | |Z21 | |Z33 | |Z13 |

Flower 2

28.1597 0.4464 28.2533 0.7545 1.2942 1.7531 2.3162 28.0022 0.4546 28.1595 0.7797 1.2164 1.7451 2.3080 28.8019 0.4202 28.1766 0.7578 1.2480 1.7409 2.3049 28.7376 0.4266 28.2351 0.7217 1.2551 1.7680 2.3197 28.2334 0.4532 28.2445 0.7375 1.2559 1.7618 2.3668

τ /μ

0.0128 0.0322 0.0015 0.0292 0.0221 0.0064 0.0108

Flower 4

53.5359 4.4651 57.4847 1.5323 6.7693 2.0172 0.9775 53.4563 4.4847 57.2078 1.5845 6.8012 2.0980 1.0067 53.1694 4.4791 57.6388 1.5973 6.7920 2.0716 0.9871 53.2595 4.4842 57.3373 1.5401 6.7374 2.0835 0.9813 53.4187 4.4915 57.4722 1.5826 6.9401 2.0295 1.0081

τ /μ

0.0030 0.0022 0.0028 0.0186 0.0114 0.0170 0.0145

PJFM invariants

0 |Z10 |

0 |Z01 |

0 |Z00 |

0 0 0 0 |Z12 | |Z21 | |Z33 | |Z13 |

Flower 2

39.8742 0.2525 37.2513 1.6863 0.2675 0.7554 0.4776 37.9226 0.2419 38.9399 1.9670 0.2212 0.7462 0.4353 39.8573 0.2549 37.2616 1.9262 0.2732 0.8581 0.4414 39.8991 0.2893 37.0060 1.7132 0.2496 0.7549 0.4727 39.3920 0.2872 36.5620 1.7816 0.2553 0.7503 0.4752

τ /μ

0.0215 0.0817 0.0242 0.0694 0.0801 0.0617 0.0442

Flower 4

67.7379 5.5426 52.1592 1.8652 4.1243 1.1457 1.8608 67.8868 5.2644 51.4544 1.7304 3.8632 1.0789 1.9969 67.2515 5.4109 52.2723 2.0329 3.4520 0.9993 1.8122 67.1184 5.5303 51.9353 1.9505 3.8360 0.9757 1.9134 68.0905 5.4653 52.0670 2.0984 3.9107 1.1154 1.9853

τ /μ

0.0062 0.0207 0.0061 0.0745 0.0634 0.0690 0.0414

Appling some properties of Jacobi polynomials, we have established a series of relationships between the proposed generalized radial polynomials and shift Jacobi polynomials, the hypergeometric function, explicit expressions, etc. Two fast and numerically stable n-recursive methods for computing the radial polynomials of G-CHFM and G-PJFM were proposed. The numerical stabilities were demonstrated by computing the MSE error for a high-order of moments. According to image reconstruction capability and invariant recognition accuracy in noise-free and noisy cases, the G-CHFM and G-PJFM 26

Table 9: The accuracies of recognition for COREL photograph database in noise environment (%) Salt-and-Pepper 5% 10%

Gaussian Variance 0.01

Methods

α

Noise-free

CHFM

0 5 10 15 25

100 100 100 100 100

73.69 80.92 87.36 91.84 93.63

68.81 75.38 79.65 81.94 88.03

72.81 76.34 82.53 88.82 92.87

0 5 10 15 25

100 100 100 100 100

72.94 81.15 87.68 90.44 95.22

69.95 73.06 78.25 80.64 87.17

71.81 77.56 84.93 90.56 93.14

100

90.23

80.24

86.07

G-CHFM PJFM G-PJFM Zernike’s

Table 10: The accuracies of recognition for MNIST database in noise environment (%) Salt-and-Pepper 5% 10%

Gaussian Variance 0.01

Methods

α

Noise-free

CHFM

0 5 10 15 25

100 100 100 100 100

79.65 81.36 86.15 90.71 97.14

72.28 75.42 79.26 86.20 91.43

78.92 80.37 88.29 91.42 94.29

0 5 10 15 25

100 100 100 100 100

78.35 80.03 88.57 91.86 95.14

73.24 77.25 80.52 86.34 90.22

78.82 79.64 83.06 90.29 94.17

100

90.14

84.76

88.70

G-CHFM PJFM G-PJFM Zernike’s

were potentially useful as feature descriptors. They gave better performances in image description than classical CHFM and PJFM. They could be used in pattern classiﬁcation, image watermarking, image retrieval, corneal surface modelling, and image data mining in the future.

27

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