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Pattern Recognition and Image Reconstruction UsingImproved Digital Zernike Moments

Huibao Lina, Jennie Sia, and Glen P. Abouslemanb

aDepartment of Electrical Engineering, Arizona State University, Tempe, AZ 85287bGeneral Dynamics C4 Systems, 8201 E. McDowell Road, Scottsdale, AZ 85257

ABSTRACT

Zernike moments are one of the most effective orthogonal, rotation-invariant moments in continuous space. Un-fortunately, the digitization process necessary for use with digital imagery results in compromised orthogonality.In this work, we introduce improved digital Zernike moments that exhibit much better orthogonality, whilepreserving their inherent invariance to rotation. We then propose a novel pattern recognition algorithm that isbased on the improved digital Zernike moments. With the improved orthogonality, targets can be representedby fewer moments, thus minimizing computational complexity. Additionally, the rotation invariance enables ouralgorithm to recognize targets with arbitrary orientation. Because our algorithm eliminates the segmentationstep that is typically applied in other techniques, it is better suited to low-quality imagery. Simulations on realimages demonstrate these aspects of the proposed algorithm.

Keywords: Moments, Zernike moments, feature, image reconstruction, pattern recognition

1. INTRODUCTION

Moments are widely used in signal processing applications because they capture the global information of thesignal.1 Among various moments that have been proposed, Zernike moments have received special attention.2–4

Zernike moments are orthogonal and rotation invariant, and are developed in continuous space. They havecomplex coefficients and are defined over the unit disk, i.e., 0 ≤ ρ ≤ 1, 0 ≤ θ < 2π. The magnitude ofthe moments is invariant to the rotation of the signal, while the orientation information is conveyed only by thephase. The rotation invariance of the magnitude is useful in pattern recognition applications because it eliminatesthe difficulty in calculating the orientation of the targets. The coefficients of Zernike moments are orthogonal toeach other, and this orthogonality has several important consequences. First, the moments are non-redundant,and thus, fewer moments are needed to represent a signal than non-orthogonal moments. Secondly, signalreconstruction is much easier for orthogonal moments. Finally, as more moments are used, the reconstructionbecomes progressively better, and consequently, every signal can be accurately reconstructed by using a set(possibly infinite) of these moments.

For applications in discrete space, Zernike moments are digitized both geometrically and numerically. Specif-ically, a set of pixels is selected to resemble the unit disk, and for each moment, a coefficient is chosen for eachpixel. The so-called “direct methods” determine the coefficients for a pixel by the direct sampling of Zernikemoments at one or several locations. For example, the simplest direct method uses orthogonal, rotation-invariantmoment (ORIM) coefficients at the pixel center.

Digital Zernike moments resulting from direct methods have been shown to be effective in many digitalimage processing applications such as image analysis,5, 6 patten recognition,7–9 texture classification,10 andtarget orientation estimation.11 They have proven to be noise resilient,6 and are used as the basis by which toconstruct moments with new properties.12–15 Various techniques for quickly computing digital Zernike momentshave been proposed in the literature,16–18 but the orthogonality of the Zernike moments is compromised duringthe digitization process, which limits their precision. For example, if a certain number of moments are used

Further author information: (Send correspondence to Jennie Si)Jennie Si: E-mail: [email protected]. Telephone: (480) 965-6133. Fax: (480) 965-2811.Huibao Lin: E-mail: [email protected]. Telephone: (480) 965-3257. Fax: (480) 965-2811.Glen P. Abousleman: [email protected]. Telephone: (480)441-2193. Fax: (480)441-3868.

Optical Pattern Recognition XVI, edited by David P. Casasent, Tien-Hsin Chao,Proceedings of SPIE Vol. 5816 (SPIE, Bellingham, WA, 2005)0277-786X/05/$15 doi: 10.1117/12.604076

211

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Order (n) Repetition (m) Number0 0 11 -1, 1 32 -2, 0, 2 63 -3, -1, 1, 3 104 -4, -2, 0, 2, 4 155 -5, -3, -1, 1, 3, 5 21

Table 1. The permissible repetition and number of moments from order 0 to 5.

to reconstruct an image, the inclusion of additional moments may degrade the result. Therefore, the digitalZernike moments from direct methods cannot represent the fine details of an image, and they cannot be used todistinguish images with subtle differences.

Liao and Pawlak suggested several reasons for the loss of accuracy of digital Zernike moments derived fromthe simplest direct methods.19–21 Fist, the total area of the chosen pixels is different from that of the unit disk(geometric error). Also, the coefficients sampled at the pixel center are different from the average coefficients inthe pixel area (numerical error). Accordingly, they proposed the use of a circle smaller than the inscribed circleof the image to select pixels, and the use of numerical integration rather than one-point sampling.

In our previous work, we used the coefficients derived from direct methods as an initial estimate, and then usednumerical optimization to improve the orthogonality, while preserving the rotation invariance.22 Because Zernikemoments can be calculated directly from the pixel values, image segmentation, which is a challenging task for low-quality imagery, is not required. Due to the orthogonality and rotation invariance of Zernike moments, targetscan be represented by a compact number of moments, which are invariant to the target orientation. Supportvector machines (SVMs)23, 24 are then used to distinguish the moments associated with targets from those ofnon-targets. Experiments show that our optimized digital Zernike moments exhibit better image reconstructionand recognition performance.

This paper is organized as follows. Section 2 introduces Zernike moments in continuous space. Section 3describes several methods, including our optimization-based method, for migrating Zernike moments from con-tinuous to discrete space. Section 4 compares the image reconstruction performances of digital Zernike momentsfrom different methods. Section 5 presents our image recognition algorithm, which is based upon support vectormachines and Zernike moments. Finally, a conclusion is given in Section 6.

2. ZERNIKE MOMENTS IN CONTINUOUS SPACE

Zernike moments are developed in continuous space over the unit disk, D, i.e., D = {(ρ, θ) | 0 ≤ ρ ≤ 1, 0 ≤ θ <2π}. The coefficients of Zernike moments are

vnm(ρ, θ) = rnm(ρ)pnm(θ), (1)

where pnm(θ) = ejmθ, and rnm(ρ) is a polynomial in ρ of degree, n, as shown in Eqn. (2):

rnm(ρ) =(n−|m|)/2∑

s=0

(−1)s(n − s)!ρn−2s

s!((n + |m|)/2 − s)!((n − |m|)/2 − s)!. (2)

n is the “order” of the moment and can be any non-negative integer. m is the “repetition” of the moment andis subject to the constraints that n − |m| is even and |m| ≤ n.

The coefficients of Zernike moments are orthogonal to each other, i.e.,∫ ∫

(ρ,θ)∈D

v∗nm(ρ, θ)vpq(ρ, θ)ρdρdθ =

π

n + 1δnpδmq, (3)

where δnp = 1 if n = p, and 0 otherwise.

212 Proc. of SPIE Vol. 5816


(a)

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

-1

1 x-1

1y

(b) (c)

(d) (e)

Figure 1. Direct methods for converting Zernike moments from continuous to discrete space. Each square in (a) representsa pixel, and each black dot represents the center of a pixel. The circle is the original definition domain, D, and is digitizedinto the shaded pixels, D̂. The coefficients for each pixel are derived by sampling Zernike moments at one or multiplelocations within the pixel. (b), (c), (d), and (e) are the sampling locations for formulas 1-D, 5-D(I), 5-D(II), and 13-D(I& II), respectively.

The moment of order, n, with repetition, m, for a continuous 2-D signal, f(ρ, θ), that vanishes outside theunit disk is

anm =n + 1

π

∫ ∫

(ρ,θ)∈D

f(ρ, θ)v∗nm(ρ, θ)ρdρdθ. (4)

Due to the orthogonality of the coefficients, vnm(ρ, θ), f(ρ, θ) can be easily reconstructed by

f̂(ρ, θ) =∑

n

∑

m

anmvnm(ρ, θ). (5)

As additional moments are included in the reconstruction, f̂(ρ, θ) becomes closer to f(ρ, θ). Furthermore, everysignal can be accurately reconstructed using a (possibly infinite) set of moments.

Rotating the signal, f(ρ, θ), does not change the magnitude of the Zernike moments. For example, let therotated signal be f(ρ, θ − φ), then the corresponding moments will be aφ

nm = anme−jmφ.7

3. MIGRATING ZERNIKE MOMENTS FROM CONTINUOUS SPACE TODISCRETE SPACE

3.1. Digital Zernike Moments by Direct Methods

The digital Zernike moments derived from direct methods are produced by digitizing Zernike moments bothgeometrically and numerically. This procedure is illustrated in Fig. 1. As seen in the figure, the unit disk,D = {(ρ, θ) | 0 ≤ ρ ≤ 1, 0 ≤ θ < 2π}, is digitized to form D̂, which consists of a set of pixels. For each moment,the coefficients values, vnm(ρ, θ), over the area of a pixel are quantized into one vnm,ij .

The simplest direct method works as follows. Consider an image of M × M pixels. The coordinates ofthe pixel center, (xi, yj), can be calculated by assuming that the image is sampled from the square defined by[−1, 1]2. The definition domain, D̂, is chosen to contain the pixels with centers falling on the unit circle, i.e.,

D̂ ={

(i, j)|√

x2i + y2

j ≤ γ}

, (6)

where γ = 1 is the radius of the circle, and (i, j) represents the pixel at the ith row and the jth column. Thecoefficients are simply the ORIM values at the pixel centers. Specifically, vnm,ij = vnm(xi, yj). Since a singlesampling is used to calculate the coefficients, this method is referred to as a “1-D” formula.

Proc. of SPIE Vol. 5816 213


The moments and signal reconstruction in discrete space are, respectively,

anm =n + 1

π

∑

(i,j)∈D̂

fij v∗nm,ij (7)

andf̂ij =

∑

n

∑

m

anm vnm,ij . (8)

From Fig. 1, it can be seen that the selected pixels in D̂ cannot exactly match the unit disk, D, and thatthe ORIM coefficients at the pixel center are usually different from the average coefficients over the entire pixelarea. Hence, geometric error and numerical error are expected.

Liao and Pawlak19 proposed an improvement to the accuracy of Zernike moments by using a smaller circlefor determining D̂, and multiple-point numerical integration formulas. The proposed radius for the circle wasstated to be

γ =√

1 − 1/M − 0.0001. (9)

The proposed formulas were 5-D(I), 5-D(II), 13-D(I), and 13-D(II). The sampling locations of these formulas areillustrated in Fig. 1, with details provided in.19

However, it was shown that with these improvements, the digitization error of the direct methods is still largeenough to deteriorate the image reconstruction.19, 22

3.2. Digital Zernike Moments by Numerical Optimization

Our proposed algorithm uses numerical optimization to improve the orthogonality of the digital Zernike momentsderived from direct methods, while preserving their rotation invariance.

For ease of discussion, we arrange the image and Zernike moments into vector or matrix forms. Let thenumber of pixels in D̂ be N . A column vector, F = {Fk|1 ≤ k ≤ N}, is used to represent the digital imagery,{fij |(i, j) ∈ D̂}. Let the number of moments be L. A matrix, P, of size L × N , is used to present the phaseof the coefficients such that each row corresponds to a moment, and each column corresponds to a pixel. Forexample, Plk = pnm,ij . Matrix R is constructed in the same format, but with normalized values:

Rlk =√

(n + 1)/π rnm,ij . (10)

The coefficients areVlk = RlkPlk, 1 ≤ l ≤ L, 1 ≤ k ≤ N. (11)

If we define the operation “◦” to be the dot product between matrices, then

V = R ◦ P. (12)

If the coefficients are orthonormal, then V∗VT = I. Thus, κ(V) = ||V∗VT − I||2F is a measure of theorthogonality of V. Here, || · ||F is the Frobenius norm, i.e.,

κ(V) = ||V∗VT − I||2F =L∑

i=1

L∑

j=1

∣∣∣∣∣

N∑

k=1

V ∗ikVjk − δij

∣∣∣∣∣

2

. (13)

The moments are thenA = V∗F. (14)

The reconstructed image would then beF̂ = VT A = VT V∗F, (15)



(a)

n = 4 8 12 16 20 24 28

1-D

5-D(I)

5-D(II)

13-D(I)

13-D(II)

Optimized

Original 1-D 5-D(II) Optimized

(b)

(c)

0 50 100 150 200 250 300 350 4000

0.05

0.1

0.15

0.2

0.25

0.3

4 8 12 16 20 24 28Order (n) of moments

Number of moments

Re

con

stru

ctio

n e

rro

r ε(F

)

1−D5−D(I)5−D(II)13−D(I)13−D(II)Optimized

Figure 2. Image reconstruction performance. (a) The original image, and the best reconstructed images by using digitalZernike moments from the 1-D formula, the 5-D(II) formula, and our optimization method. (b) The reconstructed imagesby using moments from order 0 to 4, 8, 12, 16, 20, 24, and 28. (c) The reconstruction error by using moments of order 0to 28.



(a) (b)

d2

d1

Figure 3. The critical points chosen for a target (a) and a magnified target portion (b). The critical points are chosenfrom the target boundary with step size, d1. The critical points are randomly perturbed by distance, d2, and thenconcatenated to form a deformed boundary.

where V∗ is the complex conjugate, and VT is the transpose of matrix V. The residual image is

F̂ − F = (VT V∗ − I)F. (16)

Define the reconstruction error for image F as

ε(F ) =N∑

k=1

|F̂k − Fk|2 /

N∑

k=1

F 2k . (17)

From Eqns. (16) and (17), we can see that the reconstruction error is determined both by (VT V∗ − I) and theoriginal image, F .

Defineε(V) = ||VT V∗ − I||2F . (18)

If ε(V) is small, the reconstruction error will be small regardless of F . Hence, ε(V) is used as the measure ofthe reconstruction accuracy of V. It can be proven that ε(V) and κ(V) differ only by a constant quantity.22

Specifically,ε(V) = κ(V) + N − L, if L ≤ N. (19)

Therefore, improving the reconstruction accuracy will improve the orthogonality and vice versa.

In this work, we use the limited-memory BFGS (L-BFGS) algorithm25 to optimize ε(V). The L-BFGSalgorithm has a linear convergence rate without using the Hessian matrix, and hence, is suited for large-scaleoptimization problems.

4. IMAGE RECONSTRUCTION USING DIGITAL ZERNIKE MOMENTS

The reconstruction error as defined in Eqn. (17) is an important performance measure for digital Zernike momentsbecause it reflects 1) the redundancy of the moments, and 2) the capability of the moments to describe the subtleimagery details.

To illustrate the performance of the digital Zernike coefficients calculated by different methods, severalmoments were used to reconstruct an image, and the reconstruction errors were compared. The size of thetest image was 24× 24, i.e., M = 24. The radius as defined in Eqn. (9) was used, and moments up to 28th orderwere calculated. Note that at order 28, the number of moments is greater than the number of pixels. Thus,according to Eqn. (19), it is possible to reduce the reconstruction error to zero.

The results are illustrated in Fig. 2. The original image, as well as the best reconstructed images by usingformulas 1-D, 5-D(II) (the best formula proposed by Liao and Pawlak19), and our optimized method, are shownin Fig. 2 (a). The reconstructed images by using moments of order 0 to 4, 8, 12, 16, 20, 24, and 28, are shownin Fig. 2 (b). The reconstruction errors as defined in Eqn. (17) are shown in Fig. 2 (c).



(a)

(c) (d)

(e) (f)

(b)

Figure 4. A template (a) and its boundary (b); a synthesized image with moderate boundary deformation (c), moreboundary deformation (d), more luminance variance (e), or more additive noise (f).

From Fig. 2, it can be seen that 1) for the same reconstruction accuracy, fewer moments are required for ouroptimization method than the other methods; 2) as more moments are added, the reconstructed image formedby our optimized moments becomes closer to the original, and when n = 28, the difference becomes negligible.Therefore, the coefficients that result from our optimization procedure are more compact, and are better atrepresenting the subtle details of the image.

5. PATTERN RECOGNITION BY OPTIMIZED DIGITAL ZERNIKE MOMENTS

The dimensionality of digital imagery and the variance of target features due to target orientation are twochallenges for image pattern recognition.26 To cope with the dimensionality problem, an image is typicallysegmented into several regions. A basic assumption of image segmentation is that the contrast between thetargets and the background is high. However, this assumption is typically violated in low-quality imagery, and isresponsible for segmentation-based approaches performing poorly for automatic target recognition. On the otherhand, to avoid the variation of target features due to a change in target orientation, several techniques havebeen developed to estimate the target’s orientation. Based upon the estimated orientation, the target is rotatedto a reference orientation prior to calculating the target’s features. However, most of the proposed orientationestimation techniques only focus on targets with certain shapes, and are unreliable in general.

Zernike moments, on the other hand, have proven to be superior to other techniques because they caneffectively address the two aforementioned challenges. Their inherent orthogonality enables the representationof imagery with a compact number of moments, which dramatically reduces the dimensionality. Moreover, theirinherent rotation invariance results in unique moments for a target regardless of its orientation.

In this section, we proposed an image pattern recognition algorithm that uses Zernike moments in conjunctionwith support vector machines (SVMs).24 To detect the presence of a target at a particular location, the Zernike



moments are calculated within a small window that is centered at that location. SVMs are then used to determinethe probability of target existence by comparing the calculated Zernike moments with several pre-stored samples,which include both targets and non-targets. A probability map is generated when all locations have been tested,and the locations with local maximum probability are selected as target locations.

SVMs have excellent data distinguishing performance, but they typically require large amounts of trainingdata,23, 27, 28 which is expensive to collect. We therefore employ an automatic image synthesis algorithm that isbased on the boundary deformation algorithm developed by Kamgra-Parsi et al .29

The fist step of the synthesis algorithm is to generate a deformed boundary. Referring to Fig. 3, this stepconsists of the following procedure:

Step 1.1 Extract the boundary from the prototype image.Step 1.2 Choose the critical points from the boundary by sampling the boundary with a step size of d1.Step 1.3 Perturb the critical points by a distance, d2, in a random direction.Step 1.4 Concatenate the perturbed critical points to form the deformed boundary.

Note that larger d1 or d2 results in a more deformed boundary.

We then generate a gray-scale image from the deformed boundary as follows:Step 2.1 Rotate, scale, and skew the boundary.Step 2.2 Determine the shadow area.Step 2.3 Determine the luminance values for the target, shadow, and background areas.Step 2.4 Add random noise to the luminance component of every pixel.

Example synthesized images are shown in Fig. 4.

Fig. 5 shows the recognition results for a sample image. The original image is shown in Fig. 5 (a). Theprobability map of target existence is shown in Fig. 5 (b), and is derived by calculating the Zernike moments upto 11th order, within a window of size 23 × 23, that is centered at every pixel. The locations of local maximumvalues, which indicate the existence of targets, are identified by black dots in Fig. 5 (c). Note that a target atthe bottom-left corner is too close to the boundary to completely fit within the window, and is not identified asa target. However, when this target is copied to the interior of the image, it is correctly recognized as shown inFig. 5 (d).

6. CONCLUSION

In this paper, we developed improved digital Zernike moments that exhibit excellent orthogonality, while pre-serving their inherent invariance to rotation. We then proposed a novel pattern recognition algorithm, which isbased upon the improved digital Zernike moments and support vector machines. An image synthesis algorithmwas also developed for generating the appropriate training data. The proposed pattern recognition algorithmwas verified on real imagery, and shown to exhibit excellent recognition performance.

ACKNOWLEDGMENTS

This work was supported by General Dynamics C4 Systems, and in part by the NSF under grant ECS-0002098.

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(a) (b)

(c) (d)

Figure 5. (a) Original image. (b) The probability of target existence at every pixel location. (c) The recognized targets,where each target is marked by a black dot (the image is brightened for easier viewing). Note that a target at thebottom-left corner has not been identified because it is too close to the image boundary. (d) When the missing target iscopied to the interior of the image (highlighted by a black circle), it is correctly recognized.



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