Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

FEATURE EXTRACTION USING EXTENDED CENTRAL PROJECTION

nnsm LAN1, JIANWEI YANG1, YONG JIANG1, XIAOXIA FENG2

lSchool of Math and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China2Department of Math, Zhangzhou Normal University, Zhangzhou 363000, China

E-MAIL: IrsOl06@yahoo.com.yjianw2002@yahoo.com.cn

Abstract:A novel feature extraction method is proposed in this paper..

Dislike contour-based or region-based approaches, an object isfirst converted to a closed curve by extended central projection(ECP). The derived curve not only keeps the affine transforminformation, but also is very robust to noise. Then whiteningtransform is performed to the curve such that the affine trans­formation is simplified to a rotation only. Finally, Fourier trans­form are employed to remove the rotation. Several experimentshave been conducted to evaluate the performance of the proposedmethod. Experimental results show that the proposed method hasa powerful discrimination ability, and is more robust to noise.

Keywords:Feature extraction; Affine transformation; Extended central

projection (ECP); Object recognition

1. Introduction

Feature extraction plays an important role in object recog­nition, and has found applications in many fields such as tex­ture classification, image matching, image retrieval and shaperecognition, etc. Many algorithms have been developed to ex­tract the features of objects undergoing affine transformation.Based on whether the features are extracted from the contouronly or from the whole object region, these methods can beclassified into two main categories: contour-based methods andregion-based methods [1].

In a contour-based method, the boundaries are first extractedfrom the objects that are segmented from the background. Con­tour representation provides better data reduction and the con­tour usually offers more shape information than interior con­tent [1]. Examples of these approaches include the Fourierdescriptor (FD) [2], the wavelet descriptor (WD) [3]. How­ever, contour-based methods often are strongly dependant onthe success of the boundary extraction process [4] which is still

978·1-4673·1535·7/121$31.00 ©2012 IEEE

a difficult problem in image processing.

In contrast to contour-based methods, region-based tech­niques take all pixels within a shape region into account to ob­tain the shape representation. Moment invariant methods arethe most widely used techniques. The commonly used affinemoments invariants (AMIs) [5, 6] are extensions of the classi­cal moment invariants firstly developed by Hu [7]. Recently,a novel approach called Multi-Scale Autoconvolution (MSA)was derived by Rahtu et al [4]. Unfortunately, region-basedmethods have some drawbacks too. The moment based meth­ods are sensitive to noise. Comparing with AMIs, the MSA,constructed from the gray intensity and probability density ofthe image, is much more robust than AMIs. But experimen­tal results show that MSA has large computational complexity,and is sensitive to noise in the background.

In our previous work, we have developed several featureextraction approaches by central projection transform (CPT)[8, 9]. In this paper, an image transform method named ex­tended central projection (ECP) is proposed for feature extrac­tion. Dislike CPT, ECP is performed by applying the extendedcentroid [10] as the center of the polar coordinate system. Wemay obtain different extended centroids by using different or­der. After ECP, the original object is converted to a closedcurve. Whitening transform and Fourier transform are usedto the closed curve respectively to extract affine invariant fea­tures which are called extended central projection descriptors(ECPDs). Some experiments have been conducted to evaluatethe proposed method.

The rest of the paper is organized as follows: in section 2,the ECP method is introduced. A feature extraction scheme isconstructed using ECP and whitening transform in section 3.The performance of proposed method is evaluated experimen­tally in section 4. Finally, some conclusion remarks are givenin section 5.

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Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

2. Extended central projection (ECP) 3.1. Affine Invariant Parameters

After the parameterization, we establish a one-to-one corre­spondence between points on the original curve and its affinetransformed version.

All points on the derived curve could be expressed in termsof a number of points along the curve from a specified startingpoint (arc length). With the affine transforms, the positions ofeach point changes and it is possible that the number of pointsbetween two specified points changes too on the discrete curve.Hence, the ECPC should be parameterized to establish one-to­one points correspondence between the original curve and itsaffine transformed version. In this paper, we adopt a curve nor­malization approach [3] by using enclosed area

In this section, we propose a simple but effective transform,referred to as extended central projection (ECP), for feature ex­traction. Dislike CPT, ECP is carried out by making use of theextended centroid [10]. The p-order extended centroid of theobject, h(xg, Yb), is defined as following

xP - I I xfP(x, y)dxdy 1!n _ I I yfP(x, y)dxdyo - f f fp(x, y)dxdy' 0 - f f fp(x, y)dxdy .

It can be easily verified that the extended centroid keeps theaffine transformation as the original centroid.

Let I (x, y) be the original image. To perform ECP, the Carte­sian coordinate system should firstly be converted to the polarcoordinate system whose center is taken at the extended cen­troid. We denote the transformed image as f(rp,8p) where rpand 8p are the radius and the polar angle with the extended cen­troid h(xg, Yb) as the center. The ECP of image f(rp, 8p) isdefined as

b

a = ~JIxy - yxldt.a

(2)

(3)G' = AGAT.

X'(t) = AX(t) + b,

G = E{(X - E{X})(X - E{X})T},

where E{.} denotes the expected values. It can be easily shownthat the covariance matrix of X' becomes:

To carry out a whitening transform, it is necessary to cal­culate the whitening matrix firstly, which can be obtained by

After the parameterization process, many existing schemescan be directly performed on the ECPC. In this work, we paymore attention to applying a whitening transform to feature ex­traction. Two point sets can be matched under affine transfor­mation if their canonical forms can be matched under rotationonly [11]. Furthermore, a point set is canonical if its covari­ance matrix is an identity matrix, which can be achieved by awhitening transform.

We consider a point set X(t) = [x(t), y(t)]T with parametert on an object. The point on the object under affine transforma­tion becomes:

As previously mentioned, the curve obtained by ECP pre­serves the affine transformation information such that it canbe used to extract the feature of the object undergoing affinetransformation. In this section, we will show how to derive theaffine invariant features based on the ECPC. In this paper, wemake use of whitening transform to derive the features. Andbefore performing the whitening transform, a parameterizationprocess is carried out to establish a one-to-one correspondencebetween points on the original ECPC and those on the ECPCof its affine transformed version.

Fp(8) = Jf(rp, 8p)drp. (1)3.2. Whitening Transform

After ECP, I(x, y) is converted to a curve, (Fp (8),8p ) , repre­sented in the polar coordinate system. Consequently, we callthe derived curve as extended central projection curve (ECPC).

The proposed image transform is based on the following im­portant property of affine transformation: an affine transfor­mation preserves the collinearity relation between points; i.e.,three points which lie on a line continue to be collinear afterthe transformation. Therefore, we can try to analyze the imageonly applying the analysis to points on the same line, whichcan be regarded as a sub-component of the original image. Ob­viously, the affine transform of the points on the line do notchange compared to the affine transform of the original image.Consequently, we can study the properties of Fp (8) instead ofthe original object, and it is not difficult to testify that ECPCkeeps affine transformation as the curve acquired by CPT [9].

where the nonsingular matrix A represents the scaling, rota-tion, skewing transformations and the vector b corresponds tothe translation. The covariance matrix of X is defined as fol­lows:

3. Feature extraction by ECP

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Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

different ways. In [11], Heikkila exploits the Cholesky factor- 4. Experimental resultsization of the covariance matrix such that

Since C is a symmetric matrix, C1/ 2 is a symmetric matrix too.Thus the above formula can be written as

(6)A=k(COS() -Sin())(a b)sin () cos () 0 1/a '

In this section, some experiments are carried out to illustrate(4) the performance of the proposed method by comparing it with

MSA and AMIs. The comparison methods are described in [4]and [5], and these methods are implemented as discussed inthose articles. 3 AMIs invariants and 29 MSA invariants areused as reported. The nearest neighbor classifier is applied forall the test methods.

In the experiments, affine transformations are generated bythe following transformation matrix [3]:

where k E {0.8,1.2}, () E {0,72° ,144° ,216° ,288°},b E {-3/2, -1, -1/2,0, 1/2, 1, 3/2} and a E {1,2}. There­fore, each image is transformed 140 times. The well-knownColumbia Coil-20 database [13] which includes some sets ofsimilar objects is used as the test images. In many real-liferecognition situations, noise is sometimes added to images de-

(5) liberately. So the effect of adding different kinds of noises hasbeen studied respectively. The noise is added to the affine trans­formed images before recognition. Therefore, in the followingtests, all the methods are compared in affine transformation anddifferent type of noise situations.

For the Coil-20 image database, we first add the salt & pep­per noise with intensities varying from 0.01 to 0.03 to the trans­formed images. Fig. 1 shows the classification accuracy of allmethods against the corresponding noise power. We can ob­serve that the classification accuracy of AMIs decreases rapidlyfrom noise free condition to small noise degree. MSA performsmuch better than AMIs, but still the results are not satisfactory.The fall in ECPDs' classification is actually less than three per-cent. With large noise amplitude, the ECPDs maintain highaccuracies all the time.

To make the situation more challenging, we add zero meanGaussian noise with different variances varying from 0.001 to0.003 to the transformed images. It is shown that almost everypixel is changed after adding the Gaussian noise to the image.Fig. 2 shows the classification accuracy of all methods againstthe corresponding noise power. The results indicate that AMIsand MSA are more sensitive to Gaussian noise than salt & pep­per noise. However, the classification accuracies of the ECPDsonly decreases a little.

C=ppT,

y = p-1(X - E{X}).

Y = C-1/2(X - E{X}).

3.3. Derivation of ECPDs

The corresponding whitening transform is expressed by

It is easy to show that Y is canonical with the covariance ma­trix C. Similarly, we can obtain y' by performing the sametransform on X'. Then Y and y' are geometrically congruentvia a rotation.

In [12], the covariance matrix is decomposed into the follow­ing form:

The Y and y', obtained in this way, are also geometricallycongruent by rotation.

Observing Eq. (4) and Eq. (5), we may find that the covari­ance matrix is decomposed into the form of a matrix multipliedby its transpose. It can be proofed that the whitening matri­ces used in [11] and [12], in fact, are equivalent to each other.Hence, we do not develop other forms of whitening matrices,and that applied in [12] is selected in this paper.

where P is a lower triangular matrix. Then the whitening trans­form can be expressed by

In this subsection, we will show how to derive an affine in­variant features by making use of ECP, whitening transform,and Fourier transform. As previously discussed, ECP con­verts the original object into a closed curve which preservesthe affine transformation. After the parameterization process,a whitening transform is performed on the ECPC such that theaffine transformation is simplified to a rotation only. Finally, toremove the rotation, we carry out Fourier transform to the ECPafter the whitening transform. Therefore, the final features, re­ferred to as extended central projection descriptors (ECPDs),are an affine invariant feature of the object.

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Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

Figure 1. Classification accuracies of AMIs, MSA,and ECPDs in case of affine transformation and dif­ferent intensities of salt & pepper noise.

5. Conclusions

In this paper, we described a novel approach to constructingaffine invariant features based on ECP and whitening trans­form. Prior to the construction, we first employ a ECP methodto convert the object into a closed curve. Fourier transform areapplied to the curve after performing a whitening transform.The obtained features are affine invariant, and experimentalresults also show that these features are quite robust to noise.However, the extended centroid used in ECP cannot performto binary images. Future work will try to settle this problem.

Acknowledgements

This work was supported in part by the National NaturalScience Foundation of China (Nos. 60973157 and 41174165),and the Innovation Program Award for Graduate Student inJiangsu (No. CXZZI2_0510).

Figure 2. Classification accuracies of AMIs, MSA,and ECPDs in case of affine transformation and dif­ferent intensities of Gaussian noise.

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