geometric invariant curve and surface … invariant curve and surface normalization ... abstract. in...

Geometric Invariant Curve and Surface Normalization

Sait Sener1 and Mustafa Unel2

1 Institute of Informatics, Istanbul Technical UniversityMaslak, Istanbul, Turkey

[email protected] Faculty of Engineering and Natural Sciences, Sabanci University

Orhanli-Tuzla 34956, Istanbul, [email protected]

Abstract. In this work, a geometric invariant curve and surface normalizationmethod is presented. Translation, scale and shear are normalized by PrincipalComponent Analysis (PCA) whitening. Independent Component Analysis (ICA)and the third order moments are then employed for rotation and reflection nor-malization. By applying this normalization, curves and surfaces that are relatedby geometric transformations (affine or rigid) can be transformed into a canoni-cal representation. Proposed technique is verified with several 2D and 3D objectmatching and recognition experiments.

1 Introduction

Shape provides powerful information about an object. Fourier descriptors [1,2], al-gebraic representations [3] and B-splines [4,5] are some of the shape analysis andrepresentation methods. Most of the proposed approaches extract global and/or localgeometric features of curves/surfaces for shape matching, recognition, or classifica-tion. An important problem in 2D object recognition is the recognition of various 2Dcurves captured from different viewpoints. Viewpoint transformations can be describedby affine transformations.

Affine transformations contain similarity transforms (rotation, translation, scale) andalso shear. Much recent work has focused on affine invariance for the object matchingand recognition. A similarity distance based on modified Fourier descriptors was intro-duced in [6] and affine invariant distance between 2D point sets has been proposed in[7]. [8] proposes a novel method to find affine invariant canonical curves of the alge-braic curves.

Geometric normalization represents a powerful method for the recognition of 2Dcurves and 3D surfaces. Normalization is applied directly to intensity or range data andis typically used to compare the shape similarity of two objects subject to an affineor rigid transformation. In this context, normalization can be treated as object match-ing or pose estimation. Once the correspondence between two object data is estab-lished, the pose can be determined by finding the underlying geometric transformation[9]. Without establishing the correspondence, the problem can not be solved in closedform.

Instead of using invariants of two affine related curves, normalization of curveshas been introduced as an alternative way for recognition and matching. The main

A. Campilho and M. Kamel (Eds.): ICIAR 2006, LNCS 4142, pp. 445–456, 2006.c© Springer-Verlag Berlin Heidelberg 2006

446 S. Sener and M. Unel

advantage of an affine normalization can be seen as keeping the curve unchanged sothat no information is lost. An algebraic normalization with respect to affine transformsby using area parametrization is proposed in [10]. Shen at al. developed an affine in-variant normalization in terms of the determination of the rotationally symmetric im-ages [11]. Their method normalize images instead of curves. Another novel methodfor affine invariant normalization is proposed in [12]. The moments and the complexFourier descriptors are used in the normalization procedures.

Recognition of 3D objects based on either 2D or 3D features necessitates some 3Dgeometric positioning which improves the recognition results. There are several ap-proaches proposed in the literature to solve the 3D recognition problem. A momentfunction method and scatter matrices are used in [13], higher order moments in [14],and least squares formulations based on a set of point correspondences in [15]. A modelbased approach using a point set and a surface model is proposed in [16]. These meth-ods either require an accurate point correspondence (as in least squares methods) or aresensitive to occlusion (as in scatter matrix) or have limited representation power. More-over, these methods would not be effective if the objects are spherically or cylindri-cally symmetric with some bumps. In [17], geometric matching of objects is achievedby the iterative closest point (ICP) algorithm. Although this algorithm does not needany point correspondence between objects, it does not always converge to the bestsolution.

Most of the proposed approaches try to establish interimage correspondence bymatching data or surface features derived from range image [18]. In [19], multiplerange views are matched using a heuristic search in the view transformation space.Since it searches the whole view transformation space, it is computationally very ex-pensive. [20] proposes a reverse calibration of the range-finder to determine the pointcorrespondences between the views directly. These approaches, however, do not takethe presence of noise or inaccuracies in the data and their effects on the estimated view-transformation into account. Another novel approach by employing a standard globaloptimization scheme to determine the relative positioning of the two fragments that cor-responds to their best complementary fit is proposed in [21]. This approach focuses onthe best fitting of the broken 3D models and surfaces.

This paper develops a novel geometric normalization which is useful for objectmatching and recognition. Principle Component Analysis (PCA) whitening and Inde-pendent Component Analysis (ICA) are used to normalize geometrically related objectsto a unique pose. The resulting curves/surfaces are invariant to translation, scale, shear,rotation and reflection, and are quite robust to data perturbations. The results presentedin this paper can be used as a preprocessing in numerous applications, such as indexinginto image databases and visual servoing.

The structure of this text is as follows: In section 2, our geometric invariant curveand surface normalization method, which is based on PCA whitening, ICA and thethird order moments, is presented. In section 3, results of 2D and 3D experimentsconducted to test the performance of our method with noisy and missing data aregiven along with discussions. Finally in section 4, we conclude the paper with someremarks.

Geometric Invariant Curve and Surface Normalization 447

2 Geometric Normalization Using PCA Whitening and ICA

Let X = Pi=1...N and X̄ = P̄i=1...N be two 2D or 3D data sets related by an affinetransformation A, which is defined by both a linear transformation M and a translationT , namely

P̄i = MPi + T (1)

It is well known [22] that if two affine related data sets are PCA whitened, they will beorthogonally equivalent, namely

ˆ̄X = QX̂ (2)

where X̂ and ˆ̄X are the normalized data sets obtained from X and X̄ respectively, andQ is an orthogonal matrix. By whitening transform, the affine relationship is reduced toan orthogonal one so that the data sets are shear, scale and translation invariant.

Independent Component Analysis is a statistical technique which seeks the direc-tions in feature space that are most independent from each other. ICA is a commonlyused technique for Blind Source Separation (BSS). It is a very general-purpose signalprocessing method to recover the independent resources given only sensor observationsthat are linear mixtures of independent source signals [23]. The simplest BSS model as-sumes the existence of n independent components s1, s2 . . . , sn, and the same numberof linear and instantaneous mixtures of these sources s̄1, s̄2 . . . , s̄n, that is,

s̄j = mj1s1 + mj2s2 + . . . + mjnsn, 1 ≤ j ≤ n, (3)

In vector-matrix notation, the above mixing model can be represented as

s̄ = Ms (4)

where M is a n × n square mixing matrix:

M =

⎡⎢⎢⎢⎣

m11 m12 · · · m1n

m21 m22 · · · m2n

......

. . ....

mn1 mn2 · · · mnn

⎤⎥⎥⎥⎦ (5)

The demixing process can be formulated as computing the separating matrix W , whichis the inverse of the mixing matrix M , and the independent components are obtained by

s = M−1s̄ = Ws̄ (6)

The basic idea behind our work is to consider coordinates of points on PCA whitenedoriginal data set as independent sources and the coordinates of points on the PCAwhitened transformed data set as the mixtures. In the light of (2), we know that thesedata sets are related by the orthogonal matrix Q. Recall that the determinant of anorthogonal matrix can be ±1. Q is a rotation if |Q| = +1 and a reflection if |Q| = −1.

The problem of ICA is to estimate the source signals from observed whitenedmixtures or, equivalently, to estimate the new demixing matrix Q by using some non-gaussianity measures. One of the most used solutions to the ICA estimation is the Fixed-Point Algorithm (FastICA), which is efficient, fast and computationally simple [24].


The FastICA is based on a fixed-point iteration scheme for finding a maximum of the

nongaussianity of the QT ˆ̄X . At this point, ICA behaves like the pseudo-inverse but thebig difference is that ICA does not know any information about sources:

ˆ̄X = QX̂,

QT ˆ̄X = QT Q︸︷︷︸≡I

X̂, (7)

First rotation normalization is performed on the data set ˆ̄X such that

ˆ̄XR = QT ˆ̄X (8)

In 2D, the horizontal and vertical reflections are normalized according to the third-order

central moments of ˆ̄XR, namely

ˆ̄XRR =[sgn(m1,2) 0

0 sgn(m2,1)

]ˆ̄XR (9)

where ˆ̄XR is the rotationally normalized data curve, sgn is the signum function, and

m1,2 and m2,1 are the third order moments of ˆ̄XR. For a 2D curve, the p, q-order mo-ments can be defined as

mp,q =1N

N−1∑i=0

xpi y

qi (10)

ICA starts with an initial Q(0) matrix for 2D decomposition. Because of the initial

matrix Q(0), some of the normalized data curves ˆ̄XRR can still be symmetric withrespect to the y = x line. The normalization algorithm can be made independent of the

initial matrix Q(0) by ordering the third order moments of the ˆ̄XRR, which are |m1,2|and |m2,1| respectively. In other words, if we perform the following additional step,namely

ˆ̄XN =

⎧⎨⎩

[0 11 0

]ˆ̄XRR |m1,2| > |m2,1|

ˆ̄XRR otherwise(11)

then all the affine transformed versions of an object will be aligned.In the case of 3D range data, rotation normalization is performed again using (8), and

the reflections along the yz, xz and xy coordinate planes, are normalized according to

the third-order central moments of ˆ̄XR, namely

ˆ̄XRR =

⎡⎣sgn(m3

1,1) 0 00 sgn(m3

1,2) 00 0 sgn(m3

1,3)

⎤⎦ ˆ̄XR (12)

where ˆ̄XR is the rotationally normalized range data, sgn is the signum function, and

m1,1, m1,2 and m1,3 are the third order central moments of ˆ̄XR. For 3D range data ofan object, the k-order central moments can be defined as


mk = E[( ˆ̄XR − μ)k] (13)

E is the expected value and the μ is the mean of the ˆ̄XR.

Because of the initial matrix Q(0), some of the normalized range data ˆ̄XRR can stillbe symmetric with respect to some coordinate planes such as the plane defined by theline x = y and the z axis, or the plane defined by the line y = z and the x axis, orthe plane defined by the line x = z and the y axis. The normalization algorithm can bemade independent of the initial matrix Q(0) by ordering the third order moments of theˆ̄XRR, namely |m1,1|, |m1,2| and |m1,3| respectively. In other words, if we perform the

following additional step, namely

ˆ̄XN =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎡⎣1 0 00 1 00 0 1

⎤⎦ ˆ̄XRR if |m1,1| < |m1,2| < |m1,3|

⎡⎣1 0 00 0 10 1 0

⎤⎦ ˆ̄XRR if |m1,1| < |m1,3| < |m1,2|

⎡⎣0 1 01 0 00 0 1

⎤⎦ ˆ̄XRR if |m1,2| < |m1,1| < |m1,3|

⎡⎣0 1 00 0 11 0 0

⎤⎦ ˆ̄XRR if |m1,2| < |m1,3| < |m1,1|

⎡⎣0 0 11 0 00 1 0

⎤⎦ ˆ̄XRR if |m1,3| < |m1,1| < |m1,2|

⎡⎣0 0 10 1 01 0 0

⎤⎦ ˆ̄XRR if |m1,3| < |m1,2| < |m1,1|

(14)

then all the transformed versions of a 3D object will have the same unique pose.

3 Experimental Results

An orthographic projection of an object which is almost planar, as it undergoes rigidbody motions in three dimensions will be very close to an affine transform of somereference view of the object, as long as the motions are not so large as to cause majorocclusions.

Fig. 1 depicts several objects used in our experiments.In our experiments, white noise with relatively large variance values are used to cor-

rupt object boundaries. Fig.2(a-c) show one of the object boundaries, butterfly, cor-rupted with the white noise of standard deviations 0.05, 0.1 and 0.15 respectively.Fig.2(d) shows the resulting normalized data contours. The proposed normalization al-gorithm is quite robust to the high level white noise.

The white noise is not a good model for generating deformed copies of a shape asmight be sketched by a human or as might appear after segmentation from an image


Fig. 1. 2D Object Boundary Curves used in experiments

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Fig. 2. Butterfly contours with white noise (a-c), normalized superimposed contours (d), butterflycontours with colored noise (e-g), normalized superimposed contours (h)

of an object taken under slightly different viewing conditions. Because the white noisemodel can not represent the desired shape variations, colored noise model is also usedin our normalization experiments to reach the real noisy conditions. To produce colorednoise, first a white noise sequence, equal in length to the number of data points, is gen-erated. This generated white noise is convolved with an averaging window of length0.15 times the number of data points. Generated sequence is added in the direction per-pendicular to the data at each point. Fig.2(e-g) depict colored noisy shapes of butterflywith different standard deviations. As it is seen in Fig.2(h), the normalization algorithmyields effective results and the normalized boundaries show that our normalization al-gorithm is also robust to colored noise.

A robustness test is performed with nonuniformly resampled contours. In Fig.3(a-c),the initial sample curve is affine transformed and the resulting curves are then resampledusing nonuniform sampling. Normalized curves of the aircraft (skyhawk) are depicted


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in Fig.3(d). While nonuniform resampling reduces the number of data points in differentpercentages up to 70%, all resulted normalized curves are almost the same.

Another experiment is conducted on real images taken from different view points,Fig.4(a-d) and the obtained contours in Fig.4(e-h). In this experiment, the effects of theperformance of the segmentation algorithm on the normalization scheme is exempli-fied. Hough transform is used for the segmentation process [25]. If we do not performany smoothing operation, the resulting normalized boundaries are depicted in Fig.5(a).When a noise removal procedure is employed before the curves are input to the nor-malization algorithm Fig.5(b) is obtained. These results show that the performance ofour proposed method can significantly be improved by performing a smoothing schemebefore applying the normalization procedure.

Our method can also be used for 2D object recognition. A recognition exampleis presented in Table 1. Normalized contours of the objects shown in this table werecompared using the Hausdorff distance [26] between their perspective point sets. The


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Fig. 5. (a) Normalized boundaries without smoothing initial contours. (b) Normalization resultswith noise removal and uniform sampling.

Table 1. Hausdorff distances between normalized data curves

0.0518 0.3302 0.3764 0.3670 0.3711 0.3736 0.3347

Hausdorff distance is a measurement of the similarity of point sets and it can be usedfor pattern matching and tracking. The first column in the table shows the butterfly andits colored noisy version.

Object recognition systems suffer from drastic losses in performance when the ob-ject is not correctly oriented. The normalization procedure introduced in the previoussections can be used to put the 3D shapes into a standard pose. Various experimentsare conducted to asses the robustness of our 3D normalization algorithm to high levelgaussian noise and missing data due to partial occlusion. Using different number ofpoints in the comparison of two data sets is the main advantage of the Hausdorff dis-tance algorithm. In comparisons, a lower distance value between the normalized objectsimplies similar objects and a higher distance value implies quite different objects.

In our experiments, rotation is described by

R = Rz(γ)Ry(β)Rx(α)

where α, β and γ are the rotation angles around the x, y and z coordinate axes,respectively.


(a) (b)

(c) (d)

(e) (f)

Fig. 6. (a-e) Ball joint bone surface and its transformed versions, (f) normalized superimposedsurfaces

Different rotation and translation parameters (α, β, γ, tx, ty, tz) were used to repre-sent the rigid transformations. In Fig.6(a) the surface model of a ball joint bone with137062 points is displayed. The ball-joint is rotated and translated with different para-meter sets as in Fig. 6(b,c,d,e). The resulting normalized surfaces for these rotated andtranslated objects are the same, and they are superimposed in Fig.6(f).

The range data was also perturbed with white noise and partial occlusion in our ex-periments. In Fig.7(c,e), the 3D object (Venus) was perturbed with the white noise ofstandard deviations 0.7071, and 1 respectively. Fig.7(b,d,f) shows corresponding nor-malized 3D venus objects. As shown in Fig.7(e), a noise with a standard deviation of 1produces major perturbations on the 3D object, and it is difficult to do the registration


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Fig. 7. (a,c,e) 3D Venus object and its noisy versions. (b,d,f) normalized venus objects

visually; nevertheless the orientation and the translation errors are almost 0 and averageHausdorff distance between the normalized objects is 0.1047.

Robustness to missing data due to partial occlusion is an important aspect of any3D recognition algorithm. In Table 2, a series of patches consisting of 5%, 10% and15% of the points in the horse data were discarded. Note that the tabulated distancevalues are still quite small when compared to the average Hausdorff distance, which wasabout 1.6964, between the normalized objects used in our experiments. Consideringthis average distance, we can say that the normalization algorithm is quite robust to amoderate level of missing data or occlusion.


Table 2. Recognition of partially occluded horses by Hausdorff distance

Missing Data Rate 0 % 5 %

3D ObjectHausdorff Dist. 0.0718 0.2208

Missing Data Rate 10 % 15 %

3D ObjectHausdorff Dist. 0.2532 0.2979

4 Conclusion

We have presented a novel geometric invariant normalization method based on PCAwhitening and ICA. We have shown that curves or surfaces that are related by geometrictransformations (affine or rigid) can be normalized to a canonical pose. Effectiveness ofour normalization technique has been assessed by several experiments conducted in 2Dand 3D with synthetic and real data. Experimental results have shown that the proposednormalization is quite insensitive to data variations such as high level noise, resamplingof the contours and missing data due to partial occlusion.

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