Synthesized affine invariant function for 2D shape recognition

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www.elsevier.com/locate/pr

Synthesized afﬁne invariant function for 2D shape recognition

Wei-Song Lin

∗

, Chun-Hsiung Fang

Department of Electrical Engineering, National Taiwan University, No. 1, Sector 4, Roosevelt Road, Taipei 106, Taiwan, ROC

Received 12 September 2005; received in revised form 21 February 2006; accepted 31 March 2006

Abstract

By defining the weighted wavelet synthesis, the synthesized feature signals of an interesting shape are extracted to derive the innovative synthesized affine invariant function (SAIF). The synthesized feature signals hold the shape information with minimum loss by excluding simply the translation dependent and noise-contaminated bands. The SAIF is shown excellent in the invariance property and representative in describing the original shape for automated recognition. Experimental results demonstrate that automated shape recognition based on the SAIF achieves high correctness and significantly outperforms those using conventional wavelet affine invariant functions.

Keywords: Afﬁne invariant function; Shape recognition; Wavelet transform; Synthesized feature signal; Weighted wavelet synthesis

1. Introduction

Human vision unveils that the shape feature is a signif-icant perception to distinguish different objects. This in-spires researchers to find reliable and efficient shape features for discriminating objects by machine vision. Presently, the shape features presented in literature are mainly region- or contour-based technique [1]. Region-based techniques ex-tract the key information to represent the overall region. They seem effective but arduous and domain-dependent. On the contrary, contour-based techniques simply trace the bound-ary, and ignore the content inside the region. Representing a shape with its contour is much more efficient than with the overall region. But the contour captured by a camera under arbitrary orientation may have been distorted by certain ge-ometric transformations. This distortion can be most appro-priately represented by the perspective transformation[2,3]. The perspective transformation is simplified and approxi-mated by the affine transformation if the depth of objects along the line of sight is small compared with the viewing distance. The affine transformation includes typical geomet-ric transformations as rotation, translation, scaling, skewing,

∗_{Corresponding author. Tel.: +886 2 33663638; fax: +886 2 23638247.}

E-mail addresses:weisong@cc.ee.ntu.edu.tw(W.-S. Lin), f1921049@ee.ntu.edu.tw(C.-H. Fang).

and shearing. Therefore, before the shape recognition, the afﬁne transformation must be removed from the contour sig-nal so as to obtain reliable result. Thus shape recognition algorithms usually use parameterized contour that is com-pact representational form of the geometric information and independent of the afﬁne transform[4].

A parameterized contour, which represents the origi-nal shape and is independent of the affine transformation, is called an affine invariant descriptor or affine invariant function (AIF). Area moment invariants are invariant to similarity transformation except the computing is usually time-consuming [5,6]. Except a few low order moment invariants, the affine weighted curve moments [7] and the affine invariant moment utilizing B-spline [8] are highly sensitive to noise. The affine arc length, which is calculated by the first and second derivatives of the object contour[9], is less robust in a noisy environment. The enclosed area, which is based on the property of all area being changed in the same ratio under an affine mapping, is translation depen-dent [3]. Recently, wavelet transform has been introduced in deriving the AIF and called the wavelet affine invariant function (WAIF). The WAIF usually derives with the coarse and detailed signals obtained in different wavelet transform levels. A WAIF using only one wavelet dyadic level was presented in[2,10]. This WAIF, however, is not invariant to translation unless the contour has its centroid as the origin.

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Deriving a WAIF with two or more wavelet dyadic levels is possible [1,11,12]. In these situations, the wavelet sig-nals are usually selected to be translation independent. That using three, four, or six wavelet dyadic levels is originated from one conic equation, four collinear points and two conic equations, respectively[13]. Generally speaking, the WAIF is suffered from a difﬁculty of losing important shape infor-mation in the neglected wavelet dyadic levels. Consequently, shape recognition based on the WAIF may be inaccurate and unreliable.

This paper presents the synthesized afﬁne invariant func-tion (SAIF) derived with the synthesized feature signals (SFS) of the shape. A SFS is deﬁned as a weighted wavelet synthesis of the original contour signal. A set of mutually orthogonal SFSs excluding simply the translation dependent and noise-contaminated bands are extracted from the con-tour signal to derive the SAIF. The SAIF is shown a repre-sentational and information-rich form of the original shape for automated recognition.

The remainder of this paper is organized as follows: Sec-tion 2 introduces the weighted wavelet synthesis and the SFS of a shape. Section 3 derives two SAIFs with the SFSs for automated shape recognition. Section 4 evaluates exper-imentally the performance of the SAIF and compares the results with those obtained by the WAIF[1,12]and the sum-mation invariant (SumAIF) [14]. Finally, Section 5 is the conclusion.

2. Weighted wavelet synthesis and synthesized feature signal

2.1. Weighted wavelet synthesis

In the multiresolution formulation of dyadic wavelet anal-ysis, i.e. wavelet transform, a signal or function f () ∈ VJ

decomposed to a certain wavelet scale level i can be ex-pressed as[15] f () = 2i₋₁ k=0 ci(k)2i/2(2i − k) +J j =i 2j₋₁ k=0 dj(k)2j/2(2j − k), (1) where ci’s are called the coarse coefficients at level i, d_js are the detailed coefficients at level j, j and k are integer indices, J is an integer denoting the starting or highest scale level, i is an integer denoting the end scale level,{(2i−k)|∀k ∈ N} is the set of level-i scaling functions,{(2j − k)|∀k ∈ N} is the set of level-j wavelet functions. The set of functions {(2i_{− k), (2}j_{− k)|∀k ∈ N} forms an orthogonal} ba-sis that spans V_J. In a practical application of the wavelet analysis, the signal is perhaps itself a sequence of numbers or samples of some function of a continuous variable. In this situation, the scaling coefficients at the starting level

are approximated by the corresponding numbers or samples, meaning no wavelet coefﬁcients are necessary at that level. Thus, cJ(k) = f (k), k = 0, 1, 2, . . . and dJ(k) = 0 are appro-priate at the starting level J. Expansion coefﬁcients at lower levels can then be evaluated by using the following iterative formula[16]: cj(k) = m h0(m − 2k)cj +1(m), (2) dj(k) = m h1(m − 2k)c_{j +1}(m), (3)

where h0(n) and h1(n) are called the scaling and wavelet

ﬁlters, respectively. Conversely, the wavelet synthesis, i.e. wavelet inverse transform, combines the expansion coefﬁ-cients as follows: cj +1(k) = m cj(m)h0(k − 2m) + m dj(m)h1(k − 2m). (4)

Iterative applications of (4) may ultimately reconstruct the original samples cJ(k)=f (k), k =0, 1, 2, . . . . If the original signal is some function of a continuous variable, the syn-thesis should be implemented by substituting the expansion coefﬁcients into (1).

The weighted wavelet synthesis is deﬁned in this paper by synthesizing a signal with,

Sw(f ()) = wc 2i−1 k=0 ci(k)2i/2(2i − k) +J j =i wj 2j−1 k=0 dj(k)2j/2(2j − k), (5) where wc is called the coarse weight, {w_j|j = i, . . . , J } are called the detailed weights, and w = {w_c, all wj} is the set of weights to be designed. Associating each de-signed weight in w with the corresponding term in (4) ob-tains the following iterative formula of the weighted wavelet synthesis: Sw(cj +1(k)) = wcj m Sw(cj(m))h0(k − 2m) + wj m dj(m)h1(k − 2m), (6)

where wcj = wc, as j = i otherwise wcj = 1. Apparently, setting one to each weight in w will recover the wavelet synthesis. Alternatively, the weights w can be designed so that the weighted wavelet synthesis obtains signal Sw(f ()) or Sw(cJ(k)) with interesting property such as transla-tion independent, noise free or mutually orthogonal. This characteristic is very useful in deriving the AIF for shape recognition.

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2.2. Synthesized feature signals of a shape

In the contour-based technique, a shape is described by a contour signal. Applying the weighted wavelet synthesis to the contour signal and adapting the weights may obtain a derivative contour signal simply consisting of interest-ing wavelet bands of the original contour signal. A set of such derivative contour signals with mutually orthogonal and translation independent properties and possibly noise free is called the SFS of the shape. Translation independency can be achieved by discarding the coarse weight wcin the weighted wavelet synthesis. Neglecting the weights corresponding to noise-contaminated bands can eliminate noise. In addition, since the wavelet transform with orthogonal basis is used, weight sets consist of non-overlapping bands may produce a set of orthogonal derivative signals. The orthogonal prop-erty makes the SFSs of the shape free from redundant infor-mation and therefore computationally efﬁcient.

Exclusive weighting is a simple method to design the weight sets to synthesize the SFSs of a shape. The exclusive weighting method is described as follows:

(1) Build the weight set w= {wc, all wj} associated with (5). Then construct the mother weight set by discard-ing (set to zero) the coarse weight wc for translation independency, discarding any wj corresponding to the noise-contaminated band, and adjust the rest of wj’s to the desirable strength (usually nonzero and one in our design).

(2) Split the mother weight set into constant dimensional, non-null, exclusive weight subsets as many as the de-sirable number of the SFSs. The union of these exclu-sive weight subsets must equal to the mother weight set so that no information loss is ensured. The intersection of these exclusive weight subsets is null. This ensures the SFSs obtained by applying these exclusive weight subsets to (5) are mutually orthogonal.

Consequently, except the translation dependent and nose-contaminated bands, all the shape information is kept in the SFSs or, in other words, the SFSs represent the original shape with minimum information loss.

3. Synthesized afﬁne invariant functions

3.1. Relative and absolute afﬁne invariant functions of a contour

Afﬁne transformation of an object contour can be ex-pressed as the following matrix form of mapping functions: ˜x() ˜y() = a11 a12 a21 a22 x() y() + b1 b2 = A x() y() + B, (7)

where (x(), y()) represents a point on the contour param-eterized by the arc length parameter , ( ˜x(), ˜y()) is the corresponding point after the afﬁne transformation, A is a nonsingular square matrix representing the rotation, scaling, and skewing transformations, and the vector B represents the translation. The afﬁne transformation matrix A can also be expressed by several parameters as follows:

A= scale · cos − sin sin cos 1 _xskew yskew 1 , (8)

where scale is the scale factor,  is the rotation angle, and xskew and yskew are the skewing parameters in x- and y-directions, respectively. The characteristic within the afﬁne transformation is that parallel lines map to parallel lines, i.e. a square can map onto an arbitrary parallelogram.

If is an AIF, and ˜ is the corresponding invariant function calculated by the points under the afﬁne transformation, the relation can be denoted as[1]

˜ = |A|_, ₍₉₎

where|·| denotes the determinant and is called the weight of the invariance. The AIF is called an absolute invariant as  = 0 and a relative invariant as = 0.

3.2. Deriving the synthesized afﬁne invariant functions Let (S(x), S(y)) and (S(x), S(y)) be the SFSs of a shape, then the afﬁne transformation mapping function is represented as S( ˜x) S( ˜x) S( ˜y) S( ˜y) = a11 a12 a21 a22 S(x) S(x) S(y) S(y) + S(b1) S(b1) S(b2) S (b2) . (10)

Eq. (10) can be obtained easily by applying the wavelet anal-ysis to (7) and then followed by the weighted wavelet syn-thesis. From the translation-independent condition, S(b1)= S(b1) = S(b2) = S (b2) = 0, (10) is rewritten as S( ˜x) S( ˜x) S( ˜y) S( ˜y) = a11 a12 a21 a22 S(x) S(x) S(y) S(y) . (11) Based on the determinant properties and using (9), a relative SAIF is deﬁned as

_,() = S[x()]S [y()] − S[y()]S [x()]. (12) Normalization of (12) obtains the following absolute SAIF:

ˆ_,() =, () , (∗)

= S[x()]S [y()] − S[y()]S [x()]

S[x(∗)]S [y(∗)] − S[y(∗)]S [x(∗)], (13) where_,(∗) represents the maximum of _,(), and ∗ denotes the instant of occurrence.

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Let (S(x), S(y)), (S(x), S(y)) and (S(x), S(y)) be the SFSs of a shape, then another SAIF can be derived by using the conic equation. The conic is a curve deﬁned di-rectly in terms of a projective invariant property. Given a point of the contour signal, the conic can be expressed as the quadratic form, which is symmetric, deﬁned by x() y() T v11 v12 v12 v22 x() y() = 1. (14)

The AIF is deﬁned as v11v22 − v₁₂2 [13]. Thus, a relative

SAIF with the three SFSs is deﬁned as _, ,() = v11()v22() − v122() = 1 2S[x()]S[y()] S2[y()] 1 2S [x()]S [y()] S 2[y()] 1 2S[x()]S[y()] S2[y()] × S2[x()] 2S[x()]S[y()] 1 S2[x()] 2S[x()]S[y()] 1 S2[x()] 2S[x()]S[y()] 1 − S2[x()] 1 S2[y()] S2[x()] 1 S2[y()] S2[x()] 1 S2[y()] 2 . (15)

Normalization of (15) obtains the following absolute SAIF: ˆ_, ,() = , ,()

_, ,(∗)

= v11()v22() − v212() v11(∗)v22(∗) − v₁₂2(∗)

, (16)

where_, ,(∗) represents the maximum of _, ,(), and ∗_{denotes the instant of occurrence. Theoretically, it is}

pos-sible to derive other SAIF using more than three SFSs. But the computational complexity is a concern. Since the SFSs represent the original shape with minimum information loss. Consequently the SAIF based on the SFSs is a reliable rep-resentative of the original shape for automated recognition. On the other hand, conventional WAIFs simply choose sev-eral wavelet bands to derive the invariant function. They therefore suffer from a difﬁculty of loosing important shape information in the neglected bands.

4. Experimental results

In the following experiments, the performance of the SAIF is tested by applying to 10 electronic devices and the results are compared with those obtained by conventional WAIF and the SumAIF. The SAIF using two and three SFSs are denoted by SAIF2 and SAIF3, respectively. Conventional

WAIF [1,12] using two, three and six dyadic wavelet lev-els are denoted by WAIF2, WAIF3 and WAIF6, respec-tively. The summation invariant[14]is denoted by SumAIF.

Fig. 1(a) shows a photo of the test devices. Among them the M1, M2, M3 and M4 are rectangular integrated circuit devices with different numbers of pins. The image of each device is captured by a digital camera and preprocessed for noise removal and image enhancement before the contour extraction. As illustrated inFig. 1(b), each contour signal is represented by a 256-point coordinate vector and saved in the contour database. The performances of the SAIF, WAIF and SumAIF are evaluated and compared by examining the functional trajectories, root mean square errors (RMS), lo-cal invariance measure (LIM), representative index (RI) and correctness of recognition. However, it should be noticed that the correctness of recognition is dependent of not only the AIF but also the discriminant function.

Let C be a set of contour signals taken from the same shape but undergoing different afﬁne transformations. Then for i, ref ∈ C the RMS and LIM are deﬁned as follows:

RMS=

_n

=1(AIFi() − AIFref())2

n , (17)

LIM= AIF_i− AIFref_∞ = max

|AIFi() − AIFref()|, (18) where n is the length of the contour signal, is the arc length parameter and ‘ref’ denotes the reference signal. The RMS represents general difference between the reference and an-other contour signal of the same shape. On the an-other hand, the LIM indicates the worst case difference. The RMS and LIM together can show the stability of an AIF in representing a shape undergoing variant afﬁne transformations. Smaller RMS and LIM imply better invariance property.

Let S be a set of contour signals extracted from different shapes. Then for i, target ∈ S, the RI is deﬁned as follows: RI=AIFi() − AIFtarget()2 AIFtarget()2 = (AIFi() − AIFtarget())2 AIF2target() , (19)

where “target” denotes the target shape to be recognized. The denominator can be explained as the energy of the tar-get shape. Accordingly, the numerator represents the energy difference between the target and another shape. The RI indi-cates the ability of an AIF in discriminating different shapes. Greater RI makes the target shape to be recognized easily from a set of similar shapes.

4.1. Experiment 1: invariance property of the SAIF The contour signal of the device M5 inFig. 1 is chosen arbitrarily as the reference. Five associative contour signals obtained by applying translation, scaling, rotation, skewing or general afﬁne transform to the M5 contour signal are shown as the ﬁrst column in Fig. 2. Applying the SAIF2,

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M1 M1 M2 M2 M3 M3 M4 M4 M5 M5 M6 M6 M7 M7 M8 (b) M8 (a) M9 M9 M10 M10

Fig. 1. (a) A photo of the test devices. (b) Contour signals of the test devices.

0 200 400 0 200 400 M5 0 200 400 0 200 400 0 200 400 0 200 400 0 200 400 0 200 400 0 200 400 0 200 400 0 200 400 0 200 400 -1 0 1 SAIF₂ -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 SAIF₃ -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 WAIF₂ -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 WAIF₆ -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 SumAIF -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1

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Table 1

The RMS values of the associative contour signals of the device M5

Afﬁne AIF

SAIF2 SAIF3 WAIF2 WAIF6 SumAIF

Translation 0.0000 0.0000 0.0000 0.0000 0.0000 Scaling 0.0009 0.0008 0.0000 0.0271 0.0918 Rotation 0.0053 0.0033 0.0027 0.0984 0.1282 Skewing 0.0093 0.0070 0.0046 0.1401 0.0422 General aff. 0.0122 0.0081 0.0009 0.1304 0.3366 Table 2

The LIM values of the associative contour signals of the device M5

Afﬁne AIF

Translation 0.0000 0.0000 0.0000 0.0000 0.0000 Scaling 0.0033 0.0061 0.0001 0.2694 0.3561 Rotation 0.0193 0.0203 0.0069 0.6550 0.3910 Skewing 0.0460 0.0451 0.0118 0.9826 0.1874 General aff. 0.0460 0.0654 0.0021 1.0008 0.9212

SAIF3, WAIF2, WAIF6, and SumAIF, respectively, to each

of these contour signals, the resulting trajectories are pre-sented inFig. 2 from the 2nd to the 6th columns. In one column, perfect invariance property should present identical trajectories. The similarity between the reference and other trajectories is examined with the RMS and LMI values listed inTables 1and2, respectively. Since the smaller RMS and LIM mean the better invariance property. The SAIF2, SAIF3

0 200 0 100 200 300 M1 0 200 0 100 200 300 M2 0 200 0 100 200 300 _M3 0 200 0 100 200 300 _M4 -1 0 1 SAIF₂ -1 0 1 -1 0 1 -1 0 1 -1 0 1 SAIF₃ -1 0 1 -1 0 1 -1 0 1 -1 0 1 WAIF₂ -1 0 1 -1 0 1 -1 0 1 -1 0 1 WAIF₆ -1 0 1 -1 0 1 -1 0 1 -1 0 1 SumAIF -1 0 1 -1 0 1 -1 0 1

Fig. 3. The AIF trajectories of the devices M1, M2, M3, and M4.

and WAIF2, which present all RMS and LIM values smaller

than 0.05 and 0.07, respectively are satisfactory in experi-ence and signiﬁcantly better than the WAIF6 and SumAIF.

Therefore, the SAIF2, SAIF3 and WAIF2 perform well in

these aspects.

4.2. Experiment 2: representative property of the SAIF To obtain correct recognition, a qualified AIF should pro-duce significantly representative trajectory for each shape. The RI is used to examine the significance of representa-tive of the AIF.Fig. 1shows the devices M1, M2, M3, and M4 have very similar shapes. Their corresponding AIF tra-jectories are calculated and presented inFig. 3. For an AIF, it is found the trajectories of the M1, M2, M3, and M4 are mostly too close to be discriminated by inspection. Let the target of recognition be the M1 then the RI values of the M2, M3, and M4 are calculated for each AIF.Table 3lists the results. Since the greater RI value means the better rep-resentative.Table 3shows the WAIF6outperforms the oth-ers and the WAIF2is obviously the worst and less reliable. The SAIF2, SAIF3and SumAIF perform in between and the least RI value is 0.1859. That means at least the target is different from the test by 18.59% in energy. Therefore, they are satisfactory for discriminating different shapes correctly. Since an AIF for shape recognition should be qualified both in the invariance and representative properties. An over-all evaluation drawn from the results of Experiments 1 and 2 is listed inTable 4. It is found that the WAIF may be good in either invariance or representative property but not both. The worst case condition makes the WAIF not generally

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Table 3

The representative indices of the devices M2, M3, and M4 with respect to the M1

Model AIF

M2 0.1859 0.1876 0.0673 1.4692 0.2173 M3 0.2355 0.7089 0.0293 1.6348 0.2382 M4 0.2637 0.2795 0.0169 1.3645 0.2471

appropriate for automated shape recognition. Actually, the performance of the WAIF depends closely on the selection of the frequency bands and may be different from shape to shape. Experiments 1 and 2 conclude that the SAIF2 and

SAIF3are both qualiﬁed for the invariance and

representa-tive. But the SAIF2is more computationally efﬁcient. In

gen-eral, the SAIF outperforms the WAIF2, WAIF6and SumAIF

in the overall performance.

4.3. Experiment 3: correctness of shape recognition In this experiment, the SAIF2, SAIF3, WAIF2, WAIF3,

WAIF6, and SumAIF are compared by applying to

recog-nize target shapes out of 2400 contour signals. The thou-sands of contour signals are obtained by applying 240 dif-ferent afﬁne transformations to the 10 shapes in Fig. 1. The 240 afﬁne transformations are described by (7) and (8) with scale∈ {√2, 1, 1/√2}, ∈ {0◦, 36◦, 72◦, . . . , 324◦}, skew∈ {0, 0.3}, and B = [0, 200]T. Each transformed shape is resampled to obtain a 256-point contour signal. The dis-crimination function of the shape recognition is chosen as the following similarity measure.

Similarity(AIFM, AIFT) = _N−1

k=1AIFM(k)AIFT(k) AIFM(k)2AIFT(k)2

, (20) Eq. (20) is actually a correlation function in which subscripts “M” and “T” indicate the model and the test contour signals, respectively. In the shape recognition algorithm, a test con-tour signal is associated with each of the 2400 concon-tour sig-nals to calculate each similarity measure. The model contour signal (one of the 2400 signals) with the greatest similarity measure is regarded as a match and the corresponding shape is the result of recognition. Each of the 2400 contour signals is selected in turn as the test contour signal. Then the same procedure is done for the 2400 contour signals contaminated by Gaussian white noise. The 50 and 30 dB signal to noise ratio (SNR) cases are calculated and presented.Table 5lists the statistical results of correctness of shape recognition. Apparently, the SAIF2 is the best. The SAIF3is the runner up. More multiplications and truncations in calculating the SAIF3may be the cause of worse performance. Generally speaking the high invariance and representative properties of the SAIF2 and SAIF3 make the automated recognition so accurate. On the other hand, the WAIF simply selects some frequency bands to derive the invariant function. Due

Table 4

Overall performances of the afﬁne invariant functions

Invariance Representative Overall performance

SAIF2 Good Good Good

SAIF3 Good Good Good

WAIF2 Good Bad Bad

WAIF6 Bad Excellent Bad

SumAIF Bad Good Bad

Table 5

Correctness of recognition out of 2400 contour signals

Correctness (%) Noiseless Small white noise Large white noise (50 dB SNR) (30 dB SNR) SAIF2 95.58 95.50 93.67 SAIF3 94.83 94.79 82.67 WAIF2 72.67 72.83 59.08 WAIF3 34.50 33.79 30.42 WAIF6 54.67 47.13 18.79 SumAIF 18.58 18.50 18.08

to some distinct shape information may not be contained in the selected bands. Either the invariance or representative property may be deteriorated. As a result, less correctness of recognition is obtained. Although the WAIF6, which

in-cludes six dyadic wavelet levels, should be more informa-tion rich than the WAIF2and WAIF3, the weak signals from

high dyadic wavelet levels make the WAIF6unstable. As a

consequence, the WAIF6performs no better than its

broth-ers. As for SumAIF, the results inTable 5show it seems not a good choice for automated shape recognition.

5. Conclusion

Based on the well-established technique of wavelet trans-form, the weighted wavelet synthesis was defined for extract-ing the SFSs out of a shape. Usextract-ing the SFSs, the innovative SAIF was derived to represent the original shape for auto-mated shape recognition. The SFSs of a shape was shown to hold the shape information with minimum loss by exclud-ing simply the undesirable bands. The SAIF was shown to have excellent invariance property under variant affine trans-formations. The SAIF was also shown to have nice repre-sentative property for discriminating different shapes. Three experiments have been conducted to verify the SAIF in the invariance property, representative property and reliability in the automated shape recognition, respectively. The SAIF was confirmed to be highly reliable and the automated shape recognition based on it achieved high correctness of recogni-tion. Experimental results also compared the SAIF with the conventional wavelet affine invariant function and the sum-mation invariant. The conclusion was that the power of the SAIF was excellent and second to none of the conventional WAIF and the SumAIF.

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Acknowledgements

The ﬁnancial support for this research from the National Science Council of Taiwan, ROC under grants NSC93-2218-E002-106 and NSC94-2218-E002-049 is gratefully acknowledged.

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About the Author—WEI-SONG LIN is a professor with the Department of Electrical Engineering, National Taiwan University, Taiwan, ROC. He led the sensor calibration team of Ocean Color Imager aboard FORMOSA-1 (previously called ROCSAT-1) satellite from 1997 to 2001. His research interests are computational intelligence, robotics and remote sensing.

About the Author—CHUN-HSIUNG FANG is a Ph.D. student with the Department of Electrical Engineering, National Taiwan University, Taiwan, ROC. His research interests are machine vision, pattern recognition and robotics.