A New Affine Invariant Fitting Algorithm for Algebraic Curves

Transcription

1 A New Affine Invariant Fitting Algorithm for Algebraic Curves Sait Sener and Mustafa Unel Department of Computer Engineering, Gebze Institute of Technolog Cairova Campus 44 Gebze/Kocaeli Turke {ssener, Abstract. In this paper, we present a new affine invariant curve fitting technique. Our method is based on the affine invariant Fourier descriptors and implicitization of them b matri annihilation. Eperimental results are presented to assess the stabilit and robustness of our fitting method under data perturbations. Introduction Automatic recognition of free-form objects is an important problem in pattern recognition and computer vision. Implicit algebraic models or so-called Implicit Polnomial (IP) curves and surfaces proved to be ver useful for modeling D curves and 3D surfaces [-7]. To build recognition and positioning sstems based on implicit curves and surfaces, it is imperative to solve the fitting problem. Euclidean fitting was used for the estimation of general curves and surfaces to edge and range data. Using Euclidean distance is more stable and useful than algebraic distance commonl used in least-squares fittings. Curve fitting is usuall done through non-linear optimization, which has high computational compleit and computational cost [-3]. The 3Lfitting algorithm, which is repeatable and Euclidean invariant, is another new approach to the fitting problem. This algorithm also improves implicit curve fitting in terms of accurac and stabilit [8]. Continuous improvements have been made on the stabilit of algebraic curve fitting for obtaining IP models [9]. On the other hand, an important link between the Fourier descriptor and quadratic B-spline function is established in []. A new matri annihilation method for converting parametric representations to the implicit algebraic curves has recentl been introduced in []. This method is numerical and computationall efficient. In this paper, we present a new affine invariant fitting algorithm based on the affine invariant Fourier descriptors [3, 4] and the implicitization of them b matri annihilation method []. The algorithm is applicable for all free-form shapes. The resulting implicit algebraic curves are quite robust with respect to data perturbations. Fitting is computationall efficient and allows high degree curve fitting. In Section, we review implicit polnomial curves and the fitting problem. In section 3, we de- A. Campilho, M. Kamel (Eds.): ICIAR 4, LNCS 3, pp , 4. Springer-Verlag Berlin Heidelberg 4

2 A New Affine Invariant Fitting Algorithm for Algebraic Curves 345 scribe affine invariant Fourier descriptors and their implicitization using matri annihilation method. In Section 4, we present eperimental results to assess robustness of our fitting algorithm under noise, and missing data due to partial occlusion. Finall in Section 5, we conclude with some remarks and indicate possible future directions. Fitting Algebraic Curves to Data Fitting an algebraic curve to data is basicall an error-minimization problem over the set of data points D = { z (, )} N i = i i j=, namel N min dist ( fn(, ), zi) i= where a is the coefficient vector of the curve, a () p q f (, ) = a is the two n pq p+ q n variable (binar) implicit polnomial whose zero set is the algebraic curve, and dist is an appropriate distance function. In the classical least-squares fitting, p q dist( f (, ), z ) = f ( z ) = f (, ) = a. To avoid the trivial solution, the n i n i n i i pq i i p+ q n norm of the solution vector is constrained to be, i.e. a =. Resulting fit is affine invariant, but usuall contains etra pieces and does not respect the shape continuit. Stabilit is another issue because it does not provide an stable results. Since there is no closed-form epression for computing the geometric distance of a point to the curve, a first order distance approimation, namel fn( j, j) () dist = f (, ) n j j is usuall emploed. This is a non-linear optimization problem and some important algorithms have been presented in the literature [-3]. These algorithms have high computational compleit and in most cases do not provide stable and meaningful fits. 3L algorithm [8], which is a modified least squares fitting method, is more robust and gives better curves than the classical least squares algorithms. A solution has been offered for more stabilit and robustness of 3L algorithm to find the best implicit polnomials [9]. Although resulting fits are much more stable and superior than 3L fits, the are onl rotation invariant and therefore cannot be used for images taken from different viewpoints. 3 Affine Invariant Fitting b Matri Annihilation Matri annihilation method [] established a link between the elliptic Fourier descriptors (EFDs) and implicit polnomials (IP). EFDs are arc-length parameteriza-

3 346 S. Sener and M. Unel tions, which are made invariant to changes in location, orientation and scale, that is, the are similarit invariant []. However, Euclidean arc-length is not preserved under affine transformations. Fig. depicts this fact. Both the original and the affine transformed data points are fitted b implicit curves obtained through the implicitization of EFDs b matri annihilation method. Note in particular that the implicit curve obtained b the matri annihilation method and the affine transformed version of the implicit curve in (b) are not coincided in (c). (a) (b) (c) Fig.. (a) Hat Object, (b) Data points (line), EFD (circles-line) and implicit curve (dot-line) obtained b matri annihilation method are superimposed. (c) Affined data (line), implicit curve (circles-line) obtained b matri annihilation method and affine transformed implicit curve (dot-line) in (b) are superimposed. Instead of using similarit invariant EFDs, one could start with an affine invariant Fourier parameterization, which will then ield an affine invariant implicit polnomial curve as a result of matri annihilation method. This is eactl what we are developing in this paper. B an affine invariant parameterization, we mean that the parameterization must be linear under an affine transformation and the parameterizing function must ield the same parameterization independent of the initial representation of the contour. Arbter et al. introduced affine invariant Fourier descriptors (AIFDs) b using the first derivatives of data curve as: t = det ( ), C ( ξ ξ ) dξ = ( ) ( ) ξ d C ξ ξ ξ ξ where ξ, ξ are the first derivatives of the components ( ξ ) and ( ξ ), and C is the path along the curve. This parameterization will not be invariant under translation. Translation can be eliminated b initiall moving the coordinate sstem to the area center, defined b s ( ξ ) ( ξ ξ ) d ( ξ)det ( ξ), dξ c = 3 det ( ), ξ C The area center of an affine contour is the affine transform of the area center. This is just because the affine transformation transforms areas with a constant scale det(a) [5]. To calculate the coefficients, let ( u, v ): i =,, N, be the coordinates of i i (3) (4)

4 A New Affine Invariant Fitting Algorithm for Algebraic Curves 347 the N vertices of an oriented polgon in R, and t i their parameter values. With the periodicit, ( un, vn) = ( u, v) and tn = t + T. The Fourier transformation based on area parameterization is defined as [3,4]: U N N k ui u T + i j ui+ ui ( ki, ki, )[ ( ti ti) ] ki, ( ti ti) V = + δ + δ + k ( π k) i= ti+ ti vi+ v Φ Φ + Φ i π k i= vi+ v i if ti+ = ti where Φ ki, = ep{ j π kti/ T} and δ ( ti+ ti) =. if ti+ ti In contrast to the FFT, this transformation takes the indefinite set of all points on the polgon into account, does not need constant parameter intervals and allows the transformation of discontinuous functions. The area parameterization for polgons is given b t = ; ti+ = ti + ( uv i i+ u i+ v i ), i =,,,, N ; T = tn, with u i ui uac = v i vi vac and u N N AC ui u + i+ = ( uv i i+ ui+ vi) /(3 ( uv i i+ ui+ vi)) vac i= vi + vi + i= Since this parameterization is affine invariant, we can use the coefficients of this Fourier parameterization as an input to the implicitization method based on matri annihilation ideas detailed in []. We note that the interpolated curve of affineinvariant Fourier descriptors might ehibit Gibbs phenomenon. Consider the boot depicted in Fig.. AIFD and IP curves are superimposed in Fig. along with the amplitude of AIFDs. (5) (6) (7).3 Object Contour Interpolated AIFD Implicit Polnomial.8.6 abs(u).4. abs(v).35.3 abs(u+jv) (a) (b) (c) (d) Fig.. (a) Object contour and AIFDs (dash-dot line) and IP curve (circles-line) obtained b MA method are superimposed. Amplitude spectrums of U k, V k and U k +jv k are in (b,c,d). Suppose that Uk ( ) and V( k ) are the Fourier coefficients based on the above parameterization. Matri annihilation method determines the implicit polnomials b using implicitization the Fourier representations as inputs, namel AIFD IP. So, we start with the following EFDs:

5 348 S. Sener and M. Unel n a ak bk cos kt = + c k = ck d k sin kt (8) To transform the affine invariant Fourier descriptors into (8), we separate the Fourier coefficients into real and imaginar parts as a = U, c = V, ak = Re{ U( k)}, bk = Im{ U( k)}, ck = Re{ V( k)}, dk = Im{ V( k)} (9) The separated coefficients are then taken as inputs to the matri annihilation method. Using more harmonics increases the degree of the curve which in turn will improve the accurac of the resulting fit. However, in such a case one should be careful about overfit problems. The obtained curve with k harmonics will be of degree d=k. 4 Eperimental Results Shapes from a different viewpoint are generated according to cos β sinαsin β tr A= cosα tr where α and β are the change of angles in and aes (pitch and aw), tr and tr are the translations in and directions. Fig. 3 (a) depicts the data and the IP curve obtained from our method. Using above transform matri A with α = 45 and β = 7, one can verif that the IP fit on the data contour in (b) and the affine version of the original fit in (a) are eactl the same (a) Fig. 3. (a) Object contour (line) and the implicit curve (dashed-line) obtained b our fitting algorithm are superimposed. (b) Affine transformed contour and the IP fit. (b) To realize distortions occurred during the segmentation process, we perturbed the objects before appling our fitting procedure. Eperiments are based on the white noise distortion and missing data due to partial occlusion on the object boundaries. The random white noise is added to each point in the data. Fig. 4 (a), (b) and (c) show the perturbed shapes with white noise having the standard deviations.,.5 and.5, respectivel.

6 A New Affine Invariant Fitting Algorithm for Algebraic Curves 349 Robustness to missing data due to partial occlusion cruciall depends on a good representation. Fig. 5 shows the 6th-degree IP fits on occluded data (a-d), (b-e) and (c-f) with %, %5 and % missing data, respectivel (a) (b) (c) Fig. 4. Fits under white noise with standard deviations.,.5 and.5, respectivel (a) (b) (c) (d) (e) (f) Fig. 5. The 6th-degree IP fits on occluded data (a-d), (b-e) and (c-f) with %, %5 and % missing data, respectivel (a) (b) (c) (d) Fig. 6. (a) and (b) are data points with superimposed IP curves b least-squares fit algorithm. (c) and (d) are the same data sets with superimposed IP curves b our fit algorithm. Although the classical least-squares fit algorithm is affine invariant, it does not provide meaningful fits. The resulting implicit curves do not respect data continuit and are not stable, and usuall involve unbounded etra components. Fig. 6 shows this fact b comparing the least-squares fit results with ours.

7 35 S. Sener and M. Unel Algebraic (or IP) curves obtained b our algorithm can be used in an Image- Guided Decision Support (IGDS) sstem designed to assist pathologists to discriminate abnormal blood cells. Fig. 7 shows an abnormal blood cell and the associated IP curves. The can also be ver useful in motion estimation and aircraft recognition sstems because of the affine invariant propert of our fitting algorithm. See Fig (a) (b) (c) Fig. 7. (a) blood cell, (b) the first infected part and the 6 th degree curve (circle-line), (c) the second infected part and the 6 th degree curve (circle-line) (a) (b) (c) (d) Fig. 8. A 6 th degree fit on a B aircraft contour and its affine versions. 5 Conclusion We have now presented an affine invariant fitting algorithm for modeling D freeform curves based on affine invariant Fourier descriptors and matri annihilation theor. Proposed algorithm is fast, repeatable, numericall stable, and robust to data perturbations. Acknowledgment. This research is supported from GYTE research grant BAP #3A3. References [] Taubin, G., F. Cukierman, S. Sullivan, J. Ponce and D.J. Kriegman, Parameterized Families of Polnomials for Bounded Algebraic Curve and Surface Fitting. IEEE Trans. Pattern Analsis and Machine Intelligence, 6(3):87-33, March 994.

8 A New Affine Invariant Fitting Algorithm for Algebraic Curves 35 [] Keren, D., D. Cooper and J. Subrahmonia, Describing Complicated Objects b Implicit Polnomials, IEEE Trans. Pattern Analsis and Machine Intelligence, Vol. 6, pp , 994. [3] G. Taubin, Estimation of Planar Curves, Surfaces and Nonplanar Space Curves Defined b Implicit Equations, with Applications to Edge and Range Image Segmentation, IEEE Transactions on Pattern Analsis and Machine Intelligence, vol. 3, pp. 5-38, 99. [4] Wolovich, W. A. and Mustafa Unel, The Determination of Implicit Polnomial Canonical Curves, IEEE Transactions on Pattern Analsis and Machine Intelligence, Vol. (8), October 998. [5] Unel, Mustafa and W. A. Wolovich, On the Construction of Complete Sets of Geometric Invariants for Algebraic Curves. Advances in Applied Mathematics 4, 65-87, Januar. [6] Unel, Mustafa and W. A. Wolovich, A New Representation for Quartic Curves and Complete Sets of Geometric Invariants, Int. Jour. of Pattern Recognition and Artificial Intelligence, Vol. 3 (8), 999. [7] Subrahmonia, J., D. B. Cooper, and D. Keren, Practical Reliable Baesian Recognition of D and 3D Objects using Implicit Polnomials and Algebraic Invariants, IEEE Transactions on Pattern Analsis and Machine Intelligence, 8(5):55-59, 996. [8] Lei, Z., M. M. Blane, and D. B. Cooper, 3L Fitting of Higher Degree Implicit Polnomials, In proceedings of Third IEEE Workshop on Applications of Computer Vision, pp , Florida 996. [9] T. Tasdizen, T. Tarel and D. B. Cooper, Improving the Stabilit of Algebraic Curves for Applications, IEEE Transactions in Image Processing, 9(3):45-46,. [] Wu-Chih Hu and Hsin-Teng Sheu, Quadratic B-spline for Curve Fitting, Proc. Natl. Sci. Counc. ROC(A), vol.4 No:5 pp ,. [] H. Yalcin, M. Unel, W. A. Wolovich, Implicitization of Parameteric Curves b Matri Annihilation, International Journal of Computer Vision, Vol. 54, pp. 5-5, 3. [] Kuhl, F.P. and C.R. Giardina, Elliptic Fourier Features of a Closed Contour, Computer Graphics and Image Processing, vol.8, pp.36-58, 98. [3] K. Arbter, W. E. Snder, H. Burkhardt and G. Hirzinger, Application of Affine-Invariant Fourier Descriptors to Recognition of the 3D Objects, IEEE Transactions on Pattern Analsis and Machine Intelligence, vol. (7) pp , 99. [4] K. Arbter, Affine-Invariant Fourier Descriptors, in From Piels to Features, Amsterdam, The Netherlands: Elseiver Science, 989. [5] F. E. Pollick, G. Sapiro, Constant Affine Velocit Predicts the /3 Power Law of Planar Motion Perception and Generation, Elseiver Science, Vol. 37, No, 3, pp , 997.