Fourier Descriptors of Two Dimensional Shapes - Reconstruction and Accuracy

ones proposed in this study give a better reconstruction rate than the G method for the small number of FD's. For the ZR approach the bounds on the deviation between a contour and its approximation Two kinds of Fourier shape descriptors (FD's) are based on the finite number of Fourier coefficients are considered: ZR defined by Zahn and Roskies and G derived. These bounds tell us how many FD's are defined by Granlund. In the first part of the paper needed to achieve a given accuracy in the ZR descriptors are studied. Three modifications of ZR reconstruction, the question posed in [I l l . descriptors are proposed. The new descriptors are Surprisingly, the rate of reconstruction for the ZR based on the smoothed signature, linearized smoothed approach is asymptotically the same as the one signature and the curvature function. The amplitudes obtained for the G method [I, 41. Next, three of Fourier descriptors are shown to be invariant under improvements of the ZR approach for polygonal rotations, translations, changes in size, mirror contours are proposed. The angular bend signature reflections and shifts in the starting point. In all used in [11] has jump discontinuities for polygonal cases the reconstruction accuracy in terms of the curves. I t is well known [lo] that the partial Fourier number of Fourier descriptors is studied resulting in series derived from the discontinuous function does not approximation error bounds. An efficient reconstruction converge unifomly to its limit a t the jump points method not requiring numerical integration is proposed (Gibbs phenomenon). First we replace the linearized for polygonal shapes. It also provides Polygonal signature used in [ l l ] by the smoothed signature approximation for arbitrary contours. In the second (SZR) with controlled degree of smoothing and part of the paper theoretical results are verified in linearized smoothed signature (LSZR). The first numerical experiments involving digitized patterns. signature is obtained by smoothing the angular bend function the other by linearizing SZR. Smoothing of the signature results in rounded edges near polygonal INTRODUCTION vertices. Since LSZR is continuous the reconstruction bounds known in the literature [I , 41 become In many applications of pattern recognition and applicable to it. Finally, we propose a contour digital image the of a simply connected signature (CS) based on the curvature function. Since object is represented by its ~ i f l ~ ~ ~ t for polygonal curves the curvature is zero along the approaches have been proposed for 2-D sides and infinite a t the vertices we replace the analysis, they include statistical approaches based on curvature function by its smooth approximation using a the method of moments, ~ ~ ~ r i ~ ~ descriptors [I, 5, 6, finite Fourier series. We believe that CS descriptors 8, 111, curve [7], circular autoregressive may be very useful in shape classification since they models [21, syntactic approaches [3] and relaxation are sensitive to sharp changes in the contours. or approaches. the statistical approach including all three signatures we derive Fourier descriptors and Fourier descriptors, moments and autoregressive models, rates. We Propose numerical features are computed from the complete fomulae that do not require numerical integration boundary and statistical discriminators are used to which results in considerable savings in computer time. classify contours. Among different techniques, Fourier The ZR and LSZR methods are tested and descriptors (or simple quantities derived from them) are with approach in experiments with distinguished by their invariance to afine shape digitized handwritten characters. For the purpose of transformations (scaling, rotation, translation and comparison several similarity measures are used. The mirror reflections) and to shifts in the starting point obtained results can be applied in shape recognition [b , 8, 111. In the literature the popularity of G and descriptors [4, 5, 81 far exceeds that of ZR descriptors [9, 111. One of the reasons is the discontinuity of ZR DESCRIPTORS-EFFICIENT the poly~onal signature resulting from the ZR RECONSTRUCTION AND BOUNDS approach;which causes the ~ouAer coefficients to decrease slowly. Another reason is that time-consuming numerical integration is used for the reconstruction in the ZR method. In this paper the problems mentioned above are resolved providing answers to the open questions posed in [8, 111. The results of this paper show that the methods using ZR Fourier descriptors as well as the Supported by the Natural Sciences and Engineering &search Council of Canada and Department of Education of Quebec. Zahn and Roskies [ l l ] defined Fourier descriptors as follows. Let q be a clockwise-oriented simple, closed,smooth curve of length L with parametric representation Z(O=(x(O, y(L)) where L is an arc length and 0 5 L 5 L. Also, let B(C) be the angular direction of q a t point L. The cumulative angular bend function 4(L) is defined as the net amount of angular bend between the starting point L=O and point L. So ~(O=B(O-O(O) except for possible multiples of 2* and 4(L) = -2x. In [ I l l a curve signature was defined as IAPR Workshop on CV Speaal Hardware and Industrial Applications OCT.12-14. 1988. Tokyo Clearly, d*(t) is invariant under rotation, translation and scaling, making it a good candidate for a shape signature. Expanding 4* as a Fourier series in amplitude-phase form we have and {Ak, a k ) y is the set of Fourier descriptors (ZR * descriptors) of the curve 7. Let dn denote the Fourier series in equation (2) truncated to the first n terms. For the signature in equation (1) reconstruction formula using an integral was suggested in [ l l , (eq.) 51. The use of an integral in the reconstruction formula is one of the disadvantages of the ZR approach due to the long computation time required [El. We propose a simplified reconstruction formula for an important class of polygons. Let 7 be a polygon with m vertices Vo, V ,..., V V =Vo and edges 1 m-1' m (ViWl, Vi) of length Ati, i=l,...,m (AtO=O). The angular change of direction a t a vertex Vi is Adi and m i L C A . Define ti = C At to-0, ni = i= l j=l j' 2 r i-1 r [ t i l , tJ, ci = C A$j. and assume also that 4(0) j-1 m Ab0/2. We note that d*(t) = C (t+ci) I (t), i=l "i where I (.) stands for the characteristic function of the n set n. Let f E [La, ts+ll, 80,-, m-1. We propose a fixed increment approximate reconstruction formula which can be applied to arbitrary and not necessary polygonal cuwes. where At is an arbitrary fixed length, ts -