Jian-Jiun Ding, Wei-Lun Chao, Jiun-De Huang, and Cheng-Jin Kuo

REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 22-DcontouryShapesignaturex1-Dsignature(a)(b)DFTThe contributions of this paper are summarized as follows:• A new Fourier descriptor scheme is presented to compressand reconstruct the non-closed segments, which maintainsthe compressibility and exactly preserves the end pointlocations.• The proposed Fourier descriptor scheme could also be usedon closed boundaries with a pre-segmentation process. Thismethod outperforms the general method under the samecompression rates for boundary reconstruction.• Extensive simulations on several Chinese characters and thestandard MPEG-7 CE-shape-1 database [8] are conducted.This paper is organized as follows: In Section II, we brieflyreview the concept of Fourier descriptors and previous work onnon-closed segments. In Section III, the proposed Fourierdescriptor scheme for non-closed segments is presented. Theextension of the proposed scheme towards closed boundaries isdescribed in Section IV. In Section V, several simulations areperformed to demonstrate the effectiveness of our method onboth closed and non-closed contours. Finally, in Section VI, weconclude this paper.II. REVIEW OF FOURIER DESCRIPTORSFrequencyCoefficientsA. The Basic Concept of Fourier DescriptorsAccording to existing literatures, Fourier descriptors havetwo main purposes in practice: One is to extract a compactshape descriptor for matching and retrieval [2], [4]-[5], [7]-[8],[10], [13]-[14], [18]; the other is to efficiently record andcompress the boundaries [4][11][16][19], which is the majorconcern of this paper.The Fourier descriptor algorithm consists of two stages:shape signature and discrete Fourier transform (DFT). Thefirst stage maps the 2-D boundary into a 1-D sequence, calledthe shape signature [18]; the second stage then applies the DFTon this 1-D signature, resulting in a set of DFT frequencycoefficients to represent the boundary. To describe and recordnPreservedcoefficientsFig. 2. The general procedure of Fourier descriptors: (a) the flowchart with 2stages: shape signature and the DFT; (b) an example to illustrate the inputand output of the two stages. For efficient recording and description, only thelow-frequency coefficients are preserved.kthe boundary efficiently, only the low-frequency coefficientsare kept. Fig. 2 shows the procedure of Fourier descriptors.In general, the second stage is seen as a fixed operation, soprevious work of Fourier descriptors primarily focuses on howto design the shape signatures process. For example, Zahn et al.[19] proposed the cumulative angular function along theboundary; Granlund [4] used the complex coordinate, denotedas the conventional Fourier descriptor in this paper. In [20][21]Zhang et al. gave comparative studies among several existingshape signatures on the matching performance. An importantcharacteristic of shape signatures is the reversibility: If theshape signature is not reversible from the 1-D sequence to the2-D boundary, it is not available for boundary reconstruction.There is also some work aiming to explore the RST-invariantproperties and to design the matching metric for accurate imageretrieval [1], [13]. And in [22], Zhang et al. presented a specialFourier descriptor algorithm that performs the 2-D Fouriertransform on the whole image rather than performs the 1-DFourier transform on the boundary sequence.B. Fourier Descriptors for Non-closed SegmentsThe Fourier descriptors mentioned above are mainly used forclosed boundaries. However, for non-closed segments, whichfrequently occur in character recognition and edge recording,their performances may degrade due to the non-adjacent endpoints. That is, the non-adjacent end points lead to signaldiscontinuity at the two ends of the 1-D signature, resulting insignificant high-frequency coefficients that cannot be directlyeliminated for contour recording.To solve this problem, the shape signature process should bedesigned to produce a signature with similar end point values.In previous work, Uesaka [16] computed the sequential pointdifference along a contour as the shape signature. His methoddoes reduce the high-frequency components in the shapesignature and preserve the exact end point locations, but theequal-length sampling process is not suitable for digital images.In [11], Nijim first extended the non-closed segment at the twoends, and then applied the Fourier descriptor. He claimed thatafter compression, the distortion would only appear at theextended part, and the original end points could be preserved;the extended length, however, is remained undefined. In [17],Weyland et al. proposed the generalized Fourier descriptor bymodifying the cumulative angular function, whereas it has beenmentioned in [16], [21] that this function has slow convergencespeed and is not suitable for boundary compression.Another method to solve the non-closed segment problem isto perform the discrete cosine transform (DCT) rather than theDFT on the shape signature. Namely, the shape signature is firstextended into the even-symmetric form, which indeed has sameend point values; the DFT is then applied to get the frequencycoefficients for contour recording. Although this method isefficient and could alleviate the distortions after compression,the end point locations are not guaranteed to be preserved.III. THE PROPOSED FOURIER DESCRIPTOR SCHEMEIn this paper, our main concerns are boundary compressionand reconstruction rather than the matching and retrieval