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

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REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 4snsW ns nWEOriginalsegmentLinear offsetAnti-symmetricextensionDFTresponseReconstructedsegment(2.5% compression rate)Smaller highfrequencycoefficientsFig. 5. Illustrations of the segment shapes (Row 1), DFT responses (Row 2), and reconstruction shapes with 2.5% coefficients (Row 3) afterapplying linear offset (Column 2) and anti-symmetric extension (Column 3). The original segment is 321-point long and its DFT has 321coefficients. The anti-symmetric extended segment is 640-point long then with 640 coefficients; however, only part of them are essential. Forclearness, in Row 2 only 20% of the coefficients around the zero frequency are shown; the magnitude responses are plotted in the log scale. Row3 shows the reconstructed segments with only 8 complex values (about 2.5% of the original 321 coefficients) based on each input signatures. Thered boxes in Row 2 mark the coefficients preserved, and for s W(n) and s W+E(n), 2 of the 8 remained complex values are the end point locations.an anti-symmetric segment of s W (n) behind s W (n), as illustratedin Fig. 4(b). The new (2N-2)-point segment s W+E (n), withWarping and Extension, is defined as follows:sW( n) , for n 0,1,..., N 1sWE( n) , (4) sW(2N 2 n) , for n N,...,2N 3which is closed and has continuous first-order and zero secondorderdifferences at the two end points (hence much smoother).Note that adding the anti-symmetric segment in s W+E (n) doesnot increase the amount of coefficients to preserve, since theDFT frequency response of an anti-symmetric signal is alsoanti- symmetric. Therefore, only half of the Fourier coefficientsof s W+E (n) are required to keep for lossless reconstruction.As illustrated in Fig. 5, s W+E (n) has even smaller highfrequencycomponents than s W (n); besides, the reconstructedsegment from s W+E (n) with 2.5% compression rate has betterquality — with the exact end point locations — than that froms(n) and s W (n).D. DFT and Boundary CompressionAfter linear offset and anti-symmetric extension, the resultingsequence s W+E (n) is then fed into the DFT to get the frequencycoefficients S(k):2N31j2 nk/(2N2)Sk ( ) sWE( ke ) . (5)2N 2k 0Since s W+E (n) is of zero mean and S(k) is anti-symmetric withS(k) = -S(2N-2-k), both S(0) and S(N-1) are 0. Thus to achievelossless reconstruction, only S(k) at k = 1, 2, ……, N-2 arerequired.For boundary compression, the first P coefficients (S(k) at k= 1, 2, ……, P; 0≦ P ≦ N-2) are remained, and totally P+2complex values, including the two end point locations, arerecorded for reconstruction. The compression rate (CR) utilizedin this paper is defined as:# recorded complex valuesCR 100%. (6)# original segment lengthFor the proposed scheme, CR is computed as:P 2CR 100%.(7)NE. Boundary Reconstruction from the Proposed SchemeThe reconstruction process from the preserved P coefficientsand the two end point locations is listed as follows:• (Step1): Regenerate the (2N-2)-long frequency responseS*(k) from the P preserved coefficients by zero padding andanti-symmetric extension.• (Step 2): Perform the inverse DFT to get s W+E *(n).• (Step 3): Eliminate the anti-symmetric part in s W+E *(n) to gets W *(n) and perform the inverse operations of (2) to achievethe reconstructed segment s*(n).In Fig. 6, both the boundary compression and reconstructionprocesses are summarized.F. Proof of the Exact End Point Preserving PropertyIn this subsection, we give the proof of the exact end point
REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 5yProposed Fourierdescriptor schemeSnAnti-symmetric2N 3kConventionalFourierdescriptorx SNS 0 1 0BoundarycompressionBoundaryPre-segmentationSeg-FDReconstructionwith CR = 2.5%ypreserving property of the proposed Fourier descriptor scheme:No matter how many DFT coefficients are remained, the endpoints of the reconstructed segment will coincide with those ofthe original segment. This proof can be achieved by checkings W *(0) and s W *(N-1), which are the end point locations of thereconstructed segment of s W (n). As mentioned in (2), the twoend values of s W (n) are 0; then if s W *(0) and s W *(N-1) are also 0,the reconstructed non-closed segment s*(n) after the inverseoperations of (2) will have the same end points as what s(n) has.The formulas of s W *(0) and s W *(N-1) are presented as follows:P2N3j20 k/(2N2) j20 k/(2N2) k1 k2N2PP2N3s *(0) S( k) e S( k)eWk1 k2N2PPk 1Sk ( ) Sk ( ) Sk ( ) S(2N2 k) 0,(8)P2N3j2 ( N1) k/(2N2) j2 ( N1) k/(2N2)k1 k2N2PP2N3jkSke( ) jkSke ( )k1 k2N2PPSk ( ) S(2N 2jkk) e0k 1s *( N 1) S( k) e S( k)eWxReconstruction(Step 2 & 3)S*n P coefficients +2 end point locationsReconstruction(Step 1)Anti-symmetric2N 3kZero paddingFig. 6. Illustrations of both the boundary compression and the reconstructionprocesses: After applying the proposed scheme, P of the coefficients and thetwo end point locations are remained for boundary recording. To reconstructthe non-closed segment, we first regenerate the (2N-2)-long coefficientsequence (Step 1) and then perform Steps 2 & 3 in Section III-D to achieve it.where the anti-symmetric property of S(k) is considered, whichis derived from the proposed anti-symmetric extension; withoutanti-symmetric extension, the end point preserving property isnot guaranteed after boundary compression. Note that both (8)and (9) are held under arbitrary P (0≦ P ≦ N-2).(9)ProposedFourierdescriptorFig. 7. The flowchart and illustration of the proposed Seg-FD framework forclosed boundary: The top row shows the reconstruction result using theconventional Fourier descriptor [4] with 2.5% compression rate. The bottompart shows the result of Seg-FD (the dash box). As shown, the proposedSeg-FD framework could lead to better reconstruction quality, especiallyaround the corners.IV. THE PROPOSED METHOD FOR CLOSED BOUNDARIESThe proposed Fourier descriptor scheme could directly beapplied on closed boundaries; the resulting Fourier descriptorsand reconstructed boundaries are similar to the ones achievedby applying the convectional Fourier descriptor [3]-[4], [15] —since the linear offset step mentioned in Section III-B does notchange the shapes of the originally closed boundaries.Generally, Fourier descriptors are very efficient andeffective for recording closed boundaries; under highcompression rates, however, detailed structures of the originalboundaries such as corners and sharp angles still cannot bemaintained. To alleviate this problem, we further propose aframework named Seg-FD:Seg-FD performs boundary Segmentation at first togenerate several non-closed yet smooth segments, and thenapplies the proposed Fourier Descriptor scheme to record thesesegments. The flowchart of Seg-FD is depicted in Fig. 7.A. Boundary Pre-segmentationIn order to create smooth non-closed segments, sharp anglesor corners on the closed boundary should be detected first; theclosed boundary is then divided at these positions into severalnon-closed segments containing no corners inside. There havebeen many corner detection algorithms in existing literatures,like the Harris corner detector [6], SIFT [9], and the machinelearning approach in [12]. These methods, nevertheless, aredesigned mainly for corner detection in regular images, andeither with high computational complexity or requiring alearning step. For our cases on binary contour images and forefficient detection, a simple algorithm through checking theinner products along contours (shown in Fig. 8) is utilized.Given an N-point boundary {(x n , y n )| n = 0, 1, …, N1}thiscorner detection algorithm goes through each point (x n , y n ) in
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Jian-Jiun Ding, Wei-Lun Chao, Jiun-De Huang, and Cheng-Jin Kuo

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