2-Dimensional Image Representation. Using Beta-Spline

Transcription

1 Appled Mathematcal cences, Vol. 7, 03, no. 9, HIKARI Ltd, -Dmensonal Image Representaton Usng Beta-plne Norm Abdul Had Faculty of Computer and Mathematcal cences Unverst Teknolog MARA elangor, Malaysa Arsmah Ibrahm, Fatmah Yahya Faculty of Computer and Mathematcal cences Unverst Teknolog MARA elangor, Malaysa arsmah, Jamaludn Md Al chool of Mathematcal cences cence Unversty of Malaysa 800 Pulau Pnang, Malaysa Copyrght 03 Norm Abdul Had et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Abstract Beta-splne was developed n 98 by Bran A. Barsky wth the capablty to preserve theg contnuty, whch s the man crteron n successful curve fttng. Ths curve s also well known wth ts two control parameters that can be used n controllng the curve shape. The Beta-splne curve was extended to several types to ncrease the localty control of the parameters, and gve extra flexblty to the

2 4560 Norm Abdul Had et al. curve. The preserved contnuty and shape parameters make curve and surface fttng processes usng Beta-splne smoother and more accurate. However, currently ths curve s not well-explored, and very few applcatons of ths curve are found n lterature. Ths paper proposed a new technque n -dmensonal mage reconstructon usng Beta-splne. The results show that Beta-splne gves good accuracy and has advantage n curve manpulaton Keywords: Beta-splne, Contnuty, hape parameters Introducton Beta-splne was developed n 98 by Bran A. Barsky [-3] as a part of hs PhD project. He set up a new contnuty condton called geometrc contnuty n G to replace the current parametrc contnutyc n whch s hard to acheve. Two shape parameters called bas β, and tenson β, were ntroduced to control the Beta-splne curve shape wthout changng the control ponts. These parameters are the man advantage of Beta-splne over other curves besdes the guaranteed contnutyg. The Beta-splne curve functon F (t) s wrtten as, 3 () t V b () t F () 0 () t V b () t + V b ( t) + V b ( t) V b ( t) F () where b (t) s the Beta-splne bass functon. Although the shape parameters can be used to manpulate the curve, the localty control of the parameters s stll nsuffcent snce the value of the parameters cannot be dfferent between connected curve segments to ensure the contnuty. Therefore, some ntatves to ncrease capablty of the parameters n controllng the curve are carred out. Beta-splne then s extended from unformly-shaped to contnuously and dscretely-shaped forms. The dea of contnuously-shaped Beta-splne s to rewrte the constants β, and β as functons n t. Wth the support of quntc Hermte nterpolaton, β, and β can be wrtten as,

3 -Dmensonal mage representaton ( t) α + ( α α )[ 0t 5t t ] β (3),,,, + 6 β, ( 0) α,, ( ) α, ( t) α + ( α α )[ 0t 5t t ],,,, + 6 β (4) β (5) β, ( 0) α,, ( ) α, β (6) Before any type of Beta-splne s appled n the mage, the features of the mage have to be extracted at the early stage whch s dscussed n the next secton. Then, the technque for control ponts detecton s shown. The calculated control ponts are used n the curve fttng process, and the approxmaton error s calculated subsequently. Fnally, the manpulaton of the mages usng unformly and contnuously-shaped Beta-splne are shown. Ths paper ends wth the concluson. Image Features Extracton The mage data used can be n any mage form such as btmap, jpeg, and tff. Before the data can be processed, the mage has to be converted nto boundary ponts representaton where the coordnate of each mage pxel s extracted. Further processes of the mage such as boundary smoothng and control ponts detecton are done based on the extracted ponts. Boundary extracton s a process to trace the edge of the regon of nterest (ROI). Ths process ncludes the boundary detecton, pxel coordnate extracton and arrangement. Embedded mage may consst of varous ntensty values. To smplfy the detecton, the mage has to be converted nto b-level mage whch contans only black and whte pxel[4]. A threshold, T s set to decde whether a pxel P (, j) wth (, j) locaton and I (, j) ntensty belong to black or whte group. The value of T s user-defned whether manually of automatcally[5]. Let the ROI n bnary form s n black pxels. Then the potental boundary or edge of ROI has to be extracted from the black pxels group. Ths group s called as potental boundary ponts because t ncludes avodable boundary ponts too. A black pxel pont s assgned as a potental boundary pont f at least one of ts neghbour whether rght, up, left or down s a whte pxel [6]. Fnally, the exact boundary ponts are traced. Fg. and Fg. show a font ya n bnary form, and ts extracted boundary.

4 456 Norm Abdul Had et al. Fg. The bnary mage of ya Fg. The extracted boundary of ya 3 Image Features Extracton Before Beta-splne curve can be ftted nto the mage, the mage has to be segmented n order to smplfy the curve fttng process. In D mage, the segmentaton ponts are also known as the corner ponts. Corners represent mportant features of an mage. For boundary based mage, generally a corner can be defned as an endpont of a curve segment [7-8], and also the ntersecton pont between two connected curve segments [9]. uperfluous of corner detectors for boundary based mage have been formulated and compared to select the most sutable detector for the study mage [0-]. One of the competent methods n corner detecton s usng the chord-to-pont dstance (CPD) concept []. In ths method, a boundary pont s set as a corner pont f the dstance from the pont to ts related chord s the hghest locally. The detected corners of Fg. are shown n Fg. 3. Fg. 3 Detected corners usng CPD After the mage boundary has been segmented, then the control ponts of every segment can be evaluated by error mnmzaton. nce the control ponts of a Beta-splne curve segment are related to other three consecutve segments, the evaluaton process for all segments has to be done smultaneously. Consequently, N curve segments wth N corner ponts need N control ponts. The control ponts of each curve segment for a close boundary are lsted as,

5 -Dmensonal mage representaton : : : N N N ( V, V, V3, V4 ) ( V, V3, V4, V5 ) ( V, V, V, V ) : 3 4 : ( VN, VN, VN, V ) : ( VN, VN, V, V ) ( V, V, V, V ) N (7) Every set of control ponts then s evaluated by error mnmzaton E N F ( t ) P (8) For multple curves fttng, the evaluaton process s appled to all control E ponts. Hence, each curve segment wll has four dfferent, and N curve V E segments wll produce 4N of. The control ponts then can be evaluated by V solvng the equatons smultaneously. Let all of 4N equatons n matrx form as, where, [ ] N [ M ] N N [ B] N V 4 4 (9) [ V ] [ V V V ] 3 V N (0) [ M ] [ M ] [ M ] 0 0 [ ] N M () [ B ] P b0 ( t ) P b3 ( t ) N ()

6 4564 Norm Abdul Had et al. [ M ] s a 4 4 matrx wth, m M b ) j r m ( t r ) b j ( t (3) r for, m,,3,, N and, j,,3, 4 calculated as,. The control ponts set [V] can be [ ] [ ] [ ] N B 4 N M NA 4 V (4) N [M] s not a square matrx wth dmensons N 4N determned usng the rght-nverse property. M N N. Therefore, [ ] 4 Theorem. An m n matrx A wth m < n s sad to have a rght nverse f there exsts an n m matrx B such that AB I m, and A s sad to have a left nverse wth m > n f there exsts an n m matrx C such that CA I n. Then, [V] s determned as, T ( ) T where[ ] [ M ][ M ] [ ][ M ] [ B] [ ][ M ][ M ] [ B][ M ] T T T [ ] [ B][ M ] [ M ][ M ] V (5) V (6) ( ) V (7) M s called as rght nverse matrx of [M] [3]. The control ponts of ya and epslon, and the ftted curves are shown n Fg. 4 to Fg 7. s Fg. 4 Control ponts and polygons of ya Fg 5. Ftted Beta-splnes of 'ya'

7 -Dmensonal mage representaton 4565 Fg 6. Control ponts and polygons of Fg 7. Ftted Beta-splnes of epslon epslon The approxmaton error s calculated based on the Eucldean dstance between the data pont, and the curve pont att. The approxmaton error at the data pont P P s, E F ( t ) P (8) Approxmaton errorss for all ponts n ya and epslon are shownn n Fg.8 and Fg.9. Fg.8 Approxmaton errors of 'ya' Fg.9 Approxmaton error of 'epslon'

8 4566 Norm Abdul Had et al. Maxmum error for ya and epslon usng ths method are.944 and.934 respectvely whch are qute large. Ths s due to the larger number of segments been consdered n the chapter. Large number of curve segments wll defntely ncrease the accuracy of the ftted Beta-splne curves. 4 Image Manpulaton usng hape Parameters One of the man advantages of Beta-splne s the use of shape parameters. These parameters can change the shape of ftted curve wthout changng the control ponts and contnuty. Besdes, Beta-splne also has been extended to a ratonal and contnuously-shape form to ncrease the control of the parameters. Ths chapter wll show the effects of applyng unformly and contnuously-shaped Beta-splne curves to the mage of ya and epslon. Unform Beta-splne tself can gve varous manpulated mages of the reconstructed font as shown n Fg.0 to Fg.3. Fg.0 'ya' wth β 0, β 0 Fg. 'ya' wth β, β 7 Fg. epslon wth β, β 50 Fg.3 epslon wth β, β 0

9 -Dmensonal mage representaton 4567 The shape parameter values must be same to all Beta-splne curve segments. Ths lmtaton s overcome by contnuously shape Beta-splne. Ths type of Beta-splne allows the manpulaton to be done n dfferent segments. Fg.4 to Fg.7 show the several number of segments of ya and epslon been manpulated wthα, α 0. Fg.4 ya wth 3 sequental manpulated segments Fg.5 ya wth 8 sequental manpulated segments Fg.6 'epslon' wth 38 sequental manpulated segments Fg.7 'epslon' wth 30 unsequental manpulated segments The capablty of Beta-splne n D mage reconstructon and manpulaton may gve a promsng result f t s extended to the 3D mage. 5 Concluson In ths paper, a technque for -dmensonal mage representaton usng cubc Beta-splne curve s proposed. The technque s based on least squared method

10 4568 Norm Abdul Had et al. where the control ponts are determned by error mnmzaton. From the error analyss, ths technque needs some mprovement to reduce the approxmaton error. Though, ths technque may gve promsng results besdes the reconstructed mage can be manpulated wthout changng the control ponts poston and the contnuty. Ths method s n the process to be extended to a surface reconstructon. Acknowledgement Ths study s supported by Mnstry of Hgher Educaton, Malaysa, and Unverst Teknolog MARA. References [] B. Barsky and T. DeRose, The Beta-splne: A pecal Case of the Beta-splne Curve and urface Representaton, IEEE Comput. Graph. Appl., vol. 5, 985, [] B. A. Barsky, The Beta-splne: a local representaton based on shape parameters and fundamental geometrc measures, 98 [3] B. A. Barsky, Ratonal Beta-splnes for Representng Curves and urfaces, Computer Graphcs and Applcatons, 993, 4-3. [4] F. Yahya, J. M. Al, A. A. Majd, and A. Ibrahm, An Automatc Generatons of G Curve Fttng of Arabc Characters, n Internatonal Conference on Computer Graphcs, Imagng and Vsualsaton, 006. [5] J. Ma, X. Lv, and. Yu, A mple Way to Realze the Accurate Detecton of Cells' Edge, [6] M. arfraz and A. Masood, Capturng outlnes of planar mages usng Bézer cubcs, Computers & Graphcs, vol. 3, 007, [7] A. Masood and. A. Haq, Role of Corner Detecton n Capturng hape Outlnes, n 7 th Internatonal Conference on Computer Vson, Pattern Recognton and Image Processng n Conjuncton wth 9 th Jont Conference on Informaton cences, Tawan, 006. [8] A. Masood and M. arfraz, Corner detecton by sldng rectangles along planar curves, Computers & Graphcs, vol. 3, 007, [9] D. M. Tsa, H. T. Hou, and H. J. u, Boundary-based corner detecton usng egenvalues of covarance matrces, Pattern Recognton Letters, vol. 0, 999, [0] A. H. Norm, A. Ibrahm, F. Yahya, and J. M. Al, Competent Corner Detectors for Outlne Image, n The Thrd Internatonal Conferencen on Computer Engneerng and Technology, Kuala Lumpur, Malaysa, [] A. Dutta, A. Kar, and B. N. Chatterj, Corner Detecton Algorthms for Dgtal Images n Last Three Decades, IETE Techncal Revew, vol. 5, 008, 3-3.

11 -Dmensonal mage representaton 4569 [] A. Masood and M. arfraz, Capturng outlnes of D objects wth Bézer cubc approxmaton, Image and Vson Computng, vol. 7, 009, [3] J. H. Kwak and. Hong, Lnear Algebra econd Edton. UA: prnger, 004. Receved: March, 03