Field Visualization by Image Processing

Transcription

1 The 0th International Symposium on Flow Visualiation August 6-9, 00, Kyoto, Japan F008 Fiel Visualiation by Image Processing Image refining an fiel characteristics arinova, I. *, Eno, H. *, Hayano, S. * an Saito, Y. * * Department of Electrical Apparatus, Technical University of Sofia, Sofia-756, Bulgaria Tel: / FAX: iliana@vmei.aca.bg * Faculty of Engineering, Hosei University, 3-7- Kajino, Koganei, Toyo , Japan Abstract: The purpose of this article is to apply the image processing technique for visualiation of electromagnetic fiels. An inverse approach has been evelope. The refining, reconstruction an repairing of the images are formulate as an inverse problem. For the image color moel, the Poisson equation with open bounary conition is impose. The color source ensities have been evaluate from the color s for electromagnetic fiel ata set. The Generalie Vector Sample Pattern atching metho is applie to solve an ill pose linear system of equations for the corresponing inverse problem. The new color s is generate an visualie using the obtaine color source ensities. With the propose inverse approach the quality of the images can be essentially improve. Utiliing the color source ensity, the color of the various fiel characteristics coul be obtaine. This article collects several eamples for electromagnetic fiel visualiation of the most commonly use sensors. Keywors: Image processing, Electromagnetic fiel, Visualiation, Inverse Problem. Introuction Visualiation of electromagnetic fiels an processes is very important for investigation an esign of electromagnetic evices. The fiel is require solving many ientification, NDT or EC problems. In many cases the available fiel ata set is insufficient or it is obtaine in some limite region. Using such ata the visualiation of fiel leas to poor an amage images. In orer to refine the images, recently, we propose an approach using formulation an solution of inverse problem over the image e.g., arinova et al., 00. The available fiel ata set is visualie an the image obtaine is processe. The image is consiere as a D- of color components Re, Green, an Blue RGB. The D-Poisson equation is impose to be satisfie by each of RGB color components with homogeneous open bounary conitions. In orer to evaluate the color component, we etermine the color source ensities utiliing the Green functions. The inverse problem for color source etermination is formulate. The Generalie Vector Sample Pattern atching GVSP metho is applie to solve the compose ill pose linear system of equations e.g., Seijima et al., 00. New visualiation of the color is carrie out from the evaluate color sources. Utiliing the RGB color source ensity, we can etermine the color component, which correspons to the flu ensity of the system uner consieration. In such case, the color of the various fiel characteristics coul be obtaine an eamine. Several visualiation eamples Copyright 00 by VSJ

2 emonstrate versatility an applicability of the propose inverse approach for fiel visualiation as well as for image processing.. Color Source Densities The image colors are consiere as a D- of color components RGB. A piel-oriente strategy has been utilie. The D-Poisson equation is impose to be satisfie by each of RGB color components with homogeneous open bounary conitions A = σ where A anσ are the any of the color components RGB an color source ensities, respectively. The color component A can be represente by the appropriate Green function G over image piels A = σgs 4π S. In orer to etermine the color component source ensities, we compose a system of equations CX = Y, 3 where C, Y an X are the n by m system matri, n-th orer column vector of the color component an m-th orer column vector of unnown color source ensities, respectively. To refine an improve the image, the color components as well as color source ensities have to be etermine in terms of the - an y- piels of the target area. The re, green an blue color components are istinctly processe. For each of them the amage area is etracte an systems of equations 3 in each of the color components are compose. Cause by the etracte amage area, the number of unnowns in 3 m is much lager then number of equations n. The RGB systems of equations become linear but ill pose. Thus, the inverse problem for color source etermination is formulate. In orer to solve the RGB systems of equations 3, the GVSP is applie. The color source ensities are etermine. The new visualiation of the color s is carrie out from the evaluate color source ensities. Because of the solution error of the systems of equations, it is ifficult to obtain the eact results. uality of the reconstructe image is estimate by means of correlation factor. The high quality image coul be obtaine by organiing iterative proceure. During visualiation of fiels when the potential fiel is consiere, the RGB color of the image correspons to the potential of the fiel. Applying the relation between vector potential A an flu ensity B over the piels of the image B = A, 4 we can obtain the RGB color flu ensity of the fiel uner consieration. Also, the components of the color flu ensities are etermine utiliing the color source ensities σ B = σ S, 5 S By = σ ys, 6 S where the an y represent the geometrical relations between piels for corresponing components of the flu ensity. In matri form 5 an 6 are epresse by B = D σ, B y = D y σ, where D an D y are matri with an y elements, respectively. 7 8 Copyright 00 by VSJ

3 Thus, if the mathematical moel of the processes, that are visualie, is applie to the corresponing image uring image processing, the ifferent characteristics of these processes can be obtaine. 3. Applications The inverse problem formulate above has been applie for image processing along with visualiation of the fiel of electromagnetic evices. The magnetic fiel of sensor coil has been stuie. The coil is of circular shape. The inner iameter is r = 0. 0m, the outer iameter r = m an height is h = m. The magnetic fiel uner consieration is aisymmetrical an can be easily visualie using ifferent visualiation tools. The magnetic fiel of the linear sensor compose of ferromagnetic core with imensions m, height h=0.050m an coil with inner an outer imensions =0.04m, =0.054m an height h=0.030m has been investigate. The magnetic fiel can be consiere as D-plane. 3. Image refining When the image available is amage or corrupte, it is ifficult to analye it properly. In orer to refine or repair the image, we apply the propose inverse approach. Figs. a an b show the original a an amage b images of magnetic fiel of the investigate sensor coil, respectively. The amage part is 5% of the entire image, an also low resolution. The systems of equations 3 were compose an solve for each of the three-rgb color components of the amage image. In this case, we employ the Green function in cylinrical coorinates, epresse by G = K E π. 9 a Original image b Damage image c Refine image by 9 K=0.97 Refine image by K=0.93 Fig. Original an amage images of magnetic fiel of the sensor coil. a Re b Green c Blue RGB source Fig. Color source ensity s of coil sensor. Copyright 00 by VSJ

4 The moule in 9 is = 4, 0 an K an E are the complete elliptic integrals from first an secon in, respectively;, are the raial coorinates an, are the aial coorinates of the piels of the color component A an the piel of the current integration, respectively. The amage area was etracte. The solutions of the systems represent the re, green an blue color source ensities over image incluing amage region, shown in Fig.. The new visualiation of the color is carrie out from the evaluate color sources. As a result, we have obtaine the refine image. Fig.c shows the result. The correlation coefficient between original an refine images is Fig. shows the refine image obtaine by utiliing the Green function in Cartesian coorinates, epresse by = ln π r G. The correlation factor is The results in Fig. reveal that the visualie fiel by aisymmetrical formulation in 9 is capable of proviing the better results in terms of the correlation factors. This is cause by the aisymmetrical nature of the magnetic fiel of the sensor coil uner consieration. Figs. 3a an 3b show the original an amage images of magnetic fiel of the core-coil sensor, respectively. Appling our approach, the systems of equations 3 were solve for the three-rgb color components of the amage image. In this case, as the magnetic fiel coul be consiere as a plane, we employ the Green in D Cartesian coorinates, epresse by. The RGB color source ensity is shown in Fig. 4. In Fig. 3c is shown the refine image. The calculate correlation factor is K=0.88. a Original image b Damage image c Refine image by K=0.88 Fig. 3 Original an amage images of magnetic fiel of core-coil sensor a Re b Green c Blue RGB source Fig. 4 Color source ensity s of core-coil sensor. Copyright 00 by VSJ

5 Copyright 00 by VSJ Analying the result presente in Figs -4 it was foun that uring the image processing in orer to refine the images the fairly goo result coul be obtaine utiliing the epressions that moel the visualie fiels. 3. Fiel characteristics etermination Consiering potential of the magnetic fiel an utiliing the RGB color source ensity, we can obtain the color flu ensity by 4-8. In case of cylinrical coorinate system the elements an y of the matri D an D y are = E K π, = E K y π. 3 a Re b Green c Blue RGB source Fig. 5 Color source ensities of the coil sensor a Re b Green c Blue Fig. 6 X-components of the RGB color flu ensity of coil sensor a Re b Green c Blue Fig. 7 Y-components of the RGB color flu ensity of coil sensor

6 a X-component b Y-component Fig. 8 X- an Y-components of the color flu ensity of coil sensor In case of Cartesian coorinate system the elements an y of the matri D an D y are y y y =, 4 π y = π y, 5 In -5 an y are the coorinates of the current piel an the piel of observation. Utiliing the obtaine RGB color source ensities s for coil sensor shown in Fig. 5, we can etermine the X- an Y- components of the RGB color flu ensity s, presente in Fig. 6 an Fig.7. As the magnetic fiel is aisymmetrical the epression -3 are use. The X- an Y-components of the color flu ensity s etermine by image processing of the potential of the coil sensor are shown in Fig.8. The color flu ensity s correspon to the flu ensity s of the visualie magnetic file. a Re b Green c Blue RGB source Fig. 9 Color source ensities of the core-coil sensor a Re b Green c Blue Fig. 0 X-components of the RGB color flu ensity of the core-coil sensor Copyright 00 by VSJ

7 a Re b Green c Blue Fig. Y-components of the RGB color flu ensity of core-coil sensor. a X-component b Y-component Fig. X- an Y-components of the color flu ensity of core-coil sensor For the core coil sensor, the RGB color flu ensity s are etermine by color source ensity s shown in Fig. 9. The X- an Y- components of color flu ensity s are presente in Fig. 0 an Fig.. As the magnetic fiel is D-plane, the epressions 4-5 are use. The X- an Y-components of the color flu ensity s etermine by image processing of the potential of the core-coil sensor are shown in Fig.. 4. Conclusion In this article, we have applie the inverse approach for image refining. The color source ensities s have been etermine by solving of the image inverse problem. The new RGB color s have been generate. With appropriate selection of the Green function, it is possible essentially to improve the quality of the image. Thus, it has been foun that if the relations of the mathematical moel escribing the visualie fiel are applie to the image uring the image processing, the various fiel characteristics coul be obtaine. Thereby, from potential of magnetic fiel, it is possible to visualie the flu ensity. References arinova I., Eno H., Hayano S., Saito Y., 00, Image reconstruction for electromagnetic fiel visualiation by an inverse problem solution, Applie Electromagnetics an echanics, T. Taagi an. Uesaa es. 00 JSAE, 699 Seijima D., iyahara S., Hayano S., Saito Y., 000, A stuy on the quasi-3d current estimation," Trans. IEE of Japan, Vol. 0-A, No.0, pp in Japanese. Copyright 00 by VSJ