RADIAL BASIS FUNCTION USE FOR THE RESTORATION OF DAMAGED IMAGES

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1 RADIAL BASIS FUNCION USE FOR HE RESORAION OF DAMAGED IMAGES Karel Uhlir, Vaclav Skala Uiversity of West Bohemia, Uiverziti 8, 3064 Plze, Czech Republic Abstract: Key words: Radial Basis Fuctio (RBF) ca be used for recostructio of damaged images, fillig gaps ad for restorig missig data i images. Comparisos with stadard method for image ipaitig ad experimetal results are icluded ad demostrate the feasibility of the use of the RBF method for image processig applicatios. ipaitig, radial basis fuctios, iterpolatio, image processig. INRODUCION Oe of the iterestig problems is how to recostruct a image well possible from damaged or icomplete origial as. his problem is referred to i may papers. he mai questio is: What value was i a corrupted positio ad how ca I restore it? he Radial Basis Fuctio method (RBF) is based o variatioal implicit fuctios priciple ad ca be used for iterpolatio of scattered data. he possibility of missig data restoratio (image ipaitig) by the RBF method was metioed i Kojekie & Savcheko 2. hey used this method for surface retouchig ad margially for image ipaitig as well. hey used compactly supported radial basis fuctios (CSRBF) 3 for recostructio ad octree data structure for represetatio of the parts for recostructio. he advatage of this method is that the liear system is sparse ad ca be solved easily 4. he drawback of his work was supported by the Grat No.: MSM K. Wojciechowski et al. (eds.), Computer Visio ad Graphics, Spriger. Prited i the Netherlads.

2 840 this approach is i error which ca be obtaied with a improper selectio of the radius of support of the CSRBF. I this paper we used a global radial basis fuctio for image recostructio, ipaitig ad drawig removal. 2. PROBLEM DEFINIION Let us assume that we have a image with resolutio M x N with 256 gray levels. Some pixels have icorrect values (missig or overwritte), see Fig. (a-c). We would like to restore the origial image or remove ipaitig etc. Let us assume that we ca detect missig pixels, pixels with corrupted values or ipaited pixels 5, too. For our experimets we used origial images, see Fig. d, ad oise, writig or drawig was used to corrupt them. Note that restoratio of the origial image is related to scattered data iterpolatio problem, where may poits are ot defied ad we wat to fid a value for them. Figure. Images with ipaitig, oise, scratches ad the origial oe (a, b, c, d). 3. RADIAL BASIS FUNCIONS Let us describe the RBF method ow. he RBF method may be used to iterpolate a smooth fuctio give by poits. he resultig iterpolatig fuctio thus becomes 6 : x y f ( x ) ( x c ) P( x), c c 0 (, 2) j j j i where f(c i )=h i, for i=,,, c j are give locatios of a set of iput poits (pixels), j are ukow weights, x is a particular poit ad ( x-c j ) is a radial basis fuctio, x-c j =r j is the Euclidea distace (of pixels i our i i i i i

3 Radial Basis Fuctio Use for the Restoratio of Damaged Images 84 case) ad P(x) is a polyomial of degree m depedig o the choice of. here are some popular choices for the basis fuctio, e.g. the thi-plate splie (r) =r 2 *log(r), the Gaussia (r) = exp(-r 2 ), the multiquadric (r) = (r ), biharmoic (r) = r ad triharmoic (r) = r 3 splies, where is a parameter. Now we have the liear system of equatios Eq. () with ukows j,a x,a y,a z. Natural additioal costraits for the coefficiets j must be icluded i Eq. (2) to esure orthogoality of a solutio. hese equatios ad costraits determie the liear system: h A P B, where B a 0 ad P 0 x y c c P (3) x y c c A ( c c i, j i j ), i, j,, a [ a, a, a ] x y z,, [, 2,..., ] h h, h,..., ] [ 2 he polyomial P(x) i Eq. () esures positive-defiiteess of the solutio, of matrix B 3. Afterwards, the liear equatio system Eq. (3) is solved ad the solutio vector with ad a is kow, the the fuctio f(x) ca be evaluated for a arbitrary poit x (a pixel positio i our case) 3,7,8,9. h 4. IMAGE RESORAION For image recostructio we used the RBF method metioed above ad applied it withi a 5 x 5 widow of pixels. LoadImage( ); DefieNeighborhood(5,5); Repeat For (i,j=;i<=m,j<=n;i++,j++) { if(hole(i,j)) { /* pixel [i,j] is ot defied */ K = SelectNeighborhoodOfPixel(i,j); DeleteHoles(K); /* remove all udefied pixels */ CreateSystemAdSolve(K); Pixel[i,j] = ComputeValueFromSystem(i,j);}} Util (all pixels recostructed) For this widow the RBF fuctio was computed ad used iteratively to compute the missig pixel values. he RBF fuctio differs from case to case as i the widow several pixels might be missig. Now we ca specify the proposed algorithm (above).

4 842 If there are too may udefied pixel values i the specified widow of the size 5 x 5 pixels, the the udefied pixel is ot restored ad algorithm cotiues. he missig pixels are restored i ext iteratio. 5. RESULS he LU factorizatio method was used for solvig the system ad (r) =r 2 *log(r) was used as the RBF fuctio i our experimets. For evaluatio of the proposed method we used a followig criterio: S k M N i j ( ( i, j) 2 ( i, j)) k M N (4) where: is the origial image (without corruptio), 2 is the recostructed image, k = (liear) or k =2 (quadratic differeces). We used several images that were corrupted by oise, text ad drawig ipaitig ad the oly Lea image 5 example is preseted here. he results of the recostructio of the corrupted image, see Fig. (a-c). are preseted i Fig. 2. able. presets results obtaied for the case whe oly Q [%] of pixels left from the origial image. able. Results. Lea ext (a) Radom gaps (b) Scratches (c) Q 9 % 8 % 6 % S S Figure 2. Recostructed images (from left: text, radom gaps, scratches) ad histogram with differece betwee origial ad recostructed image (a, b, c, d). It ca be see, that the results especially for case Fig. b. are very good. Also we compared our results with the result of Bertalmio et. al 0. Origial image for recostructio is preseted i Fig. 3a ad our result is i Fig. 3b.

5 Radial Basis Fuctio Use for the Restoratio of Damaged Images 843 Histogram i Fig. 2d. presets behavior of a differece betwee origial ad recostructed images. It ca be see that the proposed method has a good property. he quality of the recostructed image is preseted i Fig. 4. o a selected detail Fig. 4a, Fig. 4b presets our results ad Fig. 4c the result of Bertalmio 0. Figure 3. Ipaited image ad its recostructio. Figure 4. Scaled parts of images. Figure 5. Preset results of Bertholmio (a) ad our recostructio (b).

6 844 REFERENCES. Nikita Kozheki, Vladimir Savcheko, Michail Sei, Ichiro Hagiwara, A Approach to Surface Retouchig ad Mesh Smoothig, Iteratioal joural "he Visual Computer", A20977, Volume 9, Number 7-8, December 2003, pp Nikita Kojekie, Vladimir Savcheko, Usig CSRBFs for Surface Retouchig, Proceedigs of he 2d IASED Iteratioal Coferece Visualizatio, Imagig ad Image Processig VIIP2002, Spai, Malaga, September 9-2, Morse, B., Yoo,. S., Rheigas, P.,Che, D.., Subramaia, K. R., Iterpolatig Implicit Surfaces from Scattered Surface Data Usig Compactly Supported Radial Basis Fuctios, i Proceedigs of the Shape Modelig coferece, Geova, Italy, 89-98, May I. obor, P. Reuter, C. Schlick, Multiresolutio Recostructio of Implicit Surfaces with Attributes from Large Uorgaized Poit Sets, SMI 2004, Geova, Italy 06/ J. Ducho.: Splies miimizig rotatio-ivariat semi-orms i Sobolev space. I W.Schempp ad k.zeller, editors, Costructive heory of Fuctios of Several Variables, umber 57 i Lector Notes i Mathematics, pp , Berli, 977. Spriger Verlag. 7. urk, G., O'Brie, J.F., Modellig with Implicit Surfaces that Iterpolate, ACM rasactios o Graphics, Vol. 2, No. 4, pp , October Carr, J. C., Beatso, R. K., Cherrie, J. B.,Mitchell,. J., Fright, W. R., McCallum, B.C., Evas,. R., Recostructio ad Represetatio of 3D Objects with RadialBasis Fuctios, Computer Graphics (SIGGRAPH 200 proceedigs), pp , August Uhlir, K.: Modelig methods with implicitly defied objects, State of the Art ad Cocept of Doctoral hesis, Uiversity of West Bohemia, Czech Republic, echical Report No. DCSE/R M. Bertalmio, G. Sapiro, V. Caselles ad C. Ballester, Image Ipaitig, i Proceedigs of the ACM SIGGRAPH Coferece o Computer Graphics, 2000, SIGGRAPH 2000, 2000, pp Bloomethal, J., Polygoizaio of Implicite Surface, Computer-Aided Geometric Desig, vol. 5, o. 4, pp , Bloomethal, J., Bajaj, C., Bli, J., Cai-Gascuel, M. P., Rockwood, A., Wyvill, B.,Wyvill, G., Itroductio to Implicit Surfaces, Morga Kaufma, 997.