VISUAL HULL CONSTRUCTION FROM SEMITRANSPARENT COLOURED SILHOUETTES

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1 Iteratioal Joural of Computer Graphics & Aimatio (IJCGA) Vol.3, No.4, October 2013 VISUA HU CONSTRUCTION FROM SEMITRANSPARENT COOURED SIHOUETTES J R Raki ad M Boyapati a Trobe Uiversity, Australia ABSTRACT This paper attempts to create coloured semi-trasparet shadow images that ca be projected oto multiple screes simultaeously from differet viewpoits. The iputs to this approach are a set of coloured shadow images ad view agles, projectio iformatio ad light cofiguratios for the fial projectios. We propose a method to covert coloured semi-trasparet shadow images to a 3D visual hull. A shadowpix type method is used to icorporate varyig ratio RGB values for each voxel. This computes the desired image idepedetly for each viewpoit from a arbitrary agle. A atteuatio factor is used to curb the coloured shadow images beyod a certai distace. The ed result is a cotiuous aimated image that chages due to the rotated projectio of the trasparet visual hull. KEYWORDS Coloured object recostructio, iverse problem, visual hull, data drive model, light atteuatio factor. 1. INTRODUCTION Iverse problem research has bee udergoig tremedous research i recet years. This ivolves work particularly doe i the area of costructig 3D object models from 2D images, shadows ad silhouettes [1-6]. The applicatios were desiged either from a artistic approach or for 3D pritig for later use i visual effect computatios. I this paper we preset a similar iverse light trasport method to compute a 3D visual hull for coloured shadow images. This meas we costruct a 3D object from differet 2D semi-trasparet images from multiple viewpoits. Each coloured image is stored i a trasparet cube that is made of voxels. It also stores iformatio such as viewpoit projectio ad directio for iput light sources to compute forward rederig images for various viewig agles. Our paper has the followig cotributios to this area: a. A scee model to costruct a visual hull for coloured semi-trasparet images. b. A shadowpix type method to icorporate varyig ratio RGB values for each voxel. This computes the desired image idepedetly for each viewpoit from a arbitrary agle. c. A atteuatio factor is used to curb the coloured shadow images beyod a certai distace. DOI : /ijcga

2 Iteratioal Joural of Computer Graphics & Aimatio (IJCGA) Vol.3, No.4, October REATED RESEARCH Data drive models are the emergig iterest i recet years i the field of computer graphics. They costruct the physical objects ad maipulate them uder differet lightig coditios to get the desired effect of the user. Examples iclude [1], [7] that propose methods to get a desired caustic image usig reflectio ad refractios. Other examples iclude [8], [9] that desig 3D models from real-world physical objects ad compute methods to desig variatios by followig a set of key costraits. Aother paper with the data drive approach [10] computes shadows, trasforms ad places them as required by the user by chagig the scee geometry. This agai does t support coloured shadows. 3D modelig of 2D silhouettes usig differet viewpoits [5], [11] take lie drawigs as iput to geerate full 3D coloured objects usig CSG techiques. A similar method [12] is used to geerate free-form 3D surfaces by placig primitives of 2D image. However this model uses a sigle image with aotatios to compute the fial object. The primitives such as cyliders ad ellipsoids ca be tweaked to get a smooth model. Traslucet shadow maps [13] use efficiet methods to add shadows to arbitrary scees. They take ito accout depth ad icidet light iformatio ad provide optimizatio techiques to compute sub-surface scatterig of traslucet objects to derive the shadow maps. However the traslucecy has uiform wavelegth of light with o coloured reflectios ad trasmissios. They are similar to our approach i that they have varyig light ad material properties although for the same viewpoit. Visual hull costructio for silhouette-based images [14] started i early 90s. This paper creates a black ad white volume itersectio method based o projectios from multiple viewpoit iput images. The iput images are opaque ad the fial projected geometry is o-trasparet. The shadow art method [2] use differet biary images from differet viewpoits to create a visual hull. They the tweak ad modify the 3D hull usig some shadow costraits to get accurate descriptios of the iput biary images. Oce this process is completed they project the visual hull with poit ad directioal light sources from differet poits to cast the required shadows i differet directios. A drawback of this approach is that the shadow hull oly hadles opaque images ad coloured semi-trasparet images are icosistet with this way of hadlig images. I other words, the differet semi-trasparet coloured voxels whe projected oto the scree may get accumulated to differet colour pixels rather tha the desired image. Our method described i this paper is a extesio to this approach. Aother method [15] proposes methods to tell whether or ot a coected 3D shape ca have 3 give rectiliear silhouettes. However it does t provide suggestios to chage the cofiguratio for the 3D object. Creatig physical objects from coloured shadow images [16] was recetly doe usig a set of stacked trasparet materials to form a multi-layer atteuator. The iput to this process is the same as our method described i this paper. It takes a umber of lightig cofiguratios ad desired images to compute the result ad prit it o a trasparet media. Whe projected with the iput lights it displays the required colour shadow images. This paper varies from our approach i that it produces a shadow image of the coloured image usig a trasparet medium, whereas our method tries to produce the coloured shadow of refracted material such as glass, plastic or marbles. 58

3 Iteratioal Joural of Computer Graphics & Aimatio (IJCGA) Vol.3, No.4, October 2013 A recet algorithm called the shadowpix [17] uses differet images embedded o a sigle surface. Whe lit from differet directios, differet images are formed due to self shadowig of the surface from those iput images. We use this techique to geerate multiple coloured shadows from differet viewpoits usig iput lightig cofiguratios. However rather tha embeddig the images oto a sigle surface we sculpt them iside a trasparet cube so that rotatig the cube ca create iterestig visual patters. 3. OUR APPROACH 3.1 Settig the stage The eviromet cosists of required lightig cofiguratios o the left ad desired iput image o the right. A trasparet cube is placed i the middle to capture the target iput image. As the iput image is chaged the cube is rotated to a differet agle to capture the ew image iformatio. 3.2 Approach Imagie a trasparet cube (which we will refer to as the light cube), which is made up of 3 smaller cubes with alog each directio. Each of the smaller cubes (we shall call them voxels) has a differet colour but is trasparet. A beam of parallel rays of white light with a square cross-sectio shies oto the cube from oe side. O the far side of the cube a white scree is placed. This produces iterestig colours o the scree. Next the cube is rotated about the vertical axis through its cetre ad the colour shadows chage i appearace i a fasciatig way. 3.3 Aalysis We will cosider the light cube sittig flat at the cetre of a roud horizotal table which ca rotate but oly about the vertical axis through its cetre. Therefore the beam of light ca be divided ito horizotal sheets passig through the light cube to arrive at the differet pixel sca lies o the display scree. We ca ow cosider the light of oe horizotal sectio through the cube. Figure 1 below shows a top dow view of this two dimesioal problem for the simple case of = 3. The cross-sectio of the light cube is depicted as the square ABCD. Figure 1. Protopype model of our approach. Figure 1 above shows parallel light rays from the left strikig the scree o the right i a row of pixels. The light source is assumed to be pure white light. Some of the light rays have passed through oe or more voxels of the light cube. The light cube has bee tured through agle 59

4 Iteratioal Joural of Computer Graphics & Aimatio (IJCGA) Vol.3, No.4, October 2013 radias (about its cetre). The colour of the resultig pixel o the scree is computed based o the legth of the sectios of the beam i each voxel cell ad the colour of the correspodig voxels. Assume that cell i,j has trasparecy coefficiets k Rij for the red compoet, k Gij for the gree compoet ad k Bij for the blue compoet. It will be more useful to us to use the extictio factors: k ) Rij l( Rij Eq(1a) Gij Bij l( k Gij Eq(1b) l( k Bij Eq(1c) ) ) The beam k will produce pixel p k (of the curret sca lie) at the scree depedet o which voxels it passes through with colour compoets p Rk, p Gk, p Bk where Rij p Rk e Eq(2a) Gij p Gk e Eq(2b) Bij p Bk e Eq(2c) The quatity is the distace that beam k of the horizotal sheet of light travels through voxel i,j ad the quatity is the maximum horizotal distace through a voxel amely = 2s where s is the side legth of a voxel i.e. s = S/ where S is the side legth of the light cube. The colour compoets have values i [0,1] ad those beams k which miss the colour cube have = 0 ad therefore produce pure white pixels (1,1,1) o the scree. 3.4 Forward ad Backward Computatios The Forward Computatio is to compute the pixel value p k per sca-lie give the trasparecies k Rij, k Gij ad k Bij of each cell o a cross sectio of the light cube ad the rotatioal agle of the light cube. This computatio is straight forward to implemet though requirig a cosiderable umber of lies of code. As the light cube rotates it shows some fasciatig colour variatios o the scree. The more complex computatio is the Backward Computatio. I the Backward Computatio we are give the pixel colours p k = (p Rk, p Gk, p Bk ) for each sca-lie for each of a set of discrete rotatio agles m. The umber of agles possible (the rage of m) is proportioal to. From this data we are to compute the trasparecies for all voxels k Rij, k Gij ad k Bij for i ad j = 1 60

5 Iteratioal Joural of Computer Graphics & Aimatio (IJCGA) Vol.3, No.4, October 2013 to. Usig the extictio factors istead of the trasmissio coefficiets the equatios become liear as follows: Eq(3) fij m l( p m fk ) where f takes values of R, G ad B. Because of the liearity of these equatios we ca adapt traditioal methods for solvig the liear system. Clearly if k rages from 1 to roughly close to (but larger) the to have eough equatios to solve for the 3 2 ukows α fij we will eed close to rotatios of the light cube. This meas that the light cube of 3 voxels ca store predefied graphics images or frames of a colour aimatio. Traditioal methods will ot work directly because the matrix of coefficiets m ca possibly be sigular. Istead we adapt the Gauss-Seidel algorithm to throw out icosistet equatios ad udetermied coefficiets are set to full trasparecy. This has bee tested for the low cout case = 3 where the icosistecies are iterfereces betwee the images stored i the light cube ad they appear as oise i the reproduced images. Because the proportio of icosistecies is low compared with the umber of ukows we expect that the images redered will still be very recogizable. 4. RESUTS We iitially tested this approach with a simple 3x3x3 voxel grid for efficiecy purposes. The colours i a particular voxel ca be a combiatio of Red, Blue ad Gree values. Alpha trasparecy is added ito the fial coloured voxel to attai trasparecy. The atteuatio factors are computed as per the frequecy of the colour of the voxel accordig to Equ(1). A program was writte to modify the stadard row reductio method for solvig liear systems where the umber of ukows ad umber of equatios are uequal. The algorithm was tested by simulatig oe layer of a slowly rotatig cube with radomly set voxels ad produced the physically correct pixelizatios per agle of rotatio as depicted i Figure 2 below. Whe viewed as a 2D scree aimatio each frame is clearly oly of resolutio 5x3 which is far too small for recogizable aimatios. This experimetal prototype has however show that the computatioal formulatio is correct ad is solvable i real time for suitably slow rotatio of the cube. The Backward computatios were ot tested. This is left for future work sice we had o aimatio frames to eter. Future research will also icrease the voxel size of the colour cube to = 300 ad more for which may more frames will eed to be etered (300 i that case). This will therefore cosiderably icrease the Backward computatios. However those computatios eed oly be doe oce. We evisage that the Backward calculatios will be doe prior ad separately from the Forward computatios which show the aimatios as the colour cube rotates. We will also geerate the frames by ray-tracig a movig geometric scee ad the storig the fames i a file. This process therefore uses three programs. The first program geerates a file of frames for a aimatio by ray-tracig. The secod program iputs these frames ad computes the voxel colours for the colour cube usig the Backward algorithm described i this paper, ad stores these voxels i a file. The third program reads i the voxel file data ad produces the scree aimatio via the Forward computatio algorithm described i this paper. It will be crucial to fid out to what extet the itermediate agles betwee frames causes iterferece. It is possible that the iterferece betwee frames may make it difficult for the viewer to uderstad the aimatio. O 61

6 Iteratioal Joural of Computer Graphics & Aimatio (IJCGA) Vol.3, No.4, October 2013 the other had the iterfereces may be small as suggested by our calculatios or the betwee frames may be acceptably smooth trasitios such that the viewer has o trouble iterpretig the aimatio. This research will be reported i a later paper. 5. CONCUSION AND IMITATIONS Figure 2. Implemetatio of the coloured trasparet cube. It is easy to esure that there are more equatios tha ukows by over-defiig the data with extra iput images. The row reductio method ca be adapted to throw out icosistet equatios to reach a solutio. This however ca result i iterferece i the images stored i the light cube. The resultig reproduced images however appear to show low oise levels. Research is progressig o testig this cojecture further for higher resolutios. REFERENCES [1] M. Papas, W. Jarosz, W. Jakob, S. Rusikiewicz, W. Matusik, ad T. Weyrich, Goal-based Caustics, Computer Graphics Forum, vol. 30, o. 2, pp , [2] N. J. Mitra ad M. Pauly, Shadow Art, ACM Trasactios o Graphics, vol. 28, o. 5, p. Article 156, [3] B. Tadiaus, H. Joha, ad H. S. Seah, Caustic Object Costructio Based o Multiple Caustic Patters, Joural of WSCG, vol. 20, o. 1, pp , [4] S. N. Siha ad M. Pollefeys, Multi-View Recostructio Usig Photo-cosistecy ad Exact Silhouette Costraits: A maximum-flow Formulatio, i Proceedigs of the Teth IEEE Iteratioal Coferece o Computer Visio, 2005, pp [5] A. Rivers, F. Durad, ad T. Igarashi, 3D Modellig with Silhouettes, ACM Trasactios o Graphics, vol. 29, o. 4, p. Article 109, [6] M. Fickh, H. Dammertz, ad H. P. A. esch, Geometry costructio from caustic images, i Proceedigs of the 11th Europea coferece o Computer visio: Part V, 2010, pp [7] T. Kiser, M. Eigesatz, M. M. Nguye, P. Bompas, ad M. Pauly, Architectural Caustics Cotrollig ight with Geometry, i Advaces i architectural geometry 2012, 2012, pp

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