CS 551 Computer Graphics. Hidden Surface Elimination. Z-Buffering. Basic idea: Hidden Surface Removal

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1 CS 55 Computer Grphis Hidden Surfe Removl Hidden Surfe Elimintion Ojet preision lgorithms: determine whih ojets re in front of others Uses the Pinter s lgorithm drw visile surfes from k (frthest) to front (nerest) Resize doesn t require relultion Works for stti senes Methods: k-fe ulling, depth-sort, BSP tree Imge preision lgorithms: determine whih ojet is visile t eh pixel Resize requires relultion Works for dynmi senes Methods: z-uffering, ry tring Z-Buffering Imge preision lgorithm: Determine whih ojet is visile t eh pixel Order of polygons not ritil Works for dynmi senes Bsi ide: Rsterize (sn-onvert) eh polygon Keep trk of z vlue t eh pixel - Interpolte z vlue of polygon verties during rsteriztion Reple pixel with new olor if z vlue is smller (i.e., if ojet is loser to eye)

2 Z-Buffering: lgorithm llote z-uffer; // llote depth uffer Sme size s viewport. for eh pixel (x,y) // For eh pixel in viewport. writepixel(x,y,kgrnd); // Initilize olor. writedepth(x,y,frplne); // Initilize depth (z) uffer. for eh polygon // Drw eh polygon (in ny order). for eh pixel (x,y) in polygon // Rsterize polygon. p z = polygon s z-vlue t (x,y); // Interpolte z-vlue t (x, y). if (p z < z-uffer(x,y)) // If new depth is loser: writepixel(x,y,olor); // Write new (polygon) olor. writedepth(x,y,p z ); // Write new depth. Note: This ssumes you ve negted the z vlues! Z-Buffering: Exmple = = = = 5 5 Colors R R R R R G G G G R Depths

3 Z-Buffering: Computing Z How do you ompute the z vlue t given pixel? Sme wy you do for olor Interpolte from verties Wht is the z vlue t vertex? Cnnot use stright z vlue prior to perspetive divide Cnnot use simple perspetive projetion: - ll points re ompressed onto the view plne Need to use one of the nonil z vlues B Hs the form: z = z z Use /z where z is fter the viewing trnsform ut efore the perspetive trnsform Z-Buffering: Computing Z /Z Demo Let x = [.0,.0,.0,.0,.0,.0,.0,.0]; Let z = [-0.0, -0.0, -0.0, -0.0, -50.0, -60.0, -70.0, ]; ner = -.0 fr = Projet x nd z using projetion equtions Grph x vs. z world ND x vs. z-projeted (/z) Z-Buffering: Computing Z x vs z n x = x z x vs. z n y = y z fn z = n f z h =

4 Z-Buffering : Summry dvntges: Esy to implement in hrdwre (nd softwre!) Fst with hrdwre support Fst depth uffer memory Hrdwre supported Proess polygons in ritrry order Hndles polygon interpenetrtion trivilly Disdvntges: Lots of memory for z-uffer: - Integer depth vlues - Sn-line lgorithm Prone to lising - Super-smpling Overhed in z-heking: requires fst memory Bk-Fe Culling Don t drw surfes fing wy from viewpoint: ssumes ojets re solid polyhedr - Usully omined with dditionl method(s) Compute polygon norml n: - ssume ounter-lokwise vertex order (when n pointing towrds you) - For tringle (,, ): n = ( ) ( ) Compute vetor from viewpoint to ny point p on polygon v: - For orthogrphi projetion: v = [0 0 ] T - For perspetive projetion: v = p - eye Fing wy (don t drw) if ngle etween n nd v is less thn 90 dot produt n v > 0 z n x (or y) Pinter s lgorithm Drw surfes from k (frthest wy) to front (losest): Sort surfes/polygons y their depth (z vlue) Drw ojets in order (frthest to losest) Closer ojets pint over the top of frther wy ojets Need speil proessing if polygons overlp in depth

5 Binry Spe Prtition (BSP) Tree Reursively sudivide spe to determine depth order: Think of sene s lusters of ojets Find plne tht seprtes two lusters Cluster on sme side of plne s viewpoint n osure, ut nnot e osured y, luster on other side of plne Reursively sudivide lusters y finding pproprite plnes Store sudivision (spe-prtioning) plnes in inry tree - Node: tringle nd spe prtioning plne - Left (right) hild: sutree of ojets on negtive (positive) side of plne Very effiient for stti senes New viewpoints lulted quikly BSP Trees: Bsi Ide t t In the plne (= 0) eye Behind the plne ( side) In front of the plne ( side) BSP Trees: Bsi Ide (ont.) ssume (for now) the polygons do not ross the plne Let f (p) e the impliit eqution of plne evluted t p t t In the plne eye if (f t (eye)< 0) then drw t drw t else drw t drw t Behind the plne Works for ny eye point!! 5

6 BSP lgorithm BSP Tree: prtitions spe into positive nd negtive sides Node: tringle nd spe prtioning plne Left (right) hild: sutree of ojets on negtive (positive) side of plne Use modified in-order trversl to drw the sene: Strt nd root Compute whih side of node s plne eye is on Reursively sn onvert polygons on opposite side s eye Sn onvert node s polygon Reursively sn onvert polygons on sme side s eye eye Root polygon BSP lgorithm (ont.) eye Root polygon drw(tree, eye) { if (tree.empty()) then return; if (tree.f plne (eye) < 0) then drw(tree.plus, eye); rsterize(tree.tringle); drw(tree.minus, eye) else drw(tree.minus, eye); rsterize(tree.tringle); drw(tree.plus, eye); } Resulting drwing order:,,, Building the Tree: Plne Eqution Impliit eqution of plne f plne (p) = ( ) ( ) (p ) More fmilir formultion: ssumes onsistent f plne (p) = x By Cz D = 0 vertex ordering where [BC] T represents the norml vetor n to the plne n = ( ) ( ) Solve for D y plugging in point D = x By Cz = n So: f plne (p) = (n p) (n ) = n (p ) 6

7 Building the Tree: Psuedo-Code uildtree() { tree = node(t ); // Crete root of tree. for (i = to N) do // dd eh tringle to tree. tree.dd(t i ); } dd(tringle t) { f = f plne (t.); // Compute whih side of plne f = f plne (t.); // eh vertex is on. f = f plne (t.); if (f 0 nd f 0 nd f 0) // If ll verties on negtive side if (minus.empty()) minus = node(t); // dd t to minus side. else minus.dd(t) else if (f 0 nd f 0 nd f 0) // If ll verties on positive side.... // dd t to plus side else... } // Wht do we do here? Cutting Tringles Plne If tringle intersets plne Split B Must mintin sme vertex ordering so they ll keep the sme norml! t Plne t t B t =(,, ) t = (, B, ) t = (, B, ) Cutting Tringles (ont.) ssume we ve isolted on one side of plne nd tht f plne () > 0, then: dd t nd t to negtive sutree: minus.dd(t ) minus.dd(t ) dd t to positive sutree: plus.dd(t ) t Plne t t B t =(,, ) t = (, B, ) t = (, B, ) 7

8 Cutting Tringles (ont.) How do we find nd B? : intersetion of line etween nd with the plne f plne Use prmetri form of line: p(t) = t( ) Plug p into the plne eqution for the tringle: f plne (p) = (n p) D = n ( t( )) D Solve for t nd plug k into p(t) to get ( n ) D t = n ( ) Repet for B B We will use sme formul in ry tring!! Cutting Tringles (ont.) Wht if is not isolted? if (f * f 0) // If nd on sme side: ->; ->; ->; // Shift verties lokwise. else if (f * f 0) // If nd on sme side: ->; ->; ->; // Shift verties ounter-lokwise. Note: ug in the ook here! plne plne plne ssumes onsistent, ounter-lokwise ordering of verties Cutting Tringles: Complete lgorithm if (f * f 0) // If nd on sme side: ->; ->; ->; // Shift verties lokwise. else if (f * f 0) // If nd on sme side: ->; ->; ->; // Shift verties ounter-lokwise. // Now is isolted on one side of the plne. ompute,b; // Compute intersetions points. t = (,,); // Crete su-tringles. t = (,B,); t = (,B,); if (f plne () 0) // dd su-tringles to tree. minus.dd(t ); minus.dd(t ); plus.dd(t ); else plus.dd(t ); plus.dd(t ); minus.dd(t ); 8

9 BSP Trees: Summry How do you pik the first tringle? Tril nd error Pik few permuttions t rndom nd try them out It s pre-proess: don t worry muh out uild speed O(N) trversl on verge Cn e worse sine it retes more tringles! Eye n e pled nywhere No prolems with resizing BSP Trees: pplitions Originlly developed for hidden surfe determintion However, rrely used for this purpose now How else might we use BSP tree? Visiility ulling Ry tring Dynmi Senes Collisions BSP Trees: Representtion Our originl representtion put polygons t internl nd lef nodes eye Root polygon We n lso store polygons t lef nodes only. The BSP tree then eomes sptil sudivision dt struture 9

10 BSP Trees: Sptil Sudivision Our originl representtion put polygons t internl nd lef nodes - B eye E Root dividing plne D C E D B C We n lso store polygons t lef nodes only. The BSP tree then eomes sptil sudivision dt struture BSP Trees: View Culling Compre the frustum to the split plne If it s on one side, you n ompletely ignore the other! - BSP Trees: Sptil Suidivion- dyn. sene Wht if the sene is hnging? Ie. hrter is moving out the environment? ssume the hrter is point point CNNOT interset plne so it s either in front or ehind dd the hrter to the BSP tree - Fst only requires omprisons to the root node until you reh lef Modified in-order trversl results in the sene with the hrter drwn ppropritely wrt. the sene. 0

11 BSP Tree: Clrifition We dont relly re out lning Why? We hve to hit every node nywys to render so it s O(N) on verge whether lned or not - t lest when we re using it for k to front rendering We DO re out polygon splits good BSP tree is one tht doesn t rete lots of splits! BSP Trees & ZBuffer Initil Quke engine used BSP trees for front to k rendering Lter modified to omine BSP with Zuffer BSP the stti prt of the world When rendering, keep /z vlues (only need to write the zuffer, no heking neessry) When render the hrter lst nd z-uffer it