Extended Perspective Shadow Maps (XPSM) Vladislav Gusev, ,

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1 Extended Pespective Shadow Maps (XPSM) Vladislav Gusev,.8.27, Figue : XPSM esults (~4 objects in a scene, 536x536 shadow map). Intoduction Shadows ae one of the most impotant effects to convey ealism in compute-geneated scene. Shadow mapping [] is one of the most popula and efficient appoaches fo eal-time shadow endeing. The image-based natue makes shadow mapping extensively useful in eal-time applications, despite they pespective aliasing poblems. With shadow maps, we simply ende the scene fom the viewpoint of the light souce, ceating a depth image. The cuent depth buffe is stoed as a shadow map textue. In the second pass, when endeing each pixel in the final image, the visible point is tansfomed into the light coodinate system. If the distance in shadow map is less than distance between the point and the lighting souce, than pixel is shadowed. One well-known dawback of shadow mapping is pespective aliasing which typically happens when the use zoom into a shadow bounday so that single shadow map pixels become visible. A numbe of papes have tied to solve pespective aliasing poblem. The most pominent fo eal-time compute gaphic of these is TSM [2], LiSPSM [3], PSM [4], and Plane optimal shadows [6]. But all of these methods suffe fom some poblems and don t povide pactical dop-in solution fo high quality shadows endeing: TSM ( + : vey good quality, - : z acne poblem, high complexity, patented), LiSPSM ( + : good quality, medium complexity, - : significant degadation in wost case), PSM ( + : good quality, - : high complexity), Plane optimal shadows ( +\- : optimal quality, but only fo vey special case). What is moe distubing that the main souce of atifacts - Z bias non-lineaity in post pespective space even not addessed in pactical manne in academic papes, with only notable exception going fom Kozlov S. [5]. In this pape, qualitatively new appoach to shadow map paameteization is pesented. If you ae not familia with pojective geomety look at bief intoduction to this subject [7], [8] though it is not necessay, but will help to clealy undestand essence of this method. In tems of pojective geomety evey tansfomation can be decomposed into a vecto 4 pependicula to infinity hype plane (in R embedding 3 of P ) we name it pojection vecto and some affine tansfomation. Thus we appoach solution in two steps: at fist, find optimal waping effect by fixing pojection vecto, at second, find some affine tansfomation that don t affect optimal waping and tansfom necessay potion of waped space to device nomalized coodinates. On the othe hand pojection vecto is just fouth column of a tansfomation matix (D3D-like ow-vecto notation used though this pape). Let s take abitay pojection vecto (without loose of geneality we can always divide by 3 fouth component) this vecto define R optimization space, but afte application of necessay constaints (evey point of a light ay should poject to the same 2 shadow map point) dimension educe to R paameteization space with simple analytical stuctue, that helped to figue out suboptimal solution and completely esolve Z bias non-lineaity issue.

2 2. Geneal case analysis 2. Paameteization plane this is equivalent to fixing of the pojection vecto diection and length. 2.2 Back pojection singulaity It's easy to see that the pespective pat of (2) is equivalent to Aw X + Bw Y + k + whee k is the length of pojection of a vecto ( X, Y) (whee X, Y ae the coodinates of light ay) to some vecto of fee vaiables P ( Aw, Bw) in the xy plane. This pojection consists of two intevals, the view fustum inteval k (.. FaPlane) fo light ays that intesect fustum and the back pojection inteval k (..) fo light ays that intesect shadow castes behind a viewe. Figue 2: An example of the light space with the light diection L, view vecto V and view fustum F. In the light space the light diection is paallel to Z axis and the viewe oigin tanslated to (,,).Any othe case can be educed into this one. Let's assume an abitay tansfomation M is applied to the light space. Fo any pespective paameteization it would be necessay that each point of a light ay is pojected in the same point ( X, Y ) of a shadow map. In othe wods the point ( X, Y ) should not depend on the fee vaiable t, thus the following constaints should be satisfied Cw and C xy (,). Ax Bx ( X, Y ) ( X, Y, t,) M ( X, Y, t,) Cx Dx Axy X + Bxy Y + Cxy t + Dxy () Aw X + Bw Y + Cw t + Aw Bw Cw The basic fomula of abitay paameteization: Cw, Сxy (,) ( X, Y ) ( Axy X + Bxy Y + Dxy) (2) Aw X + Bw Y + Then the pojection vecto fo abitay 2 paameteization is ( Aw, Bw,,) whee ( Aw, Bw) R is a pai of fee vaiables defining a paameteization plane which is equivalent to the xy plane. Thus detemining of paameteization is educed to fixing of two fee vaiables in the xy plane. In pola coodinates Ay By Cy Dy Az Bz Cz Dz Figue 3: Fustum pojection to the xy plane with a vecto of fee vaiables P. Thee is a singulaity fo k, thus all light ays behind the viewe should be in the back pojection inteval k (..). In cases when shadow castes ae located fa behind the viewe, the only way is to scale the back pojection inteval into k (..) ange using vey small pojection vecto P. But this compession affects view fustum inteval too, which leads to significant degadation of shadows quality in font of the viewe. Evey PSM method in some way should handle castes behind a viewe and theefoe is subject to the given poblem. without loose of geneality we can always divide by Dw that is just change of a unit scale

3 3. Design 3. Pojection vecto diection Distibution of light ays in a view fustum xy pojection is dense and andom hence thee is no pefeable diection. Thus the only balanced choice is the diection with symmetic waping effect fo the whole fustum. The pojection of the view fustum to xy plane can be in most cases bounded by the tiangle with cental line of symmety having the same diection as pojection of view diection V to xy plane. Then the pojection vecto diection is: Vxy unitp. Vxy 3.2 Optimal waping effect In ideal case the waping effect should depend only on the distance to the viewe oigin. That is nea objects have high shadow map esolution and fa objects low. Thus in optimal case waping effect should not depend on elative viewe and light position\diection. whee coef constant tuned fo scene scale and desiable tade-off between nea\fa objects shadow map esolution. 3.3 Handling of back pojection singulaity Fo coect back pojection evey point of a back pojection inteval should be in k (..) ange. Then following algoithm computes optimal bound of a pojection vecto length:. Fo all points in Castes 2 find the minimal length of pojection to unitp, we'll efe to it as mincastespoj. mincastespoj min[ fo all p in Castes] ( px unitpx + py unitpy) 2. Fo all points in Receives 2 find the minimal length of pojection to unitp, we'll efe to it as minreceivespoj. minreceivespoj min[ fo all p in Receives] ( px unitpx + py unitpy) 3. The focus egion inteval is the intesection of the Castes and Receives intevals. Then the minimal point of the focus egion inteval is: minfocusre gionpoj max( mincastespoj, minreceivespoj ) 4. Calculate the pojection vecto maximal length. ( unitpx X + unitpy Y ) lengthp + W minfocusregionpoj maxlengthp + epsilonw epsilonw maxlengthp max( mincastespoj,minreceivespoj ) (3) Figue 4: The view fustum F, the light diection L and P the view vecto V pojection to xy plane. Let's take some point on a view vecto line with distance d fom the viewe oigin and gamma an angle between the view vecto V and its pojection P to xy plane, fom (2): Aw X + Bw Y + cos( gamma) d lengthp + Then fo optimal waping effect coef lengthp and cos(gamma) coef d coef + cos( gamma) d + cos( gamma) Figue 5: R eceive, C caste, ed and blue line is pojection intevals of eceives and castes with focus egion inteval maked by geen. 2 assume that castes and eceives tansfomed into light space

4 3.4 Minimization of shadow map waste the post pojection light space (PPLS) fom PPLS Receives and PPLS Castes, espectively. PPLS Receives Receives[ fo all points] LightSpace* PPLS Tansfom PPLS Castes Castes[ fo all points] LightSpace * PPLS Tansfom Figue 6: PPLS befoe (left) and afte (ight) applying otation Rotation aound Z axis in post pojection light space (PPLS) is applied to minimize a shadow map waste. This tansfomation aligns the pojection vecto diection with an X axis poducing optimal PPLS AABB's. unitpx unitpy ZRotation unitpy unitpx 3.5 Post pojection light space Figue 7: R AABB of eceives in PPLS, C AABB of castes in PPLS Then the focus egion is geometical intesection in xy plane of a PPLS Receives AABB and a PPLS Castes AABB. Thus the focus egion bounds a set of light ays which intesect at least one a shadow caste AND a shadow eceive. Clipping of length of the pojection vecto is applied to avoid singulaity: if (maxlengthp > and lengthp > maxlengthp) then lengthp maxlengthp Afte fixing two fee vaiables the pojection vecto is ( unitpx lengthp, unitpy lengthp,, ). And a pojective tansfomation: Pojection unitpx lengthp unitpy lengthp Then tansfomation fom the light space to the post pojection light space (PPLS) is: PPLS Tansfom Pojection ZRotation. 3.6 Focus egion Let s define Receives as set of points of shadow eceives bounding volumes and Castes as set of points of shadow castes bounding volumes. Then PPLS Receives, PPLS Castes is bounding volumes points tansfomed to the post pojection light space. Figue 8: R AABB of eceives in PPLS, C AABB of castes in PPLS 3.7 Focus egion basis Having the focus egion it is possible to constuct linea basis in the post pojection light space that tansfom the focus egion into a unit cube and doesn t affect optimal waping. Let's assume that the focus egion box in the post pojection light space is defined by the maximal MaxPointXY and the minimal MinPointXY points. To constuct basis, Z ange ( MinZ, MaxZ ) fo the focus egion calculated. Whee MinZ is minimal Z coodinate of Receives and Castes post pojection light space axis aligned bounding boxes and MaxZ is maximal Z coodinate. PPLS Receives AABB and PPLS Castes AABB is two axes aligned bounding boxes (AABB) constucted in

5 ZMin min(ppls Castes AABB MinPoint Z, PPLS Receives AABB MinPoint Z) ZMax max(ppls Castes AABB MaxPoint Z, PPLS Receives AABB MaxPoint Z) Figue 9: R AABB of eceives in PPLS, C AABB of castes in PPLS and the focus egion basis maked by geen Thus the focus egion basis that includes all necessay PPLS points can be calculated as: FocusRegio nbasis MaxPointX MinPointX MinPoint X 3.8 Unit cube MaxPointY MinPointY MinPointY The focus egion tansfomation into a unit cube is just matix invesion of the focus egion basis. Figue : PPLS (left) and the focus egion basis tansfomed into a unit cub (ight) UnitCubeT ansfom FocusRegionBasis MinZ 3.9 Final tansfomation Finally total tansfomation can be easily computed as: XPSM Tansfom View * LightSpace* PPLS Tansfom * UnitCubeTansfom * NomalizedSpace whee View tansfomation into the camea view space, LightSpace tansfomation into the light space, PPLS Tansfom tansfomation fom the light space into the post pojection light space, UnitCubeTansfom tansfomation of the focus egion basis in the post pojection light space into a unit cube, NomalizedSpace tansfomation fom the unit cube into Diect3D\OpenGL nomalized device coodinates space. 3. Coect Z bias Let s take some point P tansfom it into the light space, apply Z bias and following XPSM tansfomations, then: P biased ( P * LightSpace (,, bias,)) * * PPLSTansfom * UnitCubeTansfom * * NomalizedSpace P * LightSpace* PPLSTansfom * UnitCubeTansfom * NomalizedSpace (,, bias,) * PPLSTansfom * UnitCubeTansfom * * NomalizedSpace bias P * XPSMTansfom (,,,) And finally: Z ZBias Z, whee ZBias is W bias ZBiasD3D 2 bias ZBiasOGL (due to diffeent nomalized device coodinates) 3. Mine s lamp case Fo the case of mine s lamp, we just tun to unifom shadow maps, which is known to be ideal fo this case. UnitCubeT ansfom MaxPointX MinPointX MinPoint X MaxPointX MinPointX MaxPoint Y MinPointY MinPoint Y MaxPoint Y MinPointY MinZ Figue : Mine s lamp case

6 4. Algoithm Input: Receives and Castes the set of shadow eceives and castes bounding volumes points in the view space, L the light diection in the view space, epsilonw constant defining how fa to infinity, waping effect can be pushed in citical case, coef constant tuned fo scene scale and desiable tade-off between nea\fa objects shadow map esolution. Step Compute the light diection look-at matix to tansfom fom a view space into the light space. LightSpace LookAt((,,), L) (left-handed vesion ) Step 2 Tansfom the view vecto into the light space. V (,,) * LightSpace Step 3 Poject the view vecto into the paameteization, xy plane and nomalize the pojection vecto. Vxy unitp Vxy if ( V xy <.) then mine lamp case, use othogonal shadows with focusing. Step 4 Tansfom Castes and Receives into the light Step 4 Calculate the focus egion basis. space, poject bounding volumes points into unitp and find pojections with minimal length. ZMin min(ppls Castes AABB MinPoint Z, PPLS Receives AABB MinPoint Z) mincastespoj min[ fo all p in LSCastes] ( px unitpx + py unitpy) ZMax max(ppls Castes AABB MaxPoint Z, minreceivespoj min[ fo all p in LSReceives] ( px unitpx + py unitpy) PPLS Receives AABB MaxPoint Z) Step 5 Calculate maximum length of the pojection vecto. epsilonw maxlengthp max( mincastespoj,minreceivespoj) Step 9 Calculate basis of otation aound Z axis. unitpx unitpy ZRotation unitpy unitpx Step Compute tansfomation fom the light space into the post pojection light space. PPLS Tansfom Pojection ZRotation Step Tansfom Castes and Receives points into the post pojection light space (PPLS). PPLS Receives Receives[ fo all points] LightSpace* PPLS Tansfom PPLS Castes Castes[ fo all points] LightSpace* PPLS Tansfom Befoe final division clip points with w < epsilonw Step 2 Find AABB s in the post pojection light space fo Castes and Receives. PPLS CastesBox Build AABB (PPLS Castes) PPLS ReceivesBox Build AABB (PPLS Receives) Step 3 Find the focus egion that is intesection of the PPLS CastesBox and PPLS ReceivesBox in xy plane. FocusRegion Intesect 2D (PPLS Castes AABB, PPLS Receives AABB) FocusRegio nbasis MaxPointX MinPointX MinPoint X MaxPointY MinPointY MinPointY MinZ Step 6 Calculate optimal length of the pojection vecto. gamma angle between and. P V cosgamma Vx unitpx + Vy unitpy lengthp coef cosgamma Step 7 Clip the pojection vecto length. if (maxlengthp > and lengthp > maxlengthp) then lengthp maxlegthp Step 8 Calculate the pojection matix. unitpx lengthp unitpy lengthp Pojection Step 5 Calculate invesion of the focus egion basis that is tansfomation into a unit cube. UnitCubeT ansfom FocusRegionBasis UnitCubeT ansfom MaxPointX MinPointX MinPoint X MaxPointX MinPointX MaxPointY MinPoint Y MinPointY MaxPointY MinPointY MinZ

7 Step 6 Calculate the tansfomation fom the unit cube into the device nomalized coodinates space. Nomalized SpaceD3D Nomalized SpaceOGL 2 2 Step 7 Compute final tansfomation. Step 8 Calculate coect Z bias. Shade code: Z ZBias Z, W whee ZBias is bias ZBiasD3D 2 bias ZBiasOGL 2 2 * UnitCubeTansfom * NomalizedSpace Output: XPSM Tansfom, ZBias 4. Results and conclusion 2 XPSM Tansfom View* LightSpace * PPLS Tansfom XPSM was successfully implemented as a pat of NVIDIA PacticalPSM demo (can be downloaded fom In aveage XPSM pefom bette than PSM/LiSPSM and shadows esolution diectly compaable to TSM scenes. Moe impotant, that in wost cases XPSM consistently outpefoms PSM/LiSPSM/TSM and neve poduces any significant visual atifacts (Z acne, etc) o singulaities. These chaacteistics make XPSM a vey pactical appoach fo shadow map paameteization. If you find this eseach and development effots valuable o want to suppot, please, make donation. Refeences [] Williams L Casting cuved shadows on cuved sufaces. In Poceedings of SIGGRAPH 978. [2] Matin T., Tan T.-S. 24. Anti-aliasing and continuity with tapezoidal shadow maps. In Poceedings of Euogaphics Symposium on Rendeing 24. [3] Wimme M., Scheze D., Pugathofe W. 24. Light space pespective shadow maps. In the Euogaphics Symposium on Rendeing 24. [4] Stamminge M., Dettakis G. 22. Pespective shadow maps. In SIGGRAPH 22 Confeence Poceedings. [5] Kozlov S. 24. Pespective shadow maps: cae and feeding. GPU Gems, Addison-Wesley. [6] Chong H., Gotle S. 24. A lixel fo evey pixel. In the Euogaphics Symposium on Rendeing 24. [7] Bichfield. S. An Intoduction to Pojective Geomety (fo compute vision). ive.pdf [8] Davis T. Pojective Geomety df

8 Figue 2a: XPSM (top), PSM(bottom)

9 Figue 2b: LiSPSM (top), TSM(bottom)

10 Figue 3a: XPSM(top), PSM(bottom)

11 Figue 3b: LiSPSM(top), TSM(bottom)