COMP texture mapping Feb 26, x 4. 3 x 3 3 x 3. 3 x 3. projection plane texture space

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1 COMP exure mapping Feb 26, 25 Texure Mapping One ofen ihe o pain a urface ih a cerain paern, or a exure. In general hi i called exure mapping. The exure migh be defined by a digial phoograph T ( p, p ) here ( p, p ) are pixel in he exure image. (Thee pixel are omeime called exel.) For example, uppoe you an a ground urface o look like gra. You could model each blade of gra. Alernaively, you could ju pain a gra paern on he urface. The gra paern could be ju a phoograph of gra. Similarly for a building: raher han making all he geomeric deail of he facade, you could ju ue a ingle plane and pain he deail. Thi migh be ufficien if he building i in he background of he cene and he deail are unlikely o be cruinized. To map a digial phoograph ono a urface, one ould need o compue he ranformaion ha goe from he exure pace ( p, p ) in hich he digial phoograph i defined, all he ay o he pixel pace (x p, y p ) on he creen here he final image i diplayed. The ubcrip p denoe ha e are in dicree pixel pace, raher han a coninuum. The mapping from he 2D exure pace o he 2D pixel pace can be illuraed ih he kech belo. A he boom righ i a mapping from he exure image defined on ( p, p ) o he parameer (, ) of he urface e.g. a quare, a polygon. Then here i a mapping from hee urface parameer (, ) o orld coordinae (hich ake 2D o 3D) and a mapping from orld coordinae o camera coordinae (3D o 3D) and he projecion back o he 2D image domain (3D o 2D). Finally, here i he mapping from he image domain o he pixel (2D o 2D). You ve een mo of hee mapping before. There an objec o orld and a orld o camera ( modelvie ) and a projecion. In he lecure lide, I aumed ha orld coordinae ere he ame a camera coordinae, bu in general ha no he cae of coure, and one need o include ha ranformaion in here. Camera Coordinae World Coordinae 4 x 4 3 x 4 projecion 4 x 3 cale, ranlae cale, ranlae image on creen I(x p, y p) 3 x 3 3 x 3 3 x 3 T( p, p) projecion plane exure pace exure (e.g. digial phoograph) The projecion i a bi uble. I map from camera coordinae o clip coordinae, hich you recall are normalized device coordinae bu prior o perpecive diviion. So e have (x, y, z, ). We are dropping he 3rd coordinae here, o i i ju (x, y, ). The 3rd (z) coordinae i needed for hidden urface removal, bu e are no dealing ih ha problem here. la updaed: 27 h Apr, 25

2 COMP exure mapping Feb 26, 25 The final mapping in he equence i he indo o viepor mapping, hich e dicued in lecure 6. You can ee ha he above equence of linear mapping collape o a 3 3 marix, call i H, and o e have he map: x p p y p = H p The invere of hi map i: p p = H To exure map he image I( p, p ) ono he polygon, e do he folloing: for each pixel (x p, y p ) in he image projecion of he polygon { x p y p. } compue exel poiion uing ( p, p, ) = H (x p, y p, ) I(x p, y p ) = T ( p, p ) Noe ha he compued exel poiion ( p, p ) generally ill no be a pair of ineger and o ill no correpond o a unique exel. One herefore need o be careful hen chooing T ( p, p ) for he color o rie ino he pixel. We ill reurn o hi iue laer in he lecure. Homography The ne concep here i he 3 3 linear ranform H beeen exure coordinae ( p, p, ) and he image plane coordinae (x p, y p, ). We examine hi in more deail by looking a o componen of he mapping, namely from (,, ) ino orld coordinae and hen from orld coordinae ino he image projecion plane. The mapping from exure coordinae (,, ) ino orld coordinae i: x y z = a x b x x a y b y y a z b z z () Thi mapping ake he origin (, ) = (, ) o (x, y, z ). I ake he corner (, ) = (, ) o (x + b x, y + b y, z + b z ), and i ake he corner (, ) = (, ) o (x + a x, y + a y, z + a z ), ec. In homogeneou coordinae, i ake (,, ) = (,, ) o (x, y, z, ). I ake (,, ) = (,, ) o (x + a x + b x, y + a y + b y, z + a z + b z, ), ec. Le aume, for impliciy, ha orld coordinae are he ame a camera coordinae, i.e. here i an ideniy mapping beeen hem. Thi i ha I did in he lide. We hen projec poin la updaed: 27 h Apr, 25 2

3 COMP exure mapping Feb 26, 25 from orld/camera coordinae ino he image plane z = f by: x fx f fy = f y z z Thi give fx fy z = f f a x b x x a y b y y a z b z z We ge 3 3 marix, namely he produc of a 3 4 marix, and a 4 3 marix. Thi give a 3 3 linear ranform from (,, ) o (x, y, ). Noe ha in hi example, for impliciy I am ignoring he normalizaion marix M normalize. (Recall lecure 5.) In fac, hi i only alloed in he pecial cae ha hi marix happen o be he ideniy marix, e.g. if lef = boom = -, op = righ =. In general, an inverible 3 3 marix ha operae on 2D poin rien in homogenou coordinae i called a homography. In he cae of mapping from (, ) o (x, y) pace, e have in general x y = h h 2 h 3 h 2 h 22 h 23 h 3 h 32 h 33 Noe ha mapping from normalized coordinae (, ) o (x, y) or from pixel coordinae ( p, p ) o (x, y ) are boh homographie. Boh are given by 3 3 marice. Example: a laned ground plane Le conider a concree example o you ge a ene for ho hi ork. Conider a plane: z = z y an θ hich e ge by roaing he plane z = by θ degree abou he x axi and hen ranlaing by (,, z ). We parameerize he roaed 2D plane by (, ) uch ha he origin (, ) = (, ) i mapped o he 3D poin (x, y, z) = (,, z ), he axi i in he direcion of he x axi. By inpecion, he ranformaion from (,, ) o poin on he roaed 3D plane, and hen on o poin on he image plane i: x y = f f co θ in θ z Muliplying he o marice in (*) give he homography: f H = f co θ. in θ z la updaed: 27 h Apr, ( )

4 COMP exure mapping Feb 26, 25 The invere happen o be: H = f f co θ an θ z f The 4 3 marix mapping exure pace o 3D may be myeriou a fir glance, bu noe ha he fir mapping i ju a roaion abou he x axi, ih he hird column removed. Thi mapping can be hough of a roaing a plane (,, ), here e don boher including he z = in he mapping ince i ha no effec. Anoher ay o hink of hi mapping i o break i don ino i componen. co θ in θ z = z + co θ in θ Alhough one can inerpre he 4D vecor (,,, ) and (, co θ, in θ, ) a poin a infiniy, hi inerpreaion i no o helpful here. Raher hink of (,, ) and (, co θ, in θ) a vecor ha allo you o reach any poin in he laned plane, via linear combinaion. (You need o ue 4D vecor here becaue he laned plane i hifed aay from he origin and i no a 3D vecor pace i.e. i i no cloed under addiion.) Aliaing Recall from lecure 6 hen I dicued ho o can conver a line: m = (y - y)/(x - x) y = y for x = round(x) o round(x) riepixel(x, Round(y), rgbvalue) y = y + m For each dicree x along he line, e had o chooe a dicree value of y. To do o, e rounded off he y value. Thi gave u a e of pixel ha approximaed he original line. If e repreen a coninuou objec (line, curve, urface,.. ) ih a finie dicree objec hen e have a many-o-one mapping and e loe informaion. We ay ha aliaing occur. Here he ord aliaing i ued lighly differenly from ha you may be ued o. In programming language, aliaing refer o an objec or more generally o memory locaion ha ha o differen name. x = ne Dog() y = x The analogy in compuer graphic (or ignal in general) i ha e have one dicree objec (RGB value on a grid of pixel) ha can arie from more han one coninuou objec (RGB value defined on a coninuou parameer pace). To underand ho aliaing arie in exure mapping, uppoe I have coninuou image I(x, y) hich i verical ripe of idh <. The ripe have alernaing ineniie,,,,, over la updaed: 27 h Apr, z

5 COMP exure mapping Feb 26, 25 coninuou inerval of idh. If you no ample hi image along a ro (fixed y), here he pixel poiion are x =,, 2, 3,... hen he value ha e ge ill ypically no be alernaing and. For example, run he pyhon code: # ample from alernaing and value on inerval of idh =.38 n = 5 # number of ample reul = zero(n) for x in range(n): if (x / ) % 2 > : reul[x] = prin reul The oupu for he given parameer i: hich i clearly no periodic. If you change hen you ll ge a differen oupu. The main problem here i ha he underlying image ha rucure ha i a a finer cale han he ampling of he pixel. Thi problem ofen arie hen you ample a coninou ignal. In he lecure lide I gave ome oher example, including a Moiré paern. Ani-aliaing In he cae of can convering a line, aliaing produce a jagged edge. Example ere given in he lecure lide. One idea o ge rid of he jaggie (called ani-aliaing ) i o ake advanage of he fac ha RGB value are no binary bu raher ypically are 8 bi value (i.e. o 255). For example, uppoe ha a line i uppoed o be black (,,) on a hie (255,255,255) background. One rick o ge rid of he jaggie i o make a pixel ineniy a hade of grey, if ha pixel doen lie exacly on he line. The grey value of he pixel could depend on he cloene of he pixel o he coninuou line. (There i no unique ay o do hi, and underand ha beer or ore depend on facor ake a hile o explain including iue in viual percepion. We re no going don ha rabbi hole.) See he example belo. The fir ho he jaggie; he econd ho he ani-aliaed line. la updaed: 27 h Apr, 25 5

6 COMP exure mapping Feb 26, 25 Aliaing and ani-aliaing in exure mapping Aliaing arie in exure mapping a ell. Recall he algorihm on page 2 of oday noe. For each pixel (x p, y p ) in he image of a polygon, e map back o ome pixel poiion ( p, p ) in he original exure. The imple mehod ould be o ju round off ( p, p ) o he neare ineger and copy he (RGB) exure value T (round( p ), round( p )) ino I(x p, y p ). A alernaive approach i o conider a pixel a a mall uni quare, for example, cenered around he poin (x p, y p ), namely he quare [x p, x 2 p + ] [y 2 p, y 2 p + )]. If e ere o map he 2 corner of hi lile quare pixel back o he exure pace ( p, p ) hen e ould ge a four verex polygon (quad) in ha pace. Thee four verice ypically ould no correpond exacly o ineger coordinae of he exure. GIven hi lack of correpondence, ho hould one chooe he color for he image pixel (x p, y p )? There are o general cae o conider. Here e conider exel and pixel o be uni quare in ( p, p ) and (x p, y p ) pace, repecively. magnificaion: Thi i he cae ha a exel in ( p, p ) map forard o a region in (x p, y p ) ha i larger ( magnify ) han a pixel in (x p, y p ). Noe ha if e conider he mapping in he invere direcion, hen a quare image pixel cenered a (x p, y p ) ould map back o a region ha i maller han a exel quare in ( p, p ) pace. Magnificaion can happen, for example, if he camera i cloe o he urface ha i being exure mapped. (You may have noiced hi hen playing cheap video game, and you drive oo cloe o a all.) minificaion: Thi i he cae ha a exel in ( p, p ) map forard o a region in (x p, y p ) ha i maller ( minify ) han a pixel in (x p, y p ). In he invere direcion, a pixel in (x p, y p ) map back o a region ha i larger han a exel in ( p, p ) pace. In he cae of a laned ground plane, minificaion can happen pixel ha are near he horizon. Noice ha maller or larger i a bi ambiguou here ince here i boh an x and y direcion. I could happen ha here i a minificaion in one direcion bu a magnificaion in he oher direcion, for example, if you are cloe o a urface bu he urface i highly laned. The mehod ha one ue o chooe he color I(x p, y p ) could depend on heher one ha magnificaion or minificaion a ha pixel. For example, if magnificaion occur, hen i can eaily happen ha he invere map of an image pixel quad ill no conain a exure pixel. To chooe he color of he image pixel, one could ignore he invere map of he image pixel quad and ju conider he poiion of he invere mapped pixel cener. One could hen chooe he ineniy baed on he neare exure pixel ( p, p ) or ake a eighed average of he neare four exure pixel, hich form a quare. Thi can be done ih bi-linear inerpolaion. See he Exercie. If one i minifying (hrinking) he exure, hen he invere map of he quare image pixel migh correpond o a larger region (quad) of he exure image. In hi cae, one migh can conver hi region ue he average of he RGB ineniie. A you can image, here i grea flexibiliy in ha one doe here. There i no ingle approach ha i be, ince ome approache are more expenive han oher, and he aliaing iue ha arie depend on he exure being ued and he geomery involved. la updaed: 27 h Apr, 25 6

7 COMP exure mapping Feb 26, 25 Texure Mapping in OpenGL A he end of he lecure, I menioned he baic of ho exure mapping i done in OpenGL. Fir, you need o define a exure. For a 2D exure uch a e have been dicuing, you pecify ha i i 2D, and variou parameer uch a i ize (idh and heigh) and he daa i.e. RGB value. glteximage2d( GL_TEXTURE_2D,..., ize,.., daa ) You alo need o aociae each verex of your polygon ih a exure coordinae in (, ) pace. You need o do hi in order o define he mapping from exure coordinae pace o your polygon, a dicued a he boom of page 2 of hee noe. Thi definiion of (, ) value for each verex i occur hen you declare he polygon. The exure coordinae, like a urface normal, i an OpenGL ae. Each verex i aigned he exure coordinae a ha curren ae. If you an differen verice o have differen exure coordinae and you generally do an hi hen you do exure mapping hen you need o change he ae from verex o verex. For example, here I define he exure coordinae o be he poiion (, ) = (, ), (, ), (, ). You can define he exure coordinae o be any (, ) value hough. glbegin(gl_polygon) gltexcoord2f(, ) glverex( x, y, z ) gltexcoord2f(, ) glverex( x, y, z ) gltexcoord2f(, ) glverex( x2, y2, z2 ) glend() I ll ay more abou exure coordinae nex lecure. No all exure are 2D. You can define D and 3D exure oo. la updaed: 27 h Apr, 25 7