Spatial Transformations. Prof. George Wolberg Dept. of Computer Science City College of New York

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1 Sptil Trnsformtions Prof. George Wolberg Dept. of Compter Science City College of New York

2 Objecties In this lectre we reiew sptil trnsformtions: - Forwrd nd inerse mppings - Trnsformtions Liner Affine Perspectie Biliner - Inferring ffine nd perspectie trnsformtions 2

3 Forwrd nd Inerse Mppings A sptil trnsformtion defines geometric reltionship between ech point in the inpt nd otpt imges. Forwrd mpping: [x, y] = [X(,), Y(,)] Inerse mpping : [, ] = [U(x,y), V(x,y)] Forwrd mpping specifies otpt coordintes (x,y) for ech inpt point (,). Inerse mpping specifies inpt point (,) for ech otpt point (x,y). Inpt Otpt (ccmltor) Forwrd mpping Inpt Otpt Inerse mpping 3

4 Liner Trnsformtions [ x, y] [, ] x 2 y Aboe eqtions re liner becse they stisfy the following two conditions necessry for ny liner fnction L(x): ) L(x+y) = L(x) + L(y) 2) L(cx) = cl(x) for ny sclr c nd position ectors x nd y. Note tht liner trnsformtion re sm of scled inpt coordinte: they do not ccont for simple trnsltion. We wnt: x y [ x, y] [,,]

5 Homogeneos Coordintes To compte inerse, the trnsformtion mtrix mst be sqre. Therefore, 2 [ x, y,] [,,] All 2-D position ectors re now represented with three components: homogeneos nottion. Third component (w) refers to the plne pon which the trnsformtion opertes. [,, ] = [, ] position ector lying on w= plne. The representtion of point in the homogeneos nottion is no longer niqe: [8, 6, 2] = [4, 8, ] = [6, 32, 4]. To recoer ny 2-D position ector p[x,y] from p h =[x, y, w ], diide by the homogeneos coordinte w. ' x [ x, y] ' w ' y ' w 5

6 6 Affine Trnsformtions (),], [,], [ y x Trnsltion T T cos sin sin cos Rottion Scle S S rows Sher H colmns Sher H

7 7 Affine Trnsformtions (2) Affine trnsformtion he 6 degrees of freedom:, 2, 3, 2, 22, 23. They cn be inferred by giing the correspondence of three 2-D points between the inpt nd otpt imges. Tht is, All points lie on the sme plne. Affine trnsformtions mp tringles onto tringles. ) 3for, s:(3for 6constrint ), ( ), ( ), ( ), ( ), ( ), ( y x y x y x y x

8 Affine Trnsformtions (3) Skew (sher) Rottion/Scle Rottion 8

9 Perspectie Trnsformtions () ' 2 3 x x ' ' ' [,, ] [,, ] ' x y w w 2 22 w 23 ' y y ' 3 w 23 not necessrily s in ffine trnsformtions Withot loss of generlity, we set 33 =. This yields 8 degrees of freedom nd llows s to mp plnr qdrilterls to plnr qdrilterls (correspondence mong 4 sets of 2-D points yields 8 coordintes). Perspectie trnsformtions introdces foreshortening effects. Stright lines re presered

10 Perspectie Trnsformtions (2)

11 Biliner Trnsforms () [ x, y] [,,,] 3 2 b b b b corner mpping mong nonplnr qdrilterls (2 nd degree de to fctors). Coneniently compted sing seprbility (see p. 57-6). Preseres spcing long edges. Stright lines in the interior no longer remin stright.

12 Biliner Trnsforms (2) 2

13 Exmples Similrity trnsformtion (RST) Affine trnsformtion Perspectie trnsformtion Polynomil trnsformtion 3

14 Scnline Algorithms Prof. George Wolberg Dept. of Compter Science City College of New York

15 Objecties In this lectre we reiew scnline lgorithms: - Incrementl textre mpping - 2-pss Ctmll-Smith lgorithm Rottion Perspectie - 3-pss sher trnsformtion for rottion - Morphing 5

16 Ctmll-Smith Algorithm Two-pss trnsform First pss resmples ll rows: [, ] [x, ] [x, ] = [ F (), ] where F () = X(, ) is the forwrd mpping fct Second pss resmples ll colmns: [x, ] [x, y] [x, y] = [x, G x ()] where G x ()=Y(H x (), ) H x () is the inerse projection of x, the colmn we wish to resmple. It brings s bck from [x, ] to [, ] so tht we cn directly index into Y to get the destintion y coordintes. 6

17 7 Exmple: Rottion () cos sin cos sin into ) ( ).Sbstitte ( Compte b) cos sin sin cos tht Recll ). ( Compte ) 2 : Pss ], sin cos [ ], [ Pss: cos sin sin cos ], [ ], [ x y y H G x x H x y x x x x

18 2-Pss Rottion scle/sher rows scle/sher colmns 8

19 9 3-Pss Rottion Algorithm Rottion cn be decomposed into two scle/sher mtrices. Three pss trnsform ses on sher mtrices. Adntge of 3-pss: no scling necessry in ny pss. 2 tn sin 2 tn cos sin sin cos R cos tn sin cos cos sin sin cos R

20 3-Pss Rottion sher rows sher colmns sher rows 2

21 Softwre Implementtion The following slides contin code for initmtrix.c to prodce 3x3 perspectie trnsformtion mtrix from list of for corresponding points (e.g. imge corners). Tht mtrix is then sed in perspectie.c to resmple the imge. The code in resmple.c performs the ctl scnline resmple. 2

22 initmtrix.c () /* ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ * initmtrix: * * Gien Icorr, list of the 4 correspondence points for the corners * of imge I, compte the 3x3 perspectie mtrix in Imtrix. */ oid initmtrix(imgep I, imgep Icorr, imgep Imtrix) { int w, h; flot *p, *, 3, 23; flot x, x, x2, x3; flot y, y, y2, y3; flot dx, dx2, dx3, dy, dy2, dy3; /* init pointers */ = (flot *) Imtrix->bf[]; p = (flot *) Icorr->bf[]; /* init,,x,y rs nd print them */ x = *p++; y = *p++; x = *p++; y = *p++; x2 = *p++; y2 = *p++; x3 = *p++; y3 = *p++; 22

23 initmtrix.c (2) w = I->width; h = I->height; UI_printf("\nCorrespondence points:\n"); UI_printf("%4d %4d %6.f %6.f\n",,, x, y); UI_printf("%4d %4d %6.f %6.f\n", w,, x, y); UI_printf("%4d %4d %6.f %6.f\n", w, h, x2, y2); UI_printf("%4d %4d %6.f %6.f\n",, h, x3, y3); /* compte xiliry rs */ dx = x - x2; dx2 = x3 - x2; dx3 = x - x + x2 - x3; dy = y - y2; dy2 = y3 - y2; dy3 = y - y + y2 - y3; 23

24 initmtrix.c (3) /* compte 3x3 trnsformtion mtrix: * 2 * * */ 3 = (dx3*dy2 - dx2*dy3) / (dx*dy2 - dx2*dy); 23 = (dx*dy3 - dx3*dy) / (dx*dy2 - dx2*dy); [] = (x-x+3*x) / w; [] = (y-y+3*y) / w; [2] = 3 / w; [3] = (x3-x+23*x3) / h; [4] = (y3-y+23*y3) / h; [5] = 23 / h; [6] = x; [7] = y; [8] = ; } 24

25 perspectie.c () #define X(A, U, V) #define Y(A, U, V) #define H(A, X, V) ((A[]*U + A[3]*V + A[6]) / (A[2]*U + A[5]*V + A[8])) ((A[]*U + A[4]*V + A[7]) / (A[2]*U + A[5]*V + A[8])) ((-(A[5]*V+A[8])*X+ A[3]*V + A[6]) / (A[2]*X - A[])) /* ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ * perspectie: * * Apply perspectie imge trnsformtion on inpt I. * The 3x3 perspectie mtrix is gien in Imtrix. * The otpt is stored in I2. */ oid perspectie(imgp I, imgep Imtrix, imgep I2) { int i, w, h, ww, hh; chr *p, *p2; flot,, x, y, xmin, xmx, ymin, ymx, *, *F; imgep II; w = I->width; h = I->height; = (flot *) Imtrix->bf[]; 25

26 perspectie.c (2) xmin = xmx = X(,, ); x = X(, w, ); xmin = MIN(xmin, x); x = X(, w, h); xmin = MIN(xmin, x); x = X(,, h); xmin = MIN(xmin, x); ymin = ymx = Y(,, ); y = Y(, w, ); ymin = MIN(ymin, y); y = Y(, w, h); ymin = MIN(ymin, y); y = Y(,, h); ymin = MIN(ymin, y); xmx = MAX(xmx, x); xmx = MAX(xmx, x); xmx = MAX(xmx, x); ymx = MAX(ymx, y); ymx = MAX(ymx, y); ymx = MAX(ymx, y); ww = CEILING(xmx) - FLOOR(xmin); hh = CEILING(ymx) - FLOOR(ymin); /* llocte mpping fct bffer */ x = MAX(MAX(w, h), MAX(ww, hh)); F = (flot *) mlloc(x * sizeof(flot)); if(f == NULL) IP_bilot("perspectie: No memory"); /* llocte intermedite imge */ II = IP_llocImge(ww, h, BW_TYPE); IP_clerImge(II); p = (chr *) I->bf[]; p2 = (chr *) II->bf[]; 26

27 perspectie.c (3) /* first pss: resmple rows */ for(=; <h; ++) { /* init forwrd mpping fnction F; mp xmin to */ for(=; < w; ++) F[(int) ] = X(,, ) - xmin; } resmple(p, w,, F, p2); p += w; p2 += ww; /* disply intermedite imge */ IP_copyImge(II, NextImgeP); IP_displyImge(); /* init finl imge */ IP_copyImgeHeder(I, I2); I2->width = ww; I2->height = hh; IP_initChnnels(I2, BW_TYPE); IP_clerImge(I2); 27

28 perspectie.c (4) /* second pss: resmple colmns */ for(x=; x<ww; x++) { p = (chr *) II->bf[] + (int) x; p2 = (chr *) I2->bf[] + (int) x; /* skip pst pdding */ for(=; <h; ++,p+=ww) { if(*p) brek; /* check for nonzero pixel */ = H(, (x+xmin), ); /* else, if pixel is blck */ if(>= && <w) brek; /* then check for lid */ } /* init forwrd mpping fnction F; mp ymin to */ for(i=; <h; ++) { = H(, (x+xmin), ); = CLIP(,, w-); F[i++] = Y(,, ) - ymin; } resmple(p, i, ww, F, p2); } IP_freeImge(II); } 28

29 resmple.c () /* ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ * Resmple the len elements of src (with stride offst) into dst ccording * to the monotonic sptil mpping gien in F (len entries). * The scnline is ssmed to he been clered erlier. */ oid resmple(chr *src, int len, int offst, flot *F, chr *dst) { int,, x, xx, ix, ix, I, I, pos; doble x, x, di; if(f[] < F[len-]) pos = ; /* positie otpt stride */ else pos = ; /* negtie otpt stride */ for(=; <len-; ++) { /* index into src */ = * offst; /* otpt interl (rel nd int) for inpt pixel */ if(pos) { /* positie stride */ ix = x = F[]; ix = x = F[+]; I = src[]; I = src[+offst]; Wolberg: } Imge Processing Corse Notes 29

30 resmple.c (2) else { /* flip interl to enforce positie stride */ ix = x = F[+]; ix = x = F[]; I = src[+offst]; I = src[]; } /* index into dst */ xx = ix * offst; /* check if interl is embedded in one otpt pixel */ if(ix == ix) { dst[xx] += I * (x-x); contine; } /* else, inpt strddles more thn one otpt pixel */ /* left strddle */ dst[xx] += I * (ix+-x); 3

31 resmple.c (3) /* centrl interl */ xx += offst; di = (I-I) / (x-x); for(x=ix+; x<ix; x++,xx+=offst) dst[xx] = I + di*(x-x); } } /* right strddle */ if(x!= ix) dst[xx] += (I + di*(ix-x)) * (x-ix); 3