Incremental Image Synthesis

Transcription

1 Incremental Image Synthesis Balázs Csébfalvi Department of Control Engineering and Information Technology Web: Incremental image synthesis Shading and visibility calculation are difficult to do for general objects Coherence: Calculate for bigger components Avoid unnecessary computation: clipping Transformations: use the appropriate coordinate system for the given task An arbitrary object cannot be efficiently clipped and transformed: tesselation 2 /36 3D incremental image synthesis Lokális k.r. Világ k.r. Képernyo k.r. 3 /36

2 Tesselation Selection of surface points: r n,m = r(u n,v m ) Connection of the surface points: triangular mesh, wire frame 4 /36 Modeling transformation z y x. scaling: sx, sy, sz 2. orientation: α, β, γ 3. position: px, py, pz T M = sx sy sz R px, py, pz, 5 /36 Orientation z (R 2,R 22,R 23 ) (R 3,R 32,R 33 ) y (R,R 2,R 3 ) x Orientation: orthogonal matrix 3 degrees of freedom 3 rotation angles T M = R = cosα sinα -sinα cosα cosβ -sinβ cosγ sinγ sinβ cosβ -sinγ cosγ Roll Pitch Yaw 6 /36 2

3 Quaternion Generalization of complex numbers q = [s,x,y,z] = [s,w] = s+xi+yj+zk q +q 2 = [s +s 2, x +x 2, y +y 2, z +z 2 ] aq = [as,ax,ay,az] q = s 2 +x 2 +y 2 +z 2 Multiplication: i 2 = j 2 = k 2 = ijk = -, ij=k, ji=-k, jk=i, kj=-i, ki=j, ik=-j multiplication is associative but not commutative Unit: [,0,0,0] Inverse: q - = [s,-w]/ q 2, q - q = q q - =[,0,0,0] (q q 2 ) - =q - 2 q - 7 /36 Quaternion: rotation around an axis w q = [s,x,y,z] = [s,w] q [0,u] q - = [0,v] v is the rotation of u around w by angle α cos α = 2(x 2 +y 2 +z 2 )/(s 2 +x 2 +y 2 +z 2 ) the length of q does not matter Let the rotation be a unit-length quaternion [cosα/2, w sinα/2], w = Sequence of rotations q (q 2 [0,u] q 2 - ) q - = (q q 2 ) [0,u] (q q 2 ) - Unit-length quaternion = orientation 8 /36 Rotation around an axis w by angle α w is a vector of unit length, q = [cosα/2, w sinα/2] (0, R,R 2,R 3 ) = q [0,,0,0] q - (0, R 2,R 22,R 23 ) = q [0,0,,0] q - (0, R 3,R 32,R 33 ) = q [0,0,0,] q - Rodriquez formula glrotate(alpha, wx, wy, wz); 9 /36 3

4 Transformations Modeling transformation: [r local,] T M = [r world,] View transformation: [r world,] T v = [r screen h, h] Composite transformation: [r local,] T M T v = [r local,] T C = [r screen h, h] 0 /36 View transformation: Camera model fp vup eye z fov aspect vrp (lookat) bp y x /36 Steps of the view transformation eye y left side z z 2. Camera 4. Normalized screen y viewing angle: 90 o. World x 3. Normalized camera 5. Screen 2 /36 4

5 Camera transformation: glulookat(eye,lookat,vup) z w eye u w v vup x u v y lookat [x,y,z,] = [x,y,z,] w = (eye-lookat)/ eye-lookat u = (vup x w)/ w x vup v = w x u eye x -eye y -eye z - u x u y u z 0 v x v y v z 0 w x w y w z u x v x w x 0 u y v y w y 0 u z v z w z /36 Normalization bp*tg(fov/2) fp viewing angle: 90 o bp T norm /(tg(fov/2) aspect) /tg(fov/2) /36 Perspective transformation: gluperspective(fov, aspect, fp, bp) [-mx*z, -my*z, z] [mx, my, -] [mx, my, ] -fp -bp [mx*fp, my*fp, -fp] [mx, my, - ] [mx*bp, my*bp, -bp] [mx, my, ] [mx*fp, my*fp, -fp, ] [mx, my, -, ] a [mx*bp, my*bp, -bp, ] [mx, my,, ] b [mx*fp, my*fp, -fp, ] [mx*fp, my*fp, -fp, fp] // a = fp // b = bp 5 /36 5

6 Perspective transformation bp viewing angle: 90 o fp T persp -fp*t33+t43 = -fp -fp*t34+t44 = fp t t2 t3 t4 t2 t22 t23 t24 t3 t32 t33 t34 t4 t42 t43 t44 fp] bp] [mx*fp,my*fp,-fp,] [mx*fp, my*fp, -fp, [mx*bp,my*bp,-bp,] [mx*bp, my*bp, bp,, -bp*t33+t43= bp -bp*t34+t44 = bp (fp+bp)/(bp-fp) fp*bp/(bp-fp) 0 6 /36 y Perspective transformation: gluperspective(fov,asp,fp,bp) z /(tg(fov/2) asp) /tg(fov/2) (fp+bp)/(bp-fp) fp*bp/(bp-fp) 0 [X h,y h,z h,h] = [xc,yc,zc,] T persp [X,Y,Z,] = [X h /h, Y h /h, Z h /h,] 7 /36 Viewing pipeline Model: x,y,z T M T VIEW T PERSP Clipping frame buffer: X, Y Projection Occlusion viewport trans. Homogeneous division Skipped in case of wire frame rendering 8 /36 6

7 Clipping before the homogeneous division [,,2] [,,-2] fp= bp=3 -(fp+bp)/(bp-fp) -2fp*bp/(bp-fp) Homogeneous division: [X,Y,Z,] = [X h /h, Y h /h, Z h /h,] [,,-2, ] * [ 2] [/2 /2 /2 ] [,, 2, ] * [ -7-2] [-/2 -/2 7/2 ] 9 /36 Clipping in homogeneous coocrdinates Goal: - < X = X h /h < - < Y = Y h /h < - < Z = Z h /h < Additionally: h > 0 (h = -z in the camera coordinate system) h = X h -h < X h < h -h < Y h < h -h < Z h < h External [3, 0, 0, 2] h = 2 < X h = 3 Internal [2, 0, 0, 3] h = 3 > X h = 2 20 /36 Line/polygon clipping -h < X h < h -h < Y h < h -h < Z h < h h = X h [X h,y h,z h,h ] h = h (-t)+h 2 t = = X h = X h (-t) + X h2 t [X h2,y h2,z h2,h 2 ] X h = X h (-t) + X h2 t Y h = Y h (-t) + Y h2 t Z h = Z h (-t) + Z h2 t h = h (-t) + h 2 t t = 2 /36 7

8 Wire frame rendering (X h,y h,z h,h) Model: x,y,z T M T V clipping frame buffer X, Y color T C 2D line drawing (X,Y,Z) (X,Y) homogeneous division viewport trans 22 /36 Image synthesis with solids In the screen coordinate system the viewing direction is the z-axis Object-space algorithms: visibility does not depend on the resolution Image-space algorithms: what object is visible in the given pixel 23 /36 r 3 Back-face culling r 2 n = (r 3 - r ) (r 2 - r ) r n z < 0 n z > 0 Classification of the faces: From an external point: front face From an internal point: back face Assumption: From an external point the ordering of the vertices is clockwise 24 /36 8

9 Z-buffer algorithm /36 Z-coordinate: linear interpolation (X 2,Y 2,Z 2 ) Z Z(X,Y) = ax + by+ c (X 3,Y 3,Z 3 ) Y (X,Y,Z ) Z(X,Y) X Z(X+,Y) = Z(X,Y) + a 26 /36 Hardware for Z-interpolation X Z(X,Y) X counter Z register CLK Σ a 27 /36 9

10 Shading Coherence: shading not pixel-by-pixel Triangle by triangle: per object: own color per triangle: constant color per vertex: in between linear interpolation: Gouraud shading per pixel: the normals are interpolated: Phong shading 28 /36 Shading with own color 29 /36 Constant shading 30 /36 0

11 Gouraud shading R,G,B (X 2,Y 2,R 2 ) R(X,Y) = ax + by+ c (X 3,Y 3,R 3 ) Y (X,Y,R ) R(X,Y) X R(X+,Y) = R(X,Y) + a 3 /36 Gouraud shading specular diffuse ambient 32 /36 Phong shading L N V Y N(X,Y) X N(X,Y) = A X + BY + C L(X,Y) =... V(X,Y) = /36

12 Illumination in world coordinates View transformation Pixel by pixel: linear interpolation of the normals normalization evaluation of the shading model 34 /36 Phong shading 35 /36 Gouraud versus Phong Gouraud Gouraud Phong Phong 36 /36 2