Lecture 5: Transforms II. Computer Graphics and Imaging UC Berkeley CS184/284A

Transcription

1 Lecture 5: Transforms II Computer Graphics and Imaging UC Berkeley

2 3D Transforms

3 3D Transformations Use homogeneous coordinates again: 3D point = (x, y, z, 1) T 3D vector = (x, y, z, 0) T Use 4 4 matrices for affine transformations x y z 1 = a b c t x d e f t y g h i t z x y z 1

4 3D Transformations Scale s x S(s x,s y,s z ) = 0 s y s z Translation t x T(t x,t y,t z ) = t y t z Coordinate Change (Frame-to-world) F(u, v, w, o) = apple u v w o

5 3D Transformations Rotation around x-, y-, or z-axis y R x ( )= 0 cos sin 0 0 sin cos cos 0 sin 0 R y ( )= sin 0 cos cos sin 0 0 R z ( )= sin cos z Rotation around x-axis x

6 3D Rotations Compose any 3D rotation from Rx, Ry, Rz? R xyz (,, ) = R x ( ) R y ( ) R z ( ) So-called Euler angles Often used in flight simulators: roll, pitch, yaw

7 3D Rotations Compose any 3D rotation from Rx, Ry, Rz? R xyz (,, ) = R x ( ) R y ( ) R z ( ) So-called Euler angles Often used in flight simulators: roll, pitch, yaw

8 3D Rotation Around Arbitrary Axis Construct orthonormal frame transformation F with p, u, v, w, y v where p and w match the rotation axis w p x Apply the transform (F R z (θ) F -1 ) Interpretations (both valid): z u 2 Move to Z axis, rotate, then move back Cast w-axis rotation in new coordinate frame

9 Rodrigues Rotation Formula Rotation by angle α around axis n R(n, ) = cos( ) I + (1 cos( )) nn T + sin( ) n z n n z 0 n A x n y n x {z 0 } N How to prove this magic formula? Matrix N computes a cross-product: N x = n x Assume orthonormal system e1, e2, n Rn = n Re 1 = cos e 1 + sin e 2 Re 2 = sin e 1 + cos e 2

10 Many Other Representations of Rotations Quaternions Exponential map

11 Hierarchical Transforms

12 Skeleton - Linear Representation head torso right upper arm right lower arm right hand left upper arm left lower arm left hand right upper leg right lower leg right foot left upper leg left lower leg left foot

13 Linear Representation Each shape associated with its own transform A single edit can require updating many transforms E.g. raising arm requires updating transforms for all arm parts

14 Skeleton - Hierarchical Representation torso head right arm upper arm lower arm hand left arm upper arm lower arm hand right leg upper leg lower leg foot left leg upper leg lower leg foot

15 Hierarchical Representation Grouped representation (tree) Each group contains subgroups and/or shapes Each group is associated with a transform relative to parent group Transform on leaf-node shape is concatenation of all transforms on path from root node to leaf Changing a group s transform affects all parts Allows high level editing by changing only one node E.g. raising left arm requires changing only one transform for that group

16 Skeleton - Hierarchical Representation translate(0, 10); drawtorso(); pushmatrix(); // push a copy of transform onto stack translate(0, 5); // right-multiply onto current transform rotate(headrotation); // right-multiply onto current transform drawhead(); popmatrix(); // pop current transform off stack pushmatrix(); translate(-2, 3); rotate(rightshoulderrotation); drawupperarm(); pushmatrix(); translate(0, -3); rotate(elbowrotation); drawlowerarm(); pushmatrix(); right translate(0, -3); lower right rotate(wristrotation); arm drawhand(); hand group popmatrix(); popmatrix(); popmatrix();. right arm group

17 Skeleton - Hierarchical Representation translate(0, 10); drawtorso(); pushmatrix(); // push a copy of transform onto stack translate(0, 5); // right-multiply onto current transform rotate(headrotation); // right-multiply onto current transform drawhead(); popmatrix(); // pop current transform off stack pushmatrix(); translate(-2, 3); rotate(rightshoulderrotation); drawupperarm(); pushmatrix(); translate(0, -3); rotate(elbowrotation); drawlowerarm(); pushmatrix(); right translate(0, -3); lower right rotate(wristrotation); arm drawhand(); hand group popmatrix(); popmatrix(); popmatrix();. right arm group

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19 Viewing and Perspective

20 Viewing and Perspective Transforms? Scene modeling 3D world coordinates Rasterization 2D screen coordinates

21 Camera Space

22 Standard Camera Space y z We will use this convention for standard camera coordinates: camera located at the origin x looking down negative z-axis vertical vector is y-axis (x-axis) orthogonal to y & z

23 Standard Camera Coordinates y z y x x Resulting image (z-axis pointing away from scene)

24 Consider A Camera Pointing in The World

25 Consider A Camera Pointing in The World u = up vector v = view direction e = eye point (position of camera)

26 Camera Look-At Transformation u = up vector v = view direction e = eye point (position of camera) Input: e, u & v given in world space coordinates Output: transform matrix from world space to standard camera space

27 Camera Look-At Transformation Inverse: Matrix from standard camera to world space (Why? This is a coordinate frame transform to (e,r,u,-v)) r x u x v x e x r y u y v y e y r z u z v z e z Look-at transform is the inverse of above matrix: r x u x v x e x r y u y v y e y r z u z v z e z r x r y r z e x = u x u y u z 0 v x v y v z e y e z

28 Transform Camera Space to Image Plane? z How to transform from 3D camera space to 2D image plane? One option: orthographic projection (just delete z) Useful, e.g. for engineering drawings But is this the whole story?

29 Perspective Credit: jefflynchphoto.com

30 Perspective in Art Berlinghieri 1235

31 Perspective in Art Giotto 1290

32 Perspective in Art Giotto 1290

33 Perspective in Art Brunelleschi experiment c. 1413

34 Perspective in Art Delivery of the Keys (Sistine Chapel), Perugino, 1482

35 Perspective in Art The Last Supper, Leonardo da Vinci, 1499

36 Could An Orthographic Projection Model Leonardo s Last Supper? z

37 Pinhole Camera Model

38 Pinhole Camera

39 Projective Transform x, y (x, y, z) T Scene point Image point C -z d Inverted image (as in real pinhole camera)

40 Pinhole Camera Projective Transform x, y (x, y, z) T Image point Scene point C -z d Upright image

41 Projective Transforms Standard perspective projection Center of projection: (0, 0, 0) T Image plane at z = d x, y (x,y,z) T 0 1 ya 7! 0 1 x d/z B y d/za C -z z d z = d

42 Projective Transforms Standard perspective projection Center of projection: (0, 0, 0) T Image plane at z = d 0 1 ya 7! z 0 1 x d/z B y d/za d Perspective foreshortening Need division by z Matrix representation? Homogeneous coordinates!

43 Homogenous Coordinates (3D) p = 0 1 wx Bwy wza 0 1 x By za M = B A w /d 0 q = M 0 1 x By za = 0 x y z 1 C A 0 1 xd/z Byd/z d A Note non-zero term in final row. First time we have seen this. 1 z/d 1

44 Pinhole Camera Model This mathematical model produces all linear perspective effects! Converging lines + vanishing points Closer objects appear larger in images

45 Specifying Real Camera Perspectives

46 Perspective Composition = Camera Position + Angle of View In this sequence, angle of view decreases as distance from subject increases, to size of human subject in image. Notice the dramatic change in background perspective. From Canon EF Lens Work III

47 Perspective Composition 16 mm (110 ) Up close and zoomed wide with short focal length

48 Perspective Composition 200 mm (12 ) Walk back and zoom in with long focal length

49 Specifying Perspective Projection From Angel and Shreiner, Interactive Computer Graphics

50 Specifying Perspective Viewing Volume From Angel and Shreiner, Interactive Computer Graphics

51 Specifying Perspective Viewing Volume Parameterized by fovy aspect ratio near far : vertical angular field of view : width / height of field of view : depth of near clipping plane : depth of far clipping plane Derived quantities top bottom right left = near * tan (fovy) = top = top * aspect = right

52 Perspective Projection Implementation

53 Perspective Projection Transform (1, 1, 1) (right, top, near) (left, bottom, near) z = far z = near ( 1, 1, 1) Camera Coordinates Normalized Device Coords NDC Later we will flatten NDC to get screen coordinates

54 Perspective Projection Transform Notes: Need not be symmetric about z- axis, but for simplicity here we assume so This transform will preserve depth information (ordering) in NDC

55 Perspective Transform Matrix P = near right 0 near 0 0 top 0 0 far+near far near 2far near far near

56 Perspective Transform Matrix Example P = 2 6 = = = near near 6 4near far+near far near near near right 0 near 0 0 top 1 1 far+near far near far+near far near 7 2far near far near 2far near far near right top 7 67 near

57 Transforms Recap

58 Transforms Recap Coordinate Systems Object coordinates Apply modeling transforms World (scene) coordinates Apply viewing transform Camera (eye) coordinates Apply perspective transform + homog. divide Normalized device coordinates Apply 2D screen transform Screen coordinates

59 Transforms Recap Modeling transforms Object coords World coords

60 Transforms Recap Viewing transform World coords Camera coords

61 Transforms Recap (1, 1, 1) Perspective projection and ( 1, 1, 1) homogeneous divide Camera coords NDC

62 Transforms Recap (w, h) (1, 1, 1) Screen transform ( 1, 1, 1) NDC (0, 0) Screen coords

63 Transforms Recap (w, h) Rasterization (0, 0) Screen coords

64 Things to Remember Transform uses Basic transforms: rotate, scale, translate, Modeling, viewing, projection, perspective Change in coordinate system Hierarchical scene descriptions by push/pop Implementing transforms Linear transforms = matrices Transform composition = matrix multiplication Homogeneous coordinates for translation, projection

65 Acknowledgments Thanks to Pat Hanrahan, Kayvon Fatahalian, Mark Pauly for presentation resources.