CS559: Computer Graphics

Transcription

1 CS559: Computer Graphics Lecture 8: 3D Transforms Li Zhang Spring 28 Most Slides from Stephen Chenne

2 Finish Color space Toda 3D Transforms and Coordinate sstem Reading: Shirle ch 6

3 RGB and HSV Green(,,) Can (,,) Yellow (,,) White(,,) Blue (,,) Magenta (,,) Black (,,) Red (,,) Different was to represent/parameterize color

4 Photoshop Color Picker

5 L-A-B L-A-B Color Space L: luminance/brightness A: position between magenta and green (negative values indicate green while positive values indicate magenta) B: position between ellow and blue (negative values indicate blue and positive values indicate ellow)

6 Spatial resolution and color R G B original

7 Blurring the G component R G B original processed

8 Blurring the R component R G original processed B

9 Blurring the B component R G original processed B

10 Lab Color Component L a A rotation of the color coordinates into directions that are more perceptuall meaningful: L: luminance, a: magenta-green, b: blue-ellow b

11 Bluring L L a original processed b

12 Bluring a L a original processed b

13 Bluring b L a original processed b

14 Application to image compression (compression is about hiding differences from the true image where ou can t see them).

15 Where to now We are now done with images We will spend several weeks on the mechanics of 3D graphics 3D Transform Coordinate sstems and Viewing Drawing lines and polgons Lighting and shading We will finish the semester with modeling and some additional topics

16 3D Graphics Pipeline Modeling (Creating 3D Geometr) Rendering (Creating, shading images from geometr, lighting, materials)

17 3D Graphics Pipeline Modeling (Creating 3D Geometr) Rendering (Creating, shading images from geometr, lighting, materials) Want to place it at correct location in the world Want to view it from different angles Want to scale it to make it bigger or smaller Need transformation between coordinate sstems -- Represent transformations using matrices and matri-vector multiplications.

18 Recall: All 2D Linear Transformations Linear transformations are combinations of Scale, Rotation, Shear, and Mirror d c b a ' '

19 2D Rotation Rotate counter-clockwise about the origin b an angle cos sin sin cos

20 Rotating About An Arbitrar Point What happens when ou appl a rotation transformation to an object that is not at the origin??

21 Rotating About An Arbitrar Point What happens when ou appl a rotation transformation to an object that is not at the origin? It translates as well

22 How Do We Fi it? How do we rotate an about an arbitrar point? Hint: we know how to rotate about the origin of a coordinate sstem

23 Rotating About An Arbitrar Point

24 Scaling an Object not at the Origin What happens if ou appl the scaling transformation to an object not at the origin? Based on the rotating about a point composition, what should ou do to resize an object about its own center?

25 Back to Rotation About a Pt Sa R is the rotation matri to appl, and p is the point about which to rotate Translation to Origin: p Rotation: R R( p) R Rp Translate back: p R Rp p How to epress all the transformation using matri multiplication?

26 Homogeneous Coordinates Use three numbers to represent a point Translation can now be done with matri multiplication! b a a b a a usuall w, an w for w w w w w w / /

27 Homogeneous Coordinates Use three numbers to represent a point Translation can now be done with matri multiplication! M M M M M usuall w, an w for w w w w w w / /

28 Basic Transformations Translation: Rotation: Scaling: b b s s cos sin sin cos

29 Composing rotations, scales R S R( S ) ( RS) 3 SR 3 Rotation and scaling are not commutative.

30 Inverting Composite Transforms Sa I want to invert a combination of 3 transforms Option : Find composite matri, invert Option 2: Invert each transform and swap order M M M M M M M M ( ( ) ) M M M M M M M M

31 Inverting Composite Transforms Sa I want to invert a combination of 3 transforms Option : Find composite matri, invert Option 2: Invert each transform and swap order Obvious from properties of matrices M M M M M M M M ( ( ) ) M M M M M M M M

32 Homogeneous Transform Advantages Unified view of transformation as matri multiplication Easier in hardware and software To compose transformations, simpl multipl matrices Order matters: BA vs AB Allows for transforming directional vectors Allows for non-affine transformations: Perspective projections!

33 Directions vs. Points We have been talking about transforming points Directions are also important in graphics Viewing directions Normal vectors Ra directions (,) (-2,-) Directions are represented b vectors, like points, and can be transformed, but not like points

34 Transforming Directions Sa I define a direction as the difference of two points: d=a b This represents the direction of the line between two points Now I translate the points b the same amount: a =a+t, b =b+t d =a b =d How should I transform d?

35 Homogeneous Directions Translation does not affect directions! Homogeneous coordinates give us a ver clean wa of handling this The direction (,) becomes the homogeneous direction (,,) The correct thing happens for rotation and scaling also Scaling changes the length of the vector, but not the direction Normal vectors are slightl different we ll see more later b b

36 Transforming normal vectors normal tangent normal tangent M n T t t' Mt n ' T t' If M is a rotation, T ) ( M M n' ( n ( n T T M M )( Mt) ) T ( M ) T n

37 3D Transformations Homogeneous coordinates: (,,z)=(w,w,wz,w) Transformations are now represented as 44 matrices usuall w, an w for w wz w w z w z w z w w / / / z t t t z z

38 3D Affine Transform z t i h g t f e d t c b a z z

39 3D Rotation Rotation in 3D is about an ais in 3D space passing through the origin Using a matri representation, an matri with an orthonormal top-left 33 sub-matri is a rotation Rows/columns are mutuall orthogonal ( dot product) Determinant is Implies columns are also orthogonal, and that the transpose is equal to the inverse.,,,,, then r r r r r r r r r r r r r r r R

40 Specifing a rotation matri

41 Specifing a rotation matri Euler angles: Specif how much to rotate about X, then how much about Y, then how much about Z Hard to think about, and hard to compose

42 Alternative Representations Specif the ais and the angle (OpenGL method) Hard to compose multiple rotations A rotation b an angle is given b around ais specified b the unit vector

43 Non-Commutativit Not Commutative (unlike in 2D)!! Rotate b, then is not same as then Order of appling rotations does matter Follows from matri multiplication not commutative R * R2 is not the same as R2 * R

44 Other Rotation Issues Rotation is about an ais at the origin For rotation about an arbitrar ais, use the same trick as in 2D: Translate the ais to the origin, rotate, and translate back again

45 Transformation Leftovers Scale, shear etc etend naturall from 2D to 3D Rotation and Translation are the rigid-bod transformations: Do not change lengths or angles, so a bod does not deform when transformed

46 Coordinate Frames All of discussion in terms of operating on points But can also change coordinate sstem Eample, motion means either point moves backward, or coordinate sstem moves forward P (2,) ' P (,) P (,)

47 Coordinate Frames: Rotations P ' P cos sin sin cos R v u cos sin sin cos u v P P P'

48 Geometric Interpretation 3D Rotations Rows of matri are 3 unit vectors of new coord frame Can construct rotation matri from 3 orthonormal vectors Effectivel, projections of point into new coord frame u u zu Ruvw v v zv u u X uy zuz w w z w z u u u p v v v p w w z w z p Rp z? u p v p w p