Animating Transformations

Transcription

1 Animating Transformations Connelly Barnes CS445: Graphics Acknowledgment: slides by Jason Lawrence, Misha Kazhdan, Allison Klein, Tom Funkhouser, Adam Finkelstein and David Dobkin

2 Overview Rotations and SVD Interpolating/Approximating Points o Vectors o Unit-Vectors Interpolating/Approximating Transformations o Matrices o Rotations» SVD Factorization» Euler Angles

3 Rotations What are rotations?

4 Rotations What are rotations? A rotation R is a linear transformation that has determinant equal to one and preserves the angle between any two vectors:

5 Rotations What are rotations? A rotation R is a linear transformation that has determinant equal to one and preserves the angle between any two vectors: Recall that the dot-product between two vectors can be expressed as a matrix multiplication:

6 Rotations What are rotations? A rotation R is a linear transformation that has determinant equal to one and preserves the angle between any two vectors: This implies that:

7 Rotations What are rotations? A rotation R is a linear transformation that has determinant equal to one and preserves the angle between any two vectors: This implies that:

8 Rotations What are rotations? A rotation R is a linear transformation that has determinant equal to one and preserves the angle between any two vectors: This implies that: Since this is true for all v and w, this means that:

9 Rotations What are rotations? A rotation R is a linear transformation that has determinant equal to one and preserves the angle between any two vectors. A rotation R is a linear transformation that has determinant equal to one and whose transpose is equal to its inverse.

10 Rotations What are rotations? A rotation R is a linear transformation that has determinant equal to one and preserves the angle between any two vectors. A rotation R is a linear transformation that has determinant equal to one and whose transpose is equal to its inverse. A rotation in 3D can be specified by a 3x3 matrix.

11 Rotations What are rotations? A rotation in 3D can also be specified by: o its axis of rotation w ( w =1) and o its angle of rotation θ θ w

12 Rotations What are rotations? A rotation in 3D can also be specified by: o its axis of rotation w ( w =1) and o its angle of rotation θ Properties: o The rotation corresponding to (θ,w) is the same as the rotation corresponding to (-θ,-w).

13 Rotations What are rotations? A rotation in 3D can also be specified by: o its axis of rotation w ( w =1) and o its angle of rotation θ Properties: o The rotation corresponding to (θ,w) is the same as the rotation corresponding to (-θ,-w). o Given two rotations corresponding to (θ 1,w) and (θ 2,w), the product of the rotations corresponds to (θ 1 +θ 2,w).

14 Rotations What are rotations? A rotation in 3D can also be specified by: o its axis of rotation w ( w =1) and o its angle of rotation θ Properties: o The rotation corresponding to (θ,w) is the same as the rotation corresponding to (-θ,-w). o Given two rotations corresponding to (θ 1,w) and (θ 2,w), the product of the rotations corresponds to (θ 1 +θ 2,w). o Given a rotation corresponding (θ,w), the rotation raised to the power α corresponds to (αθ,w).

15 Rotations What are rotations? A rotation in 3D can also be specified by: o its axis of rotation w ( w =1) and o its angle of rotation θ Properties: o The rotation corresponding to (θ,w) is the same as the rotation corresponding to (-θ,-w). o Given two rotations corresponding to (θ 1,w) and (θ 2,w), the product of the rotations corresponds to (θ 1 +θ 2,w). o Given a rotation corresponding (θ,w), the rotation raised to the How power do α we corresponds define the to product (αθ,w). of rotations corresponding to (θ 1,w 1 ) and (θ 2,w 2 )?

16 SVD Any mxn matrix M can be expressed in terms of its Singular Value Decomposition as: where: o U is an nxn rotation matrix, o V is an mxm rotation matrix, and o D is an mxn diagonal matrix (i.e off-diagonals are all 0).

17 SVD Applications: Compression Model Alignment Matrix Inversion Solving Over-Constrained Linear Equations

18 SVD Matrix Inversion: If we have an nxn invertible matrix M, we can use SVD to compute the inverse of M.

19 SVD Matrix Inversion: If we have an nxn invertible matrix M, we can use SVD to compute the inverse of M. Expressing M in terms of its SVD gives: where: o U is an nxn rotation matrix, o V is an nxn rotation matrix, and o D is an nxn diagonal matrix (i.e off-diagonals are all 0).

20 SVD Matrix Inversion: If we have an nxn invertible matrix M, we can use SVD to compute the inverse of M. We can express M -1 as:

21 SVD Matrix Inversion: If we have an nxn invertible matrix M, we can use SVD to compute the inverse of M. We can express M -1 as:

22 SVD Matrix Inversion: If we have an nxn invertible matrix M, we can use SVD to compute the inverse of M. We can express M -1 as: Since: o U is a rotation, U -1 =U t. o V is a rotation, V -1 =V t.

23 SVD Matrix Inversion: If we have an nxn invertible matrix M, we can use SVD to compute the inverse of M. We can express M -1 as: Since: o D is a diagonal matrix:

24 SVD Solving Over-Constrained Linear Equations: If we have m equations in n unknowns, with m>n, the problem is over-constrained and there is no general solution.

25 However, using SVD, we can find the values of {x 1,,x n } that get us as close to {y 1,,y m } as possible. SVD Solving Over-Constrained Linear Equations: If we have m equations in n unknowns, with m n, the problem is over-constrained and there is no general solution.

26 SVD Solving Over-Constrained Linear Equations: If we express the matrix A in terms of its SVD: then we can set the matrix A * to be: where D * is the diagonal matrix with: This is called the pseudo-inverse of A. That is, we invert A as much as possible.

27 SVD Solving Over-Constrained Linear Equations: If we set: this gives us the values of {x 1,,x n } that most nearly solve the initial equation:

28 Overview Rotations and SVD Interpolating/Approximating Points o Vectors o Unit-Vectors Interpolating/Approximating Transformations o Matrices o Rotations» SVD Factorization» Euler Angles

29 Vectors Given a collection of n control points {p 0,,p n-1 }, define a curve Φ(t) that approximates/interpolates the points.

30 Vectors Given a collection of n control points {p 0,,p n-1 }, define a curve Φ(t) that approximates/interpolates the points. Linear Interpolation: o Interpolating o C 0 continuous p 1 p 2 p 6 Φ 6 (t) p 7 Φ 0 (t) Φ 1 (t) Φ 2 (t) Φ 5 (t) p 5 p 3 Φ 4 (t) p 0 Φ 3 (t) p 4

31 Vectors Given a collection of n control points {p 0,,p n-1 }, define a curve Φ(t) that approximates/interpolates the points. Catmull-Rom Splines (Cardinal Splines with t=0): o Interpolating o C 1 continuous p 1 p 2 p 6 p 7 Φ 1 (t) Φ 2 (t) Φ 5 (t) p 5 p 3 Φ 4 (t) p 0 Φ 3 (t) p 4

32 Vectors Given a collection of n control points {p 0,,p n-1 }, define a curve Φ(t) that approximates/interpolates the points. Uniform Cubic B-Splines: o Approximating o C 2 continuous p 1 p 2 p 6 p 7 Φ 1 (t) Φ 2 (t) Φ 5 (t) p 5 p 0 p 3 Φ 3 (t) p 4 Φ 4 (t)

33 Unit-Vectors What if we add the additional constraint that the points {p 0,,p n-1 } and the curve Φ(t) have to lie on the unit circle/sphere ( p i =1, Φ(t) =1)?

34 Unit-Vectors What if we add the additional constraint that the points {p 0,,p n-1 } and the curve Φ(t) have to lie on the unit circle/sphere ( p i =1, Φ(t) =1)? We can t interpolate/approximate the points as before, because the in-between points don t have to lie on the unit circle/sphere! Φ(t)=(1-t)p 0 +tp 1 p 0 p 1 p 0 p 1

35 Unit-Vectors What if we add the additional constraint that the points {p 0,,p n-1 } and the curve Φ(t) have to lie on the unit circle/sphere ( p i =1, Φ(t) =1)? We can normalize the in-between points by sending them to the closest circle/sphere point: p 0 p 1 Φ(t)=(1-t)p 0 +tp 1

36 Curve Normalization Limitations: The normalized curve is not always well defined. p 0 p 1 Φ(t)=(1-t)p 0 +tp 1

37 Curve Normalization Limitations: The normalized curve is not always well defined. Just because points are uniformly distributed on the original curve, does not mean that they will be uniformly distributed on the normalized one. p 0 p 1 Φ(t)=(1-t)p 0 +tp 1

38 Curve Normalization Limitations: The normalized curve is not always well defined. Just because points are uniformly distributed on the original curve, does not mean that they will be uniformly distributed on the normalized one. SLERP: If we set: o p 0 =(cosθ 0,sinθ 0 ) o p 1 =(cosθ 1,sinθ 1 ) p 0 p 1 We can set:

39 Overview Interpolating/Approximating Rotations and SVD Interpolating/Approximating Points o Vectors o Unit-Vectors Interpolating/Approximating Transformations o Matrices o Rotations» SVD Factorization» Euler Angles

40 Matrices Given a collection of n matrices {M 0,,M n-1 }, define a curve Φ(t) that approximates/interpolates the matrices.

41 Matrices Given a collection of n matrices {M 0,,M n-1 }, define a curve Φ(t) that approximates/interpolates the matrices. As with vectors: Linear Interpolation: Catmull-Rom Interpolation: Uniform Cubic B-Spline Approximation:

42 Rotations What if we add the additional constraint that the matrices {M 0,,M n-1 } and the values of the curve Φ(t) have to be rotations?

43 Rotations What if we add the additional constraint that the matrices {M 0,,M n-1 } and the values of the curve Φ(t) have to be rotations? We can t interpolate/approximate the matrices as before, because the in-between matrices don t have to be rotations!

44 Rotations What if we add the additional constraint that the matrices {M 0,,M n-1 } and the values of the curve Φ(t) have to be rotations? We can t interpolate/approximate the matrices as before, because the in-between matrices don t have to be rotations! We could try to normalize, by mapping every matrix Φ(t) to the nearest rotation.

45 Challenge Given a matrix M, how do you find the rotation matrix R that is closest to M?

46 SVD Factorization Given a matrix M, how do you find the rotation matrix R that is closest to M? Singular Value Decomposition (SVD) allows us to express any M as a diagonal matrix, multiplied on the left and right by the rotations R 1 and R 2 :

47 SVD Factorization Given a matrix M, how do you find the rotation matrix R that is closest to M? Singular Value Decomposition (SVD) allows us to express any M as a diagonal matrix, multiplied on the left and right by the rotations R 1 and R 2 : To be fully correct, you need to ensure that the product of sgn(λ i ) is 1. If not, you need to flip the sign of the sgn(λ i ) where λ i is smallest. The closest rotation R to M is then just the rotation:

48 Euler Angles Every rotation matrix R can be expressed as: o some rotation about the x-axis, multiplied by o some rotation about the y-axis, multiplied by o some rotation about the z-axis: The angles (θ,φ,ψ) are called the Euler angles.

49 Euler Angles Instead of blending matrices and then normalizing using SVD, we can blend the Euler angles: o For each M k, compute the Euler angles (θ k,φ k,ψ k )

50 Euler Angles Instead of blending matrices and then normalizing using SVD, we can blend the Euler angles: o For each M k, compute the Euler angles (θ k,φ k,ψ k ) o Interpolate/Approximate the Euler angles:

51 Euler Angles Instead of blending matrices and then normalizing using SVD, we can blend the Euler angles: o For each M k, compute the Euler angles (θ k,φ k,ψ k ) o Interpolate/Approximate the Euler angles:» Linear Interpolation:

52 Euler Angles Instead of blending matrices and then normalizing using SVD, we can blend the Euler angles: o For each M k, compute the Euler angles (θ k,φ k,ψ k ) o Interpolate/Approximate the Euler angles:» Linear Interpolation» Catmull-Rom Interpolation:

53 Euler Angles Instead of blending matrices and then normalizing using SVD, we can blend the Euler angles: o For each M k, compute the Euler angles (θ k,φ k,ψ k ) o Interpolate/Approximate the Euler angles:» Linear Interpolation» Catmull-Rom Interpolation» Uniform Cubic B-Spline Approximation:

54 Euler Angles Instead of blending matrices and then normalizing using SVD, we can blend the Euler angles: o For each M k, compute the Euler angles (θ k,φ k,ψ k ) o Interpolate/Approximate the Euler angles:» Linear Interpolation» Catmull-Rom Interpolation» Uniform Cubic B-Spline Approximation o Set the value of the in-between matrix to: