Wavelets For Computer Graphics

Transcription

1 {f g} := f(x) g(x) dx A collection of linearly independent functions Ψ j spanning W j are called wavelets. i J(x) := 6 x +2 x + x + x Ψ j (x) := Ψ j (2 j x i) i =,..., 2 j Res. Avge. Detail Coef 4 [ ] 2 [8 4] [ ] [6] [2] Wavelets For Computer Graphics φ 2 i (x) := φ(4 x i) /2 /2 /2 /2 /4 /2 3/4 φ (4x ) φ (4x ) φ 2 (4x 2) φ 3 (4x 3) CPSC 553/65 P

2 Sources of information: The Book: Wavelets For Computer Graphics by Eric Stollnitz Tony De Rose David Salesin Published by Morgan Kaufmann These notes shamelessly stolen from this excellent book CPSC 553/65 P 2

3 But First:... A Quick Recap on Linear Algebra Vector Spaces V is a collection of things for which addition and scalar multiplication are defined:. u+v = v+u 2. (u+v)+w = u+(v+w) 3. CV such that +u=u 4. For each u C V there is an element u C V such that u+( u) = 5. (a+b)u=au+bu 6. a(u+v)=au+av 7. (ab)u=a(bu) 8..u=u The elements of a vector space V are called vectors. is the zero vector. These vectors may be column matrices like [ ] T or they may be functions. CPSC 553/65 P 3

4 Vector Spaces Basis A collection of vectors u, u 2,... in a vector space V are said to be linearly independent if: c u +c 2 u = if and only if c = c 2 =... = A collection u, u 2,... C V of linearly independent vectors is a basis for V if every v C V can be written as v = Σ c i u i i Linear independence means that the vectors are not redundant and a basis consists of a minimal set of vectors. If a basis for V has a finite number of elements u... u m then V is finite dimensional and its dimension is m. CPSC 553/65 P 4

5 Example of basis for a Finite Dimensional Vector Space This seems to be a bit abstract until you realise all they are talking about is a more formal way of defining familiar spaces such as R 3 where e = e 2 = e 3 = form the basis for it. Example of basis for an infinite dimensional Vector Space The set of all functions continuous on [,] CPSC 553/65 P 5

6 Inner Products and Orthogonality Dot product generalises to inner product for arbitrary spaces. Formal Definition: An inner product { * * } on a vector space V is any map from VxV to R 3 that is:. symmetric {u v} = {v u} 2. bilinear {au +bv w} = a{u w} + b{v w} 3. Positive definite {u u} > for all u!= A vector space together with an inner product is called an inner product space. Examples: Dot Product on R 3 is a.b = a b +a 2 b 2 +a 3 b 3 satisfies the requirements for an inner product. CPSC 553/65 P 6

7 Inner Product on functions standard inner product on functions of the unit interval [,] {f g} := f(x) g(x) dx can be generalised to include a weight function w(x): {f g} := w(x) f(x) g(x) dx Orthogonality Two vectors u,v in an inner product space are said to be orthogonal if {u v } = A collection of mutually orthogonal vectors must be linearly independent. An orthogonal basis is one consisting of mutually orthogonal vectors. CPSC 553/65 P 7

8 Norms and normalization The 2 norm u :={u u} /2 this is one of a family of p norms defined as follows: u p :={Σ u p } /p In the limit as p goes to infinity we get what is known as the max norm: u :=max u i i i In function space (e.g. functions that are continuous on [,]) we use one of the L p norms defined as: u p :={ u(x) p dx} /p CPSC 553/65 P 8

9 Norms for functions The L 2 norm is the most frequently used norm for functions can be also be written as: u 2 :={u u} /2 A vector u with u = is said to be normalised. If we have an orthogonal basis composed of vectors that are normalised in the L2 norm, the basis is called orthonormal. a basis u, u 2... is orthonormal if {u i u j } = δ i,j The Kronecker delta, δ is defined to be if i=j and otherwise. e.g. e = e 2 = e 3 = form an orthonormal basis for the inner product space endowed with the dot product of the form: a.b = a b +a 2 b 2 +a 3 b 3 CPSC 553/65 P 9

10 Eigenvectors and Eigenvalues A column vector v i is said to be a right eigenvector of a square matrix M with associated eigenvalue l if M v i = λ i v i By convention the eigenvectors and eigenvalues are typically ordered so that λ <= λ 2 <=... <= λ n By juxtaposing the column vectors v i into a single square matrix V and by placing the eigenvalues λ i along the diagonal of a matrix K:=diag( λ i ) CPSC 553/65 P

11 Now Are We Ready For Some Wavelets Motivation Medical data containing many points describing contours in 2D Data : sequence of points Time series data: (t i, y i ) each pair gives information about the behaviour at t but nowhere else. An Image (x i, y i ) A Surface f(u,v) CPSC 553/65 P

12 A Better Way to Handle Data Fourier analysis used to convert point data to a form useful for analysing frequencies. Problem is that each Fourier coefficient contains complete info. about behaviour at only one frequency. Difficult to adapt. E.g. most time series encountered in practice are finite and aperiodic but discrete Fourier transform can only be applied to periodic functions. Instead we want: Hierarchical Representation Functions or Multiresolution Methods i.e. Representation of a function with a collection of coeficients each of which provide some limited information about the position and frequency of the function. CPSC 553/65 P2

13 Why Wavelets? Hierarchical Decomposition of a function, efficient and theoretically sound. Linear Time Complexity Transforms to and from wavelets generally done in linear time. Sparsity Many coeficients are zero or negligably small data compression and accelerate convergence of iterative solutions. Adaptability Unlike Fourier techniques, waveletes can be dapted to represent a wide variety of functions (discontiniuties, bounded domains, arbitrary topological type). Well suited to problems involving images, open or closed curves and surfaces. CPSC 553/65 P3

14 History 873 Karl Weierstrass described a family of function constructed by superimposing scaled copies of a given base function. (He did this without an Indy at home). Such functions are fractal: Everywhere continuous but nowhere differentiable. 99 Alfred Haar st ortho normal system of compactly supported functions (Haar Basis) 94 Ricker coined the term wavelet to describe the disturbance that proceeds outward from a sharp seismic impulse. 946 Dennis Gabor nonorthogonal basis of wavelets with unbounded support based on translated Gaussians. All this was done without Unix. 982 Morlet et al Used Gabor functions to model Ricker s wavelets. 986/88 Meyer and Mallat multiresolution analysis 99 s Applications to computer graphics CPSC 553/65 P4

15 Haar Basis: The simplest Wavelet Decomposition of a D function. image applications: compression, editing, querying. e.g. D image defined as: Haar Basis:average pixels pairwise to obtain a lower resolution image: Now store missing information as detail coeficients: first is : 8+() = 9 8 () = 7 second is : 4+ ( ) = 3 4 ( ) = 5 Repeat this process recursively to obtain full decomposition: The wavelet transform or wavelet decomposition of the original 4 pixel image is [ ] [8 4] [8 4] [ ] Res. Avge. Detail Coef 4 [ ] 2 [8 4] [ ] [6] [2] [6 2 ] CPSC 553/65 P5

16 Haar Basis: The simplest Wavelet Recursive averaging and taking differences is known as a filter bank. Original image has 4 coefficients so does the transform. The original image and the lower resolution versions can be reconstructed from the transform. One advantage with real images is that many of the coefficients are small and can be ignored giving a way of compressing the image (lossy compression). The image can also be thought of as piecwise constant functions on the open half interval [,) (i.e. contains all values of x in range <=x<). One pixel image is a function that is constant over [,) (i.e. a vector) V denotes the vector space of all such functions (vectors) V is the space containing the 2 pixel image functions having two constant piecesover the intervals: [, /2 ) [ /2, ) CPSC 553/65 P 6

17 Images as functions V j is the space containing all the piecewise constant functions defined on [,) with constant pieces of 2 j equal sized subintervals. Every D image with 2 j pixels is an element or vector in V j Nested Vector Spaces The vectors or functions are all defined on the unit interval. Every vector in V j is also contained in V j+ Such a function with two intervals can be described as a function with four intervals, each interval in first function corresponds to a pair of intervals in the second, the spaces are nested: V V V 2 V 3... U U U CPSC 553/65 P7

18 Basis for Each Vector Space A minimum set of vectors from which all others vectors can be generated through linear combinations. The basis functions for the spaces are called scaling functions denoted by φ A simple basis for V j is given by scaled and rotated box functions φ j i (x) := φ(2j x i) i =,..., 2 j where φ(x) := { for <= x < otherwise For an example of V 2 see next slide... CPSC 553/65 P8

19 Vector Space V 2 φ j i (x) := φ(2j x i) i =,..., 2 j { for <= x < where φ(x) := otherwise i = 2 3 /4 /2 3/4 E.g. Basis for V 2 φ 2 i (x) := φ(4 x i) φ 2 i (x) := φ(4 x i) 2 φ (4x ) φ (4x ) φ 2 (4x 2) φ 3 (4x 3) /2 /2 /2 /2 CPSC 553/65 P9

20 Inner Product Defined on Vector Space V j {f g} := f(x) g(x) dx The vectors u, v are said to be ort hogonal if {u v} = Define a new space W j as the orthogonal complement of V j in space V j+ W j is the space of all functions in V j+ that are orthogonal to all functions in V j under the chosen inner product. A collection of linearly independent functions Ψ j i spanning W j are called wavelets. These basis functions have two important properties. The basis functions Ψ j of W j together with the basis functions φ j of V j form a basis for V j+ 2. Every basis function Ψ j of W j is orthogonal to every basis function φ j of V j i under the chosen inner product. i i i CPSC 553/65 P2

21 The Haar Wavelet As with the image a veraging example th e wavelets in W j represent the part s of a function in V j+ that cannot be rep resented in V j. Haar wavelets box basis Ψ j (x) := Ψ j (2 j x i) i =,..., 2 j i { for <= x < /2 where Ψ(x) := for /2<= x < otherwise Haar Wavelet j= W Ψ (2 x ) Ψ (2 x ) i= i= /2 /4 /2 CPSC 553/65 P2

22 Example Consider the image from the earlier ex ample Express the origin al image as a linea r combination of th e box basis functio ns in V 2 Res. Avge. Detail Coef 4 [ ] 2 [8 4] [ ] [6] [2] J(x) := 9φ 2 (x) + 7φ 2 (x) + 3φ 2 (x) 5φ 2 (x) 2 3 J(x) := 9x +7x +3x /2 /2 /2 /2 +5x CPSC 553/65 P22

23 Example Cont. Consider the image from the earlier ex ample Now rewrite the exp ression for J(x) in terms of basis functions in V and W using pairwise averaging and differencing. J(x) := c φ (x) + c φ (x) + d Ψ (x) + d Ψ (x) J(x) := 8 x +4 x + x x CPSC 553/65 P23

24 Haar Basis for V 2 J(x) := c φ 2 (x) + c φ 2 (x) + c φ 2 (x) c φ 2 (x) [ ] Decomposition J(x) := c φ (x) + c φ (x) + d Ψ (x) + d Ψ (x) Decomposition Re writing as a sum of basis functions in V, W and W J(x) := c φ (x) + d Ψ (x) + d Ψ (x) + d Ψ (x) [8 4 + ] [ ] J(x) := 6 x +2 x The Haar basis for V j includes these 4 fu nctions and even narrower versions of the wavelet Ψ(x) + x + x CPSC 553/65 P24

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26 abcde f g h i j k l m n op q r s t u v w x y z αβχδε φ γ η ι ϕ κ λ µ ν ο π θ ρ σ τ υ ϖ ω ξ ψ ζ ABCDEFGHIJKLMNOPQRSTUVWX Y Z ΑΒΧ ΕΦΓΗΙϑΚΛΜΝΟΠΘΡΣΤΥςΩΞ Ψ Ζ ABCDEFGHIJKLMNOPQRSTUVWX Y Z ABCDEFGHIJKLMNOPQRSTUVWX Y Z {f g} := f(x) g(x) dx CPSC 553/65 P26