2D Wavelets. Gloria Menegaz 1

Transcription

1 D Wavelets Gloria Menegaz

2 Topics Basic issues Separable spaces and bases Separable wavelet bases D DWT Fast D DWT Lifting steps scheme JPEG000 Wavelet packets Advanced concepts Overcomplete bases Discrete wavelet frames DWF Algorithme à trous Discrete dadic wavelet frames DDWF Overview on edge sensitive wavelets Contourlets Wavelets in vision Human Visual Sstem Gloria Menegaz

3 Separable Wavelet bases To an wavelet orthonormal basis { n } n Z of L R one can associate a separable wavelet orthonormal basis of L R : The functions n and n mi informations at two different scales along and which is something that we could want to avoid Separable multiresolutions lead to another construction of separable wavelet bases with wavelets that are products of functions dilated at the same scale. Gloria Menegaz 3

4 Separable multiresolutions The notion of resolution is formalized with orthogonal proections in spaces of various sizes. The approimation of an image f at the resolution - is defined as the orthogonal proection of f on a space V that is included in L R The space V is the set of all approimations at the resolution -. When the resolution decreases the size of V decreases as well. The formal definition of a multiresolution approimation {V } Z of L R is a straightforward etension of Definition 7. that specifies multiresolutions of L R. The same causalit completeness and scaling properties must be satisfied. Gloria Menegaz 4

5 Separable spaces and bases Tensor product Used to etend spaces of D signals to spaces of multi-dimensional signals A tensor product between vectors of two Hilbert spaces H and H satisfies the following properties Linearit λ C λ = λ = λ Distributivit + + = This tensor product ields a new Hilbert space including all the vectors of the form where and combination of such vectors An inner product for H is derived as H = H H H H = as well as a linear H H Gloria Menegaz 5

6 Separable bases Theorem A.3 Let H H H. If e n n N and e n n N are Riesz bases of H and H respectivel then e e is a Riesz basis for H. If the two bases = { } { } { } n m nm N are orthonormal then the tensor product basis is also orthonormal. To an wavelet orthonormal basis one can associate a separable wavelet orthonormal basis of L R { n l m } 4 n l m Z However wavelets and mi the information at two different n lm scales along and which often we want to avoid. Gloria Menegaz 6

7 Separable Wavelet bases Separable multiresolutions lead to another construction of separable wavelet bases whose elements are products of functions dilated at the same scale. We consider the particular case of separable multiresolutions A separable D multiresolution is composed of the tensor product spaces V = V V V is the space of finite energ functions f that are linear epansions of separable functions f = a[ n] f g If is a multiresolution approimation of L R then V is a multiresolution approimation of L R. n n n f n V g n V { V } { } Z Z Gloria Menegaz 7

8 Separable bases It is possible to prove Theorem A.3 that n n m n m ϕ = ϕ ϕ = ϕ ϕ is an orthonormal basis of V. n m Z A D wavelet basis is constructed with separable products of a scaling function and a wavelet m 3 3 ϕ Gloria Menegaz 8

9 Eamples Gloria Menegaz 9

10 Separable wavelet bases A separable wavelet orthonormal basis of L R is constructed with separable products of a scaling function and a wavelet. The scaling function is associated to a one-dimensional multiresolution approimation {V } Z. Let {V } Z be the separable two-dimensional multiresolution defined b V = V V Let W be the detail space equal to the orthogonal complement of the lowerresolution approimation space V in V - : V = V W To construct a wavelet orthonormal basis of L R Theorem 7.5 builds a wavelet basis of each detail space W. Gloria Menegaz 0

11 Gloria Menegaz Separable wavelet bases Theorem 7.5 Let ϕ be a scaling function and be the corresponding wavelet generating an orthonormal basis of L R. We define three wavelets and denote for <=k<=3 The wavelet famil is an orthonormal basis of W and is an orthonormal basis of L R On the same line one can define biorthogonal D bases. 3 ϕ ϕ = = = = k k m n m n { } 3 3 Z m n m n m n m n { } 3 nm nm nm nm Z

12 Separable wavelet bases The three wavelets etract image details at different scales and in different directions. and ˆ ω ˆ ϕ ω Over positive frequencies have an energ mainl concentrated respectivelon [0π ] and [π π]. The separable wavelet epressions impl that ω = = = ˆ ˆ ˆ ω ω ϕ ω ω ˆ ˆ ˆ ω ω ω ϕ ω 3 ˆ ˆ ˆ ω ω ω ω 3 3 ϕ ω Gloria Menegaz

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14 Gloria Menegaz 4 Bi-dimensional wavelets 3 ϕ ϕ ϕ ϕ ϕ = = = =

15 Eample: Shannon wavelets ω 3 3 ϕ ω Gloria Menegaz 5

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17 Biorthogonal separable wavelets Let ϕ ϕ and be a two dual pairs of scaling functions and wavelets that generate = = = a biorthogonal wavelet basis of L. 3 The dual wavelets of and are ϕ ϕ One can verif that 3 { } and n n n n Ζ 3 { } n n n n Ζ are biorthogonal Riesz basis of L 3 3 R Gloria Menegaz 7

18 Fast D Wavelet Transform a [ n m] = f ϕ n m k k d [ n m] = f k = 3 nm Approimation at scale Details at scale 3 [ aj { d d d } J ] Wavelet representation Analsis a d d d [ n m] = a [ n m] = a [ n m] = a [ n m] = a hh[nm] hg[nm] gh[nm] gg[nm] Snthesis a [ n m] 3 = a + hh[ n m] + d + hg[ n m] + d + gh[ n m] + d + gg[ n m] Gloria Menegaz 8

19 Fast D DWT Gloria Menegaz 9

20 Finite images and compleit When al is a finite image of N=NN piels we face boundar problems when computing the convolutions A suitable processing at boundaries must be chosen For square images with NN the resulting images a and dk have samples. Thus the images of the wavelet representation include a total of N samples. If h and g have size K one can verif that K - - multiplications and additions are needed to compute the four convolutions Thus the wavelet representation is calculated with fewer than 8/3 KN operations. The reconstruction of a L b factoring the reconstruction equation requires the same number of operations. Gloria Menegaz 0

21 Matlab notations Gloria Menegaz

22 Matlab notations Gloria Menegaz

23 Eample V h H h g g h g Gloria Menegaz 3

24 Eample h g H h g h g h g h g h g Gloria Menegaz 4

25 Subband structure for images ca cd h cd h cd v cd d cd v cd d Gloria Menegaz 5