Wavelets bases in higher dimensions

Transcription

1 Wavelets bases in higher dimensions 1

2 Topics Basic issues Separable spaces and bases Separable wavelet bases D DWT Fast D DWT Lifting steps scheme JPEG000 Advanced concepts Overcomplete bases Discrete wavelet frames DWF Algorithme à trous Discrete dadic wavelet frames DDWF Hints on edge sensitive wavelets Contourlets

3 Separable Wavelet bases In general to an wavelet orthonormal basis { n } n Z of L R one can associate a separable wavelet orthonormal basis of L R : n The functions and mi information at two different scales along 1 and which is something that we could want to avoid n Separable multiresolutions lead to another construction of separable wavelet bases with wavelets that are products of functions dilated at the same scale. 3

4 Separable multiresolutions The notion of resolution is formalized with orthogonal proections in spaces of various sizes. The approimation of an image f 1 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 7.1 that specifies multiresolutions of L R. The same causalit completeness and scaling properties must be satisfied. 4

5 Separable spaces and bases Tensor product Used to etend spaces of 1D signals to spaces of multi-dimensional signals A tensor product 1 between vectors of two Hilbert spaces H 1 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 as well as a linear combination of such vectors An inner product for H is derived as H = H H H H1 = H H 1 5

6 H H H Separable bases = { 1 e } { e } { } Theorem A.3 Let 1. If n n N and n n N are Riesz bases of H 1 1 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. 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 is a multiresolution approimation of L R. n n n f n V g n V { V } { V } Z Z 7

8 Separable bases It is possible to prove Theorem A.3 that 1 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 ω m ϕ 1 1 ω 8

9 Eamples 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 -1 : V = V W 1 To construct a wavelet orthonormal basis of L R Theorem 7.5 builds a wavelet basis of each detail space W. 10

11 11 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 1<=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 1 ϕ ϕ = = = = k k m n m n 1 { } Z m n m n m n m n nm 1 nm nm 3 { } 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 respectivel in [0π ] and [π π]. The separable wavelet epressions impl that ω = = = 1 ˆ ˆ ˆ ω ω ϕ ω ω ˆ ˆ ˆ ω ω ω ϕ ω 3 ˆ ˆ ˆ ω ω ω ω ϕ 1 1 ω 1

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14 14 Bi-dimensional wavelets 3 1 ϕ ϕ ϕ ϕ ϕ = = = =

15 Eample: Shannon wavelets ω ϕ 1 ω 1 15

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17 Multiresolution vision The visual acuit is greatest at the center of the retina where the densit of receptors is maimum. When moving apart from the center the resolution decreases proportionall to the distance from the retina center A retina with a uniform resolution equal to the highest fovea resolution would require about times more photoreceptors. Such a uniform resolution retina would increase considerabl the size of the optic nerve that transmits the retina information to the visual corte and the size of the visual corte that processes this data. Active vision strategies compensate the non-uniformit of visual resolution with ee saccades which move successivel the fovea over regions of a scene with a high information content. These saccades are partl guided b the lower resolution information gathered at the peripher of the retina. This multiresolution sensor has the advantage of providing high resolution information at selected locations and a large field of view with relativel little data. 17

18 Multiresolution computer vision Multiresolution algorithms implement in software the search for important high resolution data. A uniform high resolution image is measured b camera but onl a small part of this information is processed Coarse to fine algorithms analze first the lower resolution image and selectivel increase the resolution in regions where more details are needed. Applications: obect recognition stereo calculations 18

19 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. The dual wavelets of 1 and 3 are 1 = ϕ = ϕ 3 = k 1 k n m = One can verif that { 1 3 n n n} n Ζ 3 and { 1 3 nm nm nm} nm Ζ 3 are biorthogonal Riesz basis of L R n m 19

20 Fast D Wavelet Transform a [ n m] = f ϕ n m k k d [ n m] = f k = [ aj { d d d } 1 nm J ] Approimation at scale Details at scale 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] 1 3 = a + 1 hh[ n m] + d + 1 hg[ n m] + d + 1 gh[ n m] + d + 1 gg[ n m] 0

21 Fast D DWT 1

22 Finite images and compleit When a L is a finite image of N=N 1 N piels we face boundar problems when computing the convolutions A suitable processing at boundaries must be chosen For square images with N 1 N the resulting images a and d k have N 1 N / 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/ -1 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.

23 Separable biorthogonal bases One-dimensional biorthogonal wavelet bases are etended to separable biorthogonal bases of L R following the same approach used for orthogonal bases Let ϕ ϕ be two dual pairs of scaling functions and wavelets that generate biorthogonal wavelet bases of L R. The dual wavelets of 1 3 are 1 = ϕ 1 = ϕ = 3

24 Separable biorthogonal bases One can verif that { 1 3 nm nm nm} nm Z 3 { 1 nm nm 3 nm} nm Z 3 are Riesz basis of L R 4

25 Eample V h H h g g h g 5

26 Eample h g H h g h g h g h g h g 6

27 Subband structure for images ca cd h cd 1h cd v cd d cd 1v cd 1d 7

28 Wavelet bases in higher dimensions Separable wavelet orthonormal bases of L R p are constructed for an p with a procedure similar to the two-dimensional etension. Let φ be a scaling function and a wavelet that ields an orthogonal basis of L R. We denote θ 0 =φ and θ 1 =. To an integer 0 ε< p -1 written in binar form ε=ε 1..ε n we associate the p-dimensional functions defined in = 1... p b ε = ϑ ε 1 1 ϑ ε n p For ε=0 we obtain the p-dimensional scaling function ϕ 0 = ϕ 1 ϕ p Non-zero indees ε correspond to p -1 wavelets. At an scale and for n=n 1... n p we denote ε n = 1 " 1 n 1 p ε p n p % $ # ' & 8

29 Wavelets in p-dimensions Theorem 7.5: The famil obtained b dilating and translating the p -1 wavelets for ε 0 is an orthonormal basis of L R P { } n 1 ε< p 1 n Ζ p The wavelet coefficients at scales are computed with separable convolutions and subsamplings along the p signal dimensions a d [ n] = fϕ n [ n] = f ε n for 0 < ε < p 1 9

30 Fast p-dimensional WT The fast WT is calculated with filters that are separable products of the onedimensional filters h and g. The separable p-dimensional low-pass filter is h 0 [n]=h[n 1 ]..h[n p ] 30

31 Fast p-dimensional WT 31

32 Fast p-dimensional WT 3

33 Wavelet bases in higher dimensions Theorem 7.5 The famil obtained b dilating and translating the p -1 wavelets for ε different from zero is an orthonormal basis for L R p. 3D DWT { ε n } 1 ε< p 1 n Z p 33

34 3D DWT 1D-DWT for each row 1D-DWT for each column 1D-DWT for each depth LP HHH LP HP HHL LP LP HLH HP HP HLL LP LP HP LHH LHL HP HP LP HP LLH LLL Fig. The filter architecture for 3D wavelet transform 34