INTERLEAVED DIMENSION DECOMPOSITION: A NEW DECOMPOSITION METHOD FOR WAVELETS AND ITS APPLICATION TO COMPUTER GRAPHICS

Transcription

1 INTERLEAVED DIMENSION DECOMPOSITION: A NEW DECOMPOSITION METHOD FOR WAVELETS AND ITS APPLICATION TO COMPUTER GRAPHICS Manfred Kopp and Werner Purgathofer Vienna University of Technology Institute of Coputer Graphics Karlsplatz 13/186- Vienna, Austria { opp wp ABSTRACT Wavelets in D or higher diensions are often generated by a decoposition schee fro 1D wavelets. There are two decoposition schees called the standard and the nonstandard decoposition which are used in ost applications of higher diensional wavelets. This paper introduces a new decoposition ethod, the interleaved diension decoposition and copares its advantages and disadvantages with the other decopositions. Based on the properties of the interleaved diension decoposition, applications to coputer graphics are setched including ultiresolution painting, orphing in D and 3D, and iage copression. Keywords: Wavelets, orphing, ultiresolution painting, iage copression 1. INTRODUCTION In the last years, wavelets have becoe a popular research topic in various fields of coputer graphics. The different applications of wavelets in coputer graphics include wavelet radiosity [Gortl93], ultiresolution painting [Bera94], curve design [Fine94], esh optiization [Ec95], volue visualization [Lippe95], iage searching [Jacob95], aniation control [Liu94], BRDF representation [Schrö95], and, one of the first applications in coputer graphics, iage copression [Shapi93]. As it is noticed in the preable of [Glass95], wavelets and wavelet transfors can becoe as iportant and ubiquitous in coputer graphics as spline based techniques are now. In this paper a new decoposition for separable higher diensional wavelets is introduced and copared with the standard and nonstandard decoposition. Chapter describes the wavelets in 1D. In chapter 3 traditional extensions to higher diensional wavelets including atrix dilation ethods, standard and nonstandard decoposition are discussed. Chapter 4 introduces the new interleaved diension decoposition and copares it with traditional approaches. Chapter 5 shows soe coputer graphics applications of the interleaved diension decoposition.. WAVELETS IN 1D This chapter gives a very short introduction in the theory of wavelets. A ore detailed discussion of this topic can be found in [Daube9]..1 Orthonoral Wavelets The orthonoral wavelet transfor is based on two functions ψ(x) and φ(x), which have the properties: φ () xdx ψ() = 1; xdx= (1) These functions with their translations and dilations ψ, (x) and φ, (x) build an orthonoral basis and therefore any function in L (R) can be reconstructed with these basis functions. φ(x) is called scaling- or sooth-function, and ψ(x) wavelet or detail function. ψ, (x) and φ, (x) can be constructed fro

2 their other functions ψ(x) and φ(x) in the following anner: φ x φ, ( ) = ( x ), ψ x ψ, ( ) = ( x ),, Z () {φ, Z} for an orthonoral basis of functions in vector space V. These vector spaces are nested, that is, V V 1 V V 3... Given a function f(x) over [,1], this function can be approxiated in V with scaling coefficients s : 1 f ( x) = s φ ( x) =, (3) Let f( x), g( x) denote the inner product of the function f(x) and g(x) with: + f( x), g( x) = f( x) g( x) dx (4) The scaling coefficients s can be coputed by the inner product of the function f(x) and the corresponding scaling function φ, (x): s = f( x), φ ( x) (5), Also the detail functions {ψ, Z} for an orthonoral basis of functions in the detail vector space W, which is the orthogonal copleent of V in V +1. W can be thought of as containing the detail in V +1, which can not be represented in V. The vector space V +1 can be decoposed in the following anner: V +1 = V W = V -1 W -1 W =... = V W W 1... W (6) Let d be the wavelet or detail coefficients, given through: d = f( x), ψ ( x) (7), then f (x) can be calculated fro the wavelet i coefficients { d i<} and the scaling coefficient s as follows: = f ( x) = f ( x) + d ψ ( x) 1, 1 1,, = = = s φ ( x) + d ψ ( x) The calculation of the coefficients { s, d i < i } fro the scaling (8) i<; coefficients { s < } is called wavelet transforation. The fast wavelet transforation uses a pyraid schee with two subband filters, the soothing or scaling filter (h ), and the detail or wavelet filter (g ). In one transforation step the i i scaling coefficients s are replaced by i-1 scaling coefficients s i 1 and i-1 wavelet coefficients d i 1 : s 1 1 = h s ; d = g s (9) i i i i This step is repeated on the reaining scaling coefficients, until s is coputed. The reconstruction step can be perfored using the adoint filtering operation: i i 1 i 1 s = h s + g d (1) The whole reconstruction of the scaling coefficients s i fro s and the wavelet coefficients of various resolutions is called inverse wavelet transforation.. Biorthogonal Wavelets Biorthogonal wavelets use scaling basis functions φ, (x) which are not orthogonal. Therefore the scaling coefficients can not be coputed by Eq. 5. But it can be shown, that for every basis {φ, (x)}, which is called prial basis, there is a dual basis { ~ φ, ( x ) } and the scaling coefficient can be coputed by: s = f( x), ~ φ ( x) (11), In analogy, the detail coefficient can be coputed by the inner product of the function f(x) and ~ the corresponding dual wavelet function ψ ( ), x : d = f( x), ~ ψ ( x), (1), The following conditions ust be satisfied between the prial scaling and wavelet functions and their dual counterparts: d

3 φ ( x), ~ φ ( x) = δ, l, l ψ ( x), ~ ψ ( x) = δ, l, l φ ( x), ~ ψ ( x) =, l ψ ( x), ~ φ ( x) =, l (13) The fast wavelet transforation with biorthogonal wavelets uses the dual filter pair ( h ~ ) and ( ~ g ) corresponding to the dual basis functions. Therefore a wavelet transforation step uses the following forula instead of Eq. 9: s 1 ~ 1 = h s ; d = ~ g s (14) i i i The fast inverse wavelet transforation with biorthogonal wavelets uses the prial filter pair (h ) and (g ), hence the procedure is the sae as for orthogonal wavelets (Eq. 1). If the prial and dual basis functions and filters are exchanged with each other, the result is also a valid biorthogonal wavelet. 3. WAVELETS IN HIGHER DIMENSIONS The definitions in Chapter deal with wavelets in 1D space, but for any applications of wavelets in coputer graphics transforations in D or higher diensions are needed. There are two approaches to extend the wavelet definitions to higher diensions: One is to apply tensor products of 1D wavelets and scaling functions. The two ethods entioned in literature are the standard and the nonstandard decoposition. The other ethod, called dilation atrix ethod, extends the definition of the wavelet and scaling functions by using a dilation atrix instead of a siple factor. i Figure 1: D standard Haar wavelet basis Fig. 1 shows the D standard Haar basis functions for a 4x4 iage: It contains one scaling function in the upper left corner, three R1 ψ, ( xy, ) wavelets on the upper border, three R ψ, ( x, y)wavelets on the left border and the reaining nine R 3 ψ x y,,, (, )wavelets. n 3.1 Standard Decoposition The standard wavelet basis functions, in D also called rectangular wavelet basis functions, are generated through the carthesian product of the 1D wavelet basis functions or the other scaling function φ, in every diension. In the D case, the standard basis functions are: φ( xy, ) = φ, ( x) φ 1 ψ, ( xy, ) = ψ, ( x) φ, < L (15) ψ, ( xy, ) φ, ( x) ψ = < 3 ψ n,,,( xy, ) = ψ,( x) ψn,( y) n< R R R φ( xy, ) denotes the D scaling function and the different types of wavelet functions are defined as R 1 ψ, ( xy, ), R ψ, ( x, y) and R3 ψ n x y,,, (, ). Figure : D standard decoposition The fast wavelet transforation with the standard basis wavelets, also nown as standard decoposition, is coputed by successively applying the 1D wavelet transforation to the data in every diension. In the D case, all the rows are

4 transfored first, then a 1D wavelet transforation is applied on all coluns of the interediate result. Fig. illustrates the rectangular decoposition. The wavelet coefficients of the 1D transforation steps are stored in the right (row transfor) or lower (colun transfor) part, the scaling coefficients in the left or upper part, respectively. Note that the wavelet coefficients can also have negative values - therefore the value zero of a wavelet coefficient is displayed as 5% gray. 3. Non-Standard Decoposition The nonstandard wavelet basis functions, in D also called square wavelet basis functions, are generated through the carthesian product of 1D wavelets and 1D scaling functions. In contrast to the standard basis functions, the non-standard basis functions always use tensor products of wavelet and/or scaling functions of the sae resolution level. In the D case, the nonstandard basis functions are: φ( xy, ) = φ, ( x) φ h ψ, ( x, y) = ψ, ( x) φ v < L (16) ψ, ( xy, ) = φ, ( x) ψ, d ψ, ( x, y) = ψ, ( x) ψ < Lie in the standard decoposition, it contains one D scaling function and three different types of D wavelet functions. Note that the wavelet functions d ψ, ( xy, ) are a subset of the wavelet functions R 3 ψ x y,,, (, ) in the rectangular decoposition. n five h ψ, ( xy, )wavelets in the upper part, five v ψ, ( x, y) wavelets in the left part and five d ψ, ( xy, ) wavelets in the reaining positions. The non-standard wavelet decoposition can be coputed with a siilar technique as the standard decoposition: A 1D wavelet transforation step - not the whole wavelet transforation as in the rectangular decoposition - is applied in every diension. This generates ( di -1) subbands with wavelet coefficients and one subband with scaling coefficients. This transforation schee is applied recursively on the scaling coefficients until the lowest level is reached (Fig. 4). The non-standard decoposition is slightly ore efficient to copute than the standard decoposition: For an iage only (8/3)( -1) assignents are needed, copared to 4( -) in the standard decoposition. Also the copression ratios are usually better for the square decoposition, because the support of the wavelet functions are square and support widths of the wavelet basis functions are lower or equal than their counterparts in the rectangular decoposition and therefore they exploit ore locality. Figure 4: D non-standard decoposition Figure 3: D non-standard Haar wavelet basis Fig. 3 shows the D non-standard Haar basis functions for a 4x4 iage with one scaling function, 3.3 Quincunx and other Dilation Matrix Methods One alternative way to extend the forulas to D is to directly use D wavelets and D scaling functions

5 v v with a dilation atrix D in Eq. as described in [Kovac9], instead of the siple dilation factor. The wavelets and scaling functions can be coputed fro their other D wavelet an other D scaling function by: n φ ( x): = φ( D x ) n, n ψ ( x): = ψ( D x ) (17) n, n with = ( 1,... di ); =,..., 1; 1 i di i The quincunx schee is one of these dilation atrix ethods in D with the property DZ = {(x,y); x+y Z}. The ost proising dilation atrix for the quincunx schee is D = ( ). Note that the decoposition is dependent on the ordering of the diensions. In D there are two possible perutations. Therefore the second possible choice of the basis functions in D is: φ( xy, ) = φ, ( x) φ L yx < ψn, ( xy, ) = φ, ( x) ψ + 1 n (19), x < ψ, ( xy, ) = ψ, ( x) φ + 1 n< Without loss of generality, we will refer to the first perutation in the D case defined in Eq analog assuptions can be ade with the D wavelet basis in Eq. 19. Fig. 5 shows the D interleaved diension Haar basis functions for a 4x4 iage fro Eq. 18: One drawbac of this ethod is the higher coputational cost of the wavelet transforation: n-diensional filters have to be applied instead of 1D filters in the separable ethods. Because of this drawbac, it is rarely used in the application of wavelets to coputer graphics. 4. INTERLEAVED DIMENSION DECOMPOSITION In this chapter a new type of separable wavelets for higher diensions based on tensor products of 1D wavelets called interleaved diension decoposition is introduced. The wavelet functions of the interleaved diension decoposition in N diensions are generated by the tensor products of a 1D wavelet function of level in the d th diension with 1D scaling functions of level in the diensions up to d-1 and 1D scaling functions of level +1 in the diensions higher than d. The higher diensional scaling function for the interleaved diension basis is the sae as in the standard and non-standard basis. The interleaved diension basis functions in D are defined by: φ( xy, ) = φ, ( x) φ L xy < ψn, ( xy, ) = ψ, ( x) φ + 1n, y < (18) ψ, ( xy, ) = φ, ( x) ψ + 1 n< The different wavelet functions are indexed in the left superscript by the diension of the 1D wavelet function followed by the diensions with the 1D scaling functions of the higher level. Therefore the superscript xy of the wavelet in Eq. 18 denotes a tensor product of a 1D wavelet in x-direction and a 1D scaling function in y-direction of a level one higher than the 1D wavelet. Figure 5: D interleaved diension Haar basis The wavelets x ψ, ( xy, ) are the sae as the wavelets v ψ, ( xy, ) in the non-standard decoposition. Therefore one third of the wavelet functions and also wavelet coefficients of the non-standard decoposition in D are identical to the functions and coefficients of the interleaved diension decoposition. This can be seen by the coparison of Fig. 4 and Fig. 5, where five functions on the lower left corner are atching. But except one wavelet function on the lowest ultiresolution level, there is no atch of wavelets of the interleaved diension decoposition with the standard decoposition. While ost of the standard wavelet functions are tensor products of 1D wavelets, the interleaved diension wavelet functions are tensor products of exactly one 1D wavelet function in one diension and 1D scaling functions in all other diensions.

6 The nuber of different types of wavelet functions of the interleaved diension decoposition is equal the nuber of diensions di, while the standard and the non-standard decoposition use di -1 different types of wavelet functions. So in 3D we get a basis with one sooth function and three wavelets - copared to seven types of wavelets for the standard and non-standard decoposition. If we use sequential ordering of the diensions we get in 3D the basis functions: φ( xyz,, ) = φ ( x) φ ( y) φ ( z),,, xyz n,,, + 1, n + 1, yz ψon,,( xyz,, ) = φ, ( x) ψ o φ+ 1, n( z) z ψ ( xyz,, ) = ψ ( x) φ ( y) φ ( z) ψop,, ( xyz,, ) = φ, ( x) φo ψ p, ( z) +... with < L;, o, p< ; n, < 1 () The interleaved diension decoposition is very siilar to the non-standard decoposition: One 1D wavelet transforation step is applied to the first diension. But contrary to the nonstandard decoposition only the resulting scaling coefficients are transfored with the following 1D transforation steps, not the wavelet coefficients as in the non-standard decoposition. The transforation steps are cycled in the different diensions until only one scaling coefficient reains. It is easy to see, that all the interediate scaling coefficients are the sae as in the non-standard decoposition. Fig. 6 illustrates the interleaved diension decoposition in D. The interleaved diension decoposition can be seen as an incoplete non-standard decoposition, lie it is shown in [Kopp96] that the non-standard decoposition can be viewed as an incoplete standard decoposition. Therefore also the nuber of operations needed for the decoposition is lower than in the other decopositions: The interleaved diension needs only ( -1) assignents copared to 4( -) and (8/3)( -1) assignents for the standard and the nonstandard decoposition. This efficiency of the calculation is ade at the cost that the ultidiensional wavelets contain only in one diension a 1D wavelet coponent. Therefore for copression issues, coherence is only exploited in one diension leading to a worse copression rate in ost of the cases. Figure 6: D interleaved diension decoposition It is entioned in [Daube9] that the non-standard decoposition can be generated fro a extension of the ultiresolution concept to higher diensions. The interleaved diension decoposition is an alternative for ultiresolution approaches in higher diensions: The higherdiensional ultiresolution spaces V and W and the ultidiensional scaling functions are the sae. One of the ( di -1) different wavelet functions of the nonstandard decoposition reains the sae while the others are replaced with di-1 sets of translated, ore local wavelets. An interesting feature of the interleaved diension decoposition is the ability to generate di-1 interediate ultiresolutions - let us deonstrate this property with wavelets in 3D in the ordering of Eq. : V V,z V,yz V 1 V 1,z V 1,yz V... V = V V V op V,z = V V V +1 = V { z ψ,, ( xyz,, )} (1) V,yz = V V +1 V +1 = V,z { yz ψ on,, ( xyz,, )} V +1 = V,yz { xyz ψ,, ( xyz,, )} n The corresponding wavelet spaces W, W,z and W,yz can be built by orthogonal copleent. The area/volue/hypervolue dilation fro one interediate level to the next is only copared to di for the non-standard decoposition. 5. APPLICATIONS TO COMPUTER GRAPHICS

7 A detailed discussion of the different applications, where the interleaved diension decoposition perfors better than the standard or non-standard decoposition is out of scope of this paper. But the following exaples setch applications, where the new decoposition have better perforance or generate better quality than the other decopositions. 5.1 Lossless Copression The interleaved diension decoposition exploits coherence only in one direction, thus the application to lossy iage and volue copression and related applications lie iage searching [Jacob95] are expected to perfor worse than with other decopositions. But the interleaved diension decoposition can be used to iprove lossless copression. The Reversible-Two-Six (RTS) wavelet introduced by [Zandi95] is the ost proising one for the application of lossless copression. In one transforation step of the RTS wavelet in 1D, the resulting scaling coefficients of the lower level have the sae precision as the scaling coefficients of the higher level, while the range of possible 1D wavelet coefficients is about three ties as big as the range of the scaling coefficients. Applying the non-standard decoposition in D results in /3 of the wavelet coefficients in the value range three ties of the original data and 1/3 of the wavelet coefficients in a ninefold value range, while the interleaved diension decoposition needs only a threefold value range for all wavelet coefficients. Since all possible coefficient values ust be handled by the entropy coder, the interleaved diension decoposition perfors soeties better than the non-standard decoposition. It is shown in chapter 4, that the interleaved diension decoposition can be seen as an incoplete non-standard decoposition. Therefore the adaptive basis selection in [Kopp96] can be extended with the interleaved diension decoposition. 5. Multiresolution Iaging Multiresolution iages do not have a fixed pixel size, but the resolution varies over the iage. An efficient way to ipleent ultiresolution iages in a paint syste uses wavelets as a representation of the iages. [Bera94] utilize a quadtree structure with three Haar wavelet coefficients of the nonstandard decoposition in each node. The display of the ultiresolution iage is achieved by traversing the quadtree in the specified display region by applying an inverse D decoposition step. It is easy to see that the interleaved diension decoposition can speed up this procedure because less assignents are needed to copute. Also the update fro the current level, where the user is painting, to the coarser ones can be done faster with the interleaved diension decoposition. 5.3 D and 3D Morphing The calculation of an interediate iages in iage orphing algoriths is done in three steps: firstly the deterination of feature correspondence and of an interediate feature geoetry, secondly the warping of the source and the destination iages to interediate iages that atch the feature geoetry, and finally the blending of the two warped iages. [Kopp97] uses wavelets for the blending with different blending factors for different ultiresolution levels. This gives the aniator the possibility to introduce a disparity between the orphing of global and detail features that can lead to interesting new effects. The additional interediate ultiresolution levels of the interleaved diension decoposition provide a better iage quality of the resulting orphing sequence, since the transition of one level to the next has only an area dilation factor of copared to 4 in the non-standard decoposition. An other proble occurs with volue orphing: If the orphing is done in spatial doain, distracting high frequency distortions occur in the interediate volues. In [He94] this proble is avoided by applying the orphing in the wavelet doain with the non-standard decoposition. In the interediate volues the wavelets of the higher levels, which correspond to the high frequencies, are faded off. A proble with this approach is the blocing artifact, which reveals the allignent of the coordinate axes on the interediate volue. The interediate ultiresolution levels of the interleaved diension decoposition aes the fading off of the higher levels ore soothly, therefore the blocing artifact is less visible. 6. CONCLUSION A new wavelet decoposition called interleaved diension decoposition was introduced. The coparison with the standard and non-standard decoposition showed soe interesting aspects: The interleaved diension decoposition is faster to generate, exploits ore locality and builds - lie the non-standard decoposition - higherdiensional ultiresolution spaces. But unlie the non-standard decoposition, interediate ultiresolution spaces can be built. The interleaved diension decoposition can be used to iprove soe wavelet based coputer applications. Soe exaples were setched in this paper including lossless copression, ultiresolution iaging, and orphing in D and 3D.

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