T. D. Tran, R. de Queiroz, and T. Q. Nguyen, "The generalized lapped biorthogonal transform," presented at the ICASSP, WA, May 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....lter coecients in a perfect reconstruction M channel lter bank is generally a high dimensional problem. Current methods try to overcome this problem by simplifying strategies like cosine modulation of a prototype low pass lter [8] 11] or factorizations of the associated polyphase matrix [12] [15]. Design and optimization strategies in the time domain and directly on the impulse responses of the lters are also employed [16] 18] See also [19] for a di erent approach. It is always of interest to nd computationally ecient transforms, such as lters with integer valued impulse responses ....

T. D. Tran, R. L. de Queiroz and T. Q. Nguyen, \The Generalized Lapped Biorthogonal Transform," Proc. ICASSP, Seattle, May 1998.

....Due to the fact that the best basis was chosen in order to minimize the entropy, and not the distortion, ringing artifacts are experienced, sometimes, on the borders of smooth regions. 3. 5 The Generalized Lapped Biorthogonal Transform (GLBT) Recently, a new lattice structure has been proposed [22] which spans a large class of the M channel linear phase perfect reconstruction filter banks (aka LPPRFB) The new lattice represents a family of generalized lapped biorthogonal tranform (GLBT) with arbitrary integer overlapping factor, by relaxing the orthogonality constraint. Experimental ....

Trac D. Tran, Ricardo de Queiroz, and Troung Q. Nguyen. The generalized lapped biorthogonal transform. In Proceedings IEEE ICASSP, Seattle, May 1998.

....On the other hand, research on integer coefficient channel systems is still at an early stage. It has been shown recently that multiband transforms with floating point coefficients, when appropriately designed and utilized, can outperform state of the art wavelets by significant margins [6] 7] [9], 11] The overlapping basis functions of these transforms can eliminate annoying blocking artifacts just as efficiently as any wavelet. However, the wavelet transform requires many more operations per output coefficient, and it may need a large memory buffer in its implementation. LTs are more ....

.... to channel uniform linear phase perfect reconstruction filterbanks (LPPRFBs) The generalized paraunitary LPPRFBs are called GenLOT and are presented in [8] The most general lattice for channnel linear phase lapped biorthogonal trans 1053 587X 02 17.00 2002 IEEE forms (GLBTs) is presented in [9] and [10] The polyphase matrix can be factorized as [9] 1) 2) and (3) This lattice results in all filters having length . is often called the overlapping factor. Each cascading structure increases the filter length by . All and are arbitrary invertible matrices, and they can be completely ....

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T. D. Tran, R. de Queiroz, and T. Q. Nguyen, "The generalized lapped biorthogonal transform," presented at the ICASSP, WA, May 1998.

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T. D. Tran, R. de Queiroz, and T. Q. Nguyen, "The generalized lapped biorthogonal transform," in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing, Seattle, WA, May 1997.

....Example 5 are slower than the DCT; however, they eliminate blocking artifacts at a reasonable computational overhead, much less than 100 . The LiftLT II does not even require any multiplication. Performance wise, The LiftLTI attains a coding gain very close to the 9. 63 dB level of the optimal GLBT [22], 23] The improvement in coding performance over the DCT and sometimes the 9 7 tap wavelet transform is very promising as demonstrated by the image coding experiment in the next section. B. Image Coding Experiment The same SPIHT s quantizer and entropy coder [40] are utilized to encode the ....

T. D. Tran, R. de Queiroz, and T. Q. Nguyen, \The generalized lapped biorthogonal transform," Proc. IEEE ICASSP, pp. 1441-1444, Seattle, May 1997.

....Recently a new class of LTs with symmetric 0.1. INTRODUCTION 3 bases were developed yielding the class of generalized LOTs (GenLOT) 35] 40] 37] The GenLOTs were made to have basis functions of arbitrary length (not a multiple of the block size) 56] extended to the non orthogonal case [60] and even made to have filters of different lengths [59] As aforementioned, filter banks and LTs are the same, although studied independently in the past. Because of this duality, it would be impractical to mention all related work in the field. Nevertheless, Vaidyanathan s book [62] is ....

....G(z) D T K 0 1 (z)K 0 2 (z) Delta Delta Delta K 0 N Gamma1 (z) 93) where the inverse factors are K 0 i (z) z Gamma2 K T i (z Gamma1 ) 94) whose structure is evident from Fig. 24. 0.6. 4 The general factorization: GLBT The general factorization for all symmetric LTs [60] can be viewed either as an extension of GenLOTs or as a generalization of the LBT. It can be shown that for M even, all LTs obeying (65) or (66) can be factorized as in (81) where the K i (z) factors are given in (82) with the matrices U i and V i (which compose Phi i ) being only required to ....

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T. D. Tran, R. de Queiroz, and T. Q. Nguyen, "The generalized lapped biorthogonal transform," Proc. Intl. Conf. Acoust., Speech Signal Proc., Seattle, Vol. III, pp. 1441--1444, May 1998.

....when more compression is needed, annoying blocking artifacts appear since the DCT bases are short and do not overlap, thus have discontinuities at block ends. The development of the lapped orthogonal transform [2] its generalized version GenLOT [3] and the extensions to biorthogonality [4] 5] [6] help solve the blocking problem by borrowing pixels from the adjacent blocks to produce the transform coefficients of the current block. Lapped transforms outperform the DCT on two counts: i) from the analysis viewpoint, it takes into account inter block correlation, hence, provides better ....

....representation is depicted in Figure 1 below. analysis bank synthesis bank x[n] y[n] R (z ) z 1 z 1 z 1 z 1 z 1 z 1 E (z ) M M M M M M Figure 1. Polyphase representation of an LPPRFB. The most general lattice for M channel lapped biorthogonal transforms (GLBT) is presented in [6]. The polyphase matrix E(z) can be factorized as E(z) GK Gamma1 (z) GK Gamma2 (z) Delta Delta Delta G1 (z) E0 ; 1) G i (z) 1 2 U i 0 0 V i I I I GammaI I 0 0 z Gamma1 I I I I GammaI 1 2 z 1 z 1 z 1 z 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 0 1 2 3 ....

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T. D. Tran, R. de Queiroz, and T. Q. Nguyen, "The generalized lapped biorthogonal transform," ICASSP, Seattle, May 1998.

....in Figure 1 below. analysis bank synthesis bank x[n] y[n] R (z ) z 1 z 1 z 1 z 1 z 1 z 1 E (z ) M M M M M M Figure 1. Polyphase representation of an LPPRFB. The most general lattice for M channel linear phase lapped biorthogonal transforms (GLBT) is presented in [3], 4] The polyphase matrix E(z) can be factorized as E(z) GK Gamma1 (z) GK Gamma2 (z) Delta Delta Delta G1 (z) E0 ; 1) G i (z) 1 2 U i 0 0 V i I I I GammaI I 0 0 z Gamma1 I I I I GammaI 4 = 1 2 Phi i W (z) W; and (2) 1 2 z 1 z 1 z 1 z 1 ....

....the AR(1) image model with ae = 0:95, the 8 channel LiftLT in Figure 4 achieves a coding gain of 9.54 dB. The comparison of complexity between the LiftLT and other popular transforms are tabulated in Table 1. Notice that the LiftLT s performance is already very close to that of the optimal GLBT [3], 4] 9.63 dB coding gain) whereas its complexity is the lowest amongst the transforms in comparison, excluding the DCT s. 5. APPLICATION IN IMAGE CODING To be fair, the same SPIHT s quantizer and entropy coder [7] are utilized to encode the coefficients of every transform. The transforms in ....

T. D. Tran, R. de Queiroz, and T. Q. Nguyen, "The generalized lapped biorthogonal transform," Proc. ICASSP, pp. 1441-1444, Seattle, May 1998.

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T. D. Tran, R. de Queiroz, and T. Q. Nguyen, "The generalized lapped biorthogonal transform," Proc. Intl. Conf. Acoust. Speech Signal Processing, Seattle, VOl. III, pp. 1441--1444, May 1998.

....We limit the discussions on lapped transforms to M channel uniform linear phase perfect reconstruction filter banks (LPPRFBs) where analysis and synthesis filters have the same length KM . The most general lattice for M channel linear phase lapped biorthogonal transforms (GLBT) is presented in [3, 4]. The polyphase matrix E(z) can be factorized as E(z) GK Gamma1 (z) GK Gamma2 (z) Delta Delta Delta G 1 (z) E 0 ; 1) 2 G i (z) 1 2 U i 0 0 V i # I I I GammaI # I 0 0 z Gamma1 I # I I I GammaI # 4 = 1 2 Phi i W (z) W; 2) and E 0 = 1 p 2 U 0 U 0 JM 2 ....

....structure G i (z) increases the filter length by M . All U i and V i , i = 0; 1; K Gamma 1, are arbitrary M 2 Theta M 2 invertible matrices. These free invertible matrices hold the free design parameters and they can be parameterized by the singular value decomposition (SVD) [3, 4]. The complete lattice of the analysis bank is depicted in Figure 1. 1 2 z 1 z 1 z 1 z 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 0 1 2 3 4 5 6 7 G 2 E 0 G K 1 U 0 V 0 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 0 F U 1 V 1 (z) 1 L W W (z) G 0 2 4 6 1 3 5 7 1 F ....

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T. D. Tran, R. de Queiroz, and T. Q. Nguyen, "The generalized lapped biorthogonal transform," Proc. ICASSP, pp. 1441-1444, Seattle, May 1998.

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T. D. Tran, R. de Queiroz, and T. Q. Nguyen, "The generalized lapped biorthogonal transform," Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, Seattle, May 1997.

....and references therein) On the other hand, research on integer coefficient M channel systems is still at an embryonic stage. Moreover, it has been shown recently that multi band transforms, when appropriately designed and utilized, can outperform state of the art wavelets by significant margins [3, 4, 5, 6]. The overlapping basis functions of these transforms can eliminate annoying blocking artifacts just as efficiently as any wavelets. An example of M channel transform with integer coefficients is the Walsh Hadamard transform (WHT) whose coefficients consist of either 1 or 1. However, the WHT is ....

....image coding comparison. To be fair, the same transform based progressive image coder SPIHT [9] is used in all cases. The difference lies at the transform stage where the transforms in comparison are: ffl 9 7 tap biorthogonal wavelet ffl DCT, 8 filters, all 8 tap ffl GLBT, 8 filters, all 16 tap [5] ffl IGLBT, 8 filters, some 8 tap, some 24 tap. In the latter three uniform band block transform cases, we used a modified zerotree structure where the transform coefficients are grouped in wavelet like quad trees and the DC band can have several levels of wavelet decomposition depending on the ....

T. D. Tran, R. de Queiroz, and T. Q. Nguyen, "The generalized lapped biorthogonal transform," ICASSP, Seattle, May 1998.

....the synthesis viewpoint, its overlapping basis functions decay asymptotically to zero at the ends, reducing blocking discontinuities drastically. Recent image coding results show that LTs, if designed and utilized appropriately, can outperform state of the art wavelets by significant margins [3] [4] in the same embedded coding framework [5] The most general lapped transforms reported recently in the literature are the Generalized Lapped Orthogonal Transform (GenLOT) 6] and the Generalized Lapped Biorthogonal Transform (GLBT) 4] 7] The term generalized here is somewhat misleading. The ....

.... state of the art wavelets by significant margins [3] 4] in the same embedded coding framework [5] The most general lapped transforms reported recently in the literature are the Generalized Lapped Orthogonal Transform (GenLOT) 6] and the Generalized Lapped Biorthogonal Transform (GLBT) [4], 7] The term generalized here is somewhat misleading. The elegant lattices in [4] 6] 7] can only realize systems with filter length KM . They are certainly not as general as claimed. This length restriction may confine to some extent the flexibility in the system design and implementation. ....

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T. D. Tran, R. de Queiroz, and T. Q. Nguyen, "The generalized lapped biorthogonal transform," Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, Seattle, May 1997.

....image block independently. So, inter block correlation has been completely abandoned. The development of the lapped orthogonal transform (LOT) 7] and its generalized versions: the lapped biorthogonal transform (LBT) 8] the generalized LOT (GenLOT) 9, 10, 11] and the generalized LBT (GLBT) [12, 13, 14] helps solve the blocking problem by borrowing pixels from the adjacent blocks to produce the transform coefficients of the current block. It has long been recognized that lapped transforms belong to a subclass of linear phase perfect reconstruction filter banks (LPPRFBs) M channel systems with ....

....solves the blocking problem in DCTbased coders by partly smoothing out the block boundaries. To reduce blocking effect further, longer overlaps might be needed. This motivates the development of the generalized lapped orthogonal transform (GenLOT) 9, 10, 11] and its biorthogonal versions (GLBT) [12, 13, 14]. Every FB presented in this paper has an efficient lattice structure that retains both LP and PR properties under quantization of lattice coefficients. The key idea behind the lattice structure is the factorization of the filter bank s polyphase matrix E(z) Let H k (z) and F k (z) be the ....

T. D. Tran, R. de Queiroz, and T. Q. Nguyen, "The generalized lapped biorthogonal transform, " Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, Seattle, May 1997.

....appropriately designed generalized lapped orthogonal biorthogonal transforms (GenLOT or GLBT) coupled with several levels of wavelet decomposition of the DC band if needed. The uniform band block transform can be implemented in parallel fashion with fast, robust, and efficient lattice structures [7, 8]. Keying only on the transform stage, we are able to obtain the highest performance embedded coder up to date. This result shows that lapped transforms, when carefully designed, are capable of providing superior reconstructed image quality comparing to wavelets, both objectively and subjectively. ....

....and unequal length filters for ringing artifacts suppression. A comparison of various transform characteristics is presented in Table 1. More details in transform design procedure can be found in [13] For the GLBT s lattice structure, design, and optimization, the reader is referred to [8]. 4. CODING RESULTS The objective coding results (PSNR in dB) for standard 512 Theta 512 Lena, Barbara, and Goldhill test images are tabulated in Table 2. The transforms in comparison are: the popular 8 Theta 8 DCT shown in Figure 4(a) the aforementioned 8 Theta 40 GenLOT, 8 Theta 16 GLBT, ....

T. D. Tran, R. de Queiroz, and T. Q. Nguyen, "The generalized lapped biorthogonal transform," submitted to ICASSP 98, Seattle, in Nov. 1997.

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