T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linear phase perfect reconstruction filter bank: lattice structure, design, and application in image coding," IEEE Trans. on Signal Proc., vol. 48, pp. 133--147, Jan. 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....into blocks, and transforming the blocks independently. The particular block transforms we use will be those that have previously been used for image compression; this includes the popular discrete cosine transform [46] as well as many lapped transforms introduced by Malvar [36] and Tran et al. [53]. Although zerotree coding is typically used to code wavelet transformed coefficients, it has also been applied to block transforms with some success. Recall that the zerotree technique was motivated by the multi resolution structure of the dyadic wavelet decomposition, where coefficients could ....

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen. Linear-phase perfect reconstruction filter bank: Lattice structure, design, and application in image coding. IEEE Transactions on Signal Processing, 48(1):133--147, January 2000.

....t.tanaka riken.jp) Yukihiko Yamashita is with the Graduate School of Science and Engineering, Tokyo Institute of Technology, Japan. E mail: yamasita ide.titech.ac. jp) 1 Introduction Lapped transforms are powerful tools for signal and image processing, and have been investigated extensively [1 4]. Cassereau [5] and Malvar [1] pioneered the original class of lapped transforms called the lapped orthogonal transform (LOT) The LOT has been developed as a competitive alternative of traditional block transforms like the discrete cosine transform (DCT) 6] because of its extended basis ....

.... the generalized lapped orthogonal transform (GenLOT) 3, 9] These LP FB s are restricted to PU solutions which are in a subclass of maximally decimated M channel LP perfect reconstruction (PR) filter banks (LPPRFB s) Recently, a general factorization for these LPPRFB s has been developed in [4]. From the lapped transform perspective, this FB is called the generalized lapped biorthogonal transform (GLBT) It is shown that the lattice consists of nonsingular matrices and delays of which the number is minimal [4] In all LPPRFB s with the lattice structure in the literature, however, the ....

[Article contains additional citation context not shown here]

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linear-phase perfect reconstruction filter bank: Lattice structure, design, and application in image coding," IEEE Trans. Signal Processing, vol. 48, pp. 133--147, Jan. 2000. 21

....wavelet case, moreover, the construction of locally adaptive basis functions is very difficult since double shift orthogonality biorthogonality is required for perfect reconstruction. Also, lapped transforms (LT) are powerful tools for the reduction of the blocking artifacts in image compression [11 15]. The blocking artifacts are reduced by overlap ping basis functions of which size is larger than the block size. It is however difficult to construct a space varying LT [16] since the LT has a strong constraint such that the overlapping parts of the basis functions must be orthogonal or ....

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linear-phase perfect reconstruction filter bank: Lattice structure, design, and application in image coding," IEEE Trans. Signal Process- ing, vol. 48, pp. 133-147, Jan. 2000.

.... In this paper, we extend the Group Testing for Wavelets (GTW) algorithm [1] to apply to alternative transforms which have previously been used for image compression, including the wavelet packet transform, the discrete cosine transform (DCT) and several versions of the lapped transform [4] [5]. As presented in [1] the group testing framework transforms an image and then encodes the resulting transform coefficients in a bit plane order with many different adaptive group testers. For efficient compression, the coefficients are divided into classes whose coefficients have similar ....

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linear-phase perfect reconstruction filter bank: Lattice structure, design, and application in image coding," IEEE Transactions on Signal Processing, vol. 48, no. 1, pp. 133--147, January 2000.

....form for paraunitary matrices of a given order. This is the initial motivation of this work. With constraints such as linear phase and or pairwise mirror image symmetry on subband filters, complete and minimal lattice structures of the paraunitary polyphase matrices have been developed [12] [18]. The constrained paraunitary matrix of order can be expressed as the product of order one paraunitary building blocks and an additional unitary matrix. These factorizations provide efficient structures for implementing linear phase and mirrorimage paraunitary filterbanks with length constraint. ....

....as in (9) the order of produced by this structure with arbitrary unitary matrices is not necessarily equal to , i.e. can be the zero matrix. However, the order cannot be higher than . This situation appears in other special factorization forms for special classes of filterbanks [8] 13] [18]. To ensure that , one needs to impose additional constraint on and such that diag (19) In practice, it seems that there is no reason to do so. If can take arbitrary integer values from 0 to , the whole space of produced by (9) with arbitrary unitary matrices is composed of causal paraunitary ....

T. D. Tran, "Linear phase perfect reconstruction filterbank: Theory, structure, design, and application in image compression," Ph.D. dissertation, Univ. Wisconsin, Madison, 1998.

....the noise component and attains PR. Numerical results show that this class of filter banks is e#ective in transmission of oversampled data over noisy channels. 1. INTRODUCTION Lapped transforms are powerful tools for signal and image processing due to their ability to reduce the blocking e#ects [1, 2]. Malvar et al. pioneered the original class of lapped transforms called the lapped orthogonal transform (LOT) 1] For a segmented signal with M samples, the LOT has M basis functions of length 2M and therefore generates M coe#cients. Indeed, the LOT is a subclass of maximally decimated M channel ....

.... the generalized lapped orthogonal transform (GenLOT) 4] These LP FB s are restricted to PU solutions which are in a subclass of maximally decimated M channel LP perfect reconstruction (PR) filter banks (LPPRFB s) Recently, the general factorization for these LPPRFB s has been developed in [2]. From the lapped transform perspective, this FB is called the generalized lapped biorthogonal transform (GLBT) It is shown that the lattice consists of nonsingular matrices and delays of which the number is minimal [2] In these LPPRFB s with the lattice structure, however, the number of ....

[Article contains additional citation context not shown here]

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linearphase perfect reconstruction filter bank: Lattice structure, design, and application in image coding," IEEE Trans. Signal Processing, vol. 48, pp. 133--147, Jan. 2000.

....form for paraunitary matrices of a given order. This is the initial motivation of this work. With constraints such as linear phase and or pairwise mirror image symmetry on subband lters, complete and minimal lattice structures of the paraunitary polyphase matrices 2 have been developed [12] [18]. The constrained paraunitary matrix of order K 1 can be expressed as the product of K 1 order one paraunitary building blocks and an additional unitary matrix. These factorizations provide ecient structures for implementing linear phase and mirror image paraunitary lter banks with length ....

....(9) the order of E(z) produced by this structure with arbitrary unitary matrices U k is not necessarily equal to K 1, i.e. EK 1 can be the zero matrix. However the order can not be higher than K 1. This situation appears in other special factorization forms for special classes of lter banks [8][13 18]. To ensure that EK 1 6= 0, one needs to impose additional constraint on U k and r k such that 1 Y k=K 1 fU k diag(0; I r k )gU 0 6= 0: 19) In practice, it seems that there is no reason to do so. If r k can take arbitrary integer values from 0 to M , the whole space of E(z) produced by (9) ....

T. D. Tran, Linear Phase Perfect Reconstruction Filter Bank: Theory, Structure, Design, and Application in Image Compression, Ph.D. Thesis, University of Wisconsin, May 1998.

.... H8F P l k 4 B8 ND 0LAj40A4:F9= U # k P s NB N l9g , 3 NGLPBT K h C FI=8= G k 3 H r ( 9 H H b K U # k N 7WNc r ( 7 M 8z r3NG 9 k 1. INTRODUCTION Lapped transforms are powerful tools for signal and image processing, and has been investigated extensively [1 4]. Cassereau [5] and Malvar [1] pioneered the original class of lapped transforms called the lapped orthogonal transform (LOT) The LOT has been developed as a competitive alternative of traditional block transforms like the discrete cosine transform (DCT) 6] because of its extended basis ....

.... the generalized lapped orthogonal transform (GenLOT) 3, 9] These LP FB s are restricted to PU solutions which is in a subclass of maximally decimated M channel LP perfect reconstruction (PR) filter banks (LPPRFB s) Recently, the general factorization for these LPPRFB s has been developed in [4]. From the lapped transform perspective, this FB is called the generalized lapped biorthogonal transform (GLBT) It is shown that the lattice consists of nonsingular matrices and delays of which the number is minimal [4] In all LPPRFB s with the lattice structure in the literature, however, the ....

[Article contains additional citation context not shown here]

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linearphase perfect reconstruction filter bank: Lattice structure, design, and application in image coding," IEEE Trans. Signal Processing, vol. 48, pp. 133--147, Jan. 2000.

No context found.

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linear-phase perfect reconstruction filter bank: Lattice structure, design, and application in image coding," IEEE Trans. Signal Processing, vol. 48, pp. 133--147, Jan. 2000.

....GenLOT is developed on the framework of linear phase paraunitary filterbanks (LPPUFB) 5] 6] In [7] the complete and minimal factorization of even channel LPPUFB via the lattice structure is developed. The structure is generalized to cover a large class of perfect reconstruction filterbanks in [8]. Recently, it was shown in [9] and [10] that the structures in [7] and [8] can be simplified significantly and still cover the same solution sets. Since the DCT, LOT, and GenLOT are all special cases of LPPUFB, the theory of LPPUFB is very powerful in the design and analysis of such transforms. ....

....(LPPUFB) 5] 6] In [7] the complete and minimal factorization of even channel LPPUFB via the lattice structure is developed. The structure is generalized to cover a large class of perfect reconstruction filterbanks in [8] Recently, it was shown in [9] and [10] that the structures in [7] and [8] can be simplified significantly and still cover the same solution sets. Since the DCT, LOT, and GenLOT are all special cases of LPPUFB, the theory of LPPUFB is very powerful in the design and analysis of such transforms. In fact, the LOT in [2] is an elegant fast approximation of the ....

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linear-phase perfect reconstruction filter bank: Lattice structure, design, and applications in image coding," IEEE Trans Signal Processing, vol. 48, pp. 133--147, Jan. 2000.

....that the order one factorization is complete; any causal FIR paraunitary matrix of order is allowed. Our approach is based on the singular value decomposition (SVD) of coefficient matrices. Notice that the SVD has also been used in the design of channel linear phase biorthogonal filterbanks in [26]. The difference is that it was used to represent invertible matrices that appear in the proposed lattice structure, whereas we use it to derive the structure of a given paraunitary filterbank. We develop a more efficient structure for the design and implementation in Section III. The CS ....

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linear-phase perfect reconstruction filterbank: lattice structure, design and application in image coding," IEEE Trans. Signal Processing, vol. 48, pp. 133--147, Jan. 2000.

....of the longer bases in F(z) is very different than the corresponding ones in A(z) The effect of post processing few bases is a means to construct a new LT with larger bases from an initial one. In fact it can be shown that variable length LTs can be factorized using post processing stages [59][58]. A general factorization of LTs is depicted in Fig. 16. Assume a variable length F(z) whose bases are arranged in decreasing length order. Such a PTM can be factorized as F(z) M Gamma2 Y i=0 B i (z) 0 0 I i (63) where I 0 is understood to be non existing and B i (z) has size (M Gamma ....

T. D. Tran, Linear phase perfect reconstruction filter banks: theory, structure, design, and application in image compression, Ph.D. thesis, University of Wisconsin, Madison, WI, May 1998.

.... Theta N DL and (M Gamma N) Theta (M Gamma N) DS are diagonal matrices whose entries are 1 when the corresponding filter is symmetric and Gamma1 when the corresponding filter is antisymmetric. EL (z) now forms a remarkably similar system to an N channel order (K Gamma 1) GLBT [6] From [6] [11], there always exists a factorization similar to the one shown in Eq. 1) that reduces the order of the polyphase matrix EL(z) by one. Hence, the VLGLBT s polyphase matrix E(z) can always be factorized as follows (the N 2 long symmetric filters are arranged on top) E(z) G0 (z) EK Gamma2 ....

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linear phase perfect reconstruction filter bank: lattice structure, design, and application in image coding," submitted to IEEE Trans. on Signal Processing, Apr. 1998.

....N cannot be odd. 2 Theorem II. Furthermore, half of the long filters are symmetric, and half of the short filters are symmetric. The proof is omitted here due to the lack of space. It is constructed analogously to that of orthogonal systems presented in [8] The complete proof can be found in [9] and will appear in the full version of this paper [10] 2.3. Variable length Lattice Let EL(z) be the N Theta M polyphase matrix of order (K Gamma 1) representing the long analysis filters, and ES (z) be the (M Gamma N) Theta M polyphase matrix of order (K Gamma 2) representing the ....

....= E Gamma1 0 z Gamma1 G Gamma1 1 (z) Delta Delta Delta z Gamma1 G Gamma1 K Gamma2 (z) z Gamma1 G0 (z) We should also mention that the factorization in Eq. 7) can be proven to be minimal, i.e. the resulting lattice employs the least number of delays in the implementation [9]. Of course, more VL structure G i (z) can be added to increase the frequency resolution of the long filters. Each G i (z) block increases the length of N i filters by M and leaves the rest intact. The most general lattice for the VLGLBT is shown in Figure 5. The invertible matrices U i , V i ....

T. D. Tran, Linear phase perfect reconstruction filter banks: theory, structure, design, and application in image compression, Ph.D. thesis, University of Wisconsin, Madison, WI, May 1998.

....when the DFT is combined with the 8 8 DCT, the 8 16 lapped transform, the 3 level and 4 level wavelet transform. The corresponding implementations are denoted by DCTDFT, LTDFT, WT3DFT, and WT4DFT, respectively. The lapped transform used in this work is the Lapped Bi orthogonal Transform (LBT) [28], which can completely eliminate the blocking artifact of DCT. Biorthogonal Daubechies 9 7 wavelet is used for WT3DFT and WT4DFT. We also implemented two other algorithms for comparison purposes. The rst one is the global DCT based method in [3] 4] for which we provide two results with ....

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, \Linear Phase Perfect Reconstruction Filter Bank: Lattice Structure, Design, and Application in Image Coding," IEEE Trans. on Signal Processing, Vol. 48, No. 1, pp. 133-147, Jan. 2000.

....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 ....

....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. L. de Queiroz, and T. Q. Nguyen, "Linear phase perfect reconstruction filter bank: lattice structure, design, and application in image coding," IEEE Trans. on Signal Processing, to appear.

No context found.

T. D. Tran, Linear phase perfect reconstruction filter banks: theory, structure, design, and application in image compression, Ph.D. thesis, University of Wisconsin, Madison, WI, May 1998.

No context found.

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linear phase perfect reconstruction filter bank: lattice structure, design, and application in image coding," IEEE Trans. on Signal Processing, Vol. 44, Jan. 2000.

....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 ....

[Article contains additional citation context not shown here]

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linear phase perfect reconstruction filter bank: lattice structure, design, and application in image coding," IEEE Trans. on Signal Processing, Jan. 2000.

....this result, we next derive a complete and minimal factorization for all even channel solutions. This is the most general LT ever reported in the literature. Several design examples obtained from the novel lattice structure are presented. They are compared to the GenLOT [62] and the GLBT [83] in Chapter 4 (both are special cases when fi = 0) in coding gain, stopband attenuation, and attenuation at DC as well as around mirror frequencies. 5.2 Existence conditions Throughout this chapter, the class of M channel FB under investigation still possesses all of the properties in the Problem ....

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linear phase perfect reconstruction filter bank: lattice structure, design, and application in image coding," submitted to IEEE Trans. on Signal Processing in Apr. 1998.

....filter length increment in the case of even M must be at least two taps at a time, or the number of overlapping samples must be even in the LT language. Similarly, in an oddchannel LPPR system, the number of overlapping samples must be odd. For details of the proof, the reader is referred to [8] [9]. 3. GENERAL LATTICE STRUCTURE It can be shown easily that the same structure in [4] 7] propagating the LP and PR properties) can be exploited to obtain a similar factorization of the arbitray length LT s polyphase matrices. Theorem II. Suppose there exists an M channel FIR LPPRFB with all ....

....where the propagating structure is G(z) of order N , i.e. G(z) P N i=0 A i z Gammai . Then, E(z) has LP and PR if and only if ffl G(z) is FIR invertible. ffl G(z) takes the form G(z) z GammaN D G(z Gamma1 ) D: ffl A i = D AN Gammai D: The proof of Theorem II can be found in [9] and will be presented in the final version of the paper. With this result, we can construct high order even channel LPPRFBs by cascading many basic building blocks G i (z) The polyphase matrix of the analysis bank is E(z) GK Gamma1 (z) GK Gamma2 (z) Delta Delta Delta G2 (z) G1 (z) ....

T. D. Tran, "Linear Phase Perfect Reconstruction Filter Bank: Theory, Structure, Design, and Application in Image Compression," Ph.D. Thesis, University of

.... 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. From ....

....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. From a design point of view, a large increase in length means a higher dimension non linear ....

[Article contains additional citation context not shown here]

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linear phase perfect reconstruction filter bank: lattice structure, design, and application in image coding," submitted to IEEE Trans. on Signal Processing, Apr. 1998.

....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 ....

[Article contains additional citation context not shown here]

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linear phase perfect reconstruction filter bank: lattice structure, design, and application in image coding," submitted to IEEE Trans. on Signal Processing in Apr. 1998.

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T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linear phase perfect reconstruction filter bank: lattice structure, design, and application in image coding," IEEE Trans. on Signal Proc., vol. 48, pp. 133--147, Jan. 2000.

No context found.

T. D. Tran, R. L. de Queiroz, and T. Q. Nguyen, "Linear-phase perfect reconstruction filter bank: Lattice structure, design, and application in image coding," IEEE Trans. Signal Processing, vol. 48, pp. 133--147, Jan. 2000.

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