QUANTIZATION OF ADAPTIVE 2D WAVELET ... - Semantic Scholar

QUANTIZATION OF ADAPTIVE 2D WAVELET DECOMPOSITIONS B´eatrice Pesquet-Popescu∗

Henk Heijmans, G.C.K. Abhayaratne† , Gemma Piella‡

ENST Signal and Image Proc. Dept. 46, rue Barrault, 75634 Paris, France ABSTRACT Classical linear wavelet representations of images have the drawback that they are not well-suited to represent edge information. To overcome this problem, nonlinear multiresolution decompositions are being designed that can take into account the characteristics of the input signal/image. In our previous work [1, 2] we have introduced an adaptive lifting framework, that does not require bookkeeping but has the property that it processes edges and homogeneous regions in an image in a different fashion. The current paper discusses the effects of quantization in such an adaptive wavelet decomposition. We provide conditions for recovering the original decisions at the synthesis and for relating the reconstruction error to the quantization error. Such an analysis is essential for the application of these adaptive decompositions in image compression algorithms. 1. INTRODUCTION Multiscale analyses deriving from classical linear wavelet transforms lead to a uniform smoothing of the information contents in images when going to low resolutions. There is however a strong demand from various applications in image and video processing (compression, denoising, feature extraction, ...) for a more “semantic” analysis and thereby for multiresolution (MR) decompositions that do leave intact or even enhance certain important image characteristics such as sharp transitions, edges, singularities or other regions of interest over scales. To a certain extent, this can be achieved by non-linear wavelets [3], [4], [5], [6] or by adaptive subband structures [7]. In [1, 2] we have presented a new method for the construction of adaptive wavelets using update lifting. This method exploits the properties of seminorms to build lifting structures able to choose between different update filters, the choice being triggered by the local gradient-type features of the input. Moreover, we have established conditions under which these decisions can be recovered at synthesis, without the need for transmitting additional overhead information. In [8] we have applied this general framework to construct nonlinear MRA’s for images using a quadratic seminorm. However, in order to use such adaptive wavelet schemes in lossy image compression, one needs to take into account the ef∗ The work of Heijmans and Pesquet-Popescu is partially supported by the IST Programme of the EU under contract number IST-2000-26467 (MASCOT). † The work of Abhayaratne was supported by the ERCIM fellowship programme ‡ The work of Piella is supported by the Dutch Technology Foundation STW, project number CWI 4616.

CWI Kruislaan 413, 1098 SJ Amsterdam The Netherlands fects of the quantization on the adaptive scheme. In particular, the quantization errors can dramatically affect the recovery of the switching decisions at synthesis. In this paper, we derive conditions that guarantee that we can recover the original decisions. We also analyse the relations between the different parameters of the system allowing to compute upper bounds for the reconstruction error. Rate-distortion curves of linear and adaptive schemes are compared and the reduction of visual artifacts will be investigated in a real image compression setting. The paper is organized as follows: in Section 2 we outline the framework of adaptive update lifting based on seminorms and recall the main results for perfect reconstruction. In Section 3 we extend the scheme with quantization modules for all channels and in Section 4 we present some simulation results. Concluding remarks are made in Section 5. 2. ADAPTIVE UPDATE LIFTING: A REMINDER Assume that an input signal x0 : Z2 → IR is split into two polyphase components x10 and y01 , i.e., (x10 , y01 ) = S(x0 ), where S is an invertible mapping. The channels x10 , y01 pass through a two-stage lifting system comprising an adaptive update lifting step which returns x1 (n) = x10 (n) ⊕dn Udn (y01 )(n) ,

(1)

and a fixed prediction step yielding y 1 (n) = y01 (n) − P (x1 )(n) .

(2)

2

In these expressions n ∈ Z and dn = D(x10 , y01 )(n) ∈ {0, 1} is the decision at sample n which triggers the update filter Ud and the addition ⊕d . We assume that the addition ⊕d is of the form x⊕d u = αd (x+u), with αd 6= 0, which means that the operation ⊕d is invertible. The update filter is taken to be of the form Ud (y)(n) =

N X

λd,j yj (n) ,

(3)

j=1

where yj (n) = y(n + lj ), with lj ∈ Z2 given, and N ≥ 1. A typical case, which will be studied in more detail in Section 4, is the one where the splitting operator S corresponds with the polyphase decomposition in the quincunx grid where x10 , y01 are defined at all points n = (n1 , n2 ) with n1 + n2 is even and odd, respectively. Taking N = 4, we choose lj in such a way that the samples y01 (n + lj ) correspond with the four horizontal and vertical neighbours of x10 (n1 , n2 ). From (1) and (3) we get that: X (4) βdn ,j y01 (n + lj ) , x1 (n) = αdn x10 (n) + j

where βd,j = αd λd,j . Presumed that dn is known, x10 (n) can easily be recovered by inverting (4): x10 (n) =

X
1 βdn ,j y01 (n + lj ) . x1 (n) − j α dn

x0

x1

W

(5)

Q

Unfortunately, it is not true in general that dn can be recovered from x1 (instead of x10 ) and y01 . It is this problem that makes adaptive lifting a tedious approach. In our previous work [2] we have discussed a framework for which the recovery of the decisions dn = D(x10 , y01 )(n) from x1 and y01 can be guaranteed. We assume that the value dn at analysis was computed from a decision map D given by

Q

~1 y

y1

x2

W

y2

x K-1

x0k

j = 1, . . . , N ,

p is a seminorm, and T is a threshold. It has been shown in [2] that a necessary condition for perfect reconstruction is that αd + P j βd,j is independent of d. Without loss of generality it can be assumed that X βd,j = 1 for d = 0, 1 . αd + j

It can easily be shown that the gradient vector v 1 with components vj1 (n) = x1 (n) − y01 (n + lj ) is related to v01 by means of the linear relation v 1 = Ad v01 , where Ad = I − uβdT with u = (1, . . . , 1)T and βd = (βd,1 , . . . , βd,N )T . Here the superscript ‘T ’ denotes transposition. In order to have perfect reconstruction it suffices that we can compute

xk

Γ

S

v0k

D

dk

at synthesis, e.g., through the relation dn = [p(v 1 (n)) > T 0 ] , where T 0 is a threshold, possibly different from T . In [2] we have shown that the condition p(A0 )p−1 (A1 ) ≤ 1 ,

(6)

guarantees perfect reconstruction. This means in particular that p(A0 ) and p−1 (A1 ) have to be finite. Here p(A)

=

sup{p(Av) | v ∈ IRN and p(v) = 1}

p−1 (A)

=

sup{p(v) | v ∈ IRN and p(Av) = 1} .

P

x k -1

-

y0k

Finally, we complete the scheme with a prediction lifting step P (a linear filter in most cases) which modifies the detail channel as in (2). The combination of the adaptive update lifting with the fixed prediction lifting yields an adaptive wavelet decomposition W mapping x0 into (x1 , y 1 ).

Q

y~ k

Fig. 1. The analysis part of an MR decomposition with quantization. This decomposition results from a concatenation of adaptive wavelets W (top) comprising a splitting S, a decision map D, an adaptive update lifting U and a fixed prediction P (bottom).

3. ADDING QUANTIZATION Starting from the adaptive wavelet decomposition W we build a quantized MR (multiresolution) decomposition as shown in Fig. 1. In this illustration, Γ is the gradient operator, e.g., v01 = Γ(x10 , y01 ). Applying W to x0 gives output x1 and y 1 . Then y 1 is quantized by Q yielding y˜1 = Q(y 1 ). The approximation signal x1 is decomposed again by the same W , etc. After a K-level decomposition we have K quantized detail signals y˜1 , . . . , y˜K and an approximation signal xK which is also quantized by Q to yield x ˜K . The synthesis scheme that we use is depicted in Fig. 2. Here U −1 and S −1 denote the inverse of the update lifting and the splitting, and Q−1 is a right inverse of Q, i.e., QQ−1 (t) = t for t in the range of Q. We assume that |Q−1 Q(y) − y| ≤ νq ,

dn = D(x10 , y01 )(n)

yK

U

yk

where [P ] returns 1 if the predicate P is true, and 0 otherwise; v01 (n) ∈ IRN is the gradient vector with coefficients

~

yK

D(x10 , y01 )(n) = [p(v01 (n)) > T ] ,

1 v0,j (n) = x10 (n) − y01 (n + lj ) ,

Q

Q

~

y2

x~ K

xK

W

(7)

where q is the quantization parameter. For example, if Q(z) = {z/q}, where {·} denotes rounding to the closest integer, then ν = 1/2. Henceforth we shall restrict ourselves to this particular case. Obviously, since the quantization step discards information, the synthesis scheme does not invert the analysis part. Below we derive estimates for the difference between the original signal x0 and the synthesized signal x ˆ0 . In particular, we derive conditions that guarantee that we can recover each original decision dk at synthesis. This is important, since ‘wrong’ decisions at synthesis inevitably lead to ‘bad’ reconstructions. Before we state the precise result, the proof of which can be found in [9], we give some notation. For a vector y ∈ IRN , the notation |y| denotes the `∞ -norm, i.e., |y| = max{|y1 |, . . . , |yN |}. We denote by S, SN the spaces of signals x : Z2 → IR and y : Z2 → IRN , respectively. If x ∈ S, then kxk = sup{|x(n)| : n ∈ Z2 } , denotes the sup-norm. Similarly, if y ∈ SN , then kyk = sup{|y(n)| : n ∈ Z2 } ,

~ xK

W -1

~K y

Q -1

W -1

^K y

~K-1 y

Q -1

4. A CASE STUDY

^0 x

^1 x

^K-2 x

^K-1 x

^K x

Q-1

W -1

-1 ~1 Q y

yK-1

^k x

y1

^k x 0 U -1 Γ

P

+

Q -1

~k y

^k v

D

dk

^k-1 x S-1

^k y 0

^k y

Fig. 2. The synthesis part of an MR decomposition with quantization.

also denotes the sup-norm. In the last expression, however, | · | represents the `∞ -norm. Proposition 1 Assume that ρ, πx , πy > 0 are constants such that kP (x) − P (x0 )k ≤ ρkx − x0 k , x, x0 ∈ S ,
p Γ(x0 , y 0 )(n) − Γ(x, y)(n) ≤ πx kx − x0 k + πy ky 0 − yk ,

We consider the case where the polyphase decomposition corresponds with the quincunx sampling in Z2 and where p(v) = |aT v|. In order to satisfy (8) we need that |α0 | ≤ |α1 | and βd = γd a, γd ∈ IR, for d = 0, 1. Now, (8) yields that 2τ |α1 | − |α0 | ≥ . (12) T As already mentioned in Section 2, we take N = 4 and choose lj in such a way that the samples y(n + lj ) correspond with the four horizontal and vertical neighbours of an ‘even’ sample n. The prediction of y(n+lj ) is computed by averaging its four horizontal and vertical neighbours and we get that ρ = 1 in this case. We assume that a = (1, 1, 1, 1)T and now it is easy to show that πx = πy = 4. We find that n 1−α 1 − α1 o 0 |, | | . θx = |α0 |−1 and θy = max | α0 α1 Thus µ = max{1, θx + θy }. If we assume that |α0 | < 1, then we can show that µ > 1 and hence ∆max = ∆0 . Now take α1 = 1 and 12 ≤ α0 < 1. We get that λ = 1 and µ = 2/α0 − 1, hence 1 < µ ≤ 3. Furthermore, τ = 8∆0 + 2q with ∆0 = q (1 + µ + · · · + µK ) and (12) can be rewritten as 2 T µ + 1 µK+1 − 1  ≥4· 1+2· . q µ−1 µ−1 Rate−Distortion Plots for Barbara image 704x576 45

for x, x0 ∈ S and y, y 0 ∈ SN . Define =

θy

=

1 1 , } |α0 | |α1 | PN PN j=1 |β0,j | j=1 |β1,j | max{ , }, |α0 | |α1 |

max{

40

Distortion − PSNR (dB)

θx

Non−adaptive d=0 Adaptive

and λ = max{1, θy } and µ = max{ρ, θx + θy ρ} . Furthermore, define1  q ∆k = λ(1 + µ + · · · + µK−k−1 ) + µK−k , 2 ∆max = max{∆0 , . . . , ∆K } , τ = πx ∆max + πy (ρ∆max + q/2) . Assume that −1 − p(A0 ) ≥ p(A−1 1 )

2τ , T

20

(8)

dk = [p(ˆ v k ) > Tˆ] for k = 1, . . . , K

(10)

kˆ xk − xk k ≤ ∆k for k = 0, . . . , K .

(11)

and

1 Note

that ∆k = µ∆k+1 + λq/2 with ∆K = q/2.

0

0.5

1

1.5

2 Rate (bpp)

2.5

3

3.5

4

Fig. 3. Rate-distortion curve. The solid curve corresponds with the adaptive scheme, the dashed curve corresponds with the nonadaptive scheme with d = 0.

(9)

Then we have

In the next section we work out a particular case.

30

25

and choose a threshold Tˆ such that −1 T −τ. p(A0 )T + τ ≤ Tˆ ≤ p(A−1 1 )

35

In our simulations we choose α0 = 2/3, α1 = 1, K = 2 and T = 15. Note that although the conditions in Proposition 1 are not satisfied here, the results as shown in Figures 3-4 are surprisingly good. In Fig. 3 we show the rate-distortion curve for the Barbara image. The images in Fig. 4 show the approximations at half the resolution for q = 32. The top image results from our adaptive scheme, the bottom image from the non-adaptive scheme with d = 0. Because of the adaptive filtering, the first image is less blurred than the second one, especially for high-frequency textures.

6. REFERENCES [1] G. Piella and H. J. A. M. Heijmans, “Adaptive lifting schemes with perfect reconstruction,” IEEE Transactions on Signal Processing, vol. 50, no. 7, pp. 1620–1630, July 2002. [2] H. J. A. M. Heijmans, B. Pesquet-Popescu, and G. Piella, “Building nonredundant adaptive wavelets by update lifting,” Research Report PNA-R0212, CWI, Amsterdam, 2002. [3] H. J. A. M. Heijmans and J. Goutsias, “Nonlinear multiresolution signal decomposition schemes: Part II: morphological wavelets,” IEEE Transactions on Image Processing, vol. 9, no. 11, pp. 1897–1913, 2000. [4] F. J. Hampson and J.-C. Pesquet, “A nonlinear subband decomposition with perfect reconstruction,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Atlanta, Georgia, May 7-10, 1996, pp. 1523–1526. [5] E. Le Pennec and S. Mallat, “Image compression with geometrical wavelets,” in Proceedings of the IEEE International Conference on Image Processing, Vancouver, September 2000. [6] B. Matei and A. Cohen, “Compact representations of images by edge adapted multiscale transforms,” Tech. Rep., Univ. Pierre et Marie Curie, 2002. ¨ N. Gerek and A. E. Cetin, “Adaptive polyphase sub[7] O. band decomposition structures for image compression,” IEEE Transactions on Image Processing, vol. 9, pp. 1649–1659, October 2000. [8] H. J. A. M. Heijmans, G. Piella, and B. Pesquet-Popescu, “Building adaptive 2D wavelet decompositions by update lifting,” Rochester, USA, 2002, ICIP 2002.

Fig. 4. Approximation images at half resolution for q = 32. Top image: the adaptive scheme. Bottom image: the non-adaptive scheme for d = 0.

5. CONCLUSIONS AND FUTURE WORK

In this paper we have analyzed the quantization effects in an adaptive lifting scheme that does not require bookkeeping. We have derived sufficient conditions for recovering the original decisions at the synthesis and we have shown how the reconstruction error can be bounded by a term related to the quantization error. We have also shown a rate-distortion curve that shows an improvement of the adaptive scheme for low bit rates. Perhaps even more important is that our scheme gives rise to a substantial reduction of the visual artifacts. It thus provides subbands with enhanced quality, and as such better suited for spatially scalable image compression algorithms.

[9] H. J. A. M. Heijmans, B. Pesquet-Popescu, G.C.K Abhayaratne, and G. Piella, “Quantisation for adaptive update lifting wavelets,” in preparation.