DISCRETE WAVELET TRANSFORMS THAT HAVE AN ADAPTIVE LOW PASS FILTER G. Charith K. Abhayaratne Centre for Mathematics and Computer Science (CWI) P.O. Box 94079, 1090 GB Amsterdam, The Netherlands. Email:
[email protected]. ABSTRACT The design and performance analysis of wavelet transforms containing an adaptive low pass filter using the lifting framework is presented. Such transforms are useful in applications where it is undesirable to have smoothing artifacts in the low pass signal and it requires the signal singularities, sharp transitions and image edges to be left intact for further analysis and better visual quality of such low pass subbands. We show that the wavelet coefficients after the prediction lifting step can be used to compute a weighted mean gradient for each signal component of the low pass subband, magnitude of which can be used as a metric for spatially adaptive selection of low pass filters. The same selection can be realised in the synthesis without extra information being sent to the synthesis. Two examples using the 5/3 wavelet are shown and their lossless image coding and spatial scalability performances are analysed. 1. INTRODUCTION Discrete wavelet transforms are widely considered as a powerful tool for signal representation especially in image coding, analysis and processing applications. In recent years, they have become the preferred choice of signal transform in image coding due to their ability for better signal decorrelation, energy compaction and multiresolution analysis.The last property has lead to the design of progressive and embedded image coding algorithms with both spatially and distortion wise scalable decoding capability. Linear wavelet transforms with fixed basis functions, or filter banks with fixed filter coefficients result in uniform smoothing in low pass subbands. This leads to smoothing at edges in images, sharp transitions and other singularities in signals, resulting in visually unacceptable low resolution images in applications like spatially scalable image / video coding or low pass subbands with smoothing artifacts. They are unacceptable for further analysis in transform domain. Transform domain image analysis, feature extraction This work is supported by the ERCIM postdoctoral fellowship programme.
and motion estimation / analysis are becoming useful applications as this helps performing above tasks without fully decoding or synthesis transforming a coded signal/image. Usually in such applications, it is desirable to have wavelet transforms that result in small wavelet coefficients in high pass subbands while leaving the signal singularities, transitions and edges intact in the low pass subband. In this paper we present the design of wavelet transforms that have an adaptive low pass filter that takes into account the underlying signal content. This design is based on the lifting framework [1, 2] of wavelet transforms. The lifting framework has lead to designing of adaptive and nonlinear wavelet transforms [3, 4, 5, 6, 7] recently. The lifting scheme, summarised in section 2 consists of 3 main steps: Split, Prediction and Update. In [3, 4], an adaptive prediction step, where the adaptive switching between short and long filters based on the local edges of the input signal has been considered. In this case, the update lifting step, which is fixed, precedes the adaptive prediction step, so that the preserving of the running average of the input signal is not affected by the adaptive prediction. In [5, 6], an adaptive update lifting scheme followed by a fixed prediction has been developed. The main objective of this method is to achieve adaptive smoothing in the low pass signal. However, the wavelet coefficients, i.e., the high pass subbands are affected by the adaptive update process. In both above cases, the authors used a update-first lifting scheme, which is different from the classical lifting framework. The resulting high pass filter from update-first lifting schemes is longer than the corresponding low pass filter in the analysis filter bank. It is desirable to have a shorter high pass filter in the analysis filer bank for biorthogonal wavelets in order to avoid ringing artifacts in low bit rate image coding. In this paper, the designing of wavelet transforms containing an adaptive low pass filter using the classical lifting scheme is presented. The rest of the paper is organised as follows: In section 2 wavelets and the lifting scheme are summarised. The design of the new transform is shown in section 3. Two examples using the 5/3 wavelet are shown in section 4. The performances of the new adaptive scheme are shown in section 5 followed by the conclusions in section 6.
2. WAVELETS, FILTER BANKS AND LIFTING : A REMINDER Discrete wavelet transforms are usually realised using the filter bank framework. Every FIR filter bank can be decomposed into lifting steps by factoring the polyphase matrix of the corresponding wavelet filter bank using the Euclidean factoring algorithm [2]. We demonstrate the filter bank and lifting relationship, especially for the class of wavelets known as interpolating wavelets in a 2 channel filter bank framework, as below. ) in We denote the low pass filter of length ( the wavelet filter bank as with coefficients ), where . Similarly, the high pass filter of length ( ) is denoted as with coefficients ), where
X O r4sut \| q =\W3p " ' xzy "g{ s% t w "gL sut }>s% t \| & = Z p S& (9) &' % "6 s# t By definition, $ q * and $ & | & * . This yields $ & $ | & q * . Therefore, the value W p can be subtracted from each term in the right hand side of (9) without any effect to p . Hereof, the same summation limits as above are in use unless stated otherwise. w w & S&(S w 1| & = 1Z p S& ]-W p = p p p* ] \ | q 5 = \ W ] W = & & w w w * ] & & & \| q = D G & \| = D G (10)
Using $ & | & ZPp [ S& * p , we can rewrite (7) as below. W [p * W p p
.
Update :
4.1. Example 1: Binary Choice Decisions
p
and the adaptive update step can be formulated as,
W [p * W!p n )p5 (12) where n 9 " . When n * " , it results in the usual low pass filter in the filter bank, whereas n * 9 corresponds to no updating, i.e., no low pass filtering in the W channel. The n values between 0 and " result in low pass filters with different averaging strengths. The decision of n is determined by comparing the absolute weighted mean gradient ` p ` with an user defined threshold . Since Z [ is
a priori available to both analysis and synthesis, information regarding adaptivity decision needs not be sent to the synthesis, thus with no overheads.
*
(13) (14)
p
This case is based on the for (2,2) transform. is computed using the linear interpolation of and .
Z>p[
ZPp[
Z p[ Z p[ * ] W!p ZLp ] W!p Z p] W!p S * ] !" #" ] $" (15) As shown above, " 1ZPp [ ZPp [ = yields the weighted mean gra dient as explained in the previous section. We use * " * 9 and in the adaptive update step (12) resulting in an update or no update based on p . The decision % is com` &-= at point ( , where is a threshold. puted as % * ` )px' In this example, the adaptive update scheme yields +dc C= +. adaptive wavelet, where dc *) 9 $ )p *
4.2. Example 2: Multiple Choice Decisions
dc *
In this example, multiple choices for update are considered. For example, cubic interpolation, that corresponds to can be considered as an additional choice. The weighted mean gradients, and , are computed as below for the 2 interpolating functions with vanishing moments 2 and 4, respectively.
p
(11)
/ * 6] " ] " = ZPp[ * Z p ] 1W p W p S = W3[p * W p 1Z p[ Z p[ =6=
(16) 1 Z p[ Z p[ = Z p[ = ] - 1Z p[ 3" Z p[ S = (17) , . - \Z p [ In this case, the interpolator with the lowest ` p ` yet lower than a given threshold is chosen and the update step for Wp ` /0-= & ` p 1` /2-= , then is performed as in (11). If ` p $ no update is performed, thus yielding (2,0) transform. If ei ther of ` p ` or ` p ` is less than , then (2,2) and (2,4) are performed respectively. If Both ` p ` and ` p ` are less than , then update step of the lower p is performed. In this example, the adaptive update scheme yields dc "= adaptive wavelet, where dc " *) 9 + .
p * p *
p
5. RESULTS AND DISCUSSION We use this 1-D transform as a separable transform in image coding applications. In Fig.1, we show the update maps corresponding to separable rows and columns transformations using the adaptive
update schemes and . The figures show that the
scheme performs an update only in the homogeneous re
gions, that are shown in white in the update maps. In the case, most homogeneous regions are updated using (2,2) (shown in gray). In Fig.2, we show the half resolution images obtained after 1 level of decomposition using the three transforms: (2,2),
and . Since this scheme follows the classical lifting framework, the operands in equations (6) and (7) can be rounded to achieve transforms that map integers to integers. The lossless coding performances of such transforms are shown in Table 1. Note that we used T=10 for all cases. It can be seen that the adaptive update process, the main aim of which is to preserve the edges, can be incorporated into lossless coding schemes with a slight (0.05 bpp on average) increase in the bit rate. The normalisation constants mentioned in section 2, as for the linear wavelets, can be used for the normalisation of this adaptive wavelet transform, thus can be used in embedded image coding.
Fig. 2. Half resolution images after one level of decomposition for
and respectively. (2,2),
Gold Hill Barbara1 Barbara2 Boats Black Board
(2,2) 4.707 4.957 5.066 4.235 3.891
(2,4) 4.715 4.948 5.061 4.241 3.894
(2, ) 4.732 5.061 5.160 4.260 3.902
(2, ) 4.732 5.060 5.158 4.258 3.900
Average
4.571
4.572
4.623
4.621
Table 1. Weighted zero-order entropy values in bpp adaptive update step does not affect the prediction step. Thus the existing non-adaptive high order predictions can be used. This also enables the use of the same normalisation constants as in the linear case. These transforms can easily be made to map integers to integers, thus making embedded to lossless image coding with both spatial and PSNR scalability as a possible application. 7. REFERENCES
Fig. 1. Update Maps: Row 1: Row wise transformation.Row 2: Column wise transformation. Column 1: The channel before update step. Column 2: Update map for , white corresponds to (2,2) and black to (2,0), T=10 Column 3:Update map for , white corresponds to (2,4) gray corresponds to (2,2) black corresponds to (2,0), T=10. 6. CONCLUSIONS Wavelet transforms with an adaptive low pass filter were designed using an adaptive update step in the classical lifting framework. Adaptivity is achieved by comparing the weighted mean gradients for the corresponding low pass filters resulted from update steps. Since these are computed using prediction details, that are a priori available to both analysis and synthesis, no information regarding the update choice needs to be sent to the synthesis. Further, the
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