1 Introduction
The wavelet transform (WT) has become an invaluable tool in image processing applications. However, the most commonly used implementation of DWT, is shift variant and this is the main reason why it is unsuitable for many signal analysis applications. The pre-requirement of many applications such as edge detection and texture analysis need the transforms are supposed to be, at least approximately, shift invariant. Here some of these shift invariant transforms are presented and trade-off will be discussed.
1.1 DWT and Shift-invariance
DWT is an important transform and has been got wide application now. It can provide spatial-frequency decomposition that provides a flexible multiresolution analysis of an image. Mallat’s algorithm improves the computationally efficiency of DWT transform and make it more attractive. In particular, the DWT provides a sparse time-frequency representation of the original signal that has the same number of samples as the original signal.
After almost 20 years effort, people have developed many kinds of wavelets. The wavelet transform can be looked as two sets of filters, one is high-pass filter and the other is low-pass filter. In our project, we choose DWT db2 and DWT DBSS wavelets. DWT db2 is one kind of Daubechies Wavelets [Dau92], which have no explicit expression. So here we use schematic plot to express the filter constructed by DWT db2.
Figure: DWT db2
The corresponding filters of DBSS transform is:
Transform filter : H1 = [-1 2 6 2 -1 0 0] *
G1 = [0 0 -2 4 -2 0 0] *
Inverse filter: InvH1 = [0 0 0 2 4 2 0] *
InvG1 = [0 -1 -2 6 -2 -1 0] *
But DWT still has some shortcomings and one of them is shift variant. Shift variance results from the use of critical down-sampling in the DWT. In this way, every second wavelet coefficient at each decomposition level is discarded. This is done both to reduce the amount of data that has to be analyzed and to enforce the implicit time frequency uncertainty of the analysis. It can be seen as the basis of multiresolution. However, This critical down-sampling results in wavelet coefficients that are highly dependent on their location in the sub-sampling lattice. This can lead to small shifts in the input waveform causing large changes in the wavelet coefficients, large variations in the distribution of energy at different scales, and possibly large changes in reconstructed waveforms.
Another way by which we can go through this problem is to consider the frequency response of the mother wavelets. As no realizable wavelet filter can have an ideal “brick-wall” frequency response, i.e., the attenuation in the stop band will always be finite, aliasing will be introduced. That is, when the WT sub-bands, which nominally have half the bandwidth of the original signal, are sub-sampled by a factor of two, the Nyquist criteria is strictly violated and frequency components above (or below) the cut-off frequency of the filter will be aliased into the wrong sub-band
1.2 Shift-invariant Wavelet Transforms
From the above discussion, we have discussed about the reason which cause the DWT’s shift-variance. The main reason is the down-sampling, which couldn’t tolerate the movement of the energy caused by small shifts. To solve this problem, Kingsbury proposed a set of special filters, which can produce the two fully decimated trees for even part and odd part.
The above is the flowchart of the DT-CWT.
The another method to solve the shift variance of DWT is discarding all subsampling, which can integrate the shift-invariance into a common DWT, called SIDWT (Shift-Invariant Discrete Wavelet Transform). Shift invariance can be also achieved in a DWT by doubling the sampling rate. In this project, the SIDWT we used comes from the Harr wavelet transform.
Experiment:
Here is the experiment about the shift-invariance of 1-D DT-CWT. The input is a step signal. The right is the curves of coefficients from the DT-CWT transform, while the coefficients from DWT serve as comparison at the left.
Figure. 1-D DWT and DT-CWT
We can see the curve shapes of DWT change randomly, which becomes more and more clear with the layer increase. On the contrary, the coefficients of DT-CWT shift with the input signal but the curve shape keeps the same.
It’s straightforward to think, if the image recovered from the coefficient at every level is of shift invariance, the whole image reconstructed from the coefficient should be shift-invariant. Now we use a simple 2-D image to demonstrate that.
Figure . The 2-D shift invariance
Here is the image recovered from the coefficient of level four. From the experiment result, we can see the DWT db couldn’t detect the shift at most point so the image recovered looks like stair. It’s obvious that will cause the serious distort. However, because the coefficients of SIDWT and DT-CWT are shift-invariant, the recovered biases are with smooth slope. Noted that the laplacian transform have approximately shift invariant too so the slope is smooth also but the edge is a little blurring.
Now we check the effect of the fusion by shift invariant wavelet transform. First, the fusion effects without shift are checked. The following images are the images needed fusion, each with blurring the half.
Now we study the fusion result by different algorithm:
Average PCA Morphology
Laplacian DWT db2 DT-CWT
The RSME and MI evaluation is shown in the following table:
Average / PCA / Lap / Mor / SIDWT / DWT db / DTCWT7 / 7.4293 / 4.2281 / 18.6162 / 5.0445 / 4.5041 / 3.9971
3.7876 / 3.3677 / 4.1428 / 2.4032 / 3.0358 / 3.9983 / 4.1637
0.03 / 0.391 / 0.511 / 3.244 / 2.854 / 1.512 / 2.133
The corresponding plot is:
Table: The RMSE & MI Evaluation
From the evaluation, we can see the fusion method based on the pyramid algorithm and wavelet transform are better than the average and PCA algorithm. The fused images from the different pyramid and wavelet algorithms are very near, but the RMSE and MI evaluation still show that the DTCWT has the best performance
Next, we check the fusion result with a little shift. The following are the original images:
The right image shifts 7 pixels downwards from the left image. The fused images are shown:
Average PCA Morphology
Laplacian DWT db DT-CWT
From the results, we can notice that the shift has negative effects for all algorithms. The DWT db is affected most and we can see the fault area clearly. The result image from DT-CWT algorithm is much better, but we still can found the trail of shift, especially at the area near the edge. The corresponding RSME and MI evaluation is below:
Average / PCA / Lap / Mor / SIDWT / DWT db / DTCWT19.1472 / 19.1695 / 16.3978 / 22.9925 / 18.8174 / 20.5683 / 17.7966
2.3844 / 2.333 / 2.7239 / 2.1774 / 2.2799 / 1.866 / 2.0986
0.03 / 0.54 / 0.731 / 3.525 / 6.71 / 1.002 / 2.053
The corresponding plot is:
We make comparison between the results before and after the shift. The result is:
We can reach the conclusion:
1. Comparing with the DWT transform, Shift-invariant wavelet transform can provide better quality and higher robust.
2. Because DT-CWT is only approximately shift-invariant, the shift effect couldn’t be eliminated completely.