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I read in a paper that the discrete wavelet transform (DWT) has two disadvantages The first one is the shift variance property due to the downsampling process. Could you please help me understanding why downsampling leads to shift variance? The second disadvantage is the lack of directional selectivity. Why does DWT lacks directional selectivity?

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  • $\begingroup$ Can you please provide a link to the paper? $\endgroup$ – A_A Oct 13 at 12:39
  • $\begingroup$ [link] (books.google.com.eg/…) @A_A $\endgroup$ – karem Adam Oct 13 at 15:30
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    $\begingroup$ If we are lazy and use separable transform on rows and columns then we get bad directional sensitivity. Actually, the cartesian sampling grid in itself is a pretty poor choice for directional sensitivity. We can build real 2D wavelets on more suitable grids to solve this. $\endgroup$ – mathreadler Oct 15 at 11:28
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A $2$-channel stage of a wavelet transform, combines two filters in parallel, followed by a down-sampling by two. The later is the cause for shift-invariance, as the filters are time-invariant. Signals $$x_0[n] = \{\ldots,0,1,0,1,0,1,\ldots\}$$ and $$x_1[n] = \{\ldots,1,0,1,0,1,0,\ldots\}$$ which are shifted by only one sample, yield respectively $$y_0[n] = \{\ldots,0,0,0,0,0,0,\ldots\}$$ and $$y_1[n] = \{\ldots,1,1,1,1,1,1,\ldots\}$$

Answers to down-sampling a thus not shift-invariant. This happens even with low-pass or high-pass filters. In an $L$-level wavelet decomposition, the overall wavelet filter bank is invariant to multiples of $2^L$ shift, not the intermediate one.

When applied in 2D, classical wavelet schemes, by simplicity, apply 1 DWT on rows and columns of the image, in a separate way. The resulting 2D wavelet filter is of rank one, very poor at separating directions other than vertical or horizontal. However, genuine 2D wavelets with better directional selectivity exist. The paper A panorama on multiscale geometric representations, intertwining spatial, directional and frequency selectivity, Signal Processing, 2011, is devoted to that topic.

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