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This is a conceptual doubt about the two-dimensional DWT. I am trying to understand the decomposition step of a two-dimensional DWT.

In the MATLAB explanation, the two-dimensional DWT generates 4 new images (cA, cH, cV, and cD) that represent the approximation, the horizontal details, the vertical details and the diagonal details.

I am also reading the explanation about this two-dimensional DWT. But the author shows the results as $\text{LL, LH, HL and HH}$.

I want to know if it corresponds to the same thing as cA, cH, cV, and cD. Is the following comparison correct? \begin{align} &\texttt{cA} =\text{LL} =\text{approximation}\\ &\texttt{cH} =\text{LH} =\text{horizontal}\\ &\texttt{cV} =\text{HL} =\text{vertical}\\ &\texttt{cD} =\text{HH} =\text{diagonal} \end{align}

I'm also a little confused about the $\text{LH}$ and $\text{HL}$ results. In some posts, I have read that the $\text{LH}$ represents the horizontal details and in other, I have read that it represents the vertical details.

For example, as answered here , the user explains that:

In the wavelet terminology, LL is the approxmation image, LH is the vertical details, HL is the horizontal details, and HH is the diagonal details.

But in this article the author says:

The LL band corresponds roughly to a down-sampled (by a factor of two) version of the original image. The LH band tends to preserve localized horizontal features, while the HL band tends to preserve localized vertical features in the original image. Finally, the HH band tends to isolate localized high-frequency point features in the image.

Can someone clarify this for me?

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The confusion is natural as the notations and practice are misleading, and differ across the different works on 2D wavelets.

  • First, the order of operations ("row then column" or "column then row") is important. In most cases (to the best of my knowledge), LH means low-pass on rows, then high-pass on columns (so, giving horizontal edges). But in some contexts, people use filtering tools on "columnized signals", so the first filter works on columns, then the image matrix is transposed, the second filter pass is applied, and the result is transposed back.
  • Second, from an math operator point of view, LH(I) is more often understood as applying the high-pass first, then the low-pass, enhancing the confusion.
  • Third, one may think about the active derivative: a vertical derivation yields the horizontal features.

In the following image, the rightmost letter seems to denote the first filtering operation:

wavelet decomposition 1

while the opposite convention is used in second image:

wavelet decomposition 2

This might stems from a cultural and technical background: does one read images left-right then top-bottom, or top-bottom then left-right?

The best is to test your implementation, look at the resulting images, and use a consistent notation.

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    $\begingroup$ Thanks a lot for the help. I think I understood your point. In the Matlab dwt2 function, it seems to read the image left-right then top-bottom, as the first decomposition step is applied in the image rows (downsampling the columns). As I have read, most of the authors denote the LH for the horizontal details, it is probably the most used way, and if it is correct, the comparison in the question is probably correct too. I will keep the question opened if someone has another opinion/answer. $\endgroup$ – KelvinS Jul 10 '17 at 15:26
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    $\begingroup$ I have tried to find related illustrations, and added a third point of view (on derivatives). Imagine, it is even worse in 3D... $\endgroup$ – Laurent Duval Jul 10 '17 at 16:04

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