Fundamental question in image transformation and compression

[Jessy]

Hey All,

When we apply haar discrete wavelet transformation to an image we should get four quadrants LL, LH, HL and HH; where LL represents the blurred low-frequency version of the original image. So when I do it with MATLAB using the dwt2 I get it (please see DWT2.png attached). While when I perform linear algebra transformation using the following formula HIH^T; where H is the transformation matrix, I is the original input image and H^T is the transpose of H, I don't get similar results to the dwt2 (please see Transformation.png attached) method which is about high and low pass filtering instead of matrix multiplication. I don't know what's causing such a difference and can't relate between the two!

Thanks in advance.

[thorfdbg]

When we apply haar discrete wavelet transformation to an image we should get four quadrants LL, LH, HL and HH; where LL represents the blurred low-frequency version of the original image. So when I do it with MATLAB using the dwt2 I get it (please see DWT2.png attached). While when I perform linear algebra transformation using the following formula HIH^T; where H is the transformation matrix, I is the original input image and H^T is the transpose of H, I don't get similar results to the dwt2 (please see Transformation.png attached) method which is about high and low pass filtering instead of matrix multiplication. I don't know what's causing such a difference and can't relate between the two!

You are confused. The wavelet transformation Matlab implements is of course the tensor product of two one-dimensional transformations, plus coefficient reordering. And not the application of H and its transposed. That is: What matlab does is that it first applies a one-dimensional transform on all columns of the image, thus a multiplication of all column vectors of the image with H. It then applies H again, but on the row-vectors of the image. (This is called the tensor product of the transformation H with itself, i.e. H \otimes H). Note that this is something fundamentally different than the application of H H^T! As last step, matlab reorders the coefficients. Typically, the Haar wavelet is understood as generating the low-pass at even sample positions, and the high-pass as odd sample positions. A somewhat more convenient representation is to sort all low-passes to the left (or top) and all high-passes to the right (or bottom). Then you get the typical "window cross" you see matlab generates.