# Meaning of “Base” in Walsh Hadamard Tranform in Image Processing?

Currently, I am studying a paper by K. He and J. Sun, “Computing nearest-neighbor ﬁelds via propagation-assisted KD-trees,” in Proc. IEEE Conf. Comput. Vis. Pattern Recog., 2012, pp. 111–118.

In the paper there is a sentence which i can quote as:

... We use the ﬁrst 16 WHT bases for the Y channel2 and 4 for each chrominance channel (Cb/Cr) throughout this paper. ...

Although I have made a search on the internet, I couldn't get the meaning of base.

In MATLAB 2014b There is a function fwht(x,n) where n is the order of the transform. Can this order be the base that i am trying to find the meaning of?

I believe that here bases stands for "basis vectors". From the cited paper by Y. Hel-Or and H. Hel-Or. Real time pattern matching using projection kernels, 2003, ICCV, you can see the set of $8 \times 8$ vectors from a separable Walsh-Hadamard basis:

[20] Y. Hel-Or and H. Hel-Or. Real time pattern matching using projection kernels. In ICCV, pages 1430–1445, 2003.

It consists in a tensor product of two $8$ 1D Walsh-Hadamard transforms (one horizontal, one vertical).

The sixteen 2D vectors in the left-top $4\times 4$ square are the one with the lowest sequencies (sequency is the frequency analog for Walsh-Paley-Hadamard transforms), in terms of "maximum sequency":

• Thank you for your response, i think this will help me quite a lot to find myself a better understading of the concept. – Archura Jun 18 '17 at 17:11
• One small question though, after having a glance at the paper that was posted, and the one I have quoted from, in which the sentence continues as "Thus we represent each patch by a 24-d vector." I couldn't catch the mathmetical relationship between the order, base, and dimension of the W-H matrix. – Archura Jun 18 '17 at 17:36
• 24 probably comes from 16 coefficients from the Y (luminance) channel and two times 4 for each chrominance channel : $16+4+4=24$. – Laurent Duval Jun 18 '17 at 17:43
• Indeed, so it can be simplified as: 1 base = 1 dimension = lg2(1) orders, do you believe this is correct? – Archura Jun 18 '17 at 18:33
• To get a $8\times 8$ FWT, you will apply 'fwht' with order $8$ on both the columns and rows of a $8\times 8$ image patch. – Laurent Duval Jun 18 '17 at 20:21