Looking for references about Hadamard Transform

Hi,

Would like to know whether there is any good reference for Hadamard Transform> Or is there some simple explanation for it?

Thanks for your help!

Re: Hadamard Transform

Search Google for def. of HT.

Because the rows in a Hadamard matrix are orthogonal, you can use the FHT to decompose any signal into its constituent Hadamard components. It functions just like an FFT, except the Hadamard components are based on sequency rather than frequency. Signal-compression applications, for example, commonly use an FFT as the compression agent. They perform an FFT and retain only select components. You can use the FHT in an identical fashion: retain only the key Hadamard components and throw away the rest. But because an FHT requires fewer computations (it omits the FFT's twiddle factors), you can implement the FHT on smaller, cheaper hardware.

Re: Hadamard Transform

i think u can refer any image processing book.u can get in that book.

Hadamard Transform

Hi
WHT : Walsh Hadamard Transform

I have uploaded basis images of WHT for case N=4;
we have 16 ,4x4 basis Hadamard images.
our original 4x4 image is projected on basis images & corresponding WHT coefficient are obtained.

Properties of WHT:
1)WHT is a vary fast transform i.e. O(N)
2)WHT is a popular Digital transform since its kernel function is only consisted from 0 and 1(black & white)
3)WHT is a Unitary transform i.e.
transpose(H)=H^-1
4)being unitary it has to be a Orthonormal transform
i.e. its basis vectors build a Basis in space.

5)WHT is a transform with good to very good energy compaction i.e the result of applying WHT to a set of data will have a near-diagonal covariance matrix with lots of zero!!

and finally I agrre with our friend Pmonon .
juice of classical transformation theory is to represent data in another space so that data in new space our less correlated.

for detailed discussion refer to:
1)Fundamentals of digital image processing Anil.Jain
2)Digital Image Processing Gonzalez & Woods

hope it helped
Armin

Re: Hadamard Transform

Walsh Hadamard Transform is one of the few unitary Block Transform Coding techniques used in Image Compression.

the basic process is to divide an image into small non-overlapping blocks of equal sizes of say 8x8 and processes the blocks independently using a 2-d transform.
unlike the kernels of the DFT, which are sums os sines and cosines the WHT kernels consists of alternating plus and minus 1s arranged in a chekerboard pattern.

although it does not provide energy compaction as much as the DCT its considered for simplicityof implementation.

The simplicity arrises from the structure of the Hadamard Matrix which is of type called as recursive.

consider H2 = |1 1 |
|1 -1 |
and from this H2n = |Hn Hn|
|Hn -Hn|
thus H matrix of higher dimension can be recursively formed from H-matrix of lower dimensions
now, consider H matrix for N=8, where + indicates +1, and - indicates -1
|+ + + + + + + +|
|+ - + - + - + -|
|+ + - - + + - - |
|+ - - + + - - + |
|+ ++ + - - - - |
|+ - - - - + - + |
|+ + - - - - + +|
|+ - -+ - + + - |

we define the no of sign changes along a column as a sequence, which is same as concept of freq in case of discrete fourier transform.

Here, there is no straightforward relation between the frequency variable 'u' and number of sign changes unlike DFT or DCT where increasing values of 'u' causes increasing number of frequency components.
so, we need some kind of reordering of the H matrix in order to get such simple relationship.
this modified hadamard tranform is called Walsh hadamard transform.