Orthogonality: intuition challenged.

[stabu]

I'm dealing with image transforms.These are of course 2D.

I always thought orthogonality was the same as perpendicularity, so the max number of orthogonal bases you could come up with in 2D is 2.

However, image processing is full of transforms such as Hadamard, Haar, etc. that can have often 8 different bases. Trouble is, they are described as orthogonal. How can you have 8 bases that are orthogonal to each other if we are in 2D all the time?

 

[HallsofIvy]

What do you mean by "bases" here? If you are referring to "bases" as in linear geometry, of course, you can have any number of "orthogonal bases", each containing two vectors. the vectors in different bases may not be orthogonal to one another.

 

[stabu]

Hi HallsofIvy,

Thanks for the response. Sorry, the term is basis, rather than base. In the 1D case you have a set of basis functions that can represent the original function.

I'm also talking about sampled and quantized digital image that may be represented by a 2-D matrix. It's not quite linear geometry, maybe that's why I'm finding it difficult to understand ...

I've been over and over several textbook on this orthogonality issue. One condition is that the inner product of the basis functions need be zero to be considered orthogonal. That seems to be clear .. I dunno, perhaps the digital context changes the way orthogonality can be seen.

Sorry for the surmising. I suppose I really need to go to a Digial Image Processing Forum for this one.

Many thanks anyway.