Dot product between arrays: basis representation of an image

[fisico30]

Hello Forum,

When we represent a vector X using an orthonormal basis, we express X as a linear combination of the basis vectors:

x= a1 v1 + a2 v2 + a3 v3+ ...

Each coefficient a_i is the dot product between x and each basis vector v_i.

If the vector x is not a row (or column vector), but an array (like an image) the equation is still the same: the a_i are single numbers, while the basis vectors v_i become arrays. The array x becomes the weighted sum of multiple basis arrays.

But how does the dot product between two matrices, x and v1 for example, both of size NxN, give a single number, the coefficient a1? The dimension does not seem to allow this matrix product to output a 1x1 vector (the single coefficient), does it?

a1=<x^T, v1>= dot product between the transpose of x and array v1...

thanks
fisico30

 

[HallsofIvy]

In order to define a dot product between vectors in a given basis you have to first define the dot product of the basis vectors.

How are you defining the "dot product" of the arrays you are using as basis vectors?

 

[fisico30]

Well,
I am thinking of an image, say 10x10 in size as being equal to the weighted sum of 2D Fourier components:


G = a1 Phi1+ a2 Phi2+ a3 Phi3+...


where Phi_i are the two dimensional sinusoidal signals (basis functions of the Fourier transform): Phi_i= a_i *exp [i*(kx*x+ky*y)] of size 10x10.
The Fourier basis can be made orthonormal.

I would imaging a dot product between the image G and each Phi_i to produce the single number a_i...

fisico30