transpose-operator

Reference

1. Inner product

   • https://en.wikipedia.org/wiki/Inner_product_space

2. Transpose

   • http://en.wikipedia.org/wiki/Transpose

3. Complex conjugate

   • https://en.wikipedia.org/wiki/Complex_conjugate

1) Inner Product definition

    < X, Y >  = < [x1; x2; ...; xn], [y1; y2; ...; yn] >,  

              = [x1; x2; ...; xn]' * [y1; y2; ...; yn],  

              = SUM_(i=1)^(n) xi * yi,   

              = (x1 * y1) + (x2 * y2) + ... + (xn * yn).

2) Transpose definition

If A satisfies the following relation,

    < A * X, Y > = < X, AT * Y >,  

then,

    AT is transpose of A.

3) Complex conjugate definition

    (a + ib)' = (a - ib).

Transpose for Matrix ver.

If A is defined as follow,

    A in R ^ (M, N),

then,

    AT in R ^ (N, M).

Transpose for Function ver.

1) differential function

If A(x) is defined as follow,

    A(x) = x(i+1) - x(i),

then AT(y) is that,

    AT(y) = y(i) - y(i+1).

2) Fourier transform

If A(x) is Fourier transform,

    A(x) = fftn(x)/numel(x),

then AT(y) is Inverse Fourier transform,

    AT(y) = ifftn(y).

3) Radon transform

If A(x) is Radon transform called by 'Projection',

    A(x) = radon(x, THETA)
    
    where, THETA is degrees vector.

then AT(y) is Inverse Radon transform without Filtration called by 'Backprojection',

    AT(y) = iradon(y, THETA, 'none', N).
    
    where, 'none' is filtration option and N is image size.