# How to transform basis functions

I know how to transform the basis if we have two sets of basis vectors. Now in my situation, I have two basis equation and I want to find out the transform between those basis functions. Specifically, I want to convert Polar Fourier Transform (PFT) into discrete cosine transform (DCT). I know one method can be to take the Inverse PFT (IPFT) and then taking the DCT but in this case, due to change of coordinates, it produces some black region in the image resulting in an error.

Now I am thinking to convert the result of PFT into DCT using some basis transform method. The method I am using for PFT calculation is described in this paper (https://lmb.informatik.uni-freiburg.de/papers/download/wa_report01_08.pdf).

Equation for PFT:

$$\Psi _ { k , m } ( r , \varphi ) = \sqrt { k } J _ { m } ( k r ) \Phi _ { m } ( \varphi )$$

where $$J_m$$ is an $$m_{th}$$ order Bessel function and $$\phi_m$$ is given by $$\frac{1}{2\pi}e^{im\varphi}$$. For DCT I am using MATLAB implementation.

DCT Equation:

$$y ( k ) = \sqrt { \frac { 2 } { N } } \sum _ { n = 1 } ^ { N } x ( n ) \frac { 1 } { \sqrt { 1 + \delta _ { k 1 } } } \cos \left( \frac { \pi } { 2 N } ( 2 n - 1 ) ( k - 1 ) \right)$$

I am taking PFT and DCT of an image. So let say I use only the first $$10$$ PFT values and now I want to know from those PFT values that what will be the corresponding DCT values of first $$10$$ lower frequencies. I know that which $$10$$ PFT basis I used to get those $$10$$ values and I also know the equation of those $$10$$ DCT basis that I want to get from PFT.

Maybe if the aforementioned is a bit ambiguous then, in other words, I can say that I am given with $$100$$ images. I take PFT of those images using only $$10$$ PFT basis. For every PFT basis, I get a number for every image. So for every image, I get $$10$$ values. I use the PFT basis as dimensions and value of PFT as a coordinate value. I plot the resulting point in that $$10$$ dimensional PFT space. So for $$100$$ images, I will have $$100$$ points in that $$10$$ dimensional PFT space.

Now I have another space that is DCT space. It also consists of $$10$$ dimensions and every dimension corresponds to a DCT basis. I know the equation of that basis. I want to know how to map those values in PFT space into this DCT space. What will be the coordinate values of every DCT dimension?