Squared correlation coefficient

Question

I am reading the paper multimodal image registration by maximization of the correlation ratio and maybe I am wrong but I am not sure to understand why this formula is correct

$$ \rho(X, Y)=\frac{\operatorname{Cov}(X, Y)^{2}}{\operatorname{Var}(X) \operatorname{Var}(Y)}=\frac{[E(X Y)-E(X) E(Y)]^{2}}{\left[E\left(X^{2}\right)-E(X)^{2}\right]\left[E\left(Y^{2}\right)-E(Y)^{2}\right]} $$

Then the author prove that

$$ \rho(X, Y)=\cos ^{2} \alpha $$

But is it not supposed to be ?

$$ \rho(X, Y)=\frac{\operatorname{Cov}(X, Y)}{\sqrt{\operatorname{Var}(X)} \sqrt{\operatorname{Var}(Y)}} $$

which leads to

$$ \rho(X, Y)=\cos \alpha $$

Answer 1

Yes, I think so. Looking at section 3.3 of the paper, the notation and the terminology the authors use seem to be wrong. They are talking about correlation but writing down squared correlation.