PCA whitened data

Let $$X$$ be a feature matrix of size $$m\times d$$. I understand that the standard PCA whitening process follows three steps.

1. (Centerization) $${X} \to \hat{X}:=({X} - \mu)$$, where $$\mu$$ is a matrix of size $$m \times d$$ whose $$(i,j)$$-th entry is $$\mu_{ij} = \frac{\sum_{s=1}^m X_{is}}{m}$$.
2. (Eigen-decomposition of $$\hat{X}$$) Let $$C = \hat{X}^T\hat{X} = U\Sigma U^T$$.
3. Then the whitened $$X$$ is $$\tilde{X} = \hat{X}U\Sigma^{-1/2}=(X-\mu)U\Sigma^{-1/2}$$.

Here is what I think this might be an easy way of whitening $$X$$. If we want to whiten $$X$$, our goal is to enforce $$X^TX = I$$ after some transformation. Let us consider the svd of $$X$$ and say $$X = {U'}\Sigma' V'^T.$$ By setting $$\tilde{X} = U'I_{m\times d}V'^T$$, we have $$\tilde{X}^T\tilde{X}=I_d$$. Then can we say that this new $$\tilde{X}$$ is a whitened feature of $$X$$ in some sense?