# Bounding box regression in R-CNN

In R-CNN paper, they give the definition of the target values for bounding box regression

Given that $$(P, G)$$ is a (prediction box, ground-truth box) pair of the form $$(x, y, w, h)$$ where $$x, y$$ is the center coordinate of the box, $$w, h$$ are width and height respectively.

$$t_x = (G_x - P_x) / P_w \hspace{2.0cm} t_y = (G_y - P_y) / P_h$$

$$t_w = \log(G_w / P_w) \hspace{2.0cm} t_h = \log(G_h / P_h)$$

And the goal is to find $$\textbf{w}_*$$, where $$*$$ can be $$x, y, w$$ or $$h$$, so that

$$\textbf{w}_* = \arg \min_{\hat{\textbf{w}}_*} \sum_i (t^i_* - \hat{\textbf{w}}_*^T \phi(P^i))^2 + \lambda \|\hat{\textbf{w}}_*\|^2$$ where $$\phi(P^i)$$ is the feature map given by the last pooling layer of the feature extractor after passing predicted bounding box $$P^i$$

I don't understand why they come up with this approach of bounding box regression ? Can anyone tell me about this ?

P.S: since this regression approach is used not only in R-CNN but also in later models, I really want to get a clear understanding of this