# Coordinate prediction parameterization in object detection networks

State of the art object detection networks, such as RetinaNet, Faster R-CNN, and YOLO, use a coordinate encoding where the bounding box regression is given relative to the anchor box:

Centers:
$t_x = (x-x_a)/w_a$ and $t_y = (y-y_a)/h_a$

Height and width offsets:
$t_w = \log(w/w_a)$ and $t_h = \log(h/h_a)$

Why is the width and height prediction in logarithmic format? Is there a optimization reason for this?

Using this parametrization, size of a bounding box is computed as $w=w_a\exp(t)$, where $w_a$ is the size of the anchor box and $t$ is the network output. This parametrization has some (nice) properties:
• If $t=0$, size of the predicted box is the same as the anchor box
• Values $t<0$ shrink the bounding box "slowly" (large decrease in prediction is small decrease in size)
• Values $t>0$ expand the bounding box "fast" (small increase in prediction is large increase in size)