Formal definition of the Object Detection problem

Question

For many problems in computer science, there is a formal, mathematical problem defition.
Something like: Given ..., the problem is to ...

How can the Object Detection problem (i.e. detecting objects on an image) be formally defined?

Given a set of pixels, the task is to decide

1. which pixels belong to an object at all,
2. which pixels belong to the same object.

How can this be put into a formula?

Answer 1

This is just an idea

Given a set of pixels, the task is to decide:

1. Which pixel is the center of an object?
2. What is the size of the bounding boxes with the center is the pixel in part 1?

Formula, consider this is a 2D image, call $(x,y)$ is the horizontal and vertical coordinate and $(w_i,h_i)$ is the size of bouding box of object $i$:

$\text{For }m \in[x,x+w_i] \text{ and } n\in[y,y+h_i]$

$c_i(m,n) = \begin{cases} 1, \text{if pixel at position (m,n) is belongs to object i,}\\ 0, \text{else} \end{cases}$