# Motion Analysis

Here, we briefly discuss four topics: the motion field, optical flow, tracking using flow, and structure and motion from tracked points.

## The Motion Field

The motion field is a vector field that describes the idealized motion of observed points due to motion of the observer relative to the point. A brief introduction is here.

## Optical Flow

Optical flow is the apparent motion in an image. A brief introduction is here.

## Tracking Using Optical Flow

Define a function $F(P_1, P_2) \mapsto v$ that computes the vector $v$ that represents the optical flow between two image patches $P_1$ and $P_2$ of the same size and shape. Let $R(I, c) \mapsto P$ denote sampling an image to extract a patch centered on location $c \in \Re(2).$

With these functions, we can define an iterative flow estimation method as follows

Flow(P1, I2, c)
P2 = R(I2, c)
while || P1 - P2 || > eps
c = c + [P1_u, P1_v] \ (P2 - P1)
P2 = R(I2, c)
end

We can then define two different variations on motion estimation.

First we define incremental motion estimation, which appears as follows:

$c_n = Flow(R(I_{n-1},c_{n-1}), I_n, c_{n-1})$

Closed loop tracking is defined as follows:

$c_n = Flow(R(I_{0},c_{0}), I_n, c_{n-1})$

## Factorization-based Motion and Structure Estimation

This is discussed briefly here.