---
title: Feature Tracking using Differentiation
categories: algorithm
---


# Basic feature tracking: Algorithm 4.1 (Ma 2004)

*(This is Ma's algorithm for feature tracking only.  It has been 
copy-edited to remove cross-references and references to optical flow.)*

Given an image $I(x)$ at time $t$, set a window $W$ of fixed size.

1.  Calculate the image gradients $(I_x,I_y)$ using a differentiation 
    filter, for instance the Sobel filter.
2.  Calculate the time difference $I_t$ as the difference between the
    current frame $I(x)$ and the subsequent or previous frame.
1. Compute 
   $$ G(x) = 
   \begin{bmatrix}
    \sum I_x^2   & \sum I_xI_y \\
    \sum I_xI_y  & \sum I_y^2
   \end{bmatrix}
   $$
   at every pixel $\mathbf{x}$ 
   The sums are taken over the window $W$.
2. Select a number of point features by choosing
   $\mathbf{x}_1,\mathbf{x}_2,\ldots$
   such that $G(X_i)$ is invertible.
3. Compute $$\mathbf{b}(\mathbf{x}, t) = 
   \begin{bmatrix}
    \sum I_xI_t \\
    \sum I_yI_t  
   \end{bmatrix}$$
   The sums are taken over the window $W$.
4. Compute the displacement $\mathbf{u}(x, t)$ as
   $$ \mathbf{u}(\mathbf{x}, t) = -G(\mathbf{x})^{-1}\mathbf{b}(\mathbf{x}, t)$$
6. At time $t + 1$, repeat the operation at $\mathbf{x} + \mathbf{u}(\mathbf{x} , t)$.