# Active Contours

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This post is based on Ray, N., & Acton, S. T. (2002). Active contours for cell tracking. Proceedings Fifth IEEE Southwest Symposium on Image Analysis and Interpretation, 274–278.

An active contour or snake-based method for tracking cells within a video sequence. The authors apply their cell tracking techniques to rolling leukocytes (白血细胞) observed in vivo (in living animal) from video microscopy. The analysis of leukocyte motion reveals cues about the mechanism of inflammatory disease.

The proposed snake tracker utilizes shape and size information specific to the leukocytes.

Principal contribution: introducing the shape and size contrast as a geometric primitive in the parametric snakes energy model

- the energy functional is minimized through the basic principles of the calculus of variations to obtain the Euler equations used in contour updating.
- develop a partial gradient vector flow GVF that accommodates for contrast changes and weak cell edges. The previous GVF models are sensitive to initial contour placement, the modified GVF construction with Dirichlet type boundary condition (BC) allows a snake tracker to be robust for a wide range of initial positions.

Another contribution: incorporate an energy term in the snake model that eliminates the need for explicitly resampling the snake contour intermittently (间歇地) as performed in traditional snake evolution.

Using animal experiments, compare the accuracy of the proposed snake tracker with correlation and centroid based tracker and show that the proposed tracker is superior in terms of increased number of frames tracked and reduced localization error.

## Introduction

Leukocyte tracking in vivo has gained its importance to the research groups studying inflammatory disease. The analysis of leukocyte rolling is emerging as an important tool in potential and novel anti-inflammatory treatments.

An example: E-selection inhibitors (抑制剂) are capable of decreasing the number and increasing the velocity of rolling leukocytes in living animals. The analysis of rolling velocities is quite laborious requiring tens of hours of user-interactive image processing work for each experiment. Thus, seek automated leukocyte tracking methods to obtain leukocyte positions and to compute leukocyte velocities.

- Propose a shape-size constrained active contour based tracker in this work. The associated energy has a constraint term that penalizes the deviation of the snake from a prescribed circular shape and size.
- To track fast rolling leukocytes, also propose an enhancement of the GVF flow through constraining Dirichlet BC.
- Eliminate the need for explicitly resampling the snake contour by introducing an energy functional in the snake model that keeps the neighboring snake points equally apart.

## Parametric Active Contours

A parametric active contour or snake is a curve, $C(s)$ with parameter $s\in [0, 1]$. The curve can move on the image plane under the influence of two types of forces–internal and the external forces. The former constrains the snake to be smooth while the latter guides the snake to seek desirable image properties, such as edges. Such an active contour model seeks to minimize the following functional

\[E_{snake} = \int_0^1 \frac 12 \left\{ \alpha \vert C'(s)\vert^2 + \beta \vert C''(s)\vert^2 \right\} + E_{ext}[C(s)]ds \,,\]The non-negative constants $\alpha$ and $\beta$ are the resistance to stretching and bending of the active contour. The external energy term $E_{ext}$ is usually defined as $-\vert \nabla G_\delta(x, y) I(x, y)\vert$, where $I(x, y)$ is the image intensity at $(x, y)$, $G_\sigma(x, y)$ is a 2D Gaussian kernel with standard deviation $\sigma$.

Euler or the motion equations:

\[\alpha C'(s) - \beta C''(S) - \nabla E_{ext}(C(s)) = 0\]which are called snake evolution equations and can be essentially viewed as a force balance equation, where $-\nabla E_{ext}$ can be thought of an external force $(u, v)$. Then we have the general form of force balance equations

\[\alpha C'(s) - \beta C''(s) + (u(C(s)), v(C(s))) = 0\,.\]