International Journal of Computer Vision, MICHAEL KASS, ANDREW WITKIN, and DEMETRI TERZOPOULOS Snakes: Active Contour Models

Abstract A snake is an energy-minimizing spline guided by external constraint forces and influenced by image forces. Scale-space continuation can be used to enlarge the capture region surrounding a feature. We have used snakes successfully for interactive interpretation, in which user-imposed constraint forces guide the snake near features of interest.

Basic Snake Behavior The internal spline forces serve to impose a piecewise smoothness constraint. The image forces push the snake toward salient image features. The external constraint forces are responsible for putting the snake near the desired local minimum.

Internal Energy The first-order term makes the snake act like a membrane and the second-order term makes it act like a thin plate. Setting β (s) to zero at a point allows the snake to become second – order discontinuous and develop a corner. 一次微分項代表輪廓的連續性 (continuity) 或稱為張力 (tension) ，用來驅使 snake 表現出像膜一般的運動。二次微分項指的是輪廓的平滑度或曲率 (curvature) ，在某一點的位置將權重 β (S) 給定一極小值，可表現出輪廓上 曲率較大之處 ( 如 :corner) 。

Snake Pit The interface allows a user to select starting points and exert forces on snakes interactively as they minimize their energy. The user-interface has proven very useful for semiautomatic image interpretation. Our interface allows the user to connect a spring to any point on a snake. Creating a spring between X 1 and X 2 simply adds –k(X 1 -X 2 ) 2 to external constraint energy.

Image Forces In this section, we present three different energy functionals which attract a snake to lines, edges, and terminations. The total image energy can be expressed as a weighted combination of the three energy functionals.

Line Functional 為線條函數式用於偵測影像中具有線條特徵處。 depending on the sign Of w line, the snake will be attracted either to light lines or dark lines. Subject to its other constraints, the snake will try to align itself with the lightest or darkest nearby contour.

Edge Functional 外部能量的目的在於將 snake 牽引至影像的明顯特徵處，像是影像灰階值 變化較大處 (intensity value) 或邊緣 (edge) 。 If we set then the snake is attracted to contours with large image gradients.

Scale Space 如果影像之邊界受雜訊影響而並不是非常明顯時，通常先以高斯遮罩座 旋積 (convolution) 之後在計算梯度以達到平滑效果。 Scale-space continuation can greatly enlarge the capture region around features of interest.

Termination Functional 圖中的影像邊界實際上並不存在，但是導入端點函數式以及藉由外部限 制力和內力曲線平滑特性亦可偵測出虛擬邊界。

Motion If the feature then begins to move slowly, the snake will simply track the same local minimum.