Equidistant Snakes: Accurate Irregular Shaped Contour Detection

Delineation of irregular shaped objects can be realized using active contours, called snakes[l]. However, the minimization of the potential energy pushes the control points toward the lowest potential energy regions, so most of the control points lie in those regions and a few, or even no control points lie in other parts of the contours, as shown in the Fig.1. We propose an Equidistant Snake that forces its control points to be equidistant from each other while minimizing its energy. In this way, it can be assured that the control points will lie all along the contour at equidistant intervals and not only in those regions of higher energy, allowing much more accurate delineation than using standard snakes. We also present a method to create the potential field where the minimum points are on the ridge lines of the edge image, as well as, a method to calculate the weighting factor between the internal energy and the image energy. Finally, we show the behavior of our equidistant snakes in delineating irregular boundaries of some synthesized and medical images. places where the edge intensity becomes a local minimum. So, the control points will be relatively sparse in the places where the edge intensity is low and they will not be uniformly distributed along the ridge line. The Figure 1 shows an irregular contour and the result after convergence using a standard snakes. a) Original image b) Edge Image C) Resulting standard snakes Figure 1 : Standard snakes detection I The Problem with Irregular Contours The magnitude of the edge intensity is I1 Equidistant Snakes different in each part of the true edge line, therefore, if we simply minimize the image 2.1 Energy Formulation energy, the control points will concentrate on the The equidistant snakes energy functional to be *Address : F ~ r o ~ h ~ , c i k u s a k u , Nagoya-Shi , 464-01, minimized is defined in the Same Way as the J a p a n . standard snakes. That is: e -mai l : [email protected] E(U,L)= Eima.ge(U) + h. Eint(L) (1) U is the set of all control points vectors ui=[ui,vilt9 that is, where p(u i ) is the potential energy at point ui. L is the set of all the links joining the control points, so where g(wi) is the gradient. L = (11, 12, ..) and I,(ui, uj)=ui uj, (3) a) Input image b) Edge image for all i and j joined by a link I,. So, we define the internal energy as 1 2 (4) aEint (L) ' Ta C lllmll C ) G*(edge) & edge. d) Pot. image & edge zm€L Figure 2:The Potential image. where h. is the regularization parameter[l],[2]. The internal energy causes each control point to move to the position of the gravity center of its connected control points. Finally, the potential image, for each point ui is: 2.3 Movement of Control Points 2.2 The Potential Image Using the ordinary Gaussian convolution to calculate the potential image results in the minimum points deviating from the true ridge lines whenever the curvature is large, Fig.2 c).. So, we take the minimum value among the potentials created by the surrounding pixels, Fig.2 d). We will call this resulting image the potential image. Here, the gradient image is calculated applying the Finite Difference Method (FDM) using the central differences of the intensity gray level image. Applying the Gaussian convolution, we obtain an expression for the potential p', at pixel uj: Control points move based on two considerations, the minimization of its energy functional and the equidistant restriction. Then, each point continues to move until the direction of its former motion vector becomes opposite to that of the actual motion. At this time, the point is considered to have gone through its optimum, i.e., a ridge line, then it stops its motion until all the points have gone through 'their optimum. As a step length, we use a factor a that decrements its value in half whenever all the points cross the ridge lines, instead the ordinary steepest descent factor. Thus, the motion vector for each control point, ui , could be represented by the following vector: where. -* ~ i ) + I C(Ui -Uj) (8) dl. dl. 1 u,cui and U' = {u j 13 l,(ui,uj)) 2.4 Equidistant Correction Our method of achieving equidistance is based on a very simple but important property: All the . control points are equidistant only when they are on the bisector line of its neighboring two control points. We modify the motion vector, for each control point, to ensure that they will be on the bisector line. From (7), we calculate the vector A ui, the final motion vector, as follows (Figure 3): A ~ ~ 6 r . d I edge potential I I I Figure 3: Motion correction vector equal to g and the bending angle of the contour equal 2 8. Let's, also, suppose that the control points ui, u, and uk deviate a distance a from the edge line. In this situation, the condition that ui no longer deviates from the true contour is given by: I l ~ n t l l ' ~ ~ ~ m a g e ~ ~ (1 1) And from (5) and (7) we have g sin 0 11C.t 11 = 2 2 C O S ~ * ~ ~ c m g e ~ ~ =-&g; tan 8 where ?L l 2do& Figure 4: Adjusting of 2 IV Experiments In our experiments, we defined a measure of the equidistant error for the links as 4=4 &+d , (Y, ~ , ) ' ( ~ l ~ ~ --(uI +4Rml)) (q-y) (9) Where i is the average of the link lengths, and 1% -1411 6, is the standard deviation. center j U k where ui 2 I11 Weighting Factor, 2 (1 0) We found that it was about a 60% for the case of standard snakes, compared with a 10% for our equidistant snakes. We also measured the error in accuracy. Fig.5 Let's suppose that, in figure 4, the gradient is shows the residual images for the equidistant and standard snakes. In the case of equidistant snakes, the error was less than half the error of the standard snakes (given the error as the summation of black areas over the original image area). a) Standard snakes b) Equid. snakes Figure 5 : Error in accuracy Fig.6 shows how the accuracy of snakes changes when the number of control points changes for the equidistant and non equidistant snakes. Fig.7 shows the result of using snakes in a natural image from a cell of blood. Figure 6: Snakes accuracy comparison. a) Original d ) Standard snakes e) Equid. snakes 1') Std . snakes (Zoom) g ) E q u ~ d .snakes (Zoom)Figure 7: Equidistant and standard snakesperformance in a natural image.V ConclusionWe have introduced an equidistant snake thatadjusts its control points in order to delineateirregular shaped contours more accurately. Thisequidistant snake has proven to be especiallyuseful for applications such as segmenting andtracking of medical images. We have alsopresented a method to create a potential field, aswell as, a design method to calculate theregularization factor 2 .References [ 11 M. Kass, A. Witkin, and D.Terzopoulos, "Snakes: Active Contour Models.",Proc. 1 ". Int. Conf. on Comp. Vision, 1987, pp.259-269. 121 K. F. Lai, and R. T. Chin, "OnRegularization, Formulation and Initialization ofthe Active Contour Models (Snakes).", Proc.Asian Conf. on Comp. Vision, 1993, pp. 542-545. b ) Edge lrnagcc ) Pot. image