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IEICE TRANS. INF. & SYST., VOL.E82–D, NO.2 FEBRUARY 1999

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LETTER

A Fast and Stable Method for Detecting and Tracking Medical Organs in MRI Sequences Dong Joong KANG†,†† , Chang Yong KIM† , Yang Seok SEO† , and In So KWEON†† , Nonmembers

SUMMARY A discrete dynamic model for defining contours in 2-D medical images is presented. An active contour in this objective is optimized by a dynamic programming algorithm, for which a new constraint that has fast and stable properties is introduced. The internal energy of the model depends on local behavior of the contour, while the external energy is derived from image features. The algorithm is able to rapidly detect convex and concave objects even when the image quality is poor. key words: image processing, active contours, dynamic programming

1.

Introduction

The analysis of medical organization through Magnetic Resonance Image is one of the recent noninvasive techniques developed to understand and model the human head or the heart structure. Clinical MRI studies include analysis for a large number of images, from which qualitative information can be obtained by viewing the images in some sequences. Many applications in medical image processing rely on the reliable definition of object contours [1]. If done manually, this can be tedious and time-consuming. A fast and robust automated system is required for the contour extraction. This letter presents a central element of an image processing system, capable of detecting and tracking deformable contours in large MRI image sequences. Solutions should be presented to avoid undesirable deformation effects, like shrinking and vertex clustering, which are common in existing active contour models. Eviatar et al. [2] proposed an equilibrium term to prevent the snake from collapsing to a single point. The internal force enforces the distance defined between two successive points of the snake vertex (i.e., control point) to a prescribed equilibrium value. However, the method could be highly distorted in weak boundary edges by background clutters because of absence of a curvature term. Howing et al. [3] used a prior knowledge in a probabilistic term on the shape of the expected boundManuscript received March 27, 1998. Manuscript revised August 23, 1998. † The authors are with Signal Processing Lab., Samsung Advanced Institute of Technology, P.O.Box 111, Suwon, Korea 440–600. †† The authors are with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, 207–43, Cheongryangri-dong, Dongdaemoon-gu, Seoul, Korea.

ary to detect a smooth snake contour. The curvature on the snake contour is only permitted in a predefined mean and variance. However, this method could not ensure the vertex correspondence in shape tracking as well as give high time complexity for calculation of the curvature. Basically, the vertex behavior has locally independent motion property even though the points are weakly constrained by curvature and continuity forces, imposed on the snake contour. In this letter, the proposed method well maintains the correspondence of snake vertex after and before object motion from local similarity of the vertex motion. And the method does not need curvature calculation. Hence, very fast convergence is possible. 2.

Active Contour Models

We present a contour extraction method based on the concept of active contours, which can be described as energy minimizing splines [4]. The concept of active contour is used very broadly in several previous papers and the related works [1], [5], [6]. Kass [4] has proposed a model called Snakes (i.e., active contour models) as an active spline reacting with image features. The contour is initially placed near an edge under consideration, and then image forces draw the contour to the edge in the image. As the algorithm iterates, the energy terms can be adjusted to obtain a local minimum. That is, basic snake model is a controlled continuity spline under the influence of image forces. The internal spline forces serve to impose a piecewise smoothness constraint. The image forces push the snake toward salient image features. The active contour is represented by a curve, v(s) = (x(s), y(s)) having arc length s as parameter. Energy functional for the contour is defined by:
1 Einternal (v(s)) + Eexternal (v(s))ds. (1) Esnake = 0

where Einternal represents the internal energy of the contour due to bending or discontinuities, Eexternal is the image potential (from it, image forces are obtained). The image forces can be due to various events, e.g., lines, edges, and terminations. The internal spline energy is written:

IEICE TRANS. INF. & SYST., VOL.E82–D, NO.2 FEBRUARY 1999

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Einternal = α(s)|vs (s)|2 + β(s)|vss (s)|2 .

(2)

The first-order continuity term has larger values where there is a gap in the curve, and the second-order curvature term will be large where the curve is bending rapidly. The values of α and β at a point determine the extent to which the contour is allowed to stretch or bend at that point. If α is 0 at a point, a discontinuity can occur, while if β is 0, a corner can develop. The minimum energy contour is determined by using techniques of variational calculus [4] or a neighbor region search of the snake control points [6]. 3.

Modified Internal Energy

The snake paradigm models a deformable contour as possessing internal energy in order to impart smoothness to the contour. When this contour is located on an external energy field, the contour seeks a local minimum of the energy field by moving and changing shape. Our snake is a modified version of the conventional dynamic programming method [6]. Instead of classical internal energy, we use a new energy term. Proposed internal energy enforces the same direction and displacement for control points (see Fig. 1):

active contour is minimized by the following dynamic programming algorithm: for all m, S(0, m) = 0 for n = 1, . . . , N − 1 for m = 0, . . . , M − 1 S(n, m) = min{S(n − 1, k) k

+Einternal (vn,m , vn,c , vn−1,k , vn−1,c ) +Eexternal (vn,k )} B(n, m) = k min where S(n, m) represents the accumulated minimal energy level and the back pointer B(n, m) holds the index k(k = 0, . . . , M − 1) giving minimum accumulation in each m. After all vertices have been processed, the new boundary is obtained by tracking back the pointers, beginning with the candidate that has a minimal S(N − 1, m) value. The time cost of the algorithm is a lower order O(N − 1 · M 2 ), when compared to the complexity O(N − 1 · M 3 ) of the conventional DP al-

Einternal = α · |dn+1 − dn |, where dn = vn,k − vn,c (3) where vn,k , vn,c , are control points that denote, respectively, the current search candidate position and center position before movement. First, the algorithm needs an initial polygon vn,c consisting of N vertices (n = 0, . . . , N − 1). The candidates vn,k (t), k = 0, . . . , M − 1 are uniformly sampled along a search line normal to the initial polygon and α is energy ratio constant. We avoid the partial variation of the contour by image noise, because this energy form gives a local convexity constraining the movement of points to the same direction and distance. External energy consists of image edge strength [4]: Eexternal = γ · |∇Gσ ∗ I(vn,k )|

(4)

Fig. 2 A part of brain MR image. (a) An input image with overlapped normal search lines. (b) Canny edges. (c) Contour detection with the conventional DP algorithm. (d) Contour detection with the proposed constraint.

where γ is also an energy ratio constant and Eexternal is computed by Canny edge detector [7] with an experimentally decided σ. The energy function of the

Fig. 1 Modified internal force. (a) An opposite movement increases the internal energy level. (b) The same displacement along the normal direction of the contour has less energy potential.

Fig. 3 The time required during five iteration of the DP algorithm according to the change of the number of snake’s control points.

LETTER

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Fig. 4 Comparison of boundary tracking results for application of classical and the proposed internal energies. (a) By the proposed internal energy. (b) By the proposed force with classical curvature term. (c) By only the classical curvature energy.

gorithm. 4.

Experimental Results

The algorithm has been applied successfully to detect the boundaries of the brain organs in large MRI sequences, an example of which is shown with normal search lines in Fig. 2 (a). Figure 2 (c) shows an image where a portion of the front brain has been detected by the classical DP algorithm [6]. The resulted contour is locally distorted by the noisy edge peaks and the weak boundaries as shown in the left-top part of Fig. 2 (b). The distortion is frequently come out if we do not tune the energy ratio constants carefully. Application of the proposed constraint in Eq. (3) gives to the boundary in Fig. 2 (d). It can be seen that our snake is able to bridge areas of weak image feature very well and in wide range of the ratio constants while avoiding misleading strong features to produce a contour of the desired shape. Figure 3 shows a performance of the proposed DP algorithm by the low order time complexity. The time required to converge to the image edges is far lower than the classical DP algorithm when the number of snake’s control points increases. Figure 4 presents an experimental result tracking the human skull boundaries from neighboring MRI slices of the jaws and mouth. The proposed internal force highly stabilizes the shape tracking by enforcing the local motion similarity between two successive vertex points, while the conventional DP algorithm gives a distorted and unstable vertex correspondence due to locally independent behavior of the vertex. Figures 4 (a), (b) and (c) show the tracking results for the proposed internal force, the proposed force with classical curvature energy, and the curvature term only, respectively. The first order continuity term and the external energy of Eq. (4) are commonly used in all cases. For the implementation of the conventional energy terms, we use the notations of Williams et al. [8]. In Figs. 4 (b) and (c), the weighting parameters for

the energy terms are set to an equal value in two cases to show the same experimental condition. The “+” points in the figures denote the snake vertexes, and the vertex motion is permitted in the eight neighbors of current vertex position to allow the motion of two d.o.f (i.e., degree of freedom) unlike the normal line search. 5.

Conclusions

A new shape constraint for active contours based on a dynamic programming algorithm is proposed. It allows for a fast and stable detection of object boundaries in MRI sequences with low contrast and noisy edge peaks. Because of the local similarity of vertex motion, it stabilizes the boundary tracking by well maintaining the vertex correspondence between two successive slices. Robust and fast properties of the constraint with insensitive ratio constants could be easily applied to other computer vision tasks using active contours. References [1] S. Ranganath, “Contour extraction from Cardiac MRI studies using snakes,” IEEE Trans. Medical Imaging, vol.14, no.2, 1995. [2] H. Eviatar and R.L. Somorjai, “A fast, simple active contour algorithm for biomedical images,” Pattern Recognition Letters, vol.17, pp.969–974, 1996. [3] F. Howing, D. Wermser, and L.S. Dooley, “Recognition and tracking of articulatory organs in X-ray image sequences,” Electronics Letters, vol.32, no.5, pp.444–445, 1996. [4] M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contour models,” Int. J. Comput. Vision, 1988. [5] L.D. Cohen, “On active contour models and balloons,” CVGIP: Image Understanding, vol.53, no.2, pp.211–218, 1991. [6] A. Amini, T.E. Weymouth, and R.C. Jain, “Using dynamic programming for solving variational problems in vision,” IEEE Trans. Pattern Anal. & Mach. Intell., vol.12, no.9, 1990. [7] J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. & Mach. Intell., vol.8, no.6, 1986. [8] D.J. Williams and M. Shah, “A fast algorithm for active contours and curvature estimation,” CVGIP: Image Understanding, vol.55, no.1, pp.14–26, 1992.