where The first term on the right side
is similar to the Chan-Vese’s method, but is driven by the extension field of the boundary statistics rather than by global statistics. The second term is the curvature term, where is the curvature of the level set function. We set the SDF as to be a shape constraint. The term is a controlling factor:
since becomes greater where the boundary character
is strong, weakening the constraint, and becomes small where the boundary character is weak, enforcing the constraint, it can adaptively handle both the sharp and blurred edges. and are the coefficients, controlling the weight of every term.
In the contour tracking algorithm, we first calculate the predicted shape as the initial curve, then construct the extension fields, and finally evolve the curve using the fields and local intensity. Fig. 2(d)-(f) illustrate the predicting phase, Fig. 2(g) and (h) show the extension fields, and constructed from the boundary statistics of and and Fig. 2(i) is the tracking result.
Considering the inhomogeneity of intensity across different slices, a correction factor should be added to and which is equal to the difference between the means of and in region
A Level Set Algorithm for Contour Tracking in Medical Images 141
Fig. 2. Flow chart of our contour tracking algorithm
3 Result
We applied the above strategy to the MRI liver data from the Visible Human Dataset. The size of each slice is 207 × 167 pixels, with a total of 39 slices, and 8-bit gray scale. The neighborhood of any point is chosen as a 16 × 16 square window centered around the point. The coefficients in the level set equation are:
and We chose a simple form of the linear prediction coefficients: and Fig. 3 shows the results of the 29th slice. We can see that the predicted edge is very close to the true edge.
In most slices, it takes no more than 10 iterations to converge to the target boundary. Inhomogeneity across slices can be handled with the correction factor Manual assistance was needed only in one slice (18th) in order to achieve the correct result. Fig. 4 shows the 3D distribution of the contours, and gives the reconstructed surface.
142 Y. Li and Q. Tang
Fig. 3. The 29th slice from 3-D liver images: left is the origin image; middle is the predicted initial curve (dashed line); right is the extracted contour (white line)
Fig. 4. Left: the 3-D contour set of liver. Right: the reconstructed surface from the contour set
In Fig. 5, results are shown for comparison. The images in the left column are the original images of the 26th and the 37th slice; the middle column shows the results obtained by using the gradient-based method proposed in Ref. [6];
the right column shows the results obtained by using our algorithm. From this comparison, it is evident that our method can lead to more accurate results.
To measure the similarity between the predicted shape and the final result, we define two indicators : and For and
and where is the area of (intersection),
A is the area of and is the area of (exclusive or). The mean of measured in this experiment is 0.96, and the mean of measured is 0.11, while the means of the similar indices for and are 0.93 and 0.13, showing the validity of our prediction method.
A Level Set Algorithm for Contour Tracking in Medical Images 143
Fig. 5. Left column: original images. Middle column: results obtained by using the gradient-based method in [6]. Right column: results obtained by using our algorithm. Arrows are used to point out the differences between the results of the two methods
4 Discussions and Conclusion
Our contribution in this paper is the use of boundary statistics in curve evolution and the application of SDF to shape prediction. The level set equation in this paper is partly derived from the Chan-Vese’s method. The original Chan-Vese’s method is designed to handle bimodal pictures. Our method can handle not only bimodal but also multimodal pictures, depending on the size of the observing window. In an extreme case, if the observing window is extended to the whole image domain, the boundary means and will be equal to the corresponding global statistics in their method. In our method, the boundary statistics are shown to be obtained from an observing window and vary along the curve, so it can also be viewed as a non-linear filtering method or an adaptive thresholding process. Since SDF has been an effective way to describe shapes, even a simple use of it can work well. The set of SDFs is not a linear space, since in most situations, combinations of SDFs do not produce a SDF. However, the statistical process of SDF and algebraic operation on SDF are still available. The validity of such a use can be seen in the works of Leventon et al [11] and Tsai et al [12].
Acknowledgements
We would like to sincerely thank Prof. Shangkai Gao for her instruction and Mr. Gusphyl Justin for revising this paper. We also thank the National Library of Medicine and the Visible Human Project for providing the Visible Human Dataset.
144 Y. Li and Q. Tang
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Robust Object Segmentation with
Constrained Curve Embedding Potential Field
Gary H.P. Ho and Pengcheng Shi Biomedical Research Laboratory
Department of Electrical and Electronic Engineering Hong Kong University of Science and Technology
Clear Water Bay, Kowloon, Hong Kong {garyho,eeship}@ust.hk
Abstract. We have earlier introduced an implicit vector field represen- tation for arbitrary number of curves in space, the curve embedding po- tential field (CEPF), and a general image segmentation strategy based on the detection of the CEPF distortion under the influence of vector-form image data [3]. In this paper, we present an improved CEPF framework which incorporates prior knowledge of the object boundary and has con- sistent object definition through a region growing process. The embedded implicit curves deform through the image- and model-induced changes of the CEPF, which evidently improves the segmentation accuracy under noisy and broken-edge situations. Further, the closure enforcement and the natural advection on the curves enhance the stability of CEPF evo- lution and the implementation is straightforward. Robust experimental results on cardiac and brain images are presented.
1 Introduction
A generic description of an object would be a closed contour formed around the object boundary edges. Since the original active contour model [5], many researchers have contributed to the improvement of the Snakes-type boundary segmentation paradigm by achieving better balance between the necessary struc- tural constraints to maintain model integrity and the ability to fully utilize the image information [2, 7]. More recently, the geodesic active contour strategy, which combines the active contour models and the level set methods, uses ge- ometric representation of the boundary model and thus permits greater curve deformation and topological changes [1, 6]. This improved model evolves like a waterfront that propagates along the normal direction until the whole front is blocked by object edges or reaches the border of the image space.
We have earlier proposed an implicit curve representation and segmentation strategy [3], with inspirations from the vector form level set formulation [8]. The key contribution has been that continuous curves are implicitly represented by a potential field in space, the curve embedding potential field (CEPF), and all operations are in the native vector form of edge data and CEPF (see Fig. 4 for the interaction between the gradient vector flow and CEPF). Being geometric
G.-Z. Yang and T. Jiang (Eds.): MIAR 2004, LNCS 3150, pp. 145–153, 2004.
© Springer-Verlag Berlin Heidelberg 2004
146 G.H.P. Ho and P. Shi
Fig. 1. Problems with the original CEPF (left): curve stabilized prematurely when GVF between separated objects are weak. Improved segmentation results on speckle noise-corrupted image (12.5% variance), with closure enforcement and natural curve speed but no prior edge model (right).
in nature, CEPF allows merge and break of the underlying contours and offers better handling of discontinuities. Object segmentation is achieved by iterative vector field construction, vector data-driven field evolution and regularization, and the detection of the CEPF vector crossings [3] . We have also shown that the CEPF strategy exhibits improvements on segmentation accuracy over the level set strategies on certain medical images.