## Level Sets

The mathematical foundation of deformable models represents the confluence of physics and geometry. Geometry serves to represent object shape and physics puts some constrains on how it may vary over space and time. Deformable models have had great success in imaging and computer graphics. Deformable models include snakes and active contours. Snakes are used based on the geometric properties in image data to extract objects and anatomical structures in medical imaging. After initialization, snakes evolve to get the object. The change of

Figure 9.19: Brain MRI example: (upper left) the original MR image corrupted with intensity inhomogeneities. (Upper right) crisp gray matter membership using traditional FCM. (Middle left) crisp gray matter membership using the proposed BCFCM algorithm. (Middle right) the bias-field corrected image using BCFCM. The segmented image and bias field estimate using BCFCM are shown in bottom left and bottom right, respectively.

Figure 9.20: Brain tumor MRI examples. Upper row: Original MR images corrupted with salt and pepper noise. Middle row: the segmented images using FCM without any neighborhood consideration. Bottom row: The segmented images using BCFCM (a = 0.85).

snakes with time is guided by differential equations. These equations are derived from the energy minimization concept to describe the change of snakes with time. The output obtained using snakes depends highly on the initialization. It was found that initial curve has to lie close to the final solution to obtain required results. The initialization is relatively easy in the case of 2D images but in the 3D case it is very difficult. Also the topology change of the solution needs a special regulation to the model.

Level sets were invented to handle the problem of changing topology of curves. The level sets has had great success in computer graphics and vision. Also, it was used widely in medical imaging for segmentation and shape recovery. It proved to have advantages over statistical approaches followed by mathematical morphology. In the following section we will give a brief overview on level sets and its application in image segmentation.