Zero Level Set Methods in Image Segmentation

Evolution of curve can be calculated with initial curve, but the implementation is very complex when there is topological changes. From planar geometry we can find the normal velocity of the curve if the curve is function of curve length, and the magnitude of the velocity is given by curvature of the curve.

Even calculation over curve is much much complex than the calculation over function of pixel which is function of spatial variable x, y and time t. Further some calculus gives the change in the pixel value with time can be expressed as the function of image gradient, which leads us for the easy curve evolution with zero level set of pixel function. Pixel function is moving up or down according to velocity associated with it. Zero level sets are used as curve evolution and the zero level set is found with setting pixel function moving with time up or down.

In third image, we can see black colored boundary line as segmented boundary. 

Curve Evolution and Calculus of variations in Image Processing

As better Numerical Analysis has higher accuracy in discrete image analysis, researchers in image processing are developing algorithm in continuous mathematics and implementing in discrete domain like in digital computer.

Partial Differential Equations are most recently being used in image analysis and the process as:

1. Image enhancement and deblurrng
2. Edge detection
3. Image restoration
4. Image segmentation
5. Image inpainting
6. Motion analysis

Curve evolution is the tool which is used in image processing for image inpainting, image segmentation with calculus of variations, Heat equation which is one of the partial differential equations is used for image deblurring.

Curve evolution studies about the move of the curve in plan or surface. In image segmentation, normal move of the each points in the curve is subject of interest than tangential move.

The calculus of variation studies about the extremal of the curve, During the image segmentation, minimal length curve is of interest and is the area of calculus of variations.