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If we instead consider finding objects which are the union of m objects with simple geometric structure, the problem often becomes much harder (for example, finding the maximum-weight object that is the union of two star-shaped objects is NP-hard ). It is interesting from a theoretical perspective because the decomposition constraints of the problem poses an interesting computational challenge to overcome. It is interesting in practice because an algorithm for such a problem can identify more complicated objects while still allowing control of the topology of the output object. This type of problem is very interesting from both a practical perspective as well as a theoretical perspective. A region R can be decomposed into m objects of a particular structure if and only if there exists a coloring of the grid cells of R using m colors such that each of the objects induced by the grid cells of each of the color classes have the desired structure. They are interested in finding a maximum-weight region that may not have simple geometric structure, but can be decomposed into objects with simple geometric structure. consider the maximum-weight region problem with a twist on the constraints of some previous works. Objects Decomposable into Elementary Shapes. X-monotone regions, based monotone regions, rectilinear convex regions, and star-shaped regions. Polynomial-time algorithms have been given which identify an optimal solution for the following classes of objects: Research has shown that knowledge of the geometric shape of the object that you are looking for can greatly increase an algorithm's effectiveness in practice, see for example. We are interested in computing the region R with maximum weight subject to some constraints. A region (or object ) R will be defined as any subset of grid cells, and we define the weight of R to be w(R) = p∈R w(p). We call i the x-coordinate of p and j the y-coordinate of p and let p x (resp. For 1 ≤ i ≤ √ n and 1 ≤ j ≤ √ n the grid cell p at the (i, j) position in the grid has a real value w(p) called the weight of p. Let G be an √ n× √ n four-neighborhood grid graph. Then we attempt to find some subset of the grid that optimizes an objective function subject to some constraints. we are given a weighted grid graph where each grid cell corresponds to a pixel in the original image and weights on the grid cells are related to the likelihood that the particular pixel is in the object we wish to identify (positive weights are assigned to grid cells whose corresponding pixel is likely in the object and negative weights are assigned to grid cells whose corresponding pixel is likely in the background). Finding a "good" segmentation is often treated as an optimization problem, see for example. Image Segmentation as an Optimization Problem.

There are many other applications of image segmentation including fingerprint recognition, traffic control systems and agriculture imaging.

An example, in medical imaging, image segmentation is used to help locate tumors and other pathologies, measure tissue volumes, computer-guided surgery, diagnosis, treatment planning, study of anatomical structure etc. In practice image segmentation is used to detect objects and boundaries in the image. From another view point, image segmentation is assigning labels to the pixels of an image such that the pixels with the same label define a particular object which may have certain visual characteristics. It is the process of partitioning a digital image into multiple objects for better representation and analysis of an image. An area of work that has recently attracted extensive attention in the pattern recognition and computer vision communities is image segmentation.