Image segmentation

In image analysis, segmentation is the partition of a digital image into multiple regions (sets of pixels), according to some criterion.

The goal of segmentation is typically to locate certain objects of interest which may be depicted in the image. Segmentation could therefore be seen as a computer vision problem. Unfortunately, many important segmentation algorithms are too simple to solve this problem accurately: they compensate for this limitation with their predictability, generality, and efficiency.

A simple example of segmentation is thresholding a grayscale image with a fixed threshold t: each pixel p is assigned to one of two classes, P₀ or P₁, depending on whether I(p) < t or I(p) ≥ t.

Some other segmentation algorithms are based on segmenting images into regions of similar texture according to wavelet or Fourier transforms.

Segmentation criteria can be arbitrarily complex, and take into account global as well as local criteria. A common requirement is that each region must be connected in some sense.

An example of a global segmentation criteria is the famous Mumford-Shah functional. This functional measures the degree of match between an image and its segmentation. A segmentation consists of a set of non-overlapping connected regions (the union of which is the image ___domain), each of which is smooth and each of which has a piece-wise smooth boundary. The functional penalizes deviations from the original image, deviations from smoothness within in each region and the total length of the boundaries of all the regions. Mathematically, ${\displaystyle E\left(f,\Gamma \right)={\mu }^{2}\int \!\!\!\int _{R}\!\left(f-g\right)^{2}{dx}\,{dy}+\int \!\!\!\int _{R-\Gamma }\!\left(\left|{\it {\nabla f}}\right|\right)^{2}{dx}\,{dy}+\nu \,\left|\Gamma \right|}$

Computer Vision, an external wiki