7.5 Multichannel Logical Segmentation

In this section, we outline the recent efforts of Sandberg and Chan in developing logical segmentation schemes for multichannel images [259]. Other numerous extensions and novel applications of the Mumford Shah model could be found, e.g., in [325].

In a multichannel image u( x) = ( u ₁( x), u ₂( x), , u _n( x)), a single physical object can leave different traces in different channels. For example, Figure 7.15 shows a two-channel image containing a triangle which is, however, incomplete in each individual channel. For this example, most conventional segmentation models for multichannel images [64, 65, 145, 263, 330] would output the complete triangle, i.e., the union of both channels. The union is just one of the several possible logical operations for multichannel images. For example, the intersection and the differentiation are also very common in applications, as illustrated in Figure 7.16.

Figure 7.15: A synthetic example of an object in two different channels. Notice that the lower left corner of A ₁ and the upper corner of A ₂ are missing.

Figure 7.16: Different logical combinations for the sample image: the union, the intersection, and the differentiation.

First, we define two logical variables to encode the information inside and outside the contour ? separately for each channel i:

Such different treatments are motivated by the energy minimization formulation. Intuitively speaking, in order for the active contour ?