(2)If R*(X) = 
and R*(X) ≠ U, X is called an indefinable set in R. In this case, it is possible to determine that if some elements of U belong to 
but not if any element of U belongs to .
(3)If R*(X) ≠ and R*(X) = U, X is called an outside indefinable set of R. In this case, it is possible to determine whether some elements of U belong to X but not if any element of U belongs to .
(4)If R*(X) = and R*(X) = U, X is called an R total indefinable set. In this case, it is not possible to determine if any element of U belongs to X or .
Example 6.3 Examples of a rough set
Suppose a knowledge database K = (U, R) is given, where U = [x1 x2, · · ·, x10}, R is an equivalent set, and there are equivalent classes E1 = [x0, x1}, E2 = [x2, x6, x9}, E3 = [x3,x5},E4 = [x4,x8}, and E5 = [x7, x10}.
The set X1 = [x0, x1, x4, x8} is an R definable set, because R*(X1) = R*(X1) = E1 ∪ E4.
The set X2 = [x0, x3, x4, x5, x8, x10} is an R rough definable set. It has R*(X2) = E3 ∪ E4 = [x3, x4, x5, x8}, R*(X2) = E1 ∪ E3 ∪ E4 ∪ E5 = [x0, x1, x3, x4, x5, x7, x8, x10}, BR(X2) = E1 ∪ E5 = [x0, x1, x7, x10}, and dR(X2) = 1/2.
The set X3 = [x0, x2, x3} is an indefinable set in R, because R*(X3) = , R*(X3) = E1 ∪ E2 ∪ E3 = [x0, x1, x2, x3, x5, x6, x9} ≠ U.
The set X4 = [x0, x1, x2, x3, x4, x7} is an outside indefinable set of R. It has R*(X4) = E1 = [x0, x1}, R*(X4) = U, BR(X4) = E2 ∪ E3 ∪ E4 ∪ E5 = [x2, x3, x4, x5, x6, x7, x8, x9, x10},
The set X5 = [x0, x2, x3, x4, x7} is an R total indefinable set, because R*(X5) = , R*(X5) = U.
6.4.3.3Fusion Based on a Rough Set
When applying the rough set theory to multisensor information fusion, two concepts, that is, the nuclear of a rough set and the reduction of a rough set, are used. Let R be an equivalent relation set, and R 
R, if I(R) = I(R − {R}), then R can be omitted from R (unnecessary); otherwise, R cannot be omitted from R (necessary). In the above, I(•) represents an undetermined relation. If for any R R, R was not possible to omit, then set R is independent.
If R is independent, and P R, then P is independent, too. The set of all relations that cannot be omitted is called the nuclear of P and is denoted C(P). The relation between the nuclear and reduction is
where J(P) represents all reduction sets of P. It can be seen from above that the nuclear is included in all reductions and can be calculated from the intersection set of the reduction. In the process of knowledge reduction, the nuclear is the set of knowledge characters that cannot be removed.
Suppose that S and T are equivalent relations in U. The S positive domain (the equivalent class set that can be accurately partitioned to T) of T is
The dependent relation between S and T is
It can be seen that 0 ≤ QS(T) ≤ 1. Using the dependent relation between S and T, QS(T), the consistence between the two equivalent classes S and T can be determined. When QS(T) = 1, S and T are consistent. When QS(T) = 1, S and T are not consistent. When applying the rough set theory to multisensor information fusion, the dependent relation between S and T is used to help eliminate the consistent information and determine the minimum nuclear. Once the most useful decision information has been found, the fastest fusion method can be obtained.
6-1What is the relation between active vision and active fusion?
6-2When is the objective evaluation of fusion results more appropriate than the subjective evaluation of fusion results?
6-3What is the advantage of fusion evaluation according to the fusion objectives?
6-4Select two images, and obtain their weighted average fused image with various pairs of weights. Make some discussions on the results.
6-5Select a color image and decompose it into its R, G, B components. Take two of three components and use the wavelet transform method to fuse them. Judge the fusion result with one statistics-based criterion and one information-based criterion.
6-6Suppose that P(A1) = 0.1, P(A2) = 0.2, P(A3) = 0.3, P(A4) = 0.4 in Example 6.1. What is the final fusion result?
6-7Both Bayesian methods and evidence reasoning are based on the computation of probability. What are their differences? Design a set of test data and compare the computation results.
(1)R*(X ∩ Y) ⊆ R*(X) ∩ R*(Y).
(2)R*(X ∪ Y) = R*(X) ∪ R*(Y).
(1)R*(X) ⊆ X ⊆ R*(X).
(2)R*(X ∪ Y) = R*(X) ∪ R*(Y).
(3)R*(X ∩ Y) = R*(X) ∩ R*(Y).
6-10Given a knowledge database K = (U, R), where U = {x0, x1, ···, x8}, if there are equivalent classes E1 = {x0, x4}, E2 = {x1, x2, x7}, E3 = {x3, x5, x6, x8}, determine the types of sets of E1, E2, and E3.
6-11Given a knowledge database K = (U,R), where U = {x0, x1, ···, x8}R = {R1, R2, R3}, equivalent relations R1, R2, and R3 have the following equivalent classes: R1 = {{x1, x4, x5}, {x2, x8}, {x3}, {x6, x7}}, R2 = and R has the following equivalent classes, R = {{1, x5}, {x2, x8}, {x3}, {x4}, {x6}, {x7}}. Compute the nuclear and all reduction sets.
6-12*Prove the following theorem: The set X is a R-rough and R-definable set if and only if is a R-rough and R-definable set.
1.Summary of Information Fusion
–A recent trend in information processing for the last 20 years is the fusion of information (Zhang, 2016).
–Information fusion can be carried out with different forms and data, such as audio and video, multimedia, multi-modality (Renals, 2005), etc.
–A general introduction to image fusion techniques with multiple-sensors can be found in (Zhang, 2015c).
2.Image Fusion
–Stereo vision can also be considered an image fusion process, in which views from more than two points are fused to provide the complete information (Russ, 2002).
–More examples of using wavelet transforms in fusion applications of remote sensing imaging can be found in (Bian, 2005).
–Multisensor image fusion in remote sensing is discussed in more detail in (Polhl, 1998).
–Fusion examples of the SAR and FLIR images and the related registrations can be found in (Chen, 2001).
3.Pixel-Layer Fusion
–An introduction to image fusion using the wavelet transform can be found in (Pajares, 2004).
–Detailed derivation and discussion for determining the optimal level of the wavelet decomposition in a fusion task can be found in (Li, 2005b).
4.Feature-Layer and Decision-Layer Fusions
–A tendency of fusion is going from pixels to regions (objects), and a general framework can be found in (Piella, 2003).
–More detailed information on the rough set theory and applications can be found in (Zhangwx, 2001).