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1、CIRCUITS SYSTEMS SIGNAL PROCESS VOL. 11, NO. 1, 1992 AN OVERVIEW OF MORPHOLOGICAL FILTERING* Jean Serra 1 and Luc Vincent 12 Abstract. This paper consists of a tutorial overview of morphological filtering, a theory introduced in 1988 in the context of mathematical morphology. Its first section is de
2、voted to the presentation of the lattice framework. Emphasis is put on the lattices of numerical functions in digital and continuous spaces. The basic filters, namely the openings and the closings, are then described and their various versions are listed. In the third section morphological filters a
3、re defined as increasing idempotent operators, and their laws of composition are proved. The last sections are concerned with two special classes of filters and their derivations: first, the alternating sequential filters allow us to bring into play families of operators depending on a positive scal
4、e parameter. Finally, the center and the toggle mappings modify the function under study by comparing it, at each point, with a few reference transforms. 1. Mathematical morphology for complete lattices 1.1. Introduction We define a morphological filter as an operator b, acting on a complete lattice
5、 Y 4, which: (i) preserves the ordering over J. 2. For any (finite or infinite) family (A 3 in J-, there exists: ?9 a smallest majorant v A i called the “sup“ (for supremum), ?9 a largest minorant /x Ai called the “inf“ (for infimum). In particular, J- posesses a greatest element, E, and a smallest
6、one, . In a lattice, any logical consequence of a choice of ordering remains true when we commute the symbols v and A, and _. This is called the principle of duality with respect to the order. Here is a review of a few basic lattices: 1.2.1. Boolean lattices. Start from an arbitrary set E. Obviously
7、, the set (E) of the subsets of E, which is ordered for the inclusion relationship, is a complete lattice for the operations u (union) and n (intersection). Moreover, 50 SERRA AND VINCENT with each X r (E), there exists a unique xc (E), called the complement of X, such that XcX c= and XuX c=E. (1) F
8、inally, (E) also satisfies the important property of general distributivity under which, for all T E (E) and any family (Xi) of elements of (E), we have (U Xi) Y = U (xi ( Y), (2) ( Xi) u Y = (- (X i u Y). (3) 1.2.2. Topological lattices. When E is a topological space, its open sets generate a compl
9、ete lattice for the inclusion, where the sup coincides with the union and where inf(Xi) is the interior of X i. This lattice is not complemented. It satisfies the general distributivity of type (2), but finite distributivity only of type (3). Indeed, in the general case of an infinite family (Xi), w
10、e have but only /U xl) r = U r) o o Similar structures are derived for the closed sets and the compact sets. (4) (s) 1.2.3. The convex lattice. The class of the convex sets of the Euclidean space N“ generates a complete lattice where the inf coincides with intersection and where the sup is the conve
11、x hull. 1.2.4. The partition lattice. In the set of the partitions of an arbitrary set E, we can introduce the following ordering: a partition A is smaller than a partition B when each class of A is included in a class of B. This leads to a lattice which is complete, but neither complemented nor dis
12、tributive. 1.2.5. Function lattices. Let E be an arbitrary space. The class Y of the “extended“ real-valued functions f:E is obviously ordered by the relation f 1, (f)(x) = otherwise. (9) This operation is shown on Figure 3. Figure 3. The threshold mapping /,. im 52 SERRA AND VINCENT 1- 1-2/4 . . .
13、. . . . 1-1/3 . 1 d/2 . -1 Figure 4. The family of functions (fl). In set terms, the transformation consists in intersecting the umbra U +(f) by the closed half-space E a =(x,z),xE,z 1, and in taking the upper umbra of the result U+(O(f) = U+EE1 c U+(f) w E_,. (10) If functions and upper umbrae are
14、equivalent, then the two algorithms (9) and (10) must yield the same result. Let us apply both of them to the sup of the following family (see Figure 4): fi(x) = 1 - 1/i when Ixl z. As threshold level z increases, Xz(f) decreases continuously, i.e., Xz(f)= (- Xz,(f ) or Xz(f)= J,X,(f). z_l or x=0? A
15、ctually, the maximum of such a function, although it is bounded, does not exist. Conversely, as soon as we refer to the “maximum“ of a function over a continuous space, we implicitely introduce the requirement that it is upper semicontinuous (or lower semicontinuous when looking for minima). There-
16、fore, in Section 3.6, we assume that the functions under study are lower semicontinuous. Y -1 0 1 Figure 5. A function without a maximum (it is not upper semicontinuous). 54 SERRA AND VINCENT In the lattice J, of the upper semicontinuous functions, we have inffi = fe, U+(f) sup fi = fe “5, U+(f) a notation which shows that the lattice u and that of the closed upper umbrae are iso
