Open episode, yestopologyOne of the most basic concepts in.Let A bemetric spaceA subset of X.If each point in A has aNeighborhoodIf it is contained in A, it is said that A is an open set in the metric space X.
If x ^ 2+y ^ 2=r ^ 2 is satisfied, the light is blue.If x ^ 2+y ^ 2<r ^ 2 is satisfied, it will be red.The red dots form an open set.Red and blue dottedUnionyesClosed set。
Suppose X is aaggregate, if there is a series of XsubsetMeet the following requirementscondition, then each such subset is called an open set of X, and X is calledtopological space 。
(2) The intersection of a finite number of open sets is an open set (the intersection of an infinite number of open sets may not be an open set);
(3) The sum of any number of open sets is an open set.
Open sets in metric spaces
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definition
Let A bemetric spaceA subset of X.If every point in A has a neighborhood centered on that point contained in A, that is, every point in A is A'sinterior point, then A is said to be an open set in the metric space X.In collection language, it is:
For any x ∈ A, there exists δ>0, so that B (x, δ) ⊆ A.
You can also define an open set from another perspective, that is, if a set does not containBoundary point(or without boundary points), this set is called an open set.That is to say, if A ∨ Partial A=∅, then A is an open set.
It can be proved that the two definitions are equivalent.
prove
If every point in A is an internal point, it is obvious that none of these points is a boundary point, so A ∨ Partial A=∅.On the contrary, if A ∨ ∨ A=∅, then either A is an empty set, or partial A is an empty set, or the points in A are not boundary points.When A is an empty set, according to the regulations, the empty set is an open set, so A is an open set.When partial A is an empty set, the point in X is either the interior point of A or theOuter point。Obviously, all outer points do not belong to A, so the points in A are all the inner points of A, that is, A is an open set.When all points in A are not boundary points, because all external points do not belong to A, of course, there are only internal points belonging to A, that is, A is an open set.
Closed set
Closed setOne definition is similar to another definition of an open set.If the boundary points of a set are contained in the set, that is, partial A ⊆ A, then A is said to be a closed set.Of course, there are other definitions of closed set. Please refer to relevant terms for details.
Special Open Set Topology
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definition
The real number space R is given an absolute value metric, and the corresponding open set is calledtopology。
nature
(1) Neighborhood is an open set.
The neighborhood B (p, r) with p as the center and r as the radius refers to the set of all points q satisfying | p-q |<r.Here | p-q | is the distance in the metric space X, and r is a positive number.
Let q be any point in the neighborhood B (p, r), and d be the distance between p and q, then 0 ≤ d<r, so r-d>0.Let h=r-d>0, then we can take the neighborhood B (q, h), and the distance | q-s |<h between any point s and q in this neighborhood.according toTrigonometric inequality，
That is, s ∈ B (p, r).
Because s is any point in the neighborhood B (q, h), since any point of B (q, h) belongs to B (p, r), we can see that B (q, h) ⊆ B (p, r), so q is an interior point.According to the arbitrariness of q, all points on B (p, r) are interior points, so B (p, r) is an open set.
In one-dimensional space (i.e. on the number axis), neighborhood is open interval, so open interval is open set.
(2) A is an open set if and only if itsComplementIs a closed set.
The proof shall be carried out according to other definitions of closed set, which is omitted here.
It can be seen from this property that since an empty set is an open set, its complement X is a closed set.Because X is an open set, its complement ∅ is a closed set.That is, an empty set and X are both open and closed sets.
(3) The sum of any number of open sets is an open set.
set up
Is a set of open sets, where a can be finite or infinite.And set their union
, then for any point x in E, it belongs to at least one Ea。Because EaIs an open set, so x is EaSo that δ>0 exists, so that B (x, δ) ⊆ Ea⊆ E, that is, x is the interior point of E.According to the arbitrariness of x, any point in E is an interior point, so E is an open set.
(4) The intersection of a finite number of open sets is an open set.
set up
Is a set of open sets, and i is a finite natural number.And set their intersection
If E is empty, then E is an open set and the proposition is proved.If E is not empty, then for any point x in E, it is the common point of the n open sets, so there is δi, i=1,2,3,... n, so that each neighborhood B (x, δi)⊆Ei。Take δ=min {δi}, then neighborhood B (x, δ) will be included in each EiB (x, δ) ⊆ E, so x is the interior point.According to the arbitrariness of x, any point in E is an interior point, so E is an open set.
It should be noted that the intersection of infinite open sets is not necessarily an open set. For example, open intervals (- 1/n, 1/n) are taken on the number axis. When n →∞, the intersection of these countless open intervals is {0}, and the set with only one point must be a closed set.
(5) The set formed by the interior point of A (called the interior of A, represented by the symbol A ° or int (A)) is the largest open set contained in A. In other words, the interior of A is open, and for any open set E ⊆ A, there is E ⊆ A °.
According to the definition of A °, it is soon known that A ° is an open set. Now it is proved that A ° is the largest open set in A.
If there is an open set E ⊆ A, which is larger than A °, then there must be some points x ∈ E but x ∉ A °.And because E is an open set, x is the interior point of E.According to property (3), because A=E ∨ A, x is also the interior point of A, so x ∈ A °, contradictory.
Let X be a complete set, and E and F be subsets of X.The difference set of set E and F refers to the set of all points x in X that satisfy x ∈ E and x ∉ F.
Let the complement of F in X be Fc, if x ∉ F, then x ∈ Fc, so E-F=E ∨ Fc。If E is an open set and F is a closed set, according to property (2), FcIt's an open episode.According to the property (4), E ∨ Fc=E-F is an open set.
Note that the difference set between an open set and an open set may be an open set, a closed set, or a non open and non closed set.For example:
The open interval (0,1) and open interval (2,3) on a line are both open sets, and their difference set is (0,1) or open set.
The open interval (0100) and (0,1) ∨ (99100) on a line are both open sets, and their difference sets are [1,99] closed sets.
The open interval (0100) and (0,99) on a line are both open sets, and their difference set is [99100], which is not open or closed.
Several Important Plane Point Sets
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1) Open set: If all points of the point set are interior points of, it is called open set. For example, open set
2) Closed set: If the remainder of a point set is an open set, it is called a closed set. For example, it is a closed set. It should be pointed out that it is neither an open set nor a closed set
3) Connected set: If any two points in point set E can be connected by a polyline, and the points on the polyline belong to, it is called connected set
4) Region (or open region): Connected open sets are called regions or open regions
5) Closed region: The set of open regions and their boundaries is called closed region. For example, a region is a closed region
6) Bounded set: for a plane point set, if there is a positive number, where O is the coordinate origin, it is called a bounded set
7) Unbounded set: if a set is not a bounded set, it is called an unbounded set. For example, a bounded closed region is an unbounded closed region;It is an unbounded area
Note: It should be noted that although a closed region contains boundaries, it may also be unbounded;An open domain has no boundary, but it may also be bounded[1]
Application Open Set Face Recognition
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In order to solve the problem that traditional ASM is not ideal for locating facial contour points, this paper proposes a local profile constraint ASM.This model has two improvements on the traditional ASM. First, the contour strength of candidate points is added to the local texture matching function of ASM as a self-adjusting weight, so that the best matching points are more easily attracted to the facial contour.Secondly, the total variation model (TVM) is introduced as the pre-processing before image calibration, which can enhance the contrast of contour intensity between contour points and their one-dimensional neighborhood points on the premise of retaining enough texture information for calibration.The large-scale test results on the BioID face database show that the method effectively improves the positioning accuracy of contour points, and lays a good foundation for subsequent feature ratio matching.3. Open set recognition in feature comparison is studied, and an open set based on Adaboost is proposedFace recognition algorithm。 In face image recognition system, feature alignment algorithm directly affects the performance of recognition system, and is the core problem of recognition algorithm.This paper proposes a novel solution to the open set problem in feature comparison algorithm, that is, the recognition problem with rejection.The geometric transformation of samples is used to reduce the overlapping area between positive and negative sample similarity, expand the distance between positive and negative sample sets, and then improve the general closed set face recognition method based on Adaboost.At the same time, a two-layer recognition structure and a sample transformation preprocessing strategy are used to improve the recognition speed.[2]
A new method to optimize the similarity space is proposed to improve the accuracy of open set face recognition. First, the open set recognition problem is transformed into a binary classification problem, and then an optimization method is introduced to find the optimal hyperplane to partition the similarity space. The hyperplane can partition the similarity space into acceptance space and rejection space,The position of the similarity vector in the space is used to judge whether the sample is a known class. Because the information of the vector distribution in the similarity space is used, the trained features have stronger classification ability. Experiments on different human face databases show that the method proposed in this paper can significantly improve the accuracy of open set recognition compared with traditional methods[3]
