Study of geometric measures of proximity of objects and classes in recognition systems

The task:

1. Develop a decision-making algorithm in a deterministic recognition system based on the use of known geometric measures of proximity.

2. Run the software implementation of the developed algorithm, debug the program.

3. Run control recognition of unknown objects by their feature vectors.

4. To compare the efficiency of making decisions on the assignment of unknown objects to given classes for various methods of calculating the distance between an object and a class. Determine the best combination of the proposed.

Stages of sequential work:

1. Determine the values ​​of signs of five standards for each of the four classes.

2. Display on the graph certain standards of all classes, instances of each class in the form of a point of a certain shape and color.

3. Implement the sequential interactive introduction of feature vectors of unknown objects that need to be recognized and assigned to one of four classes (feature values ​​can be determined by pressing the mouse button, asking the user or setting fixed values ​​in the code).

4. Implement the functions for calculating the degree of proximity between an unknown object and a given object of a particular class (methods for calculating the degree of proximity between two objects are different).

5. Implement the function of calculating the distance from each unknown object with a given class (methods for calculating the distance between an object and a class are different).

6. Implement the ability to select combinations of methods for calculating the degree of proximity and distance between objects.

7. Calculate the distance from the unknown object to each of the four classes.

8. Determine the class, the distance from the unknown object to which the minimum (in accordance with the selected combination of methods for calculating the measure of proximity between objects and the distance between the object and class).

9. Show the recognized object on the graph as a point, in the form and color of which coincides with the class to which it belongs.

A metric space is a set in which between any pair of elements a distance is defined that has certain properties, called a metric .

Definitions

Metric space is a pair where - many, and - numeric function, which is defined on the Cartesian product , takes values ​​in the set of real numbers, and is such that

1. ( identity axiom ).
2. ( axiom of symmetry ).
3. ( triangle axiom or triangle inequality).

Wherein

• lots of called the underlying set of metric space.
• elements of the set are called points of a metric space.
• function called metric .

Remarks

• The axioms of the distance function follow from axioms, since

  .

• If the triangle inequality is represented as

  for all and ,

then the symmetry axiom follows from the axiom of identity and inequality of a triangle.

Designations

Usually the distance between points and in metric space denoted by or .

• In metric geometry the designation is accepted. or if you need to emphasize that this is about . Less commonly used symbols and .
• In classical geometry, the notation or (dots are usually denoted by capital Latin letters).

Methods for calculating the distance between two objects in two-dimensional space.

Methods for calculating the distance between an object and a class.

An example of the implementation of the study of geometric measures of proximity objects and classes in recognition systems