CN104936287A - Indoor fingerprint location method for sensor network based on matrix completion - Google Patents

Abstract

The invention discloses a sensor network indoor fingerprint positioning method based on matrix completion is applicable to the robust indoor positioning method of the wireless sensor network WSN. The sensor network indoor fingerprint positioning method disclosed by the invention comprises steps of utilizing a matrix completion theory to recover a complete fingerprint database by only sampling part of signal fingerprints through utilizing the low-rank property of a fingerprint matrix, adopting a classic KNN algorithm to perform object positioning of an online phase after the fingerprint database is constructed, in the process of the fingerprint matrix complementation and for effectively eliminating the outlier noise and the Gauss noise contained by the signal fingerprint, respectively introducing an L1 norm regularized item and an F norm regularized item, modeling a fingerprint data recovery question into a norm regularized matrix complementation question and solving the question through an alternative direction mutiplier method algorithm The sensor network indoor fingerprint positioning method based on matrix completion can effectively reduce the workload of the information fingerprint database construction and can acquire positioning precision which is higher than the positioning precision obtained from the same kind method under various noise conditions.

Description

Indoor fingerprint location method for sensor network based on matrix completion

technical field

The present invention is a robust indoor positioning method suitable for wireless sensor network (WSN). The method utilizes the low-rank characteristic of the signal fingerprint matrix to model the fingerprint data restoration problem under noise interference as norm regularization. On this basis, the L1 norm and F norm are introduced to smooth the outlier noise and Gaussian noise, and finally it is effectively solved by the method of alternating direction multipliers. This method only needs to collect a small amount of signal fingerprint data to recover the fingerprint library relatively completely, and can obtain positioning accuracy higher than similar methods in various noise scenarios. The technology belongs to the field of wireless sensor network.

Background technique

In recent years, wireless sensor network (Wireless Sensor Network, WSN) technology has made great progress, and is widely used in military investigation, intelligent transportation, environmental monitoring and other fields, smart home system, automatic parking distance warning, medical health monitoring, forest Fire monitoring and so on are typical applications of wireless sensor networks. With the popularity of a large number of smart mobile devices carrying small sensors, the implementation forms of wireless sensor networks have been further enriched, and the coverage of wireless sensor networks has been expanded.

In wireless sensor networks, the acquisition of sensor node's own location information is crucial for various applications. With the development of mobile Internet and the popularization of smart mobile terminals, indoor positioning technology has attracted extensive attention of researchers. As the continuation of outdoor positioning technology in indoor environment, indoor positioning fills the gap of traditional positioning technology and has broad application prospects. For example, positioning specific stores in large shopping malls, intelligent shopping guide systems, and indoor personnel positioning systems in public safety incidents such as fires. In the outdoor environment, the positioning technology represented by GPS is very mature, but in buildings, due to the obstruction of walls, glass and other obstacles, the GPS signal is severely attenuated, and the ideal positioning effect cannot be achieved. In addition, due to the complex indoor environment and numerous obstacles and interference sources, multipath effects and noise interference in the signal propagation process have become common phenomena, further increasing the difficulty of indoor positioning.

Existing indoor positioning technologies are mainly based on ultrasound, RFID, UWB, ZigBee, and WLAN. Compared with other technologies, the indoor positioning technology based on WLAN signal fingerprints makes use of widely available WLAN access points, without the need for additional installation of specific equipment with high costs, and has become a relatively mature indoor positioning technology at present. However, the indoor positioning algorithm based on signal fingerprints first needs to survey the indoor environment and establish a fingerprint database, which is a huge workload; at the same time, due to the existence of many obstacles and interference sources in the room, the collected RSSI data inevitably has errors, and then The final positioning accuracy is greatly reduced. Therefore, it is urgent to design a robust indoor positioning method to obtain high positioning accuracy while reducing the workload of fingerprint library construction.

Contents of the invention

Technical problem: The purpose of the present invention is to design an indoor robust positioning method based on fingerprint matrix completion, which is suitable for robust indoor positioning of wireless sensor network (WSN). This method takes advantage of the low-rank characteristics of the signal fingerprint matrix to model the fingerprint data recovery problem under noise interference as a norm regularized matrix completion problem. On this basis, L1 norm and F norm are introduced to smooth outlier noise and Gaussian noise, which is finally efficiently solved by the method of alternating direction multipliers. The method proposed by the invention can restore the fingerprint library relatively completely only by collecting a small amount of signal fingerprint data, and can obtain positioning accuracy higher than similar methods in various noise scenarios.

Technical solution: The present invention is an indoor robust positioning method based on fingerprint matrix completion. By utilizing the low-rank property of the fingerprint matrix, the complete fingerprint library can be restored by using the matrix completion theory only by sampling part of the signal fingerprints; after the fingerprint database is constructed, the classic K-nearest neighbors (K-Nearest Neighbors, KNN) algorithm is used for Targeting in the online phase. In the process of fingerprint matrix completion, in order to effectively eliminate the outlier noise and Gaussian noise contained in the signal fingerprint, the L1 norm regularization term and the F norm regularization term are respectively introduced, and the fingerprint data recovery problem is modeled as norm regularization Matrix completion problem and its solution by the method of alternating direction multipliers. This method can effectively reduce the workload of information fingerprint library construction, and obtain higher positioning accuracy than similar methods under various noise conditions.

The indoor robust positioning method based on fingerprint matrix completion is included in the following specific steps

Initial scene setup:

Step 1) Set the positioning area as a rectangular area, and divide it into n ₁ in the horizontal direction and n ₂ in the vertical direction, then the whole area is evenly divided into n ₁ ×n ₂ =n rectangular grids, and each grid represents a Reference point, a total of n reference points;

Step 2) Randomly deploy m wireless access points in the entire area; for any access point APi, assuming that wireless signals are collected at all reference points, a fingerprint matrix can be constructed Where i=1,2,...,m, the matrix elements are the wireless signal strengths from the APi collected at each reference point;

Offline fingerprint database construction:

Step 3) In order to reduce the workload, only some reference points are randomly selected for wireless signal acquisition, and the reference point subscript set is Ω;

Step 4) Generate a fingerprint matrix with missing elements and possibly noise for each access point APi Where i=1,2,...,m, P _Ω ( ) is an orthogonal projection operator, defined as:

${<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Mi</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}{\mathrm{<span\; class="notranslate">\; Mi</span>}}_{\mathrm{<span\; class="notranslate">\; jk</span>}}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Element;</span>\mathrm{<span\; class="notranslate">\; \&Omega;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; otherwise</span>}\end{array}\right.$

Indicates that when the subscript (j,k)∈Ω, the matrix element is the wireless signal strength value collected at the location;

Step 5) Utilize the norm regularization fingerprint matrix completion algorithm to restore P _Ω (Mi) to a complete fingerprint matrix;

Step 6) construct the fingerprint library of the whole area according to m fingerprint matrices;

Online Orientation:

Step 7) set TP as the point to be located, and carry out wireless signal strength measurement to m access points at this point;

Step 8) Utilize the classic K nearest neighbor (K-Nearest Neighbors, KNN) algorithm to compare the wireless signal strength of TP and each reference point, and obtain the K reference points most similar to TP;

Step 9) The physical position of TP is the mean value of coordinates of K reference points.

Some of the key concepts involved in the above steps are as follows:

Fingerprint matrix construction

As shown in Figure 1, the bit area is set as a rectangular area, which is evenly divided into n ₁ ×n ₂ =n grids, and each grid represents a reference point. Sampling at each reference point will form a matrix of size n ₁ ×n ₂ for each AP. The elements in the matrix are the wireless signal strengths collected from the AP at the corresponding reference point at the position. The entire matrix It can be regarded as the signal fingerprint distribution map of the AP in the positioning area, which we call the fingerprint matrix. If the point-by-point sampling method is used, high positioning accuracy can be obtained, but the huge sampling workload is often unbearable; at the same time, due to the complexity of the environment, not all grid areas can be used for signal measurement, usually only A small subset of sampling points can be measured, so the fingerprint matrix is incomplete, only some elements are known, and the missing elements need to be supplemented by matrix completion algorithm. As shown in Figure 1, the gray square indicates that the actual measurement has been carried out at the reference point at this position, and the signal strength value corresponding to the reference point of the white square is obtained through the matrix completion algorithm.

Fingerprint database construction

In the process of building the offline fingerprint database, some reference points are selected for wireless signal strength collection, and for each reference point AP _i (i=1, 2, L, m) will generate a missing and noisy fingerprint moment Using the norm regularized fingerprint matrix completion algorithm proposed by the present invention, we can recover the original fingerprint matrix A _i more accurately, so as to obtain the signal strength of AP _i at the unsampled position. Expand A _i into a row vector Then the vector represents the wireless signal strength values of AP _i at all reference point locations. Based on this, we can construct a fingerprint library for the entire region

$\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; =</span>\left(\begin{array}{c}{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; m</span>}\\ {\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\end{array}\right)<span\; class="notranslate">\; =</span>\left(\begin{array}{cccc}{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 1,1</span>}}& {\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 1,2</span>}}& \mathrm{<span\; class="notranslate">\; L</span>}& {\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}\\ {\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 2,1</span>}}& {\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 2,2</span>}}& \mathrm{<span\; class="notranslate">\; L</span>}& {\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}\\ \mathrm{<span\; class="notranslate">\; m</span>}& \mathrm{<span\; class="notranslate">\; m</span>}& \mathrm{<span\; class="notranslate">\; o</span>}& \mathrm{<span\; class="notranslate">\; m</span>}\\ {\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}}& {\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}}& \mathrm{<span\; class="notranslate">\; L</span>}& {\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}\end{array}\right)$

Among them, P _{i, j} represents the wireless signal strength received from the i-th AP at the j-th reference point position. Assuming that k (k<n) reference points are selected for sampling, the workload of signal acquisition is reduced to the original k/n, which undoubtedly greatly reduces the work overhead in the construction phase of the offline fingerprint library.

Norm Regularized Fingerprint Matrix Completion Algorithm

In order to reduce the workload of constructing the fingerprint database, the algorithm proposed by the invention only collects a small amount of wireless signal fingerprints, and utilizes the low-rank characteristic of the fingerprint matrix to restore the complete fingerprint database through the matrix completion theory. In indoor positioning scenarios, there are often errors in the collected wireless signal strength values. In addition to common Gaussian noise, outlier noise (that is, data far beyond the normal range) is also a non-negligible noise component.

In order to effectively deal with noise interference in wireless signal acquisition, this paper introduces regularization technology into the matrix completion problem, and describes outlier noise and Gaussian noise by L0 norm and F norm respectively. Suppose A is a fingerprint matrix, Z is a sparse outlier matrix, and P _Ω (M) is a sampling matrix, then signal fingerprint matrix completion under the condition of outlier noise can be modeled as the following problem:

Since both the rank and the L0 norm of the matrix are non-convex functions, problem (1) is an NP-hard problem. We relax the rank function of the matrix to the matrix kernel norm, and relax the L0 norm of the matrix to the L1 norm, so the above problem can be relaxed into the following convex optimization problem:

We use the Alternating Direction Multiplier Method (ADMM) to solve the problem. First rewrite the constraints of the problem into a linear form:

The augmented Lagrange function corresponding to problem (3) is:

$\begin{array}{c}{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Z</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&lambda;</span>}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; Z</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ <span\; class="notranslate">\; +</span><span\; class="notranslate">\; \&lang;</span>\mathrm{<span\; class="notranslate">\; Y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; Z</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; \&rang;</span><span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; Z</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\end{array}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

Applying the alternating direction multiplier method to (4), and setting the initial value Z ₀ =G ₀ =E ₀ =Y ₀ =0, we can obtain the solution of problem (3) through the following iterative sequence.

After several iterations, the matrix A finally converges to its optimal value, that is, a complete and more accurate fingerprint matrix is restored.

Beneficial effects: using the indoor fingerprint positioning algorithm based on matrix completion proposed by the present invention, the corresponding scheme has the following advantages:

1. Effectively reduce the workload of offline fingerprint library construction phase

The indoor positioning algorithm based on signal fingerprints includes an offline sampling stage and an online positioning stage. The offline stage needs to survey the indoor environment and establish a fingerprint database. The traditional algorithm uses the point-by-point sampling method to construct the fingerprint database, and the workload is very huge; however, in the process of building the offline fingerprint database in this scheme, only a small number of reference points are selected for wireless signal strength collection, and the signal strength values of the remaining reference points are passed through the norm Regularized matrix completion algorithm for restoration can effectively reduce the workload in the offline stage.

2. Effectively deal with noise interference in the process of wireless signal acquisition

In the process of wireless signal collection, due to the existence of many obstacles and interference sources in the room, the collected RSSI data inevitably has errors. In addition to the common Gaussian noise, some abnormal RSSI values far beyond the normal range caused by equipment failure, personnel movement and environmental changes (we call it outlier noise, Outlier) are also noise components that cannot be ignored. The existence of these noises seriously affects the authenticity of the fingerprint database, which in turn greatly reduces the final positioning accuracy. This scheme introduces regularization technology into the matrix completion problem, and smooths the outlier noise and Gaussian noise through the L0 norm and the F norm respectively, which can effectively deal with the noise interference in the wireless signal acquisition process. Under the Gaussian noise condition Under the conditions of , outlier noise and Gaussian outlier mixed noise, the positioning accuracy can be higher than that of similar algorithms.

3. Strong adaptability

This solution can be applied to various scenarios by adjusting related parameters. When the positioning accuracy requirement is high, the number of reference points for signal acquisition can be appropriately increased; when the positioning accuracy requirement is not high, the number of reference points for signal acquisition can be reduced, thereby further reducing the workload of the offline fingerprint library construction stage. At the same time, by adjusting the penalty factor of L0 norm and F norm, this scheme can obtain higher positioning accuracy under the condition of no noise, Gaussian noise, outlier noise and Gaussian outlier mixed noise .

Description of drawings

Fig. 1 Schematic diagram of positioning area based on grid division and partial sampling,

Figure 2 is a flow chart of the scheme.

Detailed ways

The core design idea of the implementation scheme of the sensor network indoor fingerprint positioning method based on matrix completion is: apply the matrix completion theory to indoor positioning based on wireless signal fingerprints, and only need to sample part of the signal fingerprints by using the low rank of the fingerprint matrix The complete fingerprint database can be restored by using the matrix completion theory; after the fingerprint database is constructed, the classic KNN algorithm is used for target positioning in the online stage. In the process of fingerprint matrix completion, in order to effectively eliminate the outlier noise and Gaussian noise contained in the signal fingerprint, the L1 norm regularization term and the F norm regularization term are respectively introduced, and the fingerprint data recovery problem is modeled as norm regularization Matrix completion problem and its solution by the method of alternating direction multipliers. This method can effectively reduce the workload of information fingerprint library construction, and obtain higher positioning accuracy than similar methods under various noise conditions.

Specific steps include:

Initial scene setup:

Step 1) Set the positioning area as a rectangular area of 50m×100m, divide it horizontally and vertically at intervals of 2m, then the entire area is evenly divided into 25×50=1250 rectangular grids, and each grid represents a reference points, a total of 1250 reference points;

Step 2) Randomly deploy 30 wireless access points in the whole area; for any access point APi, assuming that wireless signals are collected at all reference points, a fingerprint matrix can be constructed Where i=1,2,...,30, the matrix elements are the wireless signal strengths from the APi collected at each reference point;

Offline fingerprint database construction:

Step 3) In order to reduce the workload, only some reference points are randomly selected for wireless signal acquisition, and the reference point subscript set is Ω;

Step 4) Generate a fingerprint matrix with missing elements and possibly noise for each access point APi Where i=1,2,...,m, P _Ω ( ) is an orthogonal projection operator, defined as:

${<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Mi</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}{\mathrm{<span\; class="notranslate">\; Mi</span>}}_{\mathrm{<span\; class="notranslate">\; jk</span>}}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Element;</span>\mathrm{<span\; class="notranslate">\; \&Omega;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; otherwise</span>}\end{array}\right.$

Indicates that when the subscript (j,k)∈Ω, the matrix element is the wireless signal strength value collected at the location;

Step 5) Utilize the norm regularization fingerprint matrix completion algorithm to restore P _Ω (Mi) to a complete fingerprint matrix;

Step 6) construct the fingerprint library of the whole area according to 30 fingerprint matrices;

Online Orientation:

Step 7) set TP as the point to be located, and measure the wireless signal strength of 30 access points at this point;

Step 8) Utilize the classical KNN algorithm to compare the wireless signal strength of TP and each reference point, take K=20 in the experiment, promptly obtain 20 reference points most similar to TP;

Step 9) The physical position of the TP is the mean value of the coordinates of the 20 reference points.

Claims (1)

1. a sensor network indoor fingerprint location method based on matrix completion, it is characterized in that the method comprises the following concrete steps: Initial scene setup: Step 1) Set the positioning area as a rectangular area, and divide it into n ₁ in the horizontal direction and n ₂ in the vertical direction, then the whole area is evenly divided into n ₁ ×n ₂ =n rectangular grids, and each grid represents a Reference point, a total of n reference points; Step 2) Randomly deploy m wireless access points in the entire area; for any access point APi, assuming that wireless signals are collected at all reference points, a fingerprint matrix can be constructed Where i=1,2,...,m, the matrix elements are the wireless signal strengths from the APi collected at each reference point; Offline fingerprint database construction: Step 3) In order to reduce the workload, only some reference points are randomly selected for wireless signal acquisition, and the reference point subscript set is Ω; Step 4) Generate a fingerprint matrix with missing elements and possibly noise for each access point APi Where i=1,2,...,m, P _Ω ( ) is an orthogonal projection operator, defined as: ${<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Mi</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}{\mathrm{<span\; class="notranslate">\; Mi</span>}}_{\mathrm{<span\; class="notranslate">\; jk</span>}}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Element;</span>\mathrm{<span\; class="notranslate">\; \&Omega;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; otherwise</span>}\end{array}\right.$ Indicates that when the subscript (j,k)∈Ω, the matrix element is the wireless signal strength value collected at the location; Step 5) Utilize the norm regularization fingerprint matrix completion algorithm to restore P _Ω (Mi) to a complete fingerprint matrix; Step 6) construct the fingerprint library of the whole area according to m fingerprint matrices; Online Orientation: Step 7) set TP as the point to be located, and carry out wireless signal strength measurement to m access points at this point; Step 8) Utilize the classic K nearest neighbor algorithm KNN to compare the wireless signal strength of TP and each reference point, and obtain the K reference points most similar to TP; Step 9) The physical position of TP is the mean value of coordinates of K reference points.