CN103139907A - Indoor wireless positioning method by utilizing fingerprint technique - Google Patents

Abstract

The invention discloses an indoor wireless positioning method by utilizing fingerprint technique. The indoor wireless positioning method by utilizing the fingerprint technique largely lowers the requirements for the density of sampling points, enriches distribution information of signal strength, lightens the workload, and keeps high-precision positioning results. The indoor wireless positioning method by utilizing the fingerprint technique comprises an off-line training phase and an on-line training phase. The off-line training phase includes the following steps: firstly, an indoor environment is set up and signal strength data are acquired; secondly, an original fingerprint database with a low sampling rate is constructed; and thirdly, the original fingerprint database is reconstructed through a low-order matrix fill model, the low-order matrix fill model is minrank (X) s.t.A(X) = B. In the on-line training phase, signal strength vectors used for testing and the signal strength vector of each sampling point in the reconstructed fingerprint database are compared and matched, and position coordinates of a present position are estimated by utilizing the point coordinates of the known sampling points to obtain on-line positioning coordinates.

Description

A kind of indoor wireless positioning method that utilizes fingerprint technique

Technical field

The invention belongs to the technical field of wireless location, relate to particularly a kind of indoor wireless positioning method that utilizes fingerprint technique.

Background technology

At present, location-based service and a series of application that bring thus more and more receive people's concern, and mobile subscriber's location information demand of property and instantaneity on the spot increases day by day.A lot of interested events of user, as environmental monitoring, logistics management, condition of a fire report, must combine with positional information just to have to utilize is worth.Comparatively ripe localization method has the GPS technology at present, is widely used in outdoor positioning, and positioning accuracy can reach 10 meters left and right.And indoor, due to the impact of the factors such as wall, positioning accuracy is difficult to reach people's needs.Therefore at some specific areas, be the problem that people must consider as obtaining high-precision positioning result in indoor environment.

Have widely distributedly based on the location of wifi signal strength signal intensity, obtain the advantages such as convenient, become one of focus of indoor wireless positioning method research.The method need to be set up a plurality of wireless routers at interested locating area, as accessing points (AP).Each wireless router can send signal, and generally signal strength values (RSS) can be decayed along with the increase of distance.Different positions can receive the signal that different AP sends, and the signal strength values of reception is also different.Therefore, the location based on the wifi signal strength signal intensity can position according to the signal strength values from each AP that a certain position receives.

Mainly be divided into two kinds of geometric measurement method and scene analysis methods based on the localization method of wifi signal strength signal intensity.At first geometric measurement method requires the propagation model (empirical model or Mathematical Modeling) according to radio signal, and signal strength values is mapped as the distance that signal is propagated.On two dimensional surface, according to the distance between terminal equipment and other at least three AP, the geometry principle by trilateration carries out location estimation.But due to the complexity of indoor radio wave propagation, signal strength signal intensity is subject to the impacts such as multipath transmisstion, reflection, makes in actual indoor environment to be difficult to portray with fixing Mathematical Modeling.The scene analysis method, be called again fingerprint technique, not that direct measurement with signal strength values is mapped as the signal propagation distance, but utilize the scene characteristic of observing in a certain place to infer observer's position, can be regarded as first the inherent law between signal strength signal intensity and position is learnt, and then mate with new measured value and the sample point of learning.

The method proposed in 2000 in the RADAR of Microsoft system, generally was divided into off-line measurement and two stages of online location to complete the location.Off-line measurement is according to the selected some sampled points of certain spacing distance in the zone of needs location, form the grid of a sample point, measure on these sample points positions, record is from the signal strength measurement vector of each AP, these information structures the signal strength signal intensity fingerprint base.This fingerprint base has been described the relation of signal strength signal intensity and locus in this stationary positioned environment.During online location, system adopts certainty matching algorithm K nearest-neighbors (KNN) algorithm one by one, compare according to the signal strength signal intensity that records in the signal strength signal intensity that records and database, the coordinate of that point of signal strength signal intensity mean square deviation minimum is as the position of estimating.Because the radio waves propagation model that fingerprint technique is more traditional can be described the relation of RSS and locus more accurately, and need not the prior information of AP particular location, thereby be widely used in indoor locating system based on RSS.

Setting up fingerprint base is the basis of realizing positioning function.In order to reduce the unstable impact that brings of RSS, traditional method of setting up fingerprint base is to utilize the correlation of time, repeatedly measures under same sample point and averages, as shown in Figure 1, each sample point is collected the signal strength values of each AP, directly deposits these information in fingerprint base.Shih-Hau Fang proposes a kind of dynamical system, and the time series of RSS sample is merged into a kind of state, and the use state replaces RSS directly to carry out location estimation.C.Feng has added directional information in fingerprint base, respectively 0 °, 90 °, 180 °, 270 ° are sampled, and sets up the fingerprint base on four direction, with the impact of the factors such as the hand-held sample devices of minimizing people on signal.

Xing-chuan Liu proposes signal except time correlation, and correlation is spatially also arranged, and the sampled data in certain radius is weighted on average, tries to achieve a reference point.

BinghaoLi compares the quantity of sample point by experiment, shows along with sample point increases, and the sample point interval reduces, and the precision of location estimation increases.But sample point is more, can bring linear growth to the surveying work amount.The author points out sample point Existential Space correlation, and namely when measuring the sub-fraction sample point, they not only provide the information under these positions, and the information of peripheral region also is provided, and utilize spatial coherence can obtain easily more sample point information.The author has adopted in inverse distance weighted interpolation method method (LDW) and general gram golden interpolation method (UK) to set up fingerprint base, as shown in Figure 1, utilize adjacent sample point to collect the signal strength values of each AP, the signal strength values of the sample point between estimation deposits fingerprint base in the lump in.

Mostly the data of sampling are directly built up fingerprint base after by statistics in existing indoor orientation method, however the density of sampled point and positioning accuracy contact directly, sampled point is more intensive, positioning accuracy is higher.This causes wanting to obtain the high locating effect of precision, and off-line phase hand labor workload is large, inefficiency.A part of localization method adopts interpolation method to build fingerprint base in addition, but this method, the accuracy of interpolation point information can't guarantee.

Summary of the invention

Technology of the present invention is dealt with problems and is: overcome the deficiencies in the prior art, provide a kind of and greatly reduce requirement to sampling point density, enrich signal intensity profile information, keep the indoor wireless positioning method that utilizes fingerprint technique of high-precision positioning result when reducing workload.

Technical solution of the present invention is: this indoor wireless positioning method that utilizes fingerprint technique, comprise off-line training step and online positioning stage,

Off-line training step comprises the following steps:

(1) set up indoor environment and collection signal intensity data;

(2) the original fingerprint storehouse of structure low sampling rate;

(3) by low-rank matrix fill-in model, the original fingerprint storehouse is reconstructed low-rank matrix fill-in mould

Type is formula (1)

minrank(X) s.t.A(X)=B （1）

Wherein A (X) is the template operator, X is a measurement matrix that comprises whole sampled points, each element X (i, j) represent sampled point (i, j) receive signal strength signal intensity from a certain accessing points AP, A (X)=B only has the relatively sparse effective signal strength values of the existence by the sampled point of actual measurement, A (X) is defined as the matrix Q of formula (2)

Then obtain formula (3)

B(i,j)=Q(i,j)X(i,j) （3）；

In online positioning stage, will mate for the signal strength signal intensity vector of the signal strength signal intensity vector of testing with each sampled point of the fingerprint base of reconstruct, current position coordinates utilizes each known sample point coordinate to estimate to obtain the online elements of a fix.

This method is at off-line training step low sampling rate down-sampling, by low-rank matrix fill-in model to comprising the fingerprint base of a large amount of intensive sampled point signal strength signal intensities in refactoring localization zone, original fingerprint storehouse, then the signal strength signal intensity vector of each sampled point in the fingerprint base of online positioning stage and reconstruct mates and obtains the online elements of a fix, so just greatly reduces requirement to sampling point density, enriches signal intensity profile information, keeps high-precision positioning result in the reduction workload.

Description of drawings

Fig. 1 shows the schematic diagram of setting up fingerprint base according to the interpolation method of prior art;

Fig. 2 shows the flow chart according to the emulation experiment embodiment of the indoor wireless positioning method that utilizes fingerprint technique of the present invention;

Fig. 3 shows the flow chart according to the actual environment EXPERIMENTAL EXAMPLE of the indoor wireless positioning method that utilizes fingerprint technique of the present invention.

Embodiment

This indoor wireless positioning method that utilizes fingerprint technique comprises off-line training step and online positioning stage,

Off-line training step comprises the following steps:

(1) set up indoor environment and collection signal intensity data;

(2) the original fingerprint storehouse of structure low sampling rate;

(3) by low-rank (Low-rank, LR) matrix fill-in model, the original fingerprint storehouse is reconstructed, low-rank matrix fill-in model is formula (1)

minrank(X) s.t.A(X)＝B （1）

Wherein A (X) is the template operator, X is a measurement matrix that comprises whole sampled points, each element X (i, j) represent sampled point (i, j) receive signal strength signal intensity from a certain accessing points AP, A (X)=B only has the relatively sparse effective signal strength values of the existence by the sampled point of actual measurement, A (X) is defined as the matrix Q of formula (2)

Then obtain formula (3)

B(i,j)=Q(i,j)X(i,j) （3）；

In online positioning stage, will mate for the signal strength signal intensity vector of the signal strength signal intensity vector of testing with each sampled point of the fingerprint base of reconstruct, current position coordinates utilizes each known sample point coordinate to estimate to obtain the online elements of a fix.

This method is at off-line training step low sampling rate down-sampling, by low-rank matrix fill-in model to comprising the fingerprint base of a large amount of intensive sampled point signal strength signal intensities in refactoring localization zone, original fingerprint storehouse, then the signal strength signal intensity vector of each sampled point in the fingerprint base of online positioning stage and reconstruct mates and obtains the online elements of a fix, so just greatly reduces requirement to sampling point density, enriches signal intensity profile information, keeps high-precision positioning result in the reduction workload.

Preferably, step (3) also comprises by low-rank (the Smoothing Low-Rank with smoothing, SLR) the matrix fill-in model is reconstructed the original fingerprint storehouse, is formula (4) minrank (X)+λ S (X) s.t.A (X)=B with the low-rank matrix fill-in model of smoothing

（4）

Wherein S (X) is a successional smoothing factor of expression X, and the value of S (X) is less, represents that the continuity of X is better, and λ is the coefficient of balance that is obtained by experiment.

Preferably, the difference by matrix horizontal and vertical direction defines S (X), sees formula (5)

$S\left(X\right)={\left|\right|D_x\left(X\right)\left|\right|}_F^2+{\left|\right|D_y\left(X\right)\left|\right|}_F^2---\left(5\right)$

D wherein _x(X) be that a size is N ₁* (N ₂-1) matrix represents the poor of each element horizontal direction in matrix X, sees formula (6)

D _x(i,j)=X(i,j+1)-X(i,j) （6）

D _y(X) be that a size is (N ₁-1) * N ₂Matrix, represent the poor of each element vertical direction in matrix X, see formula (7)

D _y(i,j)=X(i+1,j)-X(i,j) (7)

Operator

Represent the Frobenius norm of matrix, obtain low-rank matrix fill-in model with smoothing by formula (8) like this

$\min \mathrm{rank}\left(X\right)+\mathrm{\λ}({\left|\right|D_x\left(X\right)\left|\right|}_F^2+{\left|\right|D_y\left(X\right)\left|\right|}_F^2)$ s.t.A(X)=B （8）。

Preferably, adopt singular value decomposition (Singular Value Decomposition to low-rank matrix fill-in model with the finding the solution of low-rank matrix fill-in model of smoothing in step (3), SVD) method by replacing iterative, generates the fingerprint base of reconstruct.

Preferably, the SVD method is:

Can be divided into by formula (9) decomposition is three matrixes

X=U∑V ^T （9）

Wherein U is that a size is N ₁* N ₁Unitary matrice, V is that a size is N ₂* N ₂Unitary matrice, ∑ is that a size is N ₁* N ₂Diagonal matrix, comprise the singular value σ of descending _k, be formula (10) with matrix X factorization

X=U∑V ^T=LR ^T （10）

L=U ∑ wherein ^1/2, the R=V ∑ ^1/2, the low-rank matrix fill-in model modification of this belt transect smoothing is formula (11)

$\min \mathrm{rank}\left({\mathrm{LR}}^T\right)+\mathrm{\&lambda;}({\left|\right|D_x\left({\mathrm{LR}}^T\right)\left|\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00002814988000011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+{\left|\right|D_y\left({\mathrm{LR}}^T\right)\left|\right|}_F^2)$ s.t.A(LR ^T)=B （11）

If L is a size is N ₁The matrix of * K, R are that a size is N ₂The matrix of * K, K is the value by the order pre-estimation of matrix X here, the low-rank matrix fill-in model modification of this belt transect smoothing is formula (12)

$\min {\left|\right|L\left|\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00002814988000011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+{\left|\right|R\left|\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00002814988000011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+\mathrm{\&lambda;}({\left|\right|D_x\left({\mathrm{LR}}^T\right)\left|\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00002814988000011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+{\left|\right|D_y\left({\mathrm{LR}}^T\right)\left|\right|}_F^2)$ s.t.A(LR ^T)=B （12）

Consider that the signal strength values that mobile terminal receives is usually accurate not, and the matrix in scene relaxes A (LR not in full conformity with the low-rank characteristic ^TThe constraints of)=B, the low-rank matrix fill-in model with smoothing of conversion belt restraining is the unconfinement model, sees formula (13)

$\min {\left|\right|L\left|\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00002814988000011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+{\left|\right|R\left|\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00002814988000011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+\mathrm{\&eta;}{\left|\right|A\left({\mathrm{LR}}^T\right)-B\left|\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00002814988000011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+\mathrm{\&lambda;}({\left|\right|D_x\left({\mathrm{LR}}^T\right)\left|\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00002814988000011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+{\left|\right|D_y\left({\mathrm{LR}}^T\right)\left|\right|}_F^2)---\left(13\right)$

Wherein

Represent sampling subset B at the reconstructed error of balance weight η,

Replace iterative process by following formula and derive L and R: the initial value of our random given L and R at first, then fixed L, optimize R by least square method; Upgrade afterwards R, fixedly R, allow L as optimized variable; Repeat above alternately iterative process, until default error threshold is restrained and reached to target function.

The below illustrates an emulation experiment embodiment and an actual environment embodiment.

One, emulation experiment embodiment

Fig. 2 is the flow chart of emulation experiment embodiment of the present invention, specifically comprises:

1. the at first generation of the foundation of simulated environment and signal strength data

We are supposition 50 AP of random placement in the long 100 meters wide rectangular areas of 50 meters.Then be the interval of a meter with the horizontal and vertical step-length, design has gathered the RSS value of 5000 sampled points altogether.In order to simulate the distribution spatially of RSS signal value, following radio propagation path loss model is used for the simulate signal decay.

$P_r\left(d\right)=P_t\left(d\right)-\stackrel{\&OverBar;}{P}\left(d_0\right)-10n{\log }_{10}\left(\frac{d}{d_0}\right)-X_{\mathrm{\&sigma;}}$

P wherein _r(d) be illustrated in distance and receive the signal strength signal intensity of AP, P for the position of d _t(d) signal strength signal intensity of sending for AP, P (d ₀) be illustrated in apart from being the position average signal strength loss value of d0, be generally 1 meter.N is given value, is path loss index.X _σExpression Gaussian noise distribution N (0, σ).When path loss index was known, the RSS value can be calculated.The furthest distance of disease spread of our putative signal is 30 meters, if namely the distance between AP and reference node surpasses 30 meters, the RSS value will be set as-100.In this experiment, we are made as 4.4 with path loss index n, and average signal strength loss (1m) is made as-35dB.Noise level limit is in [0,16] interval.According to above-mentioned experimental design, all reference nodes and AP can obtain.Finally, each AP is at 100 * 50 sampling point positions.These measured values consist of the above-mentioned original measurement matrix X that mentions.

2. construct the original fingerprint storehouse of low sampling rate

In the fingerprint base reconstitution experiments, we are random 20% the sampled point of selecting usually, supposes that these sampled points are the sampled point of actual measurement.Namely form an incomplete measurement matrix B for each AP, wherein there is effective value in 20% element, represents the signal strength values that receives AP of this position, and other are 0.

3. carry out fingerprint base reconstruct by LR model and SLR model

Mainly by LR model and SLR model dual mode, incomplete measurement matrix B to be similar to recover out original measurement matrix X in the present invention.The method for solving that we decompose by SVD, alternately iteration derives L and R, until default error threshold is restrained and reached to target function.Merge the fingerprint distribution that each AP reconstructs, form new signal strength signal intensity fingerprint base.

4. utilize the signal strength signal intensity fingerprint base that reconstructs to locate online

Online positioning stage will mate for the signal strength signal intensity vector of the signal strength signal intensity vector of testing with each sampled point of the fingerprint base that newly reconstructs, and current position coordinates utilizes each known sample point coordinate to estimate, namely obtains the online elements of a fix

Two, actual environment embodiment

At first the sampling of the foundation of actual environment and signal strength data

In indoor true environment, experimental site is located at building, Beijing University of Technology's information north three floor, and long 53 meters, wide 15 meters, as shown in Figure 2.In this experiment, we sample in this zone altogether from the RSS value of 90 AP.Off-line training step one people carries mobile terminal and walks in the Experimental Area, records simultaneously RSS value and coordinate.Gather altogether 337 sampled points in experiment.For fear of systematic error, obtain accurate measured value, we have all carried out 10 samplings each sampled point.The average of 10 samples is registered as the final measured value of this sampled point.

2. construct the original fingerprint storehouse of low sampling rate

Similar to emulation experiment, we are the random part sampled point of selecting usually, because the Experimental Area of 53m * 15m is not measured fully, we select a part as known sampled point at random from 337 reference nodes, namely form an incomplete measurement matrix B for each AP, wherein there is effective value in Partial Elements, represents the signal strength values that receives AP of this position, and other are 0.

3. carry out fingerprint base reconstruct by LR model and SLR model

Mainly by LR model and SLR model dual mode, incomplete measurement matrix B to be similar to recover out original measurement matrix X in the present invention.The method for solving that we decompose by SVD, alternately iteration derives L and R, until default error threshold is restrained and reached to target function.Merge the fingerprint distribution that each AP reconstructs, form new signal strength signal intensity fingerprint base.

4. utilize the signal strength signal intensity fingerprint base that reconstructs to locate online

Online positioning stage, to mate for the signal strength signal intensity vector of the signal strength signal intensity vector that measures in real time with each sampled point of the fingerprint base that newly reconstructs, current position coordinates utilizes each known sample point coordinate to estimate, namely obtains the online elements of a fix.

In order to verify the validity of above-mentioned structure fingerprint base method, we use respectively emulated data and real data that the present invention and prior art are built at fingerprint base and compare aspect two of result and positioning results.It is mainly the visually evaluation of subjectivity of people that fingerprint base builds result, and positioning result is mainly to measure by objective position error, and unit is rice (m).Its computing formula is as follows:

$\mathrm{Error}=\left|\right|P-\hat{P}\left|\right|$

Wherein position error is actual position coordinate P and estimated position coordinate

Euclidean distance, position error is less, locating effect is better.

In order to show intuitively in said method that fingerprint base builds result, we distribute the wireless signal strength of a certain AP and represent with pseudocolour picture, and the color in image represents the signal strength values of this position.In emulation experiment, we have simulated sample rate is that in 20% situation namely 1000 sampled points, utilize the low-rank model reconstruction to go out the experiment of 5000 sampled points, the signal distribution plots (IDW, RBF) that reconstruction result and primary signal distribute and interpolation method obtains compares.We can find out that the signal distribution plots that adopts the LR method to build is not ideal enough, have certain noise, and the result that adopts SLR method in this paper to build is more approaching with the primary signal distribution map, and reconstruct is relatively better.In addition, based on the low-rank reconstructing method of smoothing than interpolation method, the noise of removing in environment is had better effect, namely have better robustness in noise circumstance.

The above; it is only preferred embodiment of the present invention; be not that the present invention is done any pro forma restriction, every foundation technical spirit of the present invention all still belongs to the protection range of technical solution of the present invention to any simple modification, equivalent variations and modification that above embodiment does.

Claims (5)

1. an indoor wireless positioning method that utilizes fingerprint technique, comprise off-line training step and online positioning stage, it is characterized in that,

Off-line training step comprises the following steps:

(1) set up indoor environment and collection signal intensity data;

(2) the original fingerprint storehouse of structure low sampling rate;

(3) by low-rank matrix fill-in model, the original fingerprint storehouse is reconstructed, low-rank matrix fill-in model is formula (1)

minrank(X) s.t.A(X)=B （1）

Wherein A (X) is the template operator, X is a measurement matrix that comprises whole sampled points, each element X (i, j) represent sampled point (i, j) receive signal strength signal intensity from a certain accessing points AP, A (X)=B only has the relatively sparse effective signal strength values of the existence by the sampled point of actual measurement, A (X) is defined as the matrix Q of formula (2)

（2）

Then obtain formula (3)

B(i,j)=Q(i,j)X(i,j) （3）；

In online positioning stage, will mate for the signal strength signal intensity vector of the signal strength signal intensity vector of testing with each sampled point of the fingerprint base of reconstruct, current position coordinates utilizes each known sample point coordinate to estimate to obtain the online elements of a fix.

2. the indoor wireless positioning method that utilizes fingerprint technique according to claim 1, it is characterized in that, step (3) also comprises by with the low-rank matrix fill-in model of smoothing, the original fingerprint storehouse being reconstructed, and is formula (4) with the low-rank matrix fill-in model of smoothing

minrank(X)+λS(X) s.t.A(X)=B （4）

Wherein S (X) is a successional smoothing factor of expression X, and the value of S (X) is less, represents that the continuity of X is better, and λ is the coefficient of balance that is obtained by experiment.

3. the indoor wireless positioning method that utilizes fingerprint technique according to claim 2, is characterized in that, the difference by matrix horizontal and vertical direction defines S (X), sees formula (5)

$S\left(X\right)={\left|\right|D_x\left(X\right)\left|\right|}_F^2+{\left|\right|D_y\left(X\right)\left|\right|}_F^2$ （5）

D wherein _x(X) be that a size is N ₁* (N ₂-1) matrix represents the poor of each element horizontal direction in matrix X, sees formula (6)

D _x(i,j)＝X(i,j+1)-X(i,j) （6）

D _y(X) be that a size is (N ₁-1) * N ₂Matrix, represent the poor of each element vertical direction in matrix X, see formula (7)

D _y(i,j)=X(i+1,j)-X(i,j) (7)

Operator Represent the Frobenius norm of matrix, obtain low-rank matrix fill-in model with smoothing by formula (8) like this

$\begin{array}{cc}\min \mathrm{rank}\left(X\right)+\mathrm{\&lambda;}({\left|\right|D_x\left(X\right)\left|\right|}_F^2+{\left|\right|D_y\left(X\right)\left|\right|}_F^2)& s.t.A\left(X\right)=B\end{array}$ （8）。

4. the indoor wireless positioning method that utilizes fingerprint technique according to claim 3, it is characterized in that, adopt singular value decomposition SVD method to low-rank matrix fill-in model with finding the solution of the low-rank matrix fill-in model of smoothing in step (3), by replacing iterative, generate the fingerprint base of reconstruct.

5. the indoor wireless positioning method that utilizes fingerprint technique according to claim 4, is characterized in that, the SVD method is:

Can be divided into by formula (9) decomposition is three matrixes

X＝U∑V ^T （9）

Wherein U is that a size is N1 * N ₁Unitary matrice, V is that a size is N ₂* N ₂Unitary matrice, ∑ is that a size is N ₁* N ₂Diagonal matrix, comprise the singular value σ of descending _k, be formula (10) with matrix X factorization

X＝U∑V ^T＝LR ^T （10）

L=U ∑ wherein ^1/2, the R=V ∑ ^1/2, the low-rank matrix fill-in model modification of this belt transect smoothing is formula (11)

$\begin{array}{cc}\min \mathrm{rank}\left({\mathrm{LR}}^T\right)+\mathrm{\&lambda;}({\left|\right|D_x\left({\mathrm{LR}}^T\right)\left|\right|}_F^2+{\left|\right|D_y\left({\mathrm{LR}}^T\right)\left|\right|}_F^2)& s.t.A\left({\mathrm{LR}}^T\right)=B\end{array}$ （11）

If L is a size is N ₁The matrix of * K, R are that a size is N ₂The matrix of * K, K is the value by the order pre-estimation of matrix X here, the low-rank matrix fill-in model modification of this belt transect smoothing is formula (12)

$\min {\left|\right|L\left|\right|}_F^2+{\left|\right|R\left|\right|}_F^2+\mathrm{\&lambda;}({\left|\right|D_x\left({\mathrm{LR}}^T\right)\left|\right|}_F^2+{\left|\right|D_y\left({\mathrm{LR}}^T\right)\left|\right|}_F^2)$ s.t.A(LR ^T)=B （12）

Consider that the signal strength values that mobile terminal receives is usually accurate not, and the matrix in scene relaxes A (LR not in full conformity with the low-rank characteristic ^TThe constraints of)=B, the low-rank matrix fill-in model with smoothing of conversion belt restraining is the unconfinement model, sees formula (13)

$\min {\left|\right|L\left|\right|}_F^2+{\left|\right|R\left|\right|}_F^2+\mathrm{\&eta;}{\left|\right|A\left({\mathrm{LR}}^T\right)-B\left|\right|}_F^2+\mathrm{\&lambda;}({\left|\right|D_x\left({\mathrm{LR}}^T\right)\left|\right|}_F^2+{\left|\right|D_y\left({\mathrm{LR}}^T\right)\left|\right|}_F^2)$ （13）

Wherein

Represent sampling subset B at the reconstructed error of balance weight η,

Replace iterative process by following formula and derive L and R: the initial value of our random given L and R at first, then fixed L, optimize R by least square method; Upgrade afterwards R, fixedly R, allow L as optimized variable; Repeat above alternately iterative process, until default error threshold is restrained and reached to target function.

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