CN101608914A - RPC parametric optimization method based on multi-collinearity analysis - Google Patents

Abstract

The invention discloses a kind of RPC parametric optimization method, at first set up the tight error equation of finding the solution the RPC parameter, multi-collinearity between analytical error equation design matrix column vector and according to the preferred RPC parameter of setting principle then, reach the purpose of eliminating the RPC dependence on parameter, the RPC parameter that adopts the least square adjustment method to find the solution at last to optimize.When the reference mark, ground is sparse, optimize 20～30 independently conspicuousness RPC parameters by this method, can effectively eliminate the oscillatory occurences that RFM occurs in the landform match, obviously improve the image geometry processing accuracy with RFM found the solution of RPC parameter; When the reference mark, ground was abundant, the result who utilizes preferred 39 the RPC parameters of this method to carry out the landform match was in full accord with the result of 78 RPC parameters enforcement landform match of finding the solution with conventional least square method.

Description

RPC parametric optimization method based on multi-collinearity analysis

Technical field

The present invention relates to the method for optimizing of RPC (Rational Polynomial Coefficients) parameter, belong to the Photogrammetry and Remote Sensing field.

Background technology

On September 24th, 1999, USA space imaging company (Space Imaging) successfully sends into the IKONOS satellite planned orbit and accepts satellite remote-sensing image, indicates the arrival in high spatial resolution satellite remote sensing epoch.Since high spatial resolution remote sense satellite IKONOS adopts rational function model (Rational FunctionModel, RFM) alternative strict geometric manipulations model based on collinearity condition equation has carried out since the image geometry processing, RFM has been subjected to the extensive concern of remote sensing educational circles, and furtherd investigate by scholars, obtained good effect.Grodecki and Tao etc. have confirmed that available RFM replaces strict geometric manipulations model in single line battle array push-broom type satellite remote sensing image geometry is handled, and can be used for image rectification, orthography making and target three-dimensional reconstruction etc.Fraser and Hu etc. have studied method how to utilize ground control point to improve the RFM precision.Grodecki and Liu Jun etc. have studied the area adjustment method based on RFM.

With regard to above application, RFM essence is that the mathematics to strict geometric manipulations model approaches.Its key is accurately to find the solution each polynomial coefficient of rational function model, and (Rational Polynomial Coefficients RPC), abbreviates the RPC parameter as.Studies show that,, adopt the irrelevant scheme of resolving of landform to set up RFM and can reach high fitting precision IKONOS and QuickBird high resolution ratio satellite remote-sensing image.Because the RFM form is simple, be applicable to various types of remote sensors, comprise novel aviation/spacer remote sensing sensor, and need not to use the various geometric parameters in the imaging process, as satellite ephemeris, sensor attitude angle and physical characteristics parameter and imaging mode etc.; The RPC parameter does not have clear physical meaning in addition, well the core information of hide sensor.Therefore, RFM is widely used in the geometric manipulations of high resolution ratio satellite remote-sensing image, has become a kind of broad sense geometric manipulations model that is independent of sensor that can substitute strict geometric manipulations model.

Yet the geometric manipulations precision for facility in the RFM application and assurance remote sensing image usually needs to adopt the landform relevant programme to find the solution the RPC parameter.Ground control point skewness or RFM overparameterization, usually cause finding the solution between the column vector of error equation design matrix of RPC parameter and have approximate linear (being referred to as multi-collinearity on the mathematics), be to have correlativity between the RPC parameter, cause seriously morbid state of the normal equation set up according to the least square adjustment principle, make and separate instability or bigger deviation occurs.Though the mountain range estimation technique that adopts can be improved the morbid state of normal equation to a certain extent at present, the precision that can reach is still very limited.Tao and Hu carry out overtesting to a scape SPOT image, have manually measured 71 ground control points, find the solution the RPC parameter according to 40 reference mark wherein by the mountain range estimation technique, in the coordinate residual error of resulting reference mark and checkpoint error be respectively ± 6.42 * 10 ^-3Pixels and ± 6.13pixels.Be not difficult to find out that thus though RFM is very high to the fitting precision at reference mark, it is still very limited that RFM is used for the precision of high resolution ratio satellite remote-sensing image geometric manipulations.

Summary of the invention

Purpose of the present invention attempts from the multi-collinearity between the error equation design matrix column vector of analysis and solution RPC parameter to set about with regard to being to overcome above-mentioned the deficiencies in the prior art, proposes a kind of RPC parametric optimization method of decorrelation.Optimize an amount of conspicuousness RPC parameter according to certain criterion, reject the RPC parameter of part correlation, make the parameter among the RFM keep relatively independent, with of the influence of null method equation morbid state to RPC parametric solution precision, be implemented in and lack under the ground control point situation, can adopt the landform relevant programme accurately to find the solution the RPC parameter, and guarantee to make up the remotely sensing image geometric processing accuracy of RFM based on this RPC parameter.

Realize that the technical scheme that the object of the invention adopts is: at first set up the tight error equation of finding the solution the RPC parameter; Multiple general character line between analytical error equation design matrix column vector then, and according to the preferred RPC parameter of certain principle; At last find the solution the RPC parameter that optimizes according to the least square adjustment principle.

The foundation of above-mentioned tight error equation obtains rational function model according to Taylor series expansion, detailed process is as follows:

Rational function model with picpointed coordinate (l, s) be expressed as contain topocentric coordinates (P, L, polynomial ratio H), that is:

$\left\{\begin{array}{c}l=\frac{N_l(P,L,H)}{D_l(P,L,H)}\\ s=\frac{N_s(P,L,H)}{D_s(P,L,H)}\end{array}\right.---\left(I\right)$

In the formula, (l, s) and (P, L H) are respectively the picpointed coordinate and the topocentric coordinates of regularization, its value all between [1,1], and

N _l(P，L，H)＝a ₁+a ₂L+a ₃P+a ₄H+a ₅LP+a ₆LH+a ₇PH+a ₈L ²+a ₉P ²+a ₁₀H ²+a ₁₁PLH+

a ₁₂L ³+a ₁₃LP ²+a ₁₄LH ²+a ₁₅L ²P+a ₁₆P ³+a ₁₇PH ²+a ₁₈L ²H+a ₁₉P ²H+a ₂₀H ³

D _l(P，L，H)＝b ₁+b ₂L+b ₃P+b ₄H+b ₅LP+b ₆LH+b ₇PH+b ₈L ²+b ₉P ²+b ₁₀H ²+b ₁₁PLH+

b ₁₂L ³+b ₁₃LP ²+b ₁₄LH ²+b ₁₅L ²P+b ₁₆P ³+b ₁₇PH ²+b ₁₈L ²H+b ₁₉P ²H+b ₂₀H ³

N _s(P，L，H)＝c ₁+c ₂L+c ₃P+c ₄H+c ₅LP+c ₆LH+c ₇PH+c ₈L ²+c ₉P ²+c ₁₀H ²+c ₁₁PLH+

c ₁₂L ³+c ₁₃LP ²+c ₁₄LH ²+c ₁₅L ²P+c ₁₆P ³+c ₁₇PH ²+c ₁₈L ²H+c ₁₉P ²H+c ₂₀H ³

D _s(P，L，H)＝d ₁+d ₂L+d ₃P+d ₄H+d ₅LP+d ₆LH+d ₇PH+d ₈L ²+d ₉P ²+d ₁₀H ²+d ₁₁PLH+

d ₁₂L ³+d ₁₃LP ²+d ₁₄LH ²+d ₁₅L ²P+d ₁₆P ³+d ₁₇PH ²+d ₁₈L ²H+d ₁₉P ²H+d ₂₀H ³

Wherein, a _i, b _i, c _i, d _i(i=1,2 ..., 20) and be the RPC parameter, set b usually ₁, d ₁Value be 1.

In order to find the solution the RPC parameter, can obtain the tight error equation of following linear forms with formula (I) according to Taylor series expansion according to the least square adjustment principle:

$\left\{\begin{array}{c}{v}_l=[\frac{1}{D_l}\frac{L}{D_l}\frac{P}{D_l}\frac{H}{D_l}...\frac{P^{2}H}{D_l}\frac{H^3}{D_l}-\frac{{\mathrm{LN}}_l}{D_l^2}-\frac{{\mathrm{PN}}_l}{D_l^2}-\frac{{\mathrm{HN}}_l}{D_l^2}...-\frac{{\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="A2009100632860017H2.png"\; state="\{\{state\}\}">P</figure-callout>}}^2{\mathrm{HN}}_l}{D_l^2}-\frac{H^3{N}_l}{D_l^2}]X-(l-l^0)\\ v_s=[\frac{1}{D_s}\frac{L}{D_s}\frac{P}{D_s}\frac{H}{D_s}...\frac{P^{2}H}{D_s}\frac{H^3}{D_s}-\frac{{\mathrm{LN}}_s}{D_s^2}-\frac{{\mathrm{PN}}_s}{D_s^2}-\frac{{\mathrm{HN}}_{\mathrm{<figure-callout\; id="2"\; label="s\; D\; s"\; filenames="A2009100632860017H2.png"\; state="\{\{state\}\}">s</figure-callout>}}}{D_s^{}}...-\frac{{\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="A2009100632860017H2.png"\; state="\{\{state\}\}">P</figure-callout>}}^2{\mathrm{HN}}_s}{D_s^2}-\frac{H^3{N}_s}{D_s^2}]Y-(s-s^0)\end{array}\right.---\left(\mathrm{II}\right)$

In the formula, X=[Δ a ₁Δ a ₂₀Δ b ₂Δ b ₂₀] ^TCorrection vector for row RPC parameter;

Y=[Δ c ₁Δ c ₂₀Δ d ₂Δ d ₂₀] ^TCorrection vector for row RPC parameter.

By formula (II) as can be known, row, column RPC parameter is separate, can separately find the solution.To find the solution capable RPC parameter is example, and first formula can be written as in the formula (II):

v _l＝BX-l _l (III)

In the formula, $B=[\frac{1}{D_l}\frac{L}{D_l}\frac{P}{D_l}\frac{H}{D_l}...\frac{P^{2}H}{D_l}\frac{H^3}{D_l}-\frac{{\mathrm{LN}}_l}{D_l^2}-\frac{{\mathrm{PN}}_l}{D_l^2}-\frac{{\mathrm{HN}}_l}{D_l^2}...-\frac{{\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="A2009100632860017H2.png"\; state="\{\{state\}\}">P</figure-callout>}}^2{\mathrm{HN}}_l}{D_l^2}-\frac{H^3{N}_l}{D_l^2}].$

Multi-collinearity between above-mentioned error equation design matrix column vector is analyzed by following method:

If unknown number x _iAnd x _j(j ≠ i) is respectively i and j element among the unknown number vector X, vectorial B _i=[B _I1, B _I2..., B _In] ^TAnd B _j=[B _J1, B _J2..., B _Jn] ^TBe respectively among the design matrix B and x _iAnd x _jCorresponding column vector.As unknown number x _iAnd x _jWhen relevant, B _iWith B _jBecome linear approximate relationship.Can analyze correlativity between unknown parameter according to the multi-collinearity between column vector among the design matrix B thus, and be used to reject the parameter of strong correlation.

Column vector B _iWith B _jBetween multi-collinearity can measure with the cosine of these two column vectors angle in n dimension Euclidean space, its value is:

$\cos (B_i,B_j)=\frac{B_i\&CenterDot;B_j}{\left|B_i\right|\&times;\left|B_j\right|}---\left(\mathrm{IV}\right)$

In the formula, $B_i\&CenterDot;B_j=\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}{B}_{\mathrm{ik}}{B}_{\mathrm{jk}};$ $\left|B_i\right|=\sqrt{\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}{B}_{\mathrm{ik}}^2};$ $\left|B_j\right|=\sqrt{\underset{k=1}{\overset{\mathrm{<figure-callout\; id="2"\; label="n\; B\; jk"\; filenames="A2009100632860017H2.png"\; state="\{\{state\}\}">n</figure-callout>}}{\mathrm{\&Sigma;}}}{B}_{\mathrm{jk}}^{}{}_2^{}}.$

The absolute value of this cosine value approaches 1 more, shows column vector B _iAnd B _jBetween multi-collinearity strong more, promptly with the corresponding unknown number x of these two column vectors _iAnd x _jBetween correlativity just strong more.

The following principle of the preferred basis of above-mentioned RPC parameter is carried out:

When finding the solution the RPC parameter according to the least square adjustment principle, ground control point non-uniform Distribution or model overparameterization can make and have multi-collinearity between the column vector of design matrix B in the formula (III), thereby cause normal equation matrix of coefficients B ^TB is unusual, the deviation that the RPC parameter estimation of trying to achieve thus is unstable or appearance is bigger.At this moment, can be according to the column vector multi-collinearity between any two of formula (IV) calculation Design matrix B.As cos (B _i, B _j) during greater than a certain given threshold value, keep i column vector and the corresponding RPC parameter thereof of design matrix B, reject j column vector and corresponding RPC parameter thereof, with the multi-collinearity between the column vector of eliminating design matrix B, thereby improve the state of normal equation, improve the solving precision of RPC parameter.

In RFM, once item is used to describe the error that optical system produces, and quadratic term is used for expressing the error that earth curvature, atmosphere refractive power and lens distortion etc. produce, and cubic term can be used for representing that some have the unknown errors of high order component, as the camera vibrations etc.Because the RPC parameter of the RPC parameter higher-order item of low order item seems even more important, therefore, when multi-collinearity appears in two column vectors of design matrix B, need the relatively order of two column vectors, keep low order item and corresponding RPC parameter thereof, reject higher order term and corresponding RPC parameter thereof.

According to above principle, through after the preferred RPC parameter, be made as B ' by the design matrix that column vector constituted that remains, should there be multi-collinearity between its column vector, there is not correlativity between the promptly preferred RPC parameter.When a plurality of ground control points participated in compensating computation, the matrix form of formula (III) was:

V＝B′X′-L (V)

In the formula, X ' is preferred RPC parameter increase vector.

According to the least square adjustment principle, formula (V) is set up normal equation just can try to achieve the valuation of preferred RPC parameters optimal:

X′ ^(s)＝X′ ^(s-1)+((B′ ^(s-1)) ^TB′ ^(s-1)) ^-1(B′ ^(s-1)) ^T?L ^(s-1) (VI)

In the formula, s is an iterations.

When adopting common 78 RPC parameters of least square adjustment method integrated solution, the non-uniform Distribution of ground control point or the overparameterization of model can cause the unstable of RPC parametric solution or bigger deviation occur.When adopting the landform relevant programme to find the solution 78 RPC parameters, this phenomenon is more obvious, make the RFM model when the landform match the precision that can reach very limited.

The present invention sets about from the multi-collinearity between analytical error equation design matrix column vector, proposes a kind of decorrelation RPC parametric optimization method.When the reference mark, ground is sparse, can eliminate correlativity between parameter well by preferred 20 ~ 30 RPC parameters, effectively eliminate the oscillatory occurences that RFM occurs in the landform match, can obviously improve the image geometry processing accuracy with RFM found the solution of RPC parameter.When the reference mark, ground was abundant, the result who utilizes the preferred RPC parameter of the inventive method to carry out the landform match was in full accord with the result of 78 RPC parameters enforcement landform match of finding the solution with conventional least square method.

Description of drawings

The present invention is further illustrated below in conjunction with the drawings and specific embodiments.

Fig. 1 is the process flow diagram of the inventive method.

Fig. 2 is the distribution plan of ground control point in the SPOT-5 image.

Fig. 3 is for finding the solution the multi-collinearity synoptic diagram of the error equation design matrix of 39 capable RPC parameters according to 40 ground control points.

Fig. 4-a and Fig. 4-b are respectively 75 picpointed coordinate residual error distribution plans when utilizing 40 ground control points to find the solution 78 RPC parameters and preferred 25 RPC parameters.

Fig. 5 is the trend map that SPOT-5 image geometry processing accuracy changes with the reference mark.

Embodiment

The flow process of the inventive method comprises as shown in Figure 1: at first set up the tight error equation of finding the solution the RPC parameter; Determine multiple general character line between the rectangular array vector by error equation then, and according to the preferred RPC parameter of certain principle; At last find the solution the RPC parameter that optimizes according to the least square adjustment principle.Its concrete operations step is:

(1) sets up the tight error equation of finding the solution the RPC parameter

Rational function model with picpointed coordinate (l, s) be expressed as contain topocentric coordinates (P, L, polynomial ratio H), that is:

$\left\{\begin{array}{c}l=\frac{N_l(P,L,H)}{D_l(P,L,H)}\\ s=\frac{N_s(P,L,H)}{D_s(P,L,H)}\end{array}\right.---\left(I\right)$

In the formula, (l, s) and (P, L H) are respectively the picpointed coordinate and the topocentric coordinates of regularization, its value all between [1,1], and

N _l(P，L，H)＝a ₁+a ₂L+a ₃P+a ₄H+a ₅LP+a ₆LH+a ₇PH+a ₈L ²+a ₉P ²+a ₁₀H ²+a ₁₁PLH+

a ₁₂L ³+a ₁₃LP ²+a ₁₄LH ²+a ₁₅L ²P+a ₁₆P ³+a ₁₇PH ²+a ₁₈L ²H+a ₁₉P ²H+a ₂₀H ³

D _l(P，L，H)＝b ₁+b ₂L+b ₃P+b ₄H+b ₅LP+b ₆LH+b ₇PH+b ₈L ²+b ₉P ²+b ₁₀H ²+b ₁₁PLH+

b ₁₂L ³+b ₁₃LP ²+b ₁₄LH ²+b ₁₅L ²P+b ₁₆P ³+b ₁₇PH ²+b ₁₈L ²H+b ₁₉P ²H+b ₂₀H ³

N _s(P，L，H)＝c ₁+c ₂L+c ₃P+c ₄H+c ₅LP+c ₆LH+c ₇PH+c ₈L ²+c ₉P ²+c ₁₀H ²+c ₁₁PLH+

c ₁₂L ³+c ₁₃LP ²+c ₁₄LH ²+c ₁₅L ²P+c ₁₆P ³+c ₁₇PH ²+c ₁₈L ²H+c ₁₉P ²H+c ₂₀H ³

D _s(P，L，H)＝d ₁+d ₂L+d ₃P+d ₄H+d ₅LP+d ₆LH+d ₇PH+d ₈L ²+d ₉P ²+d ₁₀H ²+d ₁₁PLH+

d ₁₂L ³+d ₁₃LP ²+d ₁₄LH ²+d ₁₅L ²P+d ₁₆P ³+d ₁₇PH ²+d ₁₈L ²H+d ₁₉P ²H+d ₂₀H ³

Wherein, a _i, b _i, c _i, d _i(i=1,2 ..., 20) and be the RPC parameter, set b usually ₁, d ₁Value be 1.

In order to find the solution the RPC parameter, formula (I) can be become following linear forms according to Taylor series expansion according to the least square adjustment principle:

$\left\{\begin{array}{c}{v}_l=[\frac{1}{D_l}\frac{L}{D_l}\frac{P}{D_l}\frac{H}{D_l}...\frac{P^{2}H}{D_l}\frac{H^3}{D_l}-\frac{{\mathrm{LN}}_l}{D_l^2}-\frac{{\mathrm{PN}}_l}{D_l^2}-\frac{{\mathrm{HN}}_{\mathrm{<figure-callout\; id="2"\; label="l\; D\; l"\; filenames="A2009100632860017H2.png"\; state="\{\{state\}\}">l</figure-callout>}}}{D_l^{}}...-\frac{{\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="A2009100632860017H2.png"\; state="\{\{state\}\}">P</figure-callout>}}^2{\mathrm{HN}}_l}{D_l^2}-\frac{H^3{N}_l}{D_l^2}]X-(l-l^0)\\ v_s=[\frac{1}{D_s}\frac{L}{D_s}\frac{P}{D_s}\frac{H}{D_s}...\frac{P^{2}H}{D_s}\frac{H^3}{D_s}-\frac{{\mathrm{LN}}_s}{D_s^2}-\frac{{\mathrm{PN}}_s}{D_s^2}-\frac{{\mathrm{HN}}_s}{D_s^2}...-\frac{{\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="A2009100632860017H2.png"\; state="\{\{state\}\}">P</figure-callout>}}^2{\mathrm{HN}}_s}{D_s^2}-\frac{H^3{N}_s}{D_s^2}]Y-(s-s^0)\end{array}\right.---\left(\mathrm{II}\right)$

In the formula, X=[Δ a ₁Δ a ₂₀Δ b ₂Δ b ₂₀] ^TCorrection vector for row RPC parameter;

Y=[Δ c ₁Δ c ₂₀Δ d ₂Δ d ₂₀] ^TCorrection vector for row RPC parameter.

By formula (II) as can be known, row, column RPC parameter is separate, can separately find the solution.To find the solution capable RPC parameter is example, and first formula can be written as in the formula (II):

v _l＝BX-l _l (III)

In the formula, $B=[\frac{1}{D_l}\frac{L}{D_l}\frac{P}{D_l}\frac{H}{D_l}...\frac{P^{2}H}{D_l}\frac{H^3}{D_l}-\frac{{\mathrm{LN}}_l}{D_l^2}-\frac{{\mathrm{PN}}_l}{D_l^2}-\frac{{\mathrm{HN}}_{\mathrm{<figure-callout\; id="2"\; label="l\; D\; l"\; filenames="A2009100632860017H2.png"\; state="\{\{state\}\}">l</figure-callout>}}}{D_l^{}}...-\frac{{\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="A2009100632860017H2.png"\; state="\{\{state\}\}">P</figure-callout>}}^2{\mathrm{HN}}_l}{D_l^2}-\frac{H^3{N}_l}{D_l^2}].$

(2) multi-collinearity analysis

If unknown number x _iAnd x _j(j ≠ i) is respectively i and j element among the unknown number vector X, vectorial B _i=[B _I1, B _I2..., B _In] ^TAnd B _j=[B _J1, B _J2..., B _Jn] ^TBe respectively among the design matrix B and x _iAnd x _jCorresponding column vector.As unknown number x _iAnd x _jWhen relevant, B _iWith B _jBecome linear approximate relationship.Can analyze correlativity between unknown parameter according to the multi-collinearity between column vector among the design matrix B thus, and be used to reject the parameter of strong correlation.

Column vector B _iWith B _jBetween multi-collinearity can measure with the cosine of these two column vectors angle in n dimension Euclidean space, its value is:

$\cos (B_i,B_j)=\frac{B_i\&CenterDot;B_j}{\left|B_i\right|\&times;\left|B_j\right|}---\left(\mathrm{IV}\right)$

In the formula, $B_i\&CenterDot;B_j=\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}{B}_{\mathrm{ik}}{B}_{\mathrm{jk}};$ $\left|B_i\right|=\sqrt{\underset{k=1}{\overset{\mathrm{<figure-callout\; id="2"\; label="n\; B\; ik"\; filenames="A2009100632860017H2.png"\; state="\{\{state\}\}">n</figure-callout>}}{\mathrm{\&Sigma;}}}{B}_{\mathrm{ik}}^{}{}_2^{}};$ $\left|B_j\right|=\sqrt{\underset{k=1}{\overset{\mathrm{<figure-callout\; id="2"\; label="n\; B\; jk"\; filenames="A2009100632860017H2.png"\; state="\{\{state\}\}">n</figure-callout>}}{\mathrm{\&Sigma;}}}{B}_{\mathrm{jk}}^{}{}_2^{}}.$

The absolute value of this cosine value approaches 1 more, shows column vector B _iAnd B _jBetween multi-collinearity strong more, promptly with the corresponding unknown number x of these two column vectors _iAnd x _jBetween correlativity just strong more.

(3) preferred RPC parameter

Column vector multi-collinearity between any two according to formula (IV) calculation Design matrix B.As cos (B _i, B _j) during greater than a certain given threshold value, keep i column vector and the corresponding RPC parameter thereof of design matrix B, reject j column vector and corresponding RPC parameter thereof.

When multi-collinearity appearred in two column vectors of design matrix B, relatively the order of two column vectors kept low order item and corresponding RPC parameter thereof, rejects higher order term and corresponding RPC parameter thereof.

(4) find the solution the RPC parameter that optimizes

According to above principle, through after the preferred RPC parameter, be made as B ' by the design matrix that column vector constituted that remains, there is not multi-collinearity between its column vector, there is not correlativity between the promptly preferred RPC parameter.When a plurality of ground control points participated in compensating computation, the matrix form of formula (III) was:

V＝B′X′-L (V)

In the formula, X ' is preferred RPC parameter increase vector.

According to the least square adjustment principle, formula (V) is set up normal equation just can try to achieve the valuation of preferred RPC parameters optimal:

X′ ^(s)＝X′ ^(s-1)+((B′ ^(s-1)) ^TB′ ^(s-1)) ^-1(B′ ^(s-1)) ^TL ^(s-1) (VI)

In the formula, s is an iterations.

Present embodiment selects for use area, Chinese Qianxi one scape SPOT-5 HRG 1A level image to test, and adopts the landform relevant programme preferably and find the solution the RPC parameter.This image is taken on October 21st, 2004, side-looking angle-24.04 °, and ground space resolution is 5.5m, and the image size is 12000 * 12000pixels, and the ground region that covers is 66km * 66km, the about 20m of the maximum discrepancy in elevation in ground, genus flat country.Be distributed with 75 flat high ground control points (as shown in Figure 2) in the image, its three-dimensional coordinate is to utilize 1: 60000 engineer's scale aviation image in this zone to adopt the encryption of GPS auxiliary beam method area adjustment method to obtain by the WuCAPS system.Pass point coordinate precision according to 30 ground check point evaluations: the plane is ± 2.5m that elevation is ± 2.0m.This encryption precision is better than the 0.5pixels of SPOT-5 image, fully can be as the reference mark and the checkpoint of test image geometry processing.But, because the corresponding image points of SPOT-5 stereopsis centering has been chosen at the reference mark, distributing concentrates on image inside relatively, and the ground control point of image edge is fewer.

The validity and the practicality of the RPC parameter selection method that proposes in order to verify, carried out contrast test here:

Test A adopts common 78 RPC parameters of least square adjustment method integrated solution;

The preferred RPC parameter of method that test B proposes according to the present invention is earlier found the solution the RPC parameter of reservation again according to formula (VI).

Because row RPC parameter and row RPC parameter are separate among the RFM, present embodiment adopts separately solution strategies.After obtaining the RPC parameter, utilize formula (I) with the ground coordinate back projection of 75 ground control points to image, can try to achieve the measurement coordinate of each picture point and the residual error of its calculated value.The middle error that 75 picpointed coordinate residual errors on the image count is listed in table 1.

Error in the table 1SPOT-5 image coordinate residual error

Result in the analytical table 1 is as can be seen:

When 1) adopting 78 RPC parameters of landform relevant programme integrated solution,, cause factor arrays B owing to have multi-collinearity (is example with Fig. 3) between the design matrix column vector of formula (III) ^TThe conditional number of B is up to 10 ¹¹Serious morbid state appears in the order of magnitude, normal equation, can not obtain stable separating.When 40 ground control points of operating limit are found the solution the RPC parameter, only can reach ± 10.81pixels based on the image geometry processing accuracy of RFM.Along with the increase that Ground Control is counted, the precision of RFM match landform is significantly improved.When the reference mark, ground reached 70, though error can reach ± 0.71pixels in the picpointed coordinate residual error of ground check point, the multi-collinearity between the design matrix column vector did not still have too big improvement, matrix of coefficients B ^TThe conditional number of B still is 10 ¹¹The order of magnitude, normal equation still are serious morbid state.

2) utilize the preferred RPC parameter of the inventive method after, the multi-collinearity between the design matrix column vector of formula (V) is obviously eliminated.Be equally under the situation of 40 ground control points, optimize 25 RPC parameters (seeing table 2 for details), no longer have correlativity (seeing table 3 for details) between the parameter, factor arrays B ' ^TThe conditional number of B ' has dropped to 10 ³The order of magnitude, the normal equation state obviously improves, and the RPC parameters precision of being found the solution has had and has significantly improved, and is obviously improved by the landform precision of RFM match.75 picpointed coordinate residual error distribution plans when Fig. 4 has illustrated to use 40 ground control points to find the solution the RPC parameter.From Fig. 4 (a) as can be seen, be subjected to the influence of RPC dependence on parameter, though contain the RFM of 78 parameters the reference mark had very high fitting precision, checkpoint picpointed coordinate residual error is obviously vibration; From Fig. 4 (b) as can be known, the RFM that is made of 25 independent RPC parameters goes the match landform, not only can eliminate the multi-collinearity between the design matrix column vector, improve the solving precision of RPC parameter and the geometric manipulations precision of image, and the big or small basically identical of the image coordinate residual error that can guarantee each checkpoint, no longer have systematic error.At this moment, the RFM that contains 78 parameters only is ± 8.35pixels to the processing accuracy of image line direction, and brought up to ± 0.68pixels based on the processing accuracy of RFM on the image line direction of preferred 25 parameters of the inventive method, the image geometry processing accuracy that is counted by the picpointed coordinate residual error of 35 checkpoints has improved 88.1%=(10.81-1.29)/10.81.

The RPC parameter that table 2 utilizes the inventive method to optimize

Table 3 utilizes the related coefficient between the preferably capable RPC parameter of 40 ground control points

Annotate: ${\mathrm{\&rho;}}_{\mathrm{ij}}=-\frac{Q_{\mathrm{ij}}}{\sqrt{Q_{\mathrm{ii}}\&CenterDot;Q_{\mathrm{jj}}}},$ Q _IjBe normal equation factor arrays inverse matrix Q=(B ^TB) ^-1The capable j column element of i.

3) adopt 78 RPC parameters of landform relevant programme integrated solution, when reference mark, ground number is less than 39, can't find the solution according to conventional least square adjustment method.If according to the inventive method, can come preferred RPC parameter (seeing table 2 for details) according to different ground control points, the RFM model of Jian Liing still can obtain higher image geometry processing accuracy thus.Analytical table 2 can find that along with the increase that control is counted, the correlativity between the RPC parameter weakens gradually, and optionally the RPC parameter is on the increase, but basic parameter is at a ₁～a ₁₉, b ₁₁～b ₁₈And c ₁～c ₁₉The middle selection, and selected RPC parameter mainly is a under the situation of various reference mark ₁～a ₈, a ₁₁, b ₁₁, b ₁₅, c ₁～c ₈And c ₁₁Fig. 5 has drawn the curve that SPOT-5 image geometry processing accuracy changes with the reference mark, ground.Can clearly be seen that from Fig. 5 along with the increase at reference mark, the image geometry processing accuracy is in slow lifting.When the RPC parameter was 20, the image geometry processing accuracy can reach ± 1.82pixels; But when the RPC parameter was increased to 33 by 20, the image geometry bearing accuracy had only improved ± 0.61pixels.At this moment, required Ground Control is counted and has been increased to 60 by 15, and than 45 ground control points that increase, the lifting amplitude of image geometry processing accuracy is very limited.The test findings that anatomizes table 1 and Fig. 5 also finds, based on the geometric manipulations precision of the constructed RFM model of the preferred RPC parameter of method of the present invention significantly better than utilizing whole 78 RPC parameters to form the geometric manipulations precision of RFM.Also can only reach ± 3.10pixels even 78 RPC parameters utilizing 60 ground control points to ask according to least square adjustment method global solution are implemented the precision of image geometry processing, its precision is no more than the geometric manipulations precision of 20 constructed RFM of RPC parameter that utilize 15 ground control points to optimize.This just proves absolutely, in RFM be not 78 parameters all be remarkable and necessary.Therefore, when adopting RFM to carry out the high resolution ratio satellite remote-sensing image geometric manipulations, should be according to the quantity and the distribution thereof of ground control point, the most appropriate RPC parameter of choose reasonable is necessary.

For the influence of the validity of checking the inventive method and reliability and RPC parameter, adopt here the irrelevant scheme of landform to separate again to ask that table 2 optimizes respectively organizes the RPC parameter to the image geometry processing accuracy.At this moment, used the individual virtual grid points in 500 (10 * 10 * 5) as the object space reference mark altogether.Make up the RFM model based on the listed RPC parameter of table 2, utilize formula (1) with the ground coordinate back projection of 75 ground check points to testing on the image, the image geometry processing accuracy one that is counted by its image coordinate residual error is listed in table 4.

Error in the table 4SPOT-5 image coordinate residual error

As can be seen from Table 4:

When 1) adopting the irrelevant scheme of landform to find the solution the RPC parameter, the RPC parameter of selection is many more, and RFM approaches strict geometric model more, and is also high more to the precision of landform match.When the RPC parameter was less than 25, RFM that shows on 500 virtual grid points and strict geometric model processing accuracy differed ± 1.71pixels; But when RPC parameter during more than 25, the geometric manipulations precision of two kinds of models is constant, and its gap is all less than ± 0.25pixels.

2) because there are error in satellite orbit parameter and image attitude angle, and the precision of utilizing strict geometric model that the SPOT-5 image is handled is not very high, when adopting RFM to remove to approach strict geometric model, the geometric manipulations precision naturally can be very not desirable.In the present embodiment, when the RPC parameter that optimizes during more than 25, the image geometry processing accuracy only is ± 3.51pixels, and to carry out the precision that image geometry handles with the constructed RFM of 78 parameters utilizing conventional RPC parametric solution method to obtain be on all four.This has just further proved: 25～39 RPC parameters that the inventive method optimizes are the most significant.

Claims (5)

1. based on the RPC parametric optimization method of multi-collinearity analysis, it is characterized in that: at first set up the tight error equation of finding the solution the RPC parameter; Determine multiple general character line between the rectangular array vector by error equation then, and optimize the RPC parameter; At last find the solution the RPC parameter that optimizes according to the least square adjustment principle.

2. the RPC parametric optimization method based on multi-collinearity analysis according to claim 1 is characterized in that rational function model obtaining the tight error equation of linear forms according to Taylor series expansion, and detailed process is as follows:

Rational function model with picpointed coordinate (l, s) be expressed as contain topocentric coordinates (P, L, polynomial ratio H), that is:

$\left\{\begin{array}{c}l=\frac{N_l(P,L,H)}{D_l(P,L,H)}\\ s=\frac{N_s(P,L,H)}{D_s(P,L,H)}\end{array}\right.---\left(I\right)$

In the formula, (l, s) and (P, L H) are respectively the picpointed coordinate and the topocentric coordinates of regularization, its value all between [1,1], and

N _l(P，L，H)＝a ₁+a ₂L+a ₃P+a ₄H+a ₅LP+a ₆LH+a ₇PH+a ₈L ²+a ₉P ²+a ₁₀H ²+a ₁₁PLH+a ₁₂L ³+a ₁₃LP ²+a ₁₄LH ²+a ₁₅L ²P+a ₁₆P ³+a ₁₇PH ²+a ₁₈L ²H+a ₁₉P ²H+a ₂₀H ³

D _i(P，L，H)＝b ₁+b ₂L+b ₃P+b ₄H+b ₅LP+b ₆LH+b ₇PH+b ₈L ²+b ₉P ²+b ₁₀H ²+b ₁₁PLH+b ₁₂L ³+b ₁₃LP ²+b ₁₄LH ²+b ₁₅L ²P+b ₁₆P ³+b ₁₇PH ²+b ₁₈L ²H+b ₁₉P ²H+b ₂₀H ³

N _s(P，L，H)＝c ₁+c ₂L+c ₃P+c ₄H+c ₅LP+c ₆LH+c ₇PH+c ₈L ²+c ₉P ²+c ₁₀H ²+c ₁₁PLH+c ₁₂L ³+c ₁₃LP ²+c ₁₄LH ²+c ₁₅L ²P+c ₁₆P ³+c ₁₇PH ²+c ₁₈L ²H+c ₁₉P ²H+c ₂₀H ³

D _s(P，L，H)＝d ₁+d ₂L+d ₃P+d ₄H+d ₅LP+d ₆LH+d ₇PH+d ₈L ²+d ₉P ²+d ₁₀H ²+d ₁₁PLH+d ₁₂L ³+d ₁₃LP ²+d ₁₄LH ²+d ₁₅L ²P+d ₁₆P ³+d ₁₇PH ²+d ₁₈L ²H+d ₁₉P ²H+d ₂₀H ³

Wherein, a _i, b _i, c _i, d _i(i=1,2 ..., 20) and be the RPC parameter, b ₁, d ₁Value be 1; Formula (I) according to Taylor series expansion, is obtained the tight error equation of following linear forms:

$\left\{\begin{array}{c}{v}_l=[\frac{1}{D_l}\frac{L}{D_l}\frac{P}{D_l}\frac{H}{D_l}...\frac{P^{2}H}{D_l}\frac{H^3}{D_l}-\frac{{\mathrm{LN}}_l}{D_l^2}-\frac{{\mathrm{PN}}_l}{D_l^2}-\frac{{\mathrm{HN}}_l}{D_l^2}...-\frac{P^2{\mathrm{HN}}_l}{D_l^2}-\frac{H^3{N}_l}{D_l^2}]X-(l-l^0)\\ v_s=[\frac{1}{D_s}\frac{L}{D_s}\frac{P}{D_s}\frac{H}{D_s}...\frac{P^{2}H}{D_s}\frac{H^3}{D_s}-\frac{{\mathrm{LN}}_s}{D_s^2}-\frac{{\mathrm{PN}}_s}{D_s^2}-\frac{{\mathrm{HN}}_s}{D_s^2}...-\frac{P^2{\mathrm{HN}}_s}{D_s^2}-\frac{H^3{N}_s}{D_s^2}]Y-(s-s^0)\end{array}\right.---\left(\mathrm{II}\right)$

In the formula, X=[Δ a ₁Δ a ₂₀Δ b ₂Δ b ₂₀] ^TBe the correction vector of row RPC parameter,

Y=[Δ c ₁Δ c ₂₀Δ d ₂Δ d ₂₀] ^TBe the correction vector of row RPC parameter,

By formula (II) as can be known, row, column RPC parameter is separate, and wherein, row RPC parameter is that first formula can be written as in the formula (II):

v _l＝BX-l _l (III)

In the formula, $B=[\frac{1}{D_l}\frac{L}{D_l}\frac{P}{D_l}\frac{H}{D_l}...\frac{P^{2}H}{D_l}\frac{H^3}{D_l}-\frac{{\mathrm{LN}}_l}{D_l^2}-\frac{{\mathrm{PN}}_l}{D_l^2}-\frac{{\mathrm{HN}}_l}{D_l^2}...-\frac{P^2{\mathrm{HN}}_l}{D_l^2}-\frac{H^3{N}_l}{D_l^2}].$

3. the RPC parametric optimization method based on multi-collinearity analysis according to claim 1 is characterized in that the multiple general character line between definite by the following method rectangular array vector:

If unknown number x _iAnd x _j(j ≠ i) is respectively i and j element among the unknown number vector X, vectorial B _i=[B _I1, B _I2..., B _In] ^TAnd B _j=[B _J1, B _J2..., B _Jn] ^TBe respectively among the design matrix B and x _iAnd x _jCorresponding column vector is as unknown number x _iAnd x _jWhen relevant, B _iWith B _jBecome linear approximate relationship;

Column vector B _iWith B _jBetween multi-collinearity measure with the cosine of these two column vectors angle in n dimension Euclidean space, its value is:

$\cos (B_i,B_j)=\frac{B_i\&CenterDot;B_j}{\left|B_i\right|\&times;\left|B_j\right|}---\left(\mathrm{IV}\right)$

In the formula, $B_i\&CenterDot;B_j=\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}{B}_{\mathrm{ik}}{B}_{\mathrm{jk}};$ $\left|B_i\right|=\sqrt{\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}{B}_{\mathrm{ik}}^2};$ $\left|B_j\right|=\sqrt{\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}{B}_{\mathrm{jk}}^2},$ The absolute value of this cosine value approaches 1 more, column vector B _iAnd B _jBetween multi-collinearity strong more, promptly with the corresponding unknown number x of these two column vectors _iAnd x _jBetween correlativity just strong more.

4. the RPC parametric optimization method based on multi-collinearity analysis according to claim 1 is characterized in that by the preferred RPC parameter of following principle:

According to the column vector multi-collinearity between any two of formula (IV) calculation Design matrix B, as cos (B _i, B _j) during greater than a certain given threshold value, keep i column vector and the corresponding RPC parameter thereof of design matrix B, and reject j column vector and corresponding RPC parameter thereof;

When multi-collinearity appears in two column vectors of design matrix B, get the RPC parameter than low order item and correspondence thereof of two column vectors.

5. according to claim 1 or 4 described RPC parametric optimization methods, it is characterized in that finding the solution the RPC parameter that optimizes according to the least square adjustment principle based on multi-collinearity analysis.

Through after the preferred RPC parameter, be made as B ' by the design matrix that column vector constituted that remains, do not have multi-collinearity between its column vector, do not have correlativity between the promptly preferred RPC parameter, when a plurality of ground control points participated in compensating computation, the matrix form of formula (III) was:

V＝B′X′-L (V)

In the formula, X ' is preferred RPC parameter increase vector.

According to the least square adjustment principle, formula (V) is set up normal equation just can try to achieve the valuation of preferred RPC parameters optimal:

X′ ^(s)＝X′ ^(s-1)+((B′ ^(s-1)) ^TB′ ^(s-1)) ^-1(B′ ^(s-1)) ^TL ^(s-1)(VI)

In the formula, s is an iterations.

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