Joz Wu, Chia-Jun Liu
Center for Space and Remote Sensing Research
National Central University
Chungli 320, Taiwan
Tel: (886-)3-4227151-7626 Fax: (886-)3-4254908
Keywords: Modeling, Squint Angle, Optimization, Accuracy.
By using radargrammetric (range/Doppler) equations for an SAR image, radar’s trajectory polynomial modelings require statistical evaluation in order to determine the model optimality. Thus, a least-squares estimation technique, a parametric significance hypothesis testing, and the necessary optimization criteria are introduced, in detail enough for a basic algorithmic understanding. A second-order polynomial expansion is then employed to describe the time-varying squint-angle parameter. Experiments with an airborne SAR image reveal that the time dependence of an imaging radar squint parameter can have an impact on planimetric position accuracy, as would be expected. Capability to adequately model airborne/spaceborne SAR orientation parameters still merits research and development.
A proper geometric image processing frequently relies on a sound functional model. For a side-looking SAR (synthetic aperture radar) image, consisting of sequential, scanned range-lines, a sound functional model again depends on some time-varying parameter descriptions. In most instances, time polynomial expansions can be used to describe the radar antenna’s positions/velocities along a flight path (Lee et al., 2000; Tannous and Pikeroen, 1994; Toutin and Gray, 2000). This is logically done because the unknown position/velocity parameters explicitly exist in the radargrammetric (range and Doppler) equations. Other geometric parameters, such as the squint angle, the pixel spacing and the near-range slant delay, are less obvious as far as their time-dependent modelings are concerned.
This paper addresses a time polynomial modeling for the squint parameter and reports the planimetric accurary gain that can be expected. The least-squares estimator is statistical in nature, therefore sufficient control point coordinate measurements should be made available in an airborne SAR-image. Following a brief introduction of the necessary analytic equations, estimation results will be presented and analyzed.
2. Analytic Relationships
In order to understand what our parameter estimation algorithm does, it is of interest to give the next relevant relationships.
2.1 Functional and Stochastic Models
For the geometric SAR-image processing, the range and Doppler (radargrammetric) conditions are of fundamental importance (Curlander et al., 1987; Dowman, 1992; Gelautz et al., 1998; Leberl, 1979):
where Rji (m) is the slant range between the radar antenna position (Xoj,Yoj,Zoj) at station j and the object-space target position (Xx,Yy,z) at point i ; see Figure 1 for notation illustrations (Wu and Lin, 2000). The squint angle t (deg) is such as interpreted in Figure 2, where the instantaneous along-track antenna unit velocity (ux,uy,uz ) is also shown. In Eq. (3), Mb(m/pixel) stands for the pixel spacing. ri,j (pixels) is the i-th cross-track, pertaining to station j, image coordinate measurement. Rn (m) represents a constant slant range delay.
Figure 1. Range vector r and position vectors s and p in a given topocentric rectangular system of coordinates denoted by unit vectors x, y, z
Figure 2. Range vector r* in a radar antenna system of rectangular coordinates defined by unit vectors u, v, and w; squint angle t and off-nadir look angle W after Leberl (1976)
The along-track image coordinate measurement is denoted by tj (pixels, or seconds), serving as an argument. The trajectory parameters Xoj(t), Yoj(t) and Zoj(t) are further expanded in polynomials:
The polynomial coefficients/parameters (ak,bk,ck ) with k = 0, 1, …, will be estimated in a least-squares sense, and be individually checked for their significance by using statistical hypothesis testings. Recapitulating, we now have unknown parameters (T,Mb,ak,…,bk,…,ck) and image point measurements (…,ri,tj,…) in our functional model. The measurement error variance-covariance matrix then serves as the accompanying stochastic model.
2.2 Time-Varying Squint Parameter
As a logical consequence, the squint angle t is expanded in a second-order polynomial, leading to an extension of the underlying functional model:
Here, the fixed-order polynomial modeling is rather tentative. One may notice from Eq. (4) that higher-order time polynomial terms can give rise to numerical stability problems. The acceptance of any initial convergent parameter solution is statistically based on the global model chi-square distribution test (Leick, 1995), at an a % significance level.
Poster Session 1
On Modeling of The SAR-Image Squint Parameter
3. Least-Squares Estimation
If the preceding functional and stochastic models are correct, a least-squares method yields the adjusted parameters/measurements that are the best linear unbiased estimates (Koch, 1999).
3.1 Parameter Corrections and Measurement Residuals
To begin with, the nonlinear equations (1-4) have to be linearized to form a system of error equations, of which the expansion point is at the available measurements and parameter approximations:
where the n*n coefficient matrix B contains the partial derivatives of Eqs. (1-4) with respect to the n*1 measurement vector (…,ri,tj,…). The n*1 vector v stands for the measurement residuals (…,Vri,Vtj,…). Analogously, the n*u design matrix A contains the partial derivatives with respect to the u unknown parameters. The u*1 vector x represents the parameter corrections (dt,dMb,dak,dbk,dcc) with k = 0, 1, 2, 3. The n*1 vector l is the reduced observation vector. The measurement error covariance matrix is denoted by s02, where s02Q is the a priori unit weight reference variance. A least-squares method, which requires a minimization of the quadratic form vTQ-1v, produces the parameter corrections and the measurement residuals, as follows:
where the u*u scaled covariance matrix Qx and the n*n scaled covariance matrix Qv refer to the parameter correction vector x and the measurement residual vector v, respectively. The covariance matrices can be obtained by using the law of error propagation. Both Leick (1995) and Mikhail (1976) detailed the derivation of Qx and Qv so that their explicit expressions are not repeated here. The v-vector quadratic form can lead to the a posteriori estimate s02 of a unit weight reference variance. The same v-quadratic form also leads to a chi-square (c2) test statistic:
where n-u represents the degree of freedom. a is a chosen significance level, e. g. at 5%, used to create the lower and upper bounds of the inequality equation (9), in the course of a global model hypothesis testing. If this testing fails, an analyst can state that, with a 1-a confidence level, the functional and stochastic models (1-4) are not in order.
3.2 Optimal Parameter Selection
Another statistical testing can be utilized to validate parametric significance when trajectory polynomial modelings, such as described in Eq. (4), are involved. For any element x of a parameter correction vector x (7a), an F-distribution test statistic can be given as the term on the left-hand side of the following inequality equation (Zhong, 1997):
where qx denotes the scaled variance of x. The F-distribution has (1, n-u) degrees of freedom. Upon choosing a significance level a, the upper critical value F1-a;1,n-u is read from an F-distribution look-up table. If the test quantity fulfills the inequality relationship (10), this parameter element x is considered to be insignificant. After its deletion, the new parameter set will have u-1 elements. The measurement vector and its error covariance matrix remain unchanged.
A repeated least-squares adjustment is performed by using the algorithmic equations (6-7). New u-1 parameter corrections and new n measurement residuals are estimated. Their acceptance is based on the required global model test, as indicated in Eq. (9). The following minimum criteria, all related to the quadratic form of the estimated measurement residuals (Zhong, 1997), serve as the optimization indices in order to distinguish between the old/previous and new/current estimation results:
where Eq. (11a) is identical to Eq. (8). If the new model has a better performance than the old one, the search for another possible insignificant parameter continues. This means that the significance testing, Eq. (10), will be invoked. If, on the other hand, the previous model produces more optimal results than the current model does, the old/previous functional model represents the sought-after model solution.
4. Experimentation And Analysis
4.1 Airborne Chaochou SAR-Image
The airborne SAR-image over a 5.0 km*14.0 km area near the Chaochou town, Figure 3 on the last page, was a result of the Canadian CV-580 GlobeSAR campaign in Taiwan, near the end of October, 1993 (INTERA, 1994). The nominal flying height was 7.1 km above a mean sea level, with the airplane cruising at ~120 m/s (240 knots). A ground range resolution was ~4.0 m; an azimuth resolution was ~4.0 m, too.
Ground control/check point coordinates (Xi,Yi,Zi) were digitized/interpolated from the available 1:5000 topographic photo-maps. The corresponding image point line/pixel coordinates were measured, using the ERDAS/Imagine software utilities. All the coordinates were independently measured by three operators. The averaged coordinate measurements were accepted and prepared in an input data file. In our radargrammetric processing, the measurements were treated as being independent and identically distributed.
4.2 Significant Parameters
For the monoscopic Chaochou SAR-image, its space resection deals with the determination of radar antenna’s orientation parameters. With regard to the range/Doppler and the trajectory modeling equations (1-4), significant polynomial coefficients can be identified by following the stage-by-stage significance testing and the optimality assessment algorithm, in terms of Eq. (10-11). The results are given in Table 1, according to which the optimal set of parameters produced at the second stage will have been selected.
4.3 Planimetric Accuracy
When the SAR-image orientation parameters are made available, they can be used for each image point to determine its planimetric ground coordinates (Xi,Yi) where the Zi-coordinate is assumed to be known. This point-by-point space intersection is conducted for the 30 independent check points, leading to the accuracy results in Table 2. It is made clear that a single-valued variable squint angle is more suitable than a constant zero squint. The root-mean-square errors also indicate that a tentative second-order polynomial modeling (5) of the squint parameter has the highest accuracy level, in terms of the planimetric point positioning with the airborne Chaochou SAR-image.
Table 1. Iterative optimal determination of significant orientation parameters for the Chaochou SAR-image
(besides t and M)
c0, …, c3
c0, c1, c3
|a0, a1, a3
c0, c1, c3
The SAR-image range/Doppler equations are introduced so as to recognize the geometric squint parameter. Before Table 2. Planimetric point accuracy in relation to the squint, t, parameter modeling
|X/Easting (m)||Y/Northing (m)|
|t (= 0.0 deg)||± 6.2||± 7.1|
|t as a variable ( = -0.38 deg )||± 5.4||± 6.1|
|t modeled by using a
T0 = -0.053 deg
T1= 3.26×10-3 deg/pixel
T2= 2.44×10-6 deg/pixel2
|± 4.8||± 5.3|
embarking on a polynomial modeling of the squint angle, a least-squares estimation algorithm and a parametric significance testing methodology are briefly given. They serve as a sufficient processing tool in order to obtain an optimal set of radar’s orientation parameters. In studying the space resection/intersection of the airborne SAR Chaochou image, a second-order polynomial description of the squint parameter yields an improved Easting coordinate accuracy of ± 4.8 m and an improved Northing accuracy of ± 5.3 m.
Based on the positive experimental outcome, some future SAR-image processing schemes are itemized here: (1) automated setting of a polynomial expansion order for the squint angle; (2) possibility of a first-order range-dependent modeling of the pixel-spacing parameter; (3) application of the proposed methodology to spaceborne Earth resources SAR imagery.
The writers are indebted to the Council of Agriculture for sponsoring the 1993 GlobeSAR campaign. Thanks also go to Mr. C.-T. Wang of the NSC Satellite Remote Sensing Laboratory for pre-processing the SAR image.
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Figure 3. Airborne Chaochou SAR-image (C-band, HH-polarization, ten-look) in slant-range projection, on 30 October, 1993; 30 control points shown by (?), and 30 check points by (-); terrain heights varying between 4.0 m and 84.0 m