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Particle Swarm Optimization of RFM for Georeferencing of Satellite Images Somayeh Yavari, Mohammad J. Valadan Zoej, Ali Mohammadzadeh, and Mehdi Mokhtarzade
Abstract—Rational function models (RFMs) provide one of the best methods of extracting spatial information from highresolution satellite images, particularly when sensor parameters cannot be accessed. RFM terms have no physical meaning, and hence, all of these terms are typically used in conventional solutions. As a result, more ground control points (GCPs) are required, and the model is prone to overparameterization. In this letter, a modified particle swarm optimization is applied to identify the optimal terms for RFMs. In comparison to conventional models, experimental results demonstrate how well the proposed algorithm can determine an RFM, which is optimal in both the total number of terms and the positional accuracy. The proposed algorithm is determined to be efficient when subpixel accuracy can be obtained with four GCPs in IKONOS-Geo image. Index Terms—Genetic algorithm (GA), geometric correction, high-resolution satellite images (HRSIs), particle swarm optimization (PSO), rational function model (RFM).
I. I NTRODUCTION
T
ODAY, DUE to the availability of high-resolution satellite images (HRSIs), accurate geospatial information can be extracted from these types of images based on transformation functions. There are two types of transformation, which are image-to-image transformation (coregistration) and image-toobject transformation. Although coregistration can be applied by different mathematical models [1]–[11], our study is in image-to-object transformation domain. Since the accuracy of the transformation affects the accuracy of the extracted geospatial information, the production of accurate transformations is one of the most important research topics in photogrammetry and remote sensing. Existing transformation models fall into two categories: parametric (rigorous) models such as orbital parameter models [12], [13] and nonparametric (nonrigorous) models [14]–[18] such as rational function models (RFMs). Parametric models, which reconstruct the geometry of the image at the time of imaging, require accurate sensor geometric parameters as well as satellite ephemeris data. These data are not always available from some HRSIs (e.g., IKONOS and
Manuscript received July 14, 2011; revised November 20, 2011 and February 22, 2012; accepted March 16, 2012. The authors are with the Department of Remote Sensing and Photogrammetry Engineering, Faculty of Geodesy and Geomatics Engineering, Khajeh Nasir Toosi University of Technology, Tehran 19667-15433, Iran (e-mail:
[email protected];
[email protected];
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LGRS.2012.2195153
GeoEye), and as a result, nonparametric models are often preferred. However, nonparametric models are highly dependent on the structure of the selected model and on the number, distribution, and accuracy of the ground control points (GCPs). In terms of the model structure, RFMs are regarded as the most comprehensive models that can achieve high positional accuracies [14]. In conventional RFM [14], it is common to choose the maximum order of its polynomials where the full series of lower order terms is applied. This not only increases the required GCPs but also may cause accuracy degradation which is the result of overparameterization error [18]. The reason is that the applied terms are not necessarily consistent with the true nature of distortions available in input image. Accordingly, it is motivated to choose the optimum structure of RFM via term selection of its polynomials. However, with no physical interpretation of RFM terms, selecting the optimum structure of these models is a challenge. RFM optimization selects the minimum number of terms while still preserving geopositional accuracies. When there are a large number of possible solutions, trial-and-error optimization of RFMs is not practical, and zero-order optimization algorithms [19] are regarded as the best alternatives. Among the zero-order optimization algorithms, genetic algorithm (GA) [20], [21] and particle swarm optimization (PSO) [22]–[28] are concerned as the most successful and frequently used algorithms. RFM optimization using GA has been successfully implemented in [29], and to investigate the efficiency of PSO in computational time reduction in comparison to GA, PSO is studied in this research work. In the following section, the PSO for RFM optimization is presented. Implementation and experimental results are described in Section III, and Section IV is dedicated to concluding remarks. II. PSO FOR RFM O PTIMIZATION The RFM determines the image coordinates (r, c) from the ratio of two polynomials of object coordinates (X, Y, Z) [14] as in the following: P1 (X, Y, Z) P3 (X, Y, Z) P2 (X, Y, Z) c= P4 (X, Y, Z) Pi (X, Y, Z) = ai0 + ai1 X + ai2 Y + ai3 Z + ai4 XY + ai5 XZ + ai6 Y Z + ai7 X 2
1545-598X/$31.00 © 2012 IEEE
r=
+ ai8 Y 2 + ai9 Z 2 + ai10 XY Z + · · ·
(1)
(2)
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Fig. 1. Particle scheme used in a binary PSO.
Most photogrammetric applications assume that P3 = P4 . RFM coefficients are determined using GCPs, and the accuracy of the coefficients is assessed by independent check points (ICPs). In conventional RFM, a complete series of polynomial terms is applied [14] which is not necessarily the optimum structure of this model. As mentioned earlier, using all terms is not proper so this letter uses PSO for optimum term selection of RFM polynomials, where the minimum rmse of ICPs is achieved by a limited number of GCPs. PSO was originally designed and introduced by Eberhart and Kennedy in 1995 based on the social intelligence of a group of birds or fishes. Although PSO is mainly applied in the continuous form, its binary form [23] is applied in this study in order to determine the RFM terms. In a binary PSO, a population (swarm) of particles (representative of different RFM structures) is initialized randomly with a string of binary values, indicating the presence or omission of the corresponding terms, as in Fig. 1. The 1/rmse of ICPs for each particle is considered to be its fitness value. In an iterative process, the velocity of each particle (v) is updated using its present velocity, its current position (p), its best position in all iterations (P Best), and the global best position of all particles during the whole procedure (GBest) vij (t + 1) = w(t).vij (t) + c1 .r1 . [GBesti (t) − pij (t)] + c2 .r2 . [P Besti (t) − pij (t)] .
(3)
The parameters for the PSO are as follows. 1) i is the index of particle in the population. 2) j is the index of bits in the binary string of each particle. 3) t is the iteration number. 4) r1 and r2 are two uniform random values in [0,1]. 5) c1 and c2 are two constant acceleration coefficients. 6) w(t) is time-varying inertia weight. The velocities obtained (vij ) are bounded in the range of [vmin , vmax ], indicating that lower/higher values are transferred to the vmin /vmax limits, which are defined in advance. Higher values of w(t) change the algorithm attitude to a global search, while its lower values make the algorithm more intelligent for intense local searches. As a result, w(t) is designed as a decreasing function of iterations such as (4), which is used in [26]. w(t) = wmin + (wmax − wmin ) ·
tmax − t . t
Fig. 2.
Hyperbolic tangent function. TABLE I PARAMETERS OF THE PSO U SED IN RFM O PTIMIZATION
updated as in pij (t + 1) =
1, 0,
if rij < φ (vij (t)) otherwise
(5)
where rij is a uniform random number in [0,1]. In this equation, φ(.) is a normalizing function that remaps the vij (t) values in the probability range of [0,1] and usually assumes that the function is a sigmoid function [23], [27]. If φ(.) is defined as a sigmoid function, it is more likely to map pij to zero for vij < 0 and to one for vij > 0, which means that the presence and omission of RFM terms are assigned similar probabilities. However, RFM optimization aims to minimize the number of terms where acceptable accuracies are preserved. Therefore, the normalizing function φ(.) should be designed to be more prone to omit, rather than maintain, the terms. Accordingly, in this letter, φ(.) was replaced by (6). Fig. 2 shows the designed function of tanh(x), if x > 0 φ(x) = (6) 0, otherwise. This function introduces a modified PSO specialized for RFM optimization (PSORFO). The RFM optimization is repeatedly updated until a termination condition is satisfied. Usually, this condition is evaluated by a measure of closeness between the PSO solution of the current population and its global best position which is an accepted sign of convergence. III. I MPLEMENTATION AND R ESULTS
(4)
In the aforementioned equation, wmax and wmin are two constant experimental parameters, and tmax is the maximum number of iterations. In conventional discrete PSO, vij represents the probability of assigning one to pij [23]. Accordingly, particle positions are
A. Parameter Settings In this letter, the maximum number of terms in P1 , P2 , and P3 is set to 11, 11, and 10, respectively, which means that a total of 32 terms are used in the optimization process (a30 = 1). Accordingly, 32 b is used for each particle of the PSO algorithm. It should be noted that RFMs, represented by
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TABLE II RFM R ESULTS O BTAINED BY GA, C ONVENTIONAL PSO, AND M ODIFIED PSO O PTIMIZATION OVER AN IKONOS-G EO I MAGE
TABLE III C OMPLETE S ERIES OF P OLYNOMIALS IN C ONVENTIONAL RFM OVER AN IKONOS-G EO I MAGE U SING D IFFERENT C OMBINATIONS OF GCP S AND ICP S
swarm particles, are solved using direct solutions where image and object coordinates are also normalized [14]. The PSO parameters used in this research are shown in Table I. A population size of 30 is chosen as the minimum number of particles where required accuracies are obtained in a reasonable computational time. The parameters vmax , vmin , wmax , wmin , tmax , c1 , and c2 are selected based on [23], [24], and [28] and also experimentally to balance the global and local search (randomness and intelligence) of the PSO. However, it should be noticed that based on our experimental results, PSO is rather stable to the mild changes of these parameters. The termination of the PSO is controlled both by the maximum number of generations (i.e., tmax ) and by a convergence criterion. This criterion is satisfied when the maximum difference between the fitness of GBest, the mean of P Best, and the mean of the particle fitness is less than a threshold. The appropriate threshold value is determined empirically and depends on the accuracy of GCPs and, also, the image spatial resolution. As a result, this threshold is set to 10 for IKONOS and 50 for SPOT 4 images. However, different values in the range of 10–100 are tested, and the results show that these values do not have a significant effect on the final results. The mean of P Best and the particle fitness are computed over the best half of the population to prevent the undesired effects of blunder individuals. B. Results and Analysis This letter used an IKONOS-Geo image over Hamedan City, west of Iran. The IKONOS-Geo image was acquired on July 10, 2000, with an off-nadir angle of 20.4◦ and a sun elevation
of 47.4◦ . The elevation ranged between 1700 and 1900 m. In total, 58 control points are extracted from 1:1000 scale digital maps produced by the Iranian National Cartographic Center. The points are distinct features such as buildings, pool corners, walls, and road junctions. To evaluate the efficiency of the proposed algorithm (PSORFO), different combinations of well-distributed GCPs and ICPs are used. Additionally, other RFM optimization solutions, such as conventional PSO and GA, are performed using the same set of GCPs and ICPs for comparative purposes. In RFM optimization using GA, the proposed algorithm in [29] is used when the population size is set to 50. All of these optimization algorithms are performed ten times to assess the stability of the results. It should be mentioned that not only the arrangement of selected terms but also the number of these terms are different in different runs. However, their accuracies are rather comparable (see third and fourth columns of Tables II and IV). Among them, the best one is selected which is presented in Tables II and IV. As shown in Table II, PSORFO can optimize the RFM to obtain subpixel accuracy for IKONOS image when as few as four GCPs are used. Residual vector plot diagrams show that no systematic errors have occurred. Decreasing the number of GCPs causes the optimum RFM to maintain a limited number of terms in its polynomial equations (see the fifth column of Tables II and IV). In this column, different numbers of coefficients of RFM polynomials [p1 , p2 , and p3 based on (1) and (2)] in each numerator of x and y and their same denominator are shown. According to the fifth column of Table II, PSORFO can reach the same accuracy as GA but with less total number of coefficients. This is a good point that means the results of PSORFO in comparison with GA results have higher degrees of freedom and hence are more reliable.
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TABLE IV RFM R ESULTS O BTAINED BY GA, C ONVENTIONAL PSO, AND M ODIFIED PSO O PTIMIZATION OVER A SPOT 4 L1B I MAGE
Fig. 3. Comparison between rmse values of conventional RFMs and RFMs optimized by PSORFO over an IKONOS-Geo image.
Comparisons between PSORFO and the conventional PSO demonstrate that PSORFO is more accurate, has a lower processing time, and also converged in all cases in spite of conventional PSO. In fact, PSORFO can omit a larger number of RFM terms. Therefore, the optimal RFM has a higher number of degrees of freedom, resulting in a higher accuracy. PSORFO is also more efficient in terms of computation, as it converges faster than conventional PSO models. PSORFO and GA-optimized RFM models show similar accuracy values, while PSORFO models are much faster. The longer computational time of GA model can be due to the larger population size required in order to achieve the same accuracy as the PSORFO model. Furthermore, a complete series of polynomials in conventional RFM based on [14] with different combinations of GCPs and ICPs is used and showed in Table III. According to this table, using a complete series of polynomials like [14] requires a large number of GCPs, while in the proposed method, better results are obtainable using much less GCPs. Also, to validate the effectiveness of the proposed method, a comparative study is carried out by a conventional RFM using 14 terms (i.e., P1 , P2 , and P3 : five, five, and four terms, respectively). This combination is the best one in conventional RFM based on [18]. The conventional RFM is solved and evaluated using the same set of GCPs and ICPs. Fig. 3 shows the rmse values
obtained for the conventional RFM and for the RFM from the proposed PSORFO. Due to a lack of degrees of freedom, conventional RFMs cannot be solved when only four or five GCPs are available. The comparison between the results obtained from both conventional RFMs and RFMs optimized with PSORFO demonstrates the efficiency of the method proposed in this letter. Also, for evaluation purposes, a SPOT 4 Level 1B (L1B) image over Zanjan, Iran, is tested. The cross-track angle is + 24◦ , and 37 well-distributed points are established using differential GPS techniques with the elevation ranged between 530 and 1200 m. The accuracy of these points is established to be better than 1 m. The obtained results, presented in Table IV, confirm the concluded items from Table II. IV. C ONCLUSION In this letter, a modified PSO has been introduced for rational function optimization (PSORFO) that accentuates the omission of redundant polynomial terms. The proposed PSORFO can achieve a subpixel accuracy using a limited number of GCPs for IKONOS image. A comparison of the results obtained using the proposed method with those obtained using traditional RFMs demonstrates the efficiency of PSORFO in terms of both the accuracy and the number of required GCPs. Furthermore, RFM optimization using GA is compared with the PSORFO, which confirms the advantage of the proposed method in terms of computational time. In the case of SPOT image, acquired from a region with much more relief variations (about 600 m), the results of PSORFO are comparable with those of RFM optimized by GA. However, PSORFO has superiority in terms of degrees of freedom as well as computational time. R EFERENCES [1] A. A. Goshtasby, “Transformation functions,” in 2-D and 3-D Image Registration: For Medical, Remote Sensing, and Industrial Applications. Hoboken, NJ: Wiley, 2005, ch. 5, pp. 107–141. [2] L. Zagorchev and A. Goshtasby, “A comparative study of transformation functions for non-rigid image registration,” IEEE Trans. Image Process., vol. 15, no. 3, pp. 529–538, Mar. 2006.
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