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Understanding the Rational Function Model: Methods and Applications

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Understanding the Rational Function Model: Methods and Applications UNDERSTANDING THE RATIONAL FUNCTION MODEL: METHODS AND APPLICATIONS Yong Hu, Vincent Tao, Arie Croitoru GeoICT Lab, York University, 4700 Keele Street, Toronto M3J 1P3 - {yhu, tao, ariec}@yorku.ca KEY WORDS: Photogrammetry, Remote Sensing, Sensor Model, High-resolution, Satellite Imagery ABSTRACT: The physical and generalized sensor models are two widely used imaging geometry models in the photogrammetry and remote sensing. Utilizing the rational function model (RFM) to replace physical sensor models in photogrammetric mapping is becoming a standard way for economical and fast mapping from high-resolution images. The RFM is accepted for imagery exploitation since high accuracies have been achieved in all stages of the photogrammetric process just as performed by rigorous sensor models. Thus it is likely to become a passkey in complex sensor modeling. Nowadays, commercial off-the-shelf (COTS) digital photogrammetric workstations have incorporated the RFM and related techniques. Following the increasing number of RFM related publications in recent years, this paper reviews the methods and key applications reported mainly over the past five years, and summarizes the essential progresses and address the future research directions in this field. These methods include the RFM solution, the terrain- independent and terrain-dependent computational scenarios, the direct and indirect RFM refinement methods, the photogrammetric exploitation techniques, and photogrammetric interoperability for cross sensor/platform imagery integration. Finally, several open questions regarding some aspects worth of further study are addressed. 1. INTRODUCTION commercial companies, such as Space Imaging (the first high- resolution satellite imagery vendor), to adopt the RFM scheme A sensor model describes the geometric relationship between in order to deliver the imaging geometry model has also the object space and the image space, or vice visa. It relates 3-D contributed to the wide adoption of the RFM. Consequently, object coordinates to 2-D image coordinates. The two broadly instead of delivering the interior and exterior orientation used imaging geometry models include the physical sensor geometry of the Ikonos sensor and other physical parameters model and the generalized sensor model. The physical sensor associated with the imaging process, the RFM is used as a model is used to represent the physical imaging process, making sensor model for photogrammetric exploitation. The RFM use of information on the sensor’s position and orientation. supplied is determined by a terrain-independent approach, and Classic physical sensors employed in photogrammetric missions was found to approximate the physical Ikonos sensor model are commonly modeled through the collinearity condition and very well. Generally, there are two different ways to determine the corresponding equations. By contrast, a generalized sensor the physical Ikonos sensor model, depending on the availability model does not include sensor position and orientation and usage of GCPs. Without using GCPs, the orientation information. Described in the specification of the OGC (1999a), parameters are derived from the satellite ephemeris and attitude. there are three main replacement sensor models, namely, the The satellite ephemeris is determined using on-board GPS grid interpolation model, the RFM and the universal real-time receivers and sophisticated ground processing of the GPS data. senor model (USM). These models are generic, i.e., their model The satellite attitude is determined by optimally combining star parameters do not carry physical meanings of the imaging tracker data with measurements taken by the on-board gyros. process. Use of the RFM to approximate the physical sensor With GCPs used, the modeling accuracy can be significantly models has been in practice for over a decade due to its improved (Grodecki and Dial, 2001). Digital Globe (USA) also capability of maintaining the full accuracy of different physical delivers the RFM for its imagery products with up to 0.6-m sensor models, its unique characteristic of sensor independence, resolution, in addition to the spacecraft parameters (e.g., and real-time calculation. The physical sensor model and the telemetry including refined ephemeris and attitude) and (interior RFM have their own advantages and disadvantages for different and exterior) orientations of the QuickBird sensor. mapping conditions. To be able to replace the physical sensor models for photogrammetric processing, the unknown Recently, a number of recently published papers have reported parameters of the RFM are usually determined using the the algorithms and methods in the use of RFM for physical sensor models. The USM attempts to divide an image photogrammetric processing on images acquired by different scene into more sections and fit a RFM for each section. satellite and airborne imaging sensors. Most work have focused Nevertheless, it appears that one RFM is usually sufficient for on processing the Ikonos Geo imagery (up to 1-m resolution) modeling a whole image scene with 27552 rows and 27424 supplied by Space Imaging. The facility of using the RFM to columns for a QuickBird PAN image. replace physical sensor models in photogrammetric mapping is being incorporated into many COTS software packages, and is The RFM was initially used in the U.S. military community. becoming a standard way for economical and fast mapping from Gradually, the RFM scheme is becoming well known to the remotely sensed images. To follow this trend, other imagery mapping community, largely due to its wide adoption as a new vendors possessing medium and high-resolution satellite standard. OGC has already decided (1999a) to adopt it as a part sensors, such as ORBVIEW-3 (ORBIMAGE, USA), of the standard image transfer format. The decision of RADARSAT (Canada), IRS (India), and SPOT 5 (France), may also supply RFM-enabled imagery products in the near future. denominator is of twenty-term cubic form, and the order defined in NIMA (2000) has become the de facto industry standard. This paper reviews the developments in the use of the RFM The polynomial coefficients are also called RPCs, namely, rapid mainly over the past five years to summarize the essential positioning capability, rational polynomial coefficients or progresses in this field. The methodology of developing the rational polynomial camera data. In this paper, we refer the RFM is summarized in Figure 1, where the individual processes RFM as general rational functions with some variations, such as and their interrelations are explained in following sections. subsets of polynomial coefficients, equal or unequal denominators and two transformation directions (i.e., forward and inverse equations). Nine configurations of the RFM have Imaging sensors been analyzed in Tao and Hu (2001a, 2001b). An inverse form of the RFM is described in Yang (2000), but is seldom used. The term - RPC model - often refers to a specific case of the Physical sensor No RFM that is in forward form, has third-order polynomials, and model available? is usually solved by the terrain-independent scenario. The RPC model is more concentrated because it is transferred with Space Yes Imaging and Digital Globe imagery products. Terrain-independent Terrain-dependent approach approach 1.2 RFM Solution The unknown RFCs can be solved by least-squares adjustment. The normal equation is given by Eq. 2 (Tao and Hu, 2001a), Parameters RFM solution where I is the vector of RFCs; T is the design matrix of the selection linearized observation equations (Eq. 1); W is the weight matrix for image pixel coordinates G. The covariance matrix associated RFM parameters with I is given by Eq. 3 (Hu and Tao, 2002), where R is the covariance associated with the measured image positions. TTWTI – TTWG = 0 (2) No T -1 GCPs available? P = (T WT) + R (3) Yes To tackle the possible ill-conditioning problem during the adjustment, the Tikhonov regularization technique was RFM refinement suggested to turn the normal equation into a regularized one. Then the RFCs may be solved iteratively as follows: I = I + (TTW T + h2E)-1TTW w for k = 1, 2, … (4) Refined RFMs k k-1 k-1 k-1 k-1 with I0 = 0, W0 = W(I0) = E where h is the regularization parameter; k is the iteration Mapping applications: number; Wk = W(Ik) is the weight matrix; wk = G – TIk is the Ortho-rectification, 3-D feature collection, misclosures at GCPs. DSM generation, multi-sensor integration 2. APPROACHES OF DETERMINING RFCS Figure 1. The strategy of developing the RFM The RFCs can be solved by terrain-independent scenario using 1.1 The Rational Function Model known physical sensor models or by terrain-dependent scenario without using physical sensor models. The RFM relates object point coordinates (X, Y, Z) to image pixel coordinates (l, s) or vice visa, as physical sensor models, 2.1 Terrain-independent Approach but in the form of rational functions that are ratios of polynomials. The RFM is essentially a generic form of the For the terrain-independent scenario, the RFM performs as a rigorous collinearity equations and the generalized sensor fitting function between the image grid and the object grid models including the 2-D and 3-D polynomial models, the (Yang, 2000; Tao and Hu, 2001a). In detail, an image grid projective transformation model and the (extended) direct linear covering the full extent of the image is established and its transformation model. For the ground-to-image transformation, corresponding 3-D object grid with several layers (e.g., four or the defined ratios have the forward form (OGC, 1999a): more layers for the third-order case) slicing the entire elevation range is generated. The horizontal coordinates (X, Y) of a point l = p (X , Y , Z ) / p (X , Y , Z ) n 1 n n n 2 n n n of the 3-D object grid are calculated from a point (l, s) of the s = p (X , Y , Z ) / p (X , Y , Z ) (1) n 3 n n n 4 n n n image grid using the physical sensor model with specified where (ln, sn) are the normalized line (row) and sample (column) elevation Z.
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