2004
UNDERSTANDING THE RATIONAL FUNCTION MODEL:
METHODS AND APPLICATIONS
Yong Hu, Vincent Tao, Arie Croitoru
GeolCT 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
A sensor model describes the geometric relationship between
the object space and the image space, or vice visa. It relates 3-D
object coordinates to 2-D image coordinates. The two broadly
used imaging geometry models include the physical sensor
model and the generalized sensor model. The physical sensor
model is used to represent the physical imaging process, making
use of information on the sensor’s position and orientation.
Classic physical sensors employed in photogrammetric missions
are commonly modeled through the collinearity condition and
the corresponding equations. By contrast, a generalized sensor
model does not include sensor position and orientation
information. Described in the specification of the OGC (1999a),
there are three main replacement sensor models, namely, the
grid interpolation model, the RFM and the universal real-time
senor model (USM). These models are generic, i.e., their model
parameters do not carry physical meanings of the imaging
process. Use of the RFM to approximate the physical sensor
models has been in practice for over a decade due to its
capability of maintaining the full accuracy of different physical
sensor models, its unique characteristic of sensor independence,
and real-time calculation. The physical sensor model and the
RFM have their own advantages and disadvantages for different
mapping conditions. To be able to replace the physical sensor
models for photogrammetric processing, the unknown
parameters of the RFM are usually determined using the
physical sensor models. The USM attempts to divide an image
scene into more sections and fit a RFM for each section.
Nevertheless, it appears that one RFM is usually sufficient for
modeling a whole image scene with 27552 rows and 27424
columns for a QuickBird PAN image.
The RFM was initially used in the U.S. military community.
Gradually, the RFM scheme is becoming well known to the
mapping community, largely due to its wide adoption as a new
standard. OGC has already decided (19992) to adopt it as a part
of the standard image transfer format. The decision of
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commercial companies, such as Space Imaging (the first high-
resolution satellite imagery vendor), to adopt the RFM scheme
in order to deliver the imaging geometry model has also
contributed to the wide adoption of the RFM. Consequently,
instead of delivering the interior and exterior orientation
geometry of the Ikonos sensor and other physical parameters
associated with the imaging process, the RFM is used as a
sensor model for photogrammetric exploitation. The RFM
supplied is determined by a terrain-independent approach, and
was found to approximate the physical Ikonos sensor model
very well. Generally, there are two different ways to determine
the physical Ikonos sensor model, depending on the availability
and usage of GCPs. Without using GCPs, the orientation
parameters are derived from the satellite ephemeris and attitude.
The satellite ephemeris is determined using on-board GPS
receivers and sophisticated ground processing of the GPS data.
The satellite attitude is determined by optimally combining star
tracker data with measurements taken by the on-board gyros.
With GCPs used, the modeling accuracy can be significantly
improved (Grodecki and Dial, 2001). Digital Globe (USA) also
delivers the RFM for its imagery products with up to 0.6-m
resolution, in addition to the spacecraft parameters (e.g.,
telemetry including refined ephemeris and attitude) and (interior
and exterior) orientations of the QuickBird sensor.
Recently, a number of recently published papers have reported
the algorithms and methods in the use of RFM for
photogrammetric processing on images acquired by different
satellite and airborne imaging sensors. Most work have focused
on processing the Ikonos Geo imagery (up to 1-m resolution)
supplied by Space Imaging. The facility of using the RFM to
replace physical sensor models in photogrammetric mapping is
being incorporated into many COTS software packages, and is
becoming a standard way for economical and fast mapping from
remotely sensed images. To follow this trend, other imagery
vendors possessing medium and high-resolution satellite
sensors, such as ORBVIEW-3 (ORBIMAGE, USA),
RADARSAT (Canada), IRS (India), and SPOT 5 (France), may
