Rational Polynomial Coefficient (RPC) or (RFM) Rational Functional Model

1. 02-Atiqa Ijaz Khan Sunday, March 30, 2014
Institute of Geology, University of the Punjab 1
In Terms of Literature Review:
In the research literature the acronym RPC is derived either from “Rational Polynomial
Coefficients” or “Rational Polynomial Camera model”. Other authors use the term Rational
Function. Model (RFM) to refer to the same mathematical model.
RPC File:
1. Rational Polynomial satellite sensor models are simpler empirical mathematical models
relating image space (line and column position) to latitude, longitude, and surface
elevation.
2. The name Rational Polynomial derives from the fact that the model is expressed as the
ratio of two cubic polynomial expressions.
3. This is a mathematical mapping from object space coordinates to image space
coordinates. This mapping includes non-ideal imaging effects, such as lens distortion,
light aberration, and atmospheric refraction.
4. Actually, a single image involves two such rational polynomials:
a. One for computing line position (row) and
b. One for the column position.

2. 02-Atiqa Ijaz Khan Sunday, March 30, 2014
Institute of Geology, University of the Punjab 2
5. The coefficients of these two rational polynomials are computed by the satellite company
from the:
a. Satellite’s orbital position and orientation, and
b. The rigorous physical sensor model.
6. Using the georeferenced satellite image, its rational polynomial coefficients, and a DEM
to supply the elevation values, the TNTmips (the Map and Image Processing System)
Automatic Resampling Process computes the proper geographic position for each image
cell, producing an orthorectified image.
7. It’s important to note that using the RPC’s is just one way to transform from
image coordinates to Earth surface coordinates. This process can also be done using the
Rigorous Sensor Model.
Mathematical Background:
RPCs express the normalized column and row values in an image, (cn, rn), as a ratio of
polynomials of the normalized geodetic latitude, longitude, and height, (P, L, H). Normalized
values are used instead of actual values in order to minimize numerical errors in the calculation.
The scales and offset of each parameter are selected so that all normalized values fall in the range
[-1, 1]. The normalization used is as follows:
 P = (Latitude - LAT_OFF) / LAT_SCALE
 L = (Longitude - LONG_OFF) / LONG_SCALE
 H = (Height - HEIGHT_OFF) / HEIGHT_SCALE
 rn = (ROW - LINE_OFF) / LINE_SCALE

3. 02-Atiqa Ijaz Khan Sunday, March 30, 2014
Institute of Geology, University of the Punjab 3
 cn = (Column - SAMP_OFF) / SAMP_SCALE
Each polynomial is up to third order in (P, L, H), having as many as 20 terms.
Types of RPC File:
Two types of RPC data files are included.
1. RPC Data for Each Image:
One is a set of multiple RPC data files correspond to each image data file. The origin of the
image coordinate is the upper left corner of the image and the upper left corner pixel has a
coordinate value of (1,1).
2. RPC Data for Full Image:
The other is a data file which covers full image data included in the product. The origin of
the image coordinate is the upper left corner of the leftmost image.