# Seeking references for how Rational Polynomial Coefficients(RPCs) are calculated

I wonder how RPCs are created for satellite images. I read that RPCs are used to correct for systematic error in the image which arise due to Surface relief, camera orientation wrt nadir, camera tilt and geometric errors. And that it ensures uniform scale for all pixels in the image i.e., orthorectification. Now I wonder how these rational polynomial coefficients are derived for a region on earth.

Can you give me relevant references?

An RPC model is the ratio of two polynomials which can be derived from the rigorous sensor model and the corresponding terrain information, which does not reveal the sensor parameters. High resolution satellite image vendors provide a RPC file with the image. This file consist of RPC coefficients which is used to relate coordinate in a sensor plane (2D) to object coordinate (3D). l = Numl (x, y, z)/ Denl (x, y, z) s = Nums (x, y, z)/ Dens (x, y, z) Where l and s are normalised line and sample ( row and column of 2D image in sensor plane) and x, y, z are normalised latitude, longitude and height.

Numl (x, y, z) = a0+a1x+a2y+a3z+a4xy+a5xz+a6yz+a7x2+a8y2+a9z2+a10xyz+a11x3+a12xy2+ a13xz2+a14yx2+a15y3+a16yz2+a17x2z+a18y2z+a19z3

Denl (x, y, z) = b0+b1x+b2y+b3z+b4xy+b5xz+b6yz+b7x2+b8y2+b9z2+b10xyz+b11x3+b12xy2+ b13xz2+b14yx2+b15y3+b16yz2+b17x2z+b18y2z+b19z3

Nums (x, y, z) = c0+c1x+c2y+c3z+c4xy+c5xz+c6yz+c7x2+c8y2+c9z2+c10xyz+c11x3+c12xy2+c 13xz2+c14yx2+c15y3+c16yz2+c17x2z+c18y2z+c19z3

Dens (x, y, z) = d0+d1x+d2y+d3z+d4xy+d5xz+d6yz+d7x2+d8y2+d9z2+d10xyz+d11x3+d12xy2+ d13xz2+d14yx2+d15y3+d16yz2+d17x2z+d18y2z+d19z3

In the above equations a[1-20], b[1-20], c[1-20] and d[1-20] and the RPC coefficients. Advantage of using RPC model Physical parameters of camera model remains confidential. Simple and efficient way to convert coordinate in a sensor plane (2D) to object coordinate (3D)

A RPC Model is nothing but a transformation between (PIXEL,SCAN) => (LAT , LON , HEIGHT) for any given satellite image. The concept of RPCs has recently picked up in the industry because of the competition for accurate Satellite geometry products. RPC is a standardized way to hide the physical sensor model of the imaging session but still give user the power to generate a highly accurate Image with his own post processing.

Generating RPCs is different with different players in the industry , but most of them follow some version of Non linear Least square method to fit to the polynomials discussed in the answer by bharti20. The data to fit is generated by the physical sensor model obtained during imaging session.

For example: If Kompsat was imaging Delhi , it will have certain orbit trajectory and orientation. Using that the look vector is generated and for each line of the image and each pixel of the line, what point on earth is the satellite looking at is determined. Now this scan,line->lat,lon,hei fit is done. The performance of the RPC Depends on your algorithm for the fit. Ex( Levenberg-Marquardt,Broyden,Krylov) etc.