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A Memo on How to Use the Levenberg-Marquardt Algorithm

for Refining Camera Calibration Parameters

Pradit Mittrapiyanuruk

Robot Vision Laboratory, Purdue University, West Lafayette, IN, USA

mitrapiy@purdue.edu

This short note briefly explains how to implement the Levenberg-Marquardt (LM)

algorithm in MATLAB for the minimization of the geometric distance given by Equation

(10) of the Zhang’s report. For students who are not familiar with the LM method, we

suggest you visit the Wikipedia page

http://en.wikipedia.org/wiki/Levenberg-Marquardt_algorithm

For a more elaborate discussion of the LM method, the following references may be

useful

- Appendix 6 in the Hartley & Zisserman’s Book

- Section 9.4 in the Chong & Zak’s Book (the official textbook for ECE580)

- K. Madsen et al, “Methods for non-linear least squares problems”, IMM Lecture Note

2004, available on the web at

http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/3215/pdf/imm3215.pdf

In the rest of this note, I will first demonstrate how to use the LM method to solve a

simple curve fitting problem and then I will give some clues that may help you to apply

the LM algorithm for the refinement of the camera calibration parameters in Homework

5.

Example: Curve fitting with the LM algorithm

Problem: Fit a set of point data {x

i

,y

i

} (x

i

=0, 0.1, …, 2*PI) with the function

*

f(x) = a*cos(bx)+b*sin(ax)

where a and b are the parameters that we want to estimate by the LM method. The

problem of finding the optimum can be stated as follows

,

1 2

argmin

[ ...... ] [ ( ; , )]

T

a b

T

n i i i

where d d d and d y f x a b

= = −

d d

d

*

This example, taken from Wikipedia, is interesting because the complexity of the fitted function would

preclude a linear least square solution to the problem.

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