# Estimating 3-D motion and structure parameters of a rigid planar object by establishing landmark correspondences

Thesis

12-31-1990

## Degree Name

Master of Science in Electrical Engineering - (M.S.)

## Department

Electrical and Computer Engineering

Nirwan Ansari

Andrew Ulrich Meyer

Yun Q. Shi

## Abstract

The application of a landmark based approach in recognizing 3-D rigid planar objects[3] along with two different motion and structure estimation algorithm [21][23] have been studied to perform motion and structure estimation tasks of moving objects. The recognition algorithm is based on a sphericity value derived from affine transformation that maps feature points in the object under observation from first to second view in the 2-D image space. The motion and structure estimation algorithm is based on a unique mapping given four or more feature point correspondences.

More than ten experiments have been done including small and large rigid motions. From all numerical experiments that have been tried, the recognition algorithm can handle the landmark matching tasks well if the object follows only small rigid motions. The algorithm, without too much destroying the sequential order of the original landmarks, is also capable of detecting the correct matches of landmarks when missing and some extraneous landmarks in the second view (after motion), are taken into account. The algorithm, however, will fail to detect the correct matches of landmarks if the object under observation moves such that the normal of the object surface is almost perpendicular to the optical axis.

When a motion is small, only a relative depth of the observed object can be determined. The number of solutions, aside from the scale factor for the translational vector, depend on the number of the real roots of the sixth order polynomial equation. In our experiment, we have found two real and four complex roots. From the two resulting solution only one represents the actual motion parameters. If the motion is large, the number of solutions is either one or two depending on the multiplicity of the singular values of the 3x3 real matrix consisting of the pure parameters obtained from the unique mapping.

Due to the nonlinearity of the computation procedure that have been used and roundoff error generated by the computer, the results deviate up to 10% from the predefined parameters. The motion algorithm, however, is convenient, inexpensive and not shaken by missing or extraneous landmarks as long as there are more than three landmark corresponding pairs. That is, regardless of which landmarks in the observed object being used, the estimated motion and structure parameters will remain relatively the same.

COinS