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Open Access Publications from the University of California

Image Registration using Multiquadric Functions, the Finite Element Method, Bivariate Mapping Polynomials and Thin Plate Spline (96-1)


In this report, three methods of image-to-image registration using control points are evaluated. We assume that ephemeris sensor and platform data are unavailable. These techniques are the polynomial method, the piecewise linear transformation and the multiquadric method. The motivation for this research is the need for more accurate geometric correction of digital remote sensing data. This is especially important for airborne scanned imagery which is characterized by greater distortions than satellite data.

The polynomial and piecewise linear methods were developed for use with satellite imagery and have remained popular due to their relative simplicity in theory and implementation. With respect to airborne data however, both of these methods have serious shortcomings. The polynomial method, a global model, is generally applied as a least-squares approximation to the control points. Mathematically it is unconstrained between points leading to undesirable excursions in the warp. The piecewise linear method (or finite element method), a local procedure, produces a faceted irregular warp when the distortions between the control points are highly nonlinear.

The multiquadric method is a radial basis function. Two radial basis functions show promise for image warping: the multiquadric and thin plate spline. The multiquadric method is a global technique which captures local variations and interpolates, passing through the control points. It includes a tension-like parameter which can be used to adjust its behavior relative to local distortions. The principal shortcoming of the multiquadric method is that it is quite computationally intensive. Both the multiquadric method and thin plate splines have been evaluated extensively for scattered data interpolation.

In a test application using badly warped aircraft imagery, the multiquadric method produced better results both visually, e.g. crooked lines were straightened, and quantitatively with lower residual errors. The results for the multiquadric method are encouraging for improved environmental remote sensing and geographic information systems integration. The technique may be applied to satellite data as well as to airborne scanner data. The multiquadric method may be used for warping polygons and applied to mosaicking as well. Its present functional form is flexible and may be modified quite easily to further adapt to local distortions, a task not performed for this report. Advances in the rapid evaluation of radial basis functions will make both the multiquadric and thin plate spline techniques even more attractive in the future.

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