Principal Warps: Thin-Plate Splines and the Decomposition of Deformations

One conventional tool for interpolating surfaces over scattered data, the thin-plate spline, has an elegant algebra expressing the dependence of the physical bending energy of a thin metal plate on point constraints. For interpolation of a surface over a fixed set of nodes in the plane, the bending energy is a quadrat ic form in the heights assigned to the surface. The spline is the superposition of eigenvectors of the bending energy matrix, of successively larger physical scales, over a tilted flat plane having no bending energy a t all. When these splines a re paired, one representing the x-coordinate of another form and the other the y-coordinate, they aid greatly in the modeling of biological shape change as deformation. I n this context, the pair becomes an interpolation map from RZ to R2 relating two sets of landmark points. The spline maps decompose, in the same way as the spline surfaces, into a linear par t (an affine transformation) together with the superposition of principal warps, which a re geometrically independent, affine-free deformations of progressively smaller geometrical scales. The warps decompose an empirical deformation into orthogonal features more or less as a conventional orthogonal functional analysis decomposes the single scene. This paper demonstrates the decomposition of deformations by principal warps, extends the method to deal with curving edges between landmarks, relates this formalism to other applications of splines current in computer vision, and indicates how they might aid in the extraction of features for analysis, comparison, and diagnosis of biological and medical images. lndex Terms-Affine transformations, biharmonic equation, biomedical image analysis, deformation, principal warps, quadratic variation, shape, thin-plate splines, warping. I. THE THIN-PLATE SPLINE AS AN INTERPOLANT A . The Function U ( r ) HIS paper proposes an algebraic approach to the deT scription of deformations specified by finitely many point-correspondences in an irregular spacing. At the root of the analysis is the special function sketched in Fig. 1. This is the surface z(x, y ) = U ( r ) = r2 log r , 2 where r is the distance from the Cartesian origin. The minus sign is for ease of reading the form of this surface: in this pose, it appears to be a slightly dented but otherwise convex surface viewed from above. The surface incorporates the point (0, 0, 0), as marked by the X in the figure. Also, the function is zero along the indicated Manuscript received July 17, 1987; revised August 2, 1988. Recommended for acceptance by W. E. L. Grimson. This work was supported in part by the National Institutes of Health under Grant GM-37251. The author is with the Center for Human Growth and Development, University of Michigan, Ann Arbor, MI 48109. IEEE Log Number 8927505. Fig. 1 . Fundamental solution of the biharmonic equation: a circular fragment of the surface z ( x , y ) = r 2 log r’ viewed from above. The X is at ( 0 , 0.0) ; the remaining zeros of the function are on the circle of radius 1 drawn. circle, where r = 1. The maximum of the surface is achieved all along a circle of radius l /& 0.607 concentric with the circle of radius 1 that is drawn. The function U ( r ) satisfies the equation The right-hand side of this expression is proportional to the “generalized function” lie, o ) zero everywhere except at the origin but having an integral equal to 1. That is, U is a so-called fundamental solution of the biharmonic equation A2U = 0, the equation for the shape of a thin steel plate lofted as a function z(x, y ) above the (x, y ) plane. This basis function is the natural generalization to two dimensions of the function I x l3 that underlies the familiar one-dimensional cubic spline. B. Bounded Linear Combinations of Terms U ( r ) Fig. 2 is a mathematical model of a thin steel plate which should be imagined as extending to infinity in all directions. Passing through the plate is a rigid armature in the form of a square of side A, drawn in perspective view as the rhombus at the center of the figure. The steel plate is tacked (fixed in position) some distance above two diagonally opposite comers of the square, and the same distance below the other two corners of the square. In the figure, this tacking is indicated by the X ’ s , which are to be taken as lying exactly upon the steel sheet but also as rigidly welded, via their “stalks,” to the corresponding comers of the underlying square. 0162-8828/89/0600-0567$01 .OO