Landmark Warp Glossary

Affine Transformation

a N-dimensional transformation that allows for an arbitrary N-dimensional linear transformation and translation.

Basis Function

a function that is used for computing spline-based warps. In general, it is b:R^N to R^NxN. This function characterizes the behavior of the transformation. Some basis functions are found as the Green's function of a PDE, while other are defined explicitly.

Bending Energy

a measure of the non-linear deformation of a transformation. Thin plate splines minimize this quantity. In 2-D, it is

Bending Energy = ∫ ∫ (∂2f/∂x2)² + 2*(∂2f/∂x∂y)² + (∂2f/∂y2)² dxdy

                  R2

Best fit plane

a plane that minimizes some error measure of a set of points. In this library, the error measure is the sum of the squared orthogonal distances to the plane.

Dual Number Quaternion method

there is a method for computing the rigid transformation between landmarks using quaternions. The implementation is a port of Matlab code written by Yonggang Shi.

Elastic Warp

a non-linear spline-based warp that models the deformation material as a linear elastic medium. There are elasticity parameters that change "how elastic" the medium is. The transformation will satisfy the Cauchy-Navier operator. There are several types of force terms, which change the behavior of the transformation. See "Elastic Registration of Medical Images Using Radial Basis Functions with Compact Support" (1999) by Mike Fornefett, Karl Rohr, and H. Siegfried Stiehl

Inverse Multiquadratic Warp

a non-linear spline-based warp that uses the an inverse multiquadratic basis. See "Free Form Deformation with Scattered Data Interpolation Methods" (1993) Detlef Ruprecht, Heinrich Müller

General Position

generally, a set of D-dimensional points is in general position if no D+1 points lie in a D-1 dimensional plane. However, in the context of this framework, a set of points is in general position if there is a subset that are in general position (by the above definition).

Landmark

a point from R^N that is considered a 'feature'. These are used to define the transformations.

Landmark file

an ASCII file format that encodes K landmarks of dimension N. The landmarks must be approximately unique. Each landmark is line-delimited and each coordinate is space-delimited. The filename usually has the form '*.lm'

Linear Least Squares Method

a method of estimating parameters of a model which solves a linear equation by minimizing the sum of the squares of the differences between the solution's values and the desired function's values.

Linear Transformation

a transformation that preserves some geometry of the space that is transformed. Strictly speaking, this transformation will be invariant to addition and scalar multiplication. Rotation, scaling, and skew are linear transformations. If we extend R^N to the projective-space P^N, a translation can be considered a linear transformation

Procrustes Method

a method of estimating the rigid transformation between two sets of points, known as shapes in the parlance of the Procrustes method. The translation of each shape to the normalized shape space is derived from the mean of the shape points. Once in the normalized shape space, the rotation between the shapes is estimated in a least squares fashion. The rotation, and, for each shape, the scaling and translation are combined to find a single transformation that takes one shape to the other. Also see Generalized Procrustes Analysis

Generalized Procrustes Method

a method of computing a similarity transformation, i.e. a transformation defined by a scale, rotation, and translation. The translation and rotation are computed as in the Procrustes Method. The scaling of each shape to the normalized shape space is derived from the norm of the shapes data matrix. In practice, the scaling is constrained to be positive. See Procrustes Method.

Reference

the input space of a warping function.

R^N

an N-dimensional real-valued space. R1 is the real line; R2 is the real plane, etc.

Spline Order

a measure of the polynomials used when resampling contours. 1 is linear, 2 is cubic, and 3 is quintic. This 'spline' is distinct from the usage in SplineWarp.

Radial Basis Function

a basis function which is only dependent on the magnitude of the input vector. One should note that the dimension of the problem still matters, even if the function takes only a scalar. For example, thin plate splines use radial basis functions, and the basis functions for R2 and R3 are different. The basis function associated with a radial basis function will have output of the form rbf(mag(x))*I, where I is the identity matrix.

Radial Gaussian Warp

a non-linear spline-based warp that uses the a radial Gaussian basis. See "Image Warping Using Few Anchor Functions Points and Radial Functions" (1995) by Nur Arad and Daniel Reisfeld

Rigid Transformation

a linear transformation that allows for rotation and translation.

Similarity Transformation

a linear transformation that allows for rotation, translation and scaling.

Spline-based Warp

a warp that has non-linear terms in addition to an affine part. The non-linear terms are a weighted sum of basis functions. The basis function will determine the behavior of the warp. Unlike a linear warp, these will typically match landmarks exactly.

Target

the output space of the warping function.

Thin Plate Warp

a non-linear spline-based warp that models the deformation material as a metal sheet. This warp uses a radial basis function with no parameters. The warp will satisfy the homogenous biharmonic equation. Reference landmarks will be mapped exactly to target landmarks. This is implemented in two and three dimensions.

Warp

a function that takes points from R^N to R^N. This can be linear or non-linear. This definition is most likely broader than other sources. In this context, a warp does not necessarily map homologous points exactly to one another, e.g. a Procrustes warp computes the best fit rigid transformation. See the description of the warp to determine the constraints on linearity and landmark matching.

Warp File Format

a file format that encodes every parameter of a warp function. Specifically, the parameters are the coefficients of the bases of the warp. The size of the file is linearly proportional to the number of landmarks. This is an ASCII format that effectively serializes a warping object. The filename will have the form '*.lmw'.

Wendland Warp

a non-linear spline-based warp that uses the Wendland basis. This basis is notable for its compact support. In other words, the warp will have a limited effect on regions outside a certain neighborhood of a landmark. There is a parameter for this model that determines the size of this neighborhood. See "Meshless Galerkin Methods using radial basis functions" (1991) by Holger Wendland.