Smooth Surface Modeling of DEMs Based on a Regularized Least Squares Method of Thin Plate Spline

Abstract

Thin plate spline (TPS) has been widely accepted as a method for smooth fitting of noisy data. However, the classical TPS always has an ill-conditioning problem when two sample points are very close. Although the modified orthogonal least squares-based TPS (TPS-M) avoids this ill-conditioning problem, it is not completely immune to over-fitting when sample points are noisy. In this paper, a regularized least squares method of thin plate spline (TPS-RLS) was developed, which adds a weight penalty term to the error criterion of orthogonal least squares (OLS). TPS-RLS combines the advantages of both regularization and OLS, which avoid the over-fitting and the ill-conditioning problems simultaneously. Numerical tests indicate that irrespective of the standard deviation of sampling errors and the number of knots, TPS-RLS is always more accurate than TPS-M for smooth fitting of noisy data, whereas TPS-M would have a serious over-fitting problem if the optimal number of knots were not determined in advance. The real-world example of fitting total station instrument data shows that among the classical interpolation methods including IDW, natural neighbor and ordinary kriging, TPS-RLS has the highest accuracy for a series of DEMs with different resolutions, especially for the coarse one. Surface modeling of DEMs with contour lines demonstrate that TPS-RLS has a better performance than the classical methods in terms of both root mean squared error and relief shaded map appearance.

Appendices

Appendix A: Regularization Parameter Deduction Based on Bayesian Interpretation

Based on Bayesian interpretation, the error criterion Eq. (11) can be transformed into the form

$$\begin{aligned} \beta \mathbf{v}^\mathrm{T}\mathbf{v}+\alpha {\varvec{\uptheta }}^\mathrm{T}\theta , \end{aligned}$$

(22)

where $\alpha$ represents a regularizing constant; $\beta =1/\sigma _\mathrm{e}^{2};\rho =\alpha /\beta ;\sigma _\mathrm{e}$ is the standard deviation of sampling errors. Based on Eq. (4.6) in the paper of MacKay (1992), the log evidence for $\alpha$ and $\beta$ can be expressed as

$$\begin{aligned} \log P(D\vert \alpha ,\beta ,\mathrm{A},\mathfrak {R})&= -\frac{1}{2}\beta \mathbf{v}^\mathrm{T}\mathbf{v}-\frac{1}{2}\alpha {\varvec{\uptheta }}^\mathrm{T}\theta -\frac{1}{2}\log \det \mathbf{A} \nonumber \\&+\frac{M}{2}\log \alpha +\frac{N}{2}\log \beta -\frac{N}{2}\log 2\pi , \end{aligned}$$

(23)

where $\mathbf{A}$ is the second derivative of Eq. (22) with respect to $\theta$, namely

$$\begin{aligned} \mathbf{A}=\alpha \mathbf{I}+\beta \mathbf{Q}^\mathrm{T} \mathbf{Q}=\hbox {diag}(\alpha +\beta \mathbf{q}_1^\mathrm{T}\mathbf{q}_1,\ldots ,\alpha +\beta \mathbf{q}_{M}^\mathrm{T}\mathbf{q}_M). \end{aligned}$$

(24)

Setting the first derivatives of Eq. (23) with respect to $\alpha$ and $\beta$ to zeros, we can respectively obtain the following equations

$$\begin{aligned}&-\sum _{i=1}^M {\frac{1}{\alpha +\beta \mathbf{q}_{i}^\mathrm{T} \mathbf{q}_i}+\frac{M}{\alpha }}={\varvec{\uptheta }}^\mathrm{T}{\varvec{\uptheta }}, \end{aligned}$$

(25)

$$\begin{aligned}&-\sum _{i=1}^M {\frac{\mathbf{q}_i^\mathrm{T}\mathbf{q}_i}{\alpha +\beta \mathbf{q}_{i}^\mathrm{T}\mathbf{q}_i}+\frac{N}{\beta }}=\mathbf{v}^\mathrm{T}\mathbf{v}. \end{aligned}$$

(26)

Hence

$$\begin{aligned} \frac{{\varvec{\uptheta }}^\mathrm{T}{\varvec{\uptheta }} }{\mathbf{v}^\mathrm{T}\mathbf{v}}=\frac{-\sum _{i=1}^M {\frac{1}{\alpha +\beta \mathbf{q}_i^\mathrm{T}\mathbf{q}_i}+\frac{M}{\alpha }} }{-\sum _{i=1}^M {\frac{\mathbf{q}_i^\mathrm{T}\mathbf{q}_i}{\alpha +\beta \mathbf{q}_i^\mathrm{T} \mathbf{q}_i}+\frac{N}{\beta }}}. \end{aligned}$$

(27)

or

$$\begin{aligned} \frac{{\varvec{\uptheta }}^\mathrm{T}{\varvec{\uptheta }}}{\mathbf{v}^\mathrm{T}\mathbf{v}}&= \frac{-\sum _{i=1}^M {\frac{1}{\rho +\mathbf{q}_i^\mathrm{T}\mathbf{q}_i}+\frac{M}{\rho }} }{-\sum _{i=1}^M {\frac{\mathbf{q}_i^\mathrm{T}\mathbf{q}_i }{\rho +\mathbf{q}_i^\mathrm{T}\mathbf{q}_i}+N}}\nonumber \\&= \frac{\sum _{i=1}^M {\frac{\mathbf{q}_i^\mathrm{T} \mathbf{q}_i}{\rho (\rho +\mathbf{q}_i^\mathrm{T}\mathbf{q}_i)}} }{-\sum _{i=1}^M {\frac{\mathbf{q}_i^\mathrm{T}\mathbf{q}_i }{\rho +\mathbf{q}_i^\mathrm{T}\mathbf{q}_i}+N}}. \end{aligned}$$

(28)

Letting $\gamma =\sum _{i=1}^M {\frac{\mathbf{q}_i^\mathrm{T} \mathbf{q}_i}{\rho +\mathbf{q}_i^\mathrm{T}\mathbf{q}_i}}$, Eq. (28) can be transformed into the expression

$$\begin{aligned} \frac{{\varvec{\uptheta }}^\mathrm{T}{\varvec{\uptheta }}}{\mathbf{v}^\mathrm{T}\mathbf{v}}=\frac{\gamma }{(N-\gamma )\rho }. \end{aligned}$$

(29)

Namely

$$\begin{aligned} \rho =\frac{\gamma }{(N-\gamma )}\frac{\mathbf{v}^\mathrm{T} \mathbf{v}}{{\varvec{\uptheta }}^\mathrm{T}\theta }. \end{aligned}$$

(30)

Appendix B: Procedure of TPS-RLS

Repeat the above procedure until $\rho$ converges to a fixed value.